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Differential Game Theory Approach To Modeling Dynamic Imperfect Market Processes

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Differential Game Theory Approach To Modeling Dynamic Imperfect Market Processes

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Content DIFFERENTIAL GAME THEORY APPROACH TO MODELING DYNAMIC IM PERFECT • MARKET PROCESSES by G eorge Sae-Ho Chung A D issertatio n P re se n ted to the FACULTY O F THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA in P a rtia l F ulfillm ent of the R equirem ents for the D egree DOCTOR OF PHILOSOPHY (Econom ics) June 1972 INFORMATION TO USERS This dissertation was produced from a m icrofilm copy of th e original docum ent. While the m ost advanced technological m eans to photograph and reproduce this docum ent have been used, the quality is heavily dependent upon the quality of the original subm itted. The following explanation of techniques is provided to help you understand markings or patterns which may appear on this reproduction. 1. The sign or "ta rg et" fo r pages apparently lacking from the docum ent photographed is "Missing Page(s)". If it was possible to obtain the missing page(s) or section, th ey are spliced into th e film along with adjacent pages. This may have necessitated cutting thru an image and duplicating adjacent pages to insure you com plete continuity. 2. When an image on th e film is obliterated w ith a large round black mark, it is an indication th a t th e photographer suspected th a t the copy may have m oved during exposure and thus cause a blurred image. You will find a good image of th e page in the adjacent frame. 3. When a map, drawing or chart, etc., was part of the material being p h o to g ra p h e d th e photographer follow ed a definite m ethod in "sectioning" th e m aterial. It is custom ary to begin photoing at the upper left hand corner of a large sheet and to continue photoing from left to right in equal sections w ith a small overlap. If necessary, sectioning is continued again — beginning below the first row and continuing on until com plete. 4. The m ajority of users indicate th a t the textual content is of greatest value, however, a som ew hat higher quality reproduction could be made from "p h o to g rap h s" if essential to th e understanding of the dissertation. Silver prints of "photographs" may be ordered at additional charge by w riting th e O rder D epartm ent, giving the catalog num ber, title, au th o r and specific pages you wish reproduced. University Microfilms 300 North Zeeb Road Ann Arbor, Michigan 48106 A Xerox Education Company CHUNG, George Sae-Ho, 1930- DIFFERENTIAL GAME THEORY APPROACH TO MODELING DYNAMIC IMPERFECT MARKET PROCESSES. University of Southern California, Ph.D., 1972 Economics, theory University Microfilms, A X E R O X Company, Ann Arbor, Michigan hi n nntnr UTnnnt’TT T?VAPr PT.V AQ R P P 'P T V 'P 'n UNIVERSITY O F SOUTHERN CALIFORNIA THE GRADUATE SCHOOL UNIVERSITY PARK LOS ANGELES, CALIFORNIA 9 0 0 0 7 This dissertation, written by under the direction of hi& . Dissertation Com mittee, and approved by all its members, has been presented to and accepted by The Graduate School, in partial fulfillment of requirements of the degree of GEORGE SAE-HO CHUNG Dean Date. June 1972 DISSERTATION COMMT££EE ruJLJ^Z . f PLEASE NOTE: Some pages may have indistinct print. Filmed as received. University Microfilms, A Xerox Education Company TABLE O F CONTENTS LIST OF F IG U R E S .................................................... iv Chapter I. IN T R O D U C T IO N ........................................................................ 1 H isto rical Background P ro p o sa l G eneral O rganization of the D issertation II. NONZERO-SUM DIFFERENTIAL GAME AND OPTIMALITY PRIN CIPLE .................................................. 12 P ro p e rtie s of Nonzero-Sum Games T heories of Solution for N onzero-Sum G am es F o rm a l Definition of Solutions III. METHODS OF SOLUTION FOR NONZERO-SUM DIFFERENTIAL GAMES ..................................................... 23 F orm ulation of N onzero-sum D ifferential Game Nash E quilibrium Solution P a reto -O p tim a l Solution IV. PR IC E COMPETITION M O D E L .......................................... 30 F orm ulation of Model Solution of the Game Optimal Strategies Solution of H am ilton-Jacobi Equations F irm l 's Optimal Shadow P ric e s F ir m 2 's Optimal Shadow P ric e s E quilibrium Switching S trategies and T rajec to rie s of State V ariables ii V. INTER-REGIONAL T R A D E .................................................. 79 Introduc tion F orm ulation of Model Modified Model Solution of the Modified Game Solution of Cooperative Game VI. UNCERTAINTY IN DIFFERENTIAL G A M ES................ 103 Sufficient Conditions for Solution of G eneral Stochastic D ifferential Game C lass of L inear Stochastic D ifferential G am es of Z ero-S um V ariety A Stochastic Im perfect M arket P ro c ess VII. SUMMARY AND CONCLUSIONS......................................... 139 Sum m ary of the D issertation Suggestions for F u rth e r Work B IB L IO G R A P H Y ........................................................................................... 151 LIST O F FIGURES F igure 4-1 Behavior of Function R .......................................................... 48 4-2 Optim al T rajec to rie s of F irm 1 ....................................... 63 4-3 Initial Tendencies of S2 (t) and G ( t ) N ear r = 0 . . . 67 4-4 Optimal T ra je c to rie s of S2 (t) and G (t) in the Region S2 < ( 3 G .................................................. ............ 70 4-5 Optim al T ra je c to rie s of F irm 2 in the Region S2 > ( 3 G ....................................................................................... 75 iv C H A P T E R 1 INTRODUCTION H isto rical Background The work of A. Cournot (1897) on monopoly, duopoly, and oligopoly m ust be considered a c la ssic a l m a ste rp ie ce yet to be su rp assed that provides insight into the nature of im p erfect m ark et organizations. P e rh ap s the m o st im p o rtan t contribution of Cournot's work is his recognition of the stra te g ic interdependence of com peti to rs in oligopolistic m a rk e ts. A few m odifications have since been m ade [B ertrand (1883), Robinson (1930), C ham berlain (1930), F elln er (1949)], butno'one has rea lly com e up with g re a te r insight into the perplexing problem of im p erfect m a rk e t organizations. Both C ham berlain and M rs. Robinson considered m o re rea listic ally additional strategic v ariab les such as product differentiation and advertising, but they ignored both stra te g ic interlinkage among com p etito rs and the possibility of collusion. F e lln e r's treatm ent of oligopoly, on the other hand, s tre s s e d the sem i-cooperative aspects ra th e r than the non-cooperative and equilibrium featu res of the m arket, while B ertrand replaced quantity (Cournot's Theory) with price as the strategic variable. Thus, their th eo ries are conceptually no different from Cournot's in so far as the behavior of com petitors is concerned. C ournot's th eo ry su ffe rs, however, from its fundamental assum ption that the individual firm takes the outputs of all his com p etito rs as given and chooses h is own output in such a m anner as to m axim ize his profit. T his "reactio n curve" hypothesis seem s unreasonable, for it im plies that com petitors do not m ake their m oves sim ultaneously. In m any re a l-w o rld situations, however, com petitors often have to m ake th eir m oves sim ultaneously without any knowledge of the m oves of the other com petitors. The great work of Edgeworth (1881) also fails to account for the realistic interaction of strateg ies. The game theory of Von Neum ann and M orgenstern (1944) avoids the "reaction c u rv e " hypothesis of Cournot oligopoly. In addition various econom ic theories about different m ark ets can be unified within the analytic fram ew o rk of the gam e theory, i . e . , the dichotomy between pure com petition and im p erfect m ark e ts is no longer n e ce ssa ry inasm uch as the gam e theory can accommodate the different degrees of com petition, cooperation, and collusion, ranging from c u t-th ro at com petition to pure monopoly. The game theory of Von Neumann and M o rg en stern gave us such e sse n tia l item s as the concept of strategy, the value of a gam e, and a workable and sound delineation of optim al s tra te g ie s of pure o r m ixed variety. It suffers, however, fro m the following two w eaknesses: (1) it is a static theory and as such is unable to explain the dynamic aspects of m arkets, and (2 ) its bim atrix method is not a sa tisfac to ry tool for solving practical gam es, though it is highly useful fo r theoretical development and pedagogical p urposes. 3 Tintner and N ied erco rn (unpublished; p. 11-13) gave their dynamic generalization of C ournot oligopoly in a form that m ay be considered as an n -player, m u ltistag e d iscrete game, i . e . , each oligopolist trie s to m axim ize his cum ulative pro fit over the entire game period by m anipulating h is stra te g y variable, output decision. T heir model is obviously m o re re a lis tic v is -a -v is the static Cournot oligopoly theory. The dynamic oligopoly game, however, m akes the sam e basic hypothesis on the behavior of com petitors as does Cournot oligopoly. Thus, the m odel fails to consider to the full extent the intricate and com plex in teractio n s of firm s ' strateg ies over time. Shubik (1959) also investigated dynamic gam es in his gam es of econom ic survival as an extension of the static gam es of Von Neumann and M orgenstern. He placed em phasis on the im po r tance of a firm 's a s s e t position for the range of strateg ies open to it, its survival and growth in dynam ic econom ic gam es. His m odel was, however, of a conceptual n atu re ra th e r than a m ethod to solve any practical dynamic gam e. In the m eantim e se v e ra l im portant developm ents have appeared in the field of engineering and applied m athem atics. Isaacs (1965) introduced the theory of differential game as an extension of both the static game theory and the m odern optim al control theory. Isa a c s' theory is that of z e r o -s u m tw o-player variety, i;:e., there is a single c rite rio n functional which one player trie s to m inim ize at the sam e tim e that the other trie s to m axim ize by m anipulating their respective stra te g ie s over tim e. H ere the in te re sts of competing p a rties are assum ed d ia m e tric a lly opposed to each other. Because of this property, the theory of z e ro -s u m differential game h as been applied alm ost exclusively to w ar gam es. Unlike the bim atrix m ethod of the static game theory, Isa a c s' theory of differential gam es has an inner logical p a tte rn that lends itse lf readily to m athem atical analysis. It is not so m uch a theory but is ra th e r a method or procedure to obtain a solution of continuous gam es of long duration. [It was the control theory and the dynamic program m ing (Bellman, 1957) that m ade it possible to develop the theory of differential gam es.] The th eo ry of z e ro -s u m differential game is gen eralized to include n o n zero -su m n -p lay e r gam es by Case (1969). Case developed a method of solving the n o n zero -su m m any-player differential gam es for a particular type of solution, i .e . , Nash equilibrium solution. The general n o n z e ro -su m differential game is c h aracterized by the following: (1) it m ay have m o re than two players, each of which trie s to m axim ize his own objective c rite rio n functional by m anipu lating his own set of stra te g y v ariab les, and (2) individual p la y e rs' goals are not e n tire ly conflicting. Because of this property, nonzero- sum differential gam es a re highly relevant to economic gam es. Unlike the z e ro -s u m gam e, the n o n zero -su m game finds m any types of solutions relevant, depending on the application. S tarr and Ho (1969) investigated three in te restin g types of solutions for the nonzero-sum d ifferential gam e, i. e. , the Nash equilibrium , the P a re to-optim al, and the m axim in. The Nash equilibrium solution and the P a re to-optim al solution a re p articu larly relevant to the economic gam es of im p e rfe c t m a rk e ts. Rhodes and L uen berger (1969) and Behn and Ho (1968) studied the stochastic differential game of z ero -su m variety. They assum ed different d eg rees of inform ation perfection for different players to study the effects of uncertainty on the optim al stra te g ie s and the payoff under the various inform ation states. The theory of the stochastic d ifferen tial game is developed only fo r the gam es involving linear sy ste m s p rim a rily because the n e c e ssa ry dynamic estim atio n m ethod is not available for nonlinear system s. K alm an (Kalman and Busy 1961) developed a method of estim ating in re a l time the state of a lin e a r sy ste m fro m nonstationary-tim e se rie s of o b s e r vations, thus paving the way for the development of the stochastic differential gam e. The uncertainty in the context of the differential game re fe rs to the ignorance of players about the state of the system , e .g ., the state of m a rk e t such as relative m ark et share, demand, cost, etc. Thus, it is the Knightian "ignorance" (Knight 1933) that constitutes the u n certain ty in the differential game. The Knightian "behavioristic u n c ertain ty " is, on the other hand, not treated as uncertainty in the differential game, i . e . , each player is assu m ed to make h is move in total ignorance of his opponents' m oves. PROPOSAL The p re s e n t re s e a rc h is m otivated by the dynamic g e n e ra l ization of Cournot oligopoly by T intner and N iedercorn, and C a se 's work on the n o n zero -su m , m any-player differential gam es (Case 1969). The author proposes to dem onstrate the applicability and u sefu ln ess of the n o n zero -su m differential game as a tool for analyzing those im p erfect m ark e ts that to date have stubbornly re s is te d theoretical analysis. In the m ain body of the d isse rta tio n it is c le a rly shown that the theory of nonzero-su m differential gam es can give invaluable new insights into the optim al behavior of firm s o v e r tim e in im p erfect m a rk e ts. The solution of such re a listic econom ic gam es as the following (highly relevant to the oligopolistic m a rk e t of today) is well within the capability of the n o n zero -su m d ifferen tial gam e. The oligopoly firm s are assum ed to m anufacture and sell v e rs io n s of a single product that can be substituted fo r each other but a re not identical. Because of this situation, the relative m a rk e t s h a re of each individual firm depends largely on the relativ e p rices it c h a rg e s. It is also assum ed that the technology of this indu stry and the availability of facto rs of production a re such that the quantity produced by an individual firm depends only on the f ir m 's capital, i . e . , o th er facto rs of production are assum ed away in a c e te ris p a rib u s condition. F irm s sell their products at a m a rk e t clearing p ric e which is determ ined by the aggregate amount of products the com peting firm s decide to offer for sale. Borrowing fro m an outside so u rce is not allowed, so the firm 's capital source is confined to its own p ro fits. Under these assum ptions the individual firm trie s to m ax im iz e its b h are h o ld er' s utility functional by m anipulating its 7 policy variable, i . e . , dividend rate {t) at tim e t, / T >4 exP dt + K.(T) exp [-p.(T-t0)] Subject to: K.(t) = f. (Kv K2, . . ., Kn ) > 0 , K. 2 K. i * i i f. = Q .(K .)P . [ q ^ K j) , Q2 (K2 ), . . ., Qn (Kn )] -C. (K.) l i' w here KL(t) = i ^ fir m 's capital level on time t; K°^ denotes the initial am ount of capital at the beginning of the gam e. th f. = i firm 's net profit function th Q. = production function for i firm th = m ark et clearing price for the product of i firm th (L = production cost function fo r i firm 8 e ^ = discount factor; p is the in te re s t rate T -t0 = tim e duration of game This m odel re p re se n ts a highly re a lis tic m o d ern oligopoly situation where the industry is dom inated by a few larg e publicly owned corporations, each of which striv e s for both the satisfaction of sh a re-h o ld ers and the long-run growth of the company. Baumol (1959) discussed business behavior sim ila r to that given h e re . G eneral O rganization of the D issertatio n Chapter II briefly review s the difference betw een the z ero - sum and n o n zero -su m gam es. The im plications of dropping the z e ro -su m hypothesis in the differential game a re studied. The p ro p erties of highly relevant (with re s p e c t to econom ic applications) Nash equilibrium and P a reto -o p tim a l solutions of the n o n zero -su m differential gam e are exam ined by use of b im a trix and d iscrete m ultistage gam es of the type p resented by Luce and Raiffa (1958). Some existing econom ic theories of im p erfect m a rk e ts a re reviewed to dem onstrate the relevance of the Nash equilibrium and P a re to - optim al solutions. In Chapter III the general n o n z ero -su m differential game is form ulated. The procedure to obtain a Nash eq u ilib riu m solution is discussed in some detail. The relation sh ip betw een the Nash equilib riu m and P a reto -o p tim a l solutions of the n o n z ero -su m differential game is derived. A sim ple but interesting im perfect m a rk e t p ro c e ss is c a st as a non zero -su m tw o-player differential game in C hapter IV. A com plete Nash equilibrium solution for the game is given with the in te rp retatio n of economic im plications. The m odel co ncerns a m a rk e t situation that is brought about by the decision of a new firm to em bark on an aggressive "price w ar" against the w ell-established firm in an attem pt to share the profitable m ark et with the virtual monopoly firm . The two firm s m anipulate both p ric e and production decision as strategy v ariables to m axim ize their resp ectiv e cum ula tive profits over the entire game period. In this p rice com petition gam e, the payoff consists of the value of the te rm in a l m a rk e t share, the value of the term inal stockpile of goods for future dem ands, and the cum ulative sum of the p resen t revenues fro m sa les over the e n tire period of game. The solution of this m odel shows how firm s shift their policies from tim e to time during the gam e to reflect the changing state of the m ark et, i. e. , inventory level of goods and m a rk e t share, to m axim ize the respective aggregate payoffs. In Chapter V an unconventional but re a lis tic in te r-re g io n a l trade between a developed region and an underdeveloped region is c a st as a non zero -su m tw o-player differential gam e. The under developed region indirectly exports cheap labor to the developed region through the finished goods that it m anufactures out of the interm ediate goods im ported from the developed region; w hereas the developed region indirectly exports technology to the underdeveloped region through sem i-finished goods of high technological content. The n o n z ero -su m game m ay be solved for a Nash eq u ilib riu m solution, and the existence of this solution is assum ed. The original trade p ro b lem is then slightly modified to dem onstrate that under certain d e sirab le conditions, the Nash equilibrium solution of this in te r regional trade game is also P areto -o p tim al, i . e . , the Nash equilib riu m solution of the original game is identical under certain conditions with the P a reto -o p tim a l solution of an equivalent com pletely cooperative game. This approach was m otivated by the observation that the in te r-re g io n a l trade under consideration is of a highly com plem entary nature. This im plies that the conditions of our in te r-re g io n a l trade a re such that the com bined welfare of both regions is m axim ized even though each region pursues only its own in te re s t. The im plications and the n e c e ssa ry conditions of the P a re to-optim al solution are carefully studied p a rticu la rly from the view point of regional planning. The uncertainty in the differential gam e is investigated in C hapter VI. The im portance and the n ecessity of accounting for uncertainty in game situations a re em phasized. The lite ra tu re on the stochastic differential game is review ed in detail. It is shown that the certainty-equivalence principle holds for an im portan t class of z e ro -s u m linear stochastic differential g am es. Rhodes and Luenberger&s sufficiency theorem (1969) on the existence of the optim al solution to the general stochastic differential gam e of z e ro - sum v a rie ty is extended to the nonzero-sum gam es. An in terestin g im p e rfe c t com petition situation involving product styling, advertising, and other m arketing p ractices is cast as a lin e a r stochastic nonzero- sum differential gam e. A prom ising solution algorithm for the gam e 11 is proposed that is based on the tra n sfo rm a tio n of the stochastic game to a corresponding determ inistic gam e by m eans of the expected value operator. The economic im plications of form ulating the im perfect com petition as a linear stochastic d ifferen tial gam e are discussed. In the la s t chapter the p re s e n t re s e a rc h work is briefly sum m arized. The following three m a jo r d irectio n s for fu rth er work a re suggested: (1) extension of the p re s e n t capabilities of the differ ential game theory both as a th eo re tic al and operational tool for the analysis of im perfect m ark ets, (2 ) developm ent of m o re com plicated and rea listic differential game m odels along the lines of the sim ple m odels of the p resen t resea rch , and, (3) e m p iric a l studies such as experim ental gam es and com puter sim ulation to v erify and supplem ent the differential gam e theory. CHAJf/TERnll NONZERO-SUM D IFFEREN TIA L GAME AND OPTIMALITY PRIN CIPLE P ro p e rtie s of N onzero-S um G am es Is a a c 's differential gam e (Isaac 1965) has the z e ro -su m tw o-player property and thus enjoys a unique optim al solution, i . e . , the saddle-point solution. In the case of the g en eral nonzero-sum , n -p lay e r differential gam es of Case (1969), however, each player trie s to m axim ize his own objective c rite rio n by controlling his own se t of stra te g y v ariables on a single dynam ic sy ste m w here individual p la y e rs ' goals are not e n tirely conflicting. It is, therefore, no longer obvious what should be req u ired of an optim al solution for such n o n zero -su m differential gam es. V arious types of solutions are relevant, depending upon the application. Inasm uch as the sum of all the p la y e rs' payoffs in the n o n zero -su m gam e is not zero nor is it constant, the players are not always antagonistic toward each other and m ay som etim es d e sire to cooperate for m utual benefits. P e rh a p s the m o st im p o rtan t im p lica tion of the extension of gam es fro m tw o-player to m an y -p lay er is the possib ility of coalitions among groups of p lay e rs. Without the stric t 12 13 rules postulated to govern the fo rm ation of coalitions, v ery little can be said about coalitions as there are m any possible ones depending upon the circu m stan ces. The g e n eral coalition problem of m any- player gam es has stubbornly re s iste d theoretical analysis. One special case of sim ple coalition of all the p layers, however, lends itself readily to a th eo retical an alysis. We will have m ore to say la te r on the sim ple coalition. Before considering the various types of solutions of in te re s t for the n o n zero -su m differential gam es, let us review som e of the salient conceptual differences between z e ro -su m gam es and n o nzero-sum gam es by m ean s of sim ple b im atrix gam es. PLAYER 1 f*LAYER 2 PLAYER 2 < r '0 1 ' 02 ' 01 02 10, -1 0 0 ,0 * a 1 PLAYER 1 °2 -1 0 , 10 0 , 0* 20, -2 0 - 2 0 ,2 0 20. -2 0 -2 0 . 20 GAME 1 GAME 2 In Game 1, P la y e r 1 chooses between stra te g ie s and while P la y e r 2 sim ultaneously chooses (3 ^ or ^2* The corresponding e n tries in the m a trix give the payoffs for the two p lay ers. Notice that for each strateg y p a ir the sum of the corresponding payoffs is zero so that the game is a z e ro -s u m gam e. We assum e that each player w ishes to m axim ize his own payoff. If he is rational, P la y e r 2 will always choose an^ P la y e r 1 will play a realizing P la y e r 2 's 14 strateg y . This saddle-point solution is the only reasonable one in this gam e. In Game 2 there ex ists no saddle-point solution; but P la y e r 1 can m inim ize his m axim um possible lo ss by choosing stra te g y if he assu m es that P la y e r 2 attem pts to do the m axim um dam age to P la y e r 1 at the expense of ignoring his own payoff. By the sam e reasoning, P la y e r 2 chooses (3^. Thus (a^, p^) i® a naaximin solution. We note h e re that in the z e ro -s u m gam es there can be no m utual in te rest, for what is a gain to one player is a loss to the other. PLAYER 1 °7 PLAYER 2 PLAYER 2 h h 5,10 20. 20* a 1 PLAYER 1 20,20 io o ,io 20, 20* 10, 15 ia , 100 50, 60 GAME 3 GAME 4 Game 3 is a n o nzero -sum game, i . e . , for each strateg y p a ir the sum of the corresponding payoffs of both players is not zero. If P la y e r 1 announces in advance his intention to play stra te g y then P la y e r 2 has no choice but to play (3^. Likew ise if P la y e r 2 announces his choice of in advance, P la y e r 1 h a s no choice but to play a j. If, on the other hand, both players try sim ultaneously to m axim ize their respective payoffs without any com m unication between them, the two players will receive low er payoffs. Thus, in this case, it is advantageous to disclose one's stra te g y in advance, although this is not always true. P ric e lead ersh ip in an oligopolistic m a rk e t m ay 15 be considered as an econom ic exam ple of such a gam e. The dominant firm sets the m ark et price, and eq u ilib riu m com es about by the others adjusting to this price. Notice that th ere a re two Nash equilibrium solutions in this game. They are eq u ilib riu m solutions in the sense that no player can achieve a b e tter re s u lt by deviating from his Nash strateg y as long as the other player uses his Nash strategy. A Nash strateg y is defined as the best stra te g y that a player can play against the b e st stra te g ie s of h is opponents (see C hapter III for a form al definition). Game 4 is the so-called " c la ssic a l p ris o n e r's dilem m a. " The only Nash equilibrium solution is (a^, (3^), yet (a p^) gives a b e tter re s u lt for both p lay ers. The solution (a^, p^) is, however, vulnerable to cheating by one p lay er, while (a^, p^) is not. This illu stra te s the non-optim ality of the N ash equilibrium solution in the n o n zero -su m game. Although the N ash equilibrium solution is a stable solution, it probably is not a lo n g -ru n solution, for were the gam e played fo r a long duration, the stra te g ie s of P la y e r 1 would be come known to P la y er 2, and vice v e rs a , and it would be unwise of them to continue to play gam es when they could m axim ize their respective profits by sim ply cooperating with each other. The re a l difficulty lies in the fact that they m ay not tru s t each othe’ F. Even this difficulty can be overcom e if enough tim e is given for them to le a rn fro m experience. In the z e ro -su m gam es, on the other hand, there is no point in the two players entering into com m unication or cooperation, because their in te re sts are d ia m e tric a lly opposite. If two firm s are 16 struggling for an over sa tu rated m a rk e t in a duopoly m ark et situation, then the z e ro -s u m gam e m ay apply, but this would be a ra re econom ic circum stance. In alm o st all econom ic situations, the fortunes of the firm s are not d ia m e tric a lly opposed. Some fo rm of cooperation with or without side paym ents is often desirable in m any econom ic situ a tions. It follows, th ere fo re , that alm ost all m a rk e t p ro c e sse s m ust be considered as n o n z e ro -su m gam es. T heories of Solution for N onzero-Sum Gam es In the lite ra tu re on the game theory the three m ost im portant theories of solution for n o n zero -su m gam es are the Nash theory of noncooperative gam es, the Von Neumann and M orgenstern theory of cooperative gam es, and the Nash theory of cooperative gam es with and without side paym ents. The Nash equilibrium solu tion of noncooperative gam es (Nash 1950; p. 155-162) is perhaps the m ost relevant optim ality principle for the econom ic applications of the n o n zero -su m gam e. It gives the best stra te g y one can em ploy against the b e st stra te g ie s of his opponents. It a ssu m es the rational behavior of all the firm s . The Nash equilibrium has the sam e sta bility as the Cournot o r pure com petition equilibrium . Given the Nash stra te g ie s of com petitors, no firm is m otivated to change its strateg y fro m the equilibrium . In som e sense, Sw eezy's kinky oligopoly demand curve (Sweezy 1939) and C h am b erlain's sm all group com peti tion (Cham berlain 1950) re p re se n t equilibrium solutions of the Nash variety. The eq u ilib riu m in these c ases is enforced by the knowledge 17 that an effective retaliation will deter any firm fro m departing fro m the equilibrium . It is obvious that, because of these d esirable p ro p e rtie s, one would seek a Nash equilibrium solution in a m a rk e t situation w here outright cooperation among firm s is e ith er prohibited or difficult to enforce. The Von Neumann and M orgenstern theory of cooperative gam es co ncerns an oligopolistic m arket situation w here firm s agree to m axim ize their joint re tu rn and then settle between them selves by m eans of a side payment. The Von Neum ann and M o rg en stern solu tion is P a re to -o p tim a l in the sense that firm s cannot sim ultaneously im prove their lot. In term s of E dgew orth's tw o-player, two- com m odity bargain box (Edgeworth 1881), it co rresp o n d s to E d gew orth's u tilita rian point on the co n tract curve. Since Von Neum ann and M orgenstern assum e that individual u tilities are co m p arab le, the settlem ent by a side paym ent of m oney or "utility" poses no th eo retical problem once the joint profit is m axim ized. I s a r d 's "in crem ax " principle (Isard 1969) suggests v ario u s schem es of achieving "fair" and w orkable division of the joint re tu rn among cooperating firm s . The increm ax principle also suggests in terestin g sch em es of reaching the P a re to-optim al point through m ultiphase ba rg a in s after an initial equilibrium is achieved. N ash's theory of cooperative gam es (Nash 1953) differs fro m that of Von Neumann and M orgenstern in that the role of th reats is explicitly brought out to account for bargaining situations. The N ash solution also m axim izes the joint re tu rn of all the firm s , but the division of profit will depend on the ultim ate rela tiv e positions 18 of individual firm s. Nash claim s that h is schem e of division is "fair" because it takes into account both the am ount firm s can obtain by cooperation and the damage they could do to each other if their threats w ere c a rrie d out. a ssu m e s that players will jointly m ax im ize their profits and then split them up in a m anner which g u aran tees each player at le a st as m uch as he can guarantee him self. It also a ssu m e s that the m ark et situation is such that ag reem ents a re honored and no policing of enforcem ent is n ecessary . We now. form ally define Nash equilibrium , m axim in, and P a re to-optim al solutions for the n o n z e ro -su m differential game. a re utility functions for P la y e rs 1 , -------, N, respectively, then the The cooperative solution of Von Neum ann and M orgenstern F o rm al Definition of Solutions Definition. If U^(o^, a^, Un (°'i > Q'2* ■ * *' ^N^ stra te g y vector ( o. 1 * aZ ’ • • • f ) is a N ash equilibrium point if, for i = 1, • • • 9 N • • • I w here a is an adm issible stra te g y fo r P la y e r i. In a game situation w here a player cannot be sure how his opponents select their stra te g ie s, he m ay w ish to choose a strategy 19 to m axim ize his utility against the w orst possible situation, i.e., to play a m axim in strategy. % Definition. A stra te g y o '., is the m axim in strategy for P la y e r i if, for all adm issible stra te g ie s (a^, a . . ., o^) ! M i n U ^ ( Q ’j , « • . , o \ j , • • • » — h h . n l h ( Q j , . . . , • • • » w here the m inim um is taken with re sp e c t to all strateg ies except the strateg y of P lay er i. This is equivalent to finding the saddle-point solution of a z e ro -su m differential gam e, w here the opponent of P la y er i chooses all the strateg ies except that of the i * " * 1 player and tries to m inim ize IL. Because the m axim in solution is excessively conservative, and fails to take into consideration the other p la y e rs' utility functions, it is not a satisfacto ry solution in m o st of the n o n zero-sum economic gam es. Definition. The stra te g y vector ', . . . , ') is a P a re to-optim al solution if, for any other adm issible strateg ies (ttj, . . . , and for i = 1, . . . , N, > ! c s j e . • . , " ^ i( ° l ’ • • * * ^ jj s only if lL(a^, . . . , q -^) = , ...» orN ). 20 The P a re to -o p tim a l solution can be obtained by solving the equivalent optim al control problem , fo r som e (pj, . . ., p.^) satisfying N Z p. = 1, p. > 0, i = 1, . . ., N . 1 X i = 1 The determ ination of acceptable relative weights j p^ | belongs to the negotiation problem . In the p ris o n e r's dilem m a situation we noticed that the Nash eq u ilib riu m solution is not P a re to-optim al. The N ash equilibrium solutions can, however, be P a re to-optim al under suitable conditions. The P a re to -o p tim a l solution demands some form of cooperation among p la y e rs. Although the relationship between the P a re to-o ptim al and the N ash equilibrium solutions is obvious in the sim ple b im a trix gam es, it becom es v ery ambiguous and in tricate in dynam ic gam es. In the sequel we will use a sim ple m ultistage d iscrete gam e to show som e in te restin g relationships between the two types of solutions. C onsider the following sin g le-state, tw o-stage d isc rete gam e. 21 GAME 6 T here are two players in this dynam ic gam e, each with two stra te g ie s, and accordingly each of the four possible com binations of stra te g ie s m ay be played at each stage. The payoffs for a given pair of stra te g ie s are included in the corresponding c irc le . The strateg y p a ir (0, 1) gives a Nash equilibrium solution at both stages and is also P a re to-optim al at each stage. The p ris o n e r's dilem m a does not, therefore, occur at eith er stage. The stra te g y sequence (0, 1), (0, 1) gives the global Nash equilibrium solution o v er two stages. Yet this Nash solution is not P a re to-optim al globally over two stages, for the stra te g y sequence (0, 0), (1, 1), for instance, gives higher payoffs for both p lay ers. The strategy sequence (1, 1), (0, 0) is another P a re to - optim al solution over two stages. To obtain the P a re to -o p tim a l payoffs (17, 17) by the sequence (0, 0), (1, 1), for instance, P la y er 2 m u st tru s t P la y e r 1 not to try to optim ize by playing stra te g y 0 at stage t = 1. The above sim ple game illu s tra te s the following two basic p ro p e rtie s of nonzero-su m m ultistage g a m e s: (1) the absence of a 22 p ris o n e r's dilem m a at every stage does not guarantee that the Nash solution is P a reto -o p tim a l over all stages, and (2) P a re to-optim al solutions generally require cooperation among players not only at the p resen t stage but at all future stages as well. CH A PTER III METHODS OF SOLUTION FOR NONZERO-SUM D IFFEREN TIA L GAMES F orm ulation of N o n z ero -su m D ifferential Game In the g en eral n -p la y e r, nonzero-sum differential game, til the i p lay er m anipulates his stra te g y o\ in such a way as to m a x i m ize his utility functional T U. = K.[x(T), t ] + f L .[t, x(t), tt]L (t), «ft(t)]dt i = 1, 2, . . . , n (3-1) subject to the state equation com m on to all p layers, ■ — = x = f(x, t, ttj, ..., o -n ) . (3-2) The state x of the gam e m ay be constrained by v arious th inequalities and equalities. The strategy o\ of the i player is assum ed to be a continuously differentiable function of tim e t and the 23 24 state x, i. e . , = < j\ (x, t). may also be constrained by v ario us inequalities and equalities. The adm issible strateg ies j a^, i = 1, . . n | are that se t o£ stra te g ie s which allows the state x(t) to rem ain within its own c o n strain ts for t S t ^ T, beginning a t any adm issible in itial point ( x q , tQ). The function f m ust be everyw here continuously d ifferen ti able in each of its argum ents, t and T are, resp ectiv ely , the tim es at which the gam e hegins and ends. The f ir s t te rm of the utility functional (3-1) is called the term inal payoff because it depends on the te rm in a l state of the game, and the second te rm is called the in te g ral payoff. Equation (3-2) rep re se n ts the dynam ics of the gam e that govern the state of the gam e x(t) over tim e. Notice in E qua tions (3-l);and (3-2) that the general differential gam e red u c es to the optim al control problem if n = 1, and to the tw o-player, z e ro -s u m d ifferen tial gam e if n = 2 and U j = -U2* ^ *s * " ° ob serv ed that in the n o n z e ro -su m differential game both the state of the gam e and the payoff of a given player a re dependent upon stra te g ie s of all the p lay ers, i . e . , in Equation (3-2), stra te g ie s of all the p la y e rs com e into play in the determ ination of the state x(t) at any point of tim e, and th the payoff of the i player (3-1) depends on the s tra te g ie s of the other p la y e rs as well as on his own. Since the game is of a n o nzero-sum v ariety, one m u st specify what p ro p e rtie s the solution should have before the gam e could be solved. In the sequel the solution m ethods a re d isc u sse d for two types of solutions, i. e . , Nash equilibrium and P a re to -o p tim a l. Nash E quilibrium Solution The following solution a lg o rith m fo r the n o n z ero -su m differential g a m e — Equations (3-1) and (3-2) —is due to Jam es H. Case (1969). To solve this constrained m ax im izatio n prob lem , we fir s t obtain the Ham iltonian, H. = (t, x(t), a1(t), . . . , an (t) + \ j f (X , t, • • • I i = 1, 2, . . . , n (3-3) th w here is the L agrangian m u ltip lier fo r the i p layer. Then for each player we m axim ize his H am iltonian H^ with re s p e c t to his own th strag eg y a^ to obtain the optimal stra te g y fo r the i p lay er as a function of x, t, O',, . . . , a i. e . , if the functions L. and f are I n ’ 1 continuously differentiable in | , the optim al stra te g y o\ m ay be found by solving 3 H. _ _1 9 a. to obtain 26 Let us define the value functions V^, V^, . . . , by V. (x , t ) = U. (x , t ) , i = 1, 2, . . ., n i' o’ o l o o' ’ ’ Then X . = 3V. /9x. The value function so defined is a piecew ise 1 1 continuously differentiable function that m u st satisfy the Hamilton- Jacobi system of p artial differential equations 3V. * * 3V.\ i Max T T /, i . ,, at " " «i il X’ a l ’ '• * ’ “ i - r ai® i+ l’ a rx ’ a x ' ( ' with the initial condition V\(x(T), T) = IC(x(T)) 1 — 1 j 2, a a a j n a It follows fro m the definition of the Nash equilibrium solution given in C hapter II that the Nash equilibrium stra te g y for the th ^ i player is the stra te g y which achieves the m axim um in (3-5). We m ust now solve the H am ilton-Jacobi equations for ‘ V X (x, t), i = 1, 2, . . . , n fro m which we finally obtain the Nash equilibrium solution = a\(x, t) by substituting 3VX/3x into $344). The second th eo rem of C a se 's paper (1969; p. 189-191) proves that the se t of stra te g ie s | , i = 1, 2, . . n} so obtained is indeed the Nash equilibrium point. If we now substitute the Nash stra te g ie s in (3-2) and differentiate, we obtain a unique trajec to ry of x(t) for every possible starting point. We also obtain the value 27 function V(xq, t ) along such a trajectory. The negative sign on the right-hand side of Equation (3-5) im plies that the H am ilton-Jacobi equations m u st be solved backw ards from the term inal point by m eans of the dynam ic p rogram m ing method. P areto -O p tim al Solution The P a r e to-optim al solution to the general differential game (3-1) - (3-2) m ay be obtained by solving N N M a x 23 ^i Ui ’ 53 ^i = 1 ’ ^i > 0 r r a V ■ • •' “ n | i = 1 i = 1 Subject to the co n strain t (3-2). The H am ilton-Jacobi equation for this cooperative game is given by aW - Max =7/ 8W \ at " ..., ttn| ( ’ x ’ r “ n’ a x ’ t 1/ (3"7) with the initial condition n W(x (T), T) = 2 3 K; (x (T), T) i = 1 1 1 28 W here W = Value function ( J - = (m -j > M '2* * * • > * (3~8) Thus, it follows from the optim al con trol theory that the P a re to-optim al solution can be obtained by the w ell known solution m ethod of the optim al control problem . Now, suppose the Nash equilibrium solution is the sam e as the P a re to -o p tim a l solution for a given jx. Then we have n W = J L S; V. . (3-9) i = 1 And it follows fro m Equations (3-8) and (3-9) that n H = ] £ (3-10) i = 1 Thus, if the Nash solution is P a reto optim al, the im plication is that at each tim e t during the game, the set of Nash stra te g ie s which m ax im izes the H am iltonians (H^, . . . . , Hn ) also m ax im izes some w eighted lin e ar com bination of {t^, i = 1, 2, . . . , n The solution of the optim al control problem is e a s ie r to obtain than that of the n o n z e ro -su m differential game, which m eans that one can solve the corresponding optim al control problem to obtain the Nash equilibrium solution fo r the n o n zero -su m game if Equation (3-10) holds for the gam e under consideration fo r som e app ropriate weights p. This relatio n sh ip between the N ash equilibrium and P a re to -o p tim a l solu tions will be exploited in Chapter V to estab lish that in the trade game co nsid ered there, the Nash equilibrium stra te g ie s produce a P a re to - optim al re s u lt under suitable conditions. CH A PTER IV PRICE COM PETITION MODEL F o rm u latio n of Model Consider a local m a rk e t for a consum er product, fo r which the demand is inelastic with re s p e c t to the price, and the total local demand is fixed, i . e . , tim e -in v a ria n t. C onsider also that this m a rk e t is initially enjoyed by o n e -firm m onopoly whose technology is constant for all practical p urposes in the fo reseeab le future due to eith er the low technological content of the product o r the inherent static nature of the production technology involved. O ne-firm monopoly m ay have come about h isto ric a lly e ith er through the locally well known nam e brand of the m o n o p o list's product o r because the transpo rtation cost involved is g re a t owing to the physical separation of the m ark et fro m the outside world, or both. We also assum e that the technology of the industry is e sse n tia lly of the Von Neumann- type, i . e . , the product of the production p ro c e ss is an essen tial ingredient to the production of the sam e product. Inasm uch as the monopoly firm has been the only f irm in the local m arket, a re a so n able assum ption would be that the fir m 's production capacity is large enough to m eet the total m a rk e t demand. 30 31 Suppose, now, that another firm trie s to b reak into the m onopolist's m ark et by offering e sse n tia lly the sam e product at a substantially low er p ric e in the hope that it can share the lucrative m ark e t in the long-run. The second firm is assum ed to be constrained by essentially the sam e production technology except that as a late co m er it has m ore m o d ern m ac h in e ry and facilities and hence is m ore efficient. It is also assu m e d that the second firm 's reso u rce condi tions are such that it can rea d ily expand its production capacity without any difficulty if and when its expanding m a rk e t ju stifies such an increase. Although one m ay be h a rd put to find in the rea l world an im perfectly com petitive m a rk e t fitting exactly the description given above, price com petition is a ra th e r com m on phenomenon in today's m arketplace. In the m a rk e t of such daily consum er products as beer, packaged food, e tc ., new firm s do often successfully try, by m eans of vigorous price com petition to get established in a m ark et virtually dominated by a few nam e b ran d firm s . Typically in the e a rly phase of such competition, p ric e is d ra s tic a lly reduced to get the product introduced into the new m a rk e t. As the new product is accepted, and once the new firm 's m a rk e t sh are rea ch e s a c ertain com fortable level, the price is raised to the lev el of that of the established firm s . The lo st revenue during the p rice cam paign is typically reg ard ed by the new firm as an im portant and im p erativ e investm ent to acquire a la rg e r m arket share, for it is c le a r that the reasonable level of m a rk e t share is n e c e ssa ry fo r the new firm for its v e ry survival and profitability in the long run. 32 The im p e rfe c t m a rk e t p ro ce ss under discussion h ere offers an excellent exam ple of a n o n z ero -su m differential game. F o r the sake of sim plicity, let us consider a tw o-player, non zero -su m gam e. If Sj (t) denotes F ir m l 's stock of its products on day t, a denotes constant production coefficient, d^ (t) m a rk e t demand on day t, (t) the amount of stock to be allocated to the production p ro cess on day t, and V the m axim um am ount of stock that can be used as input to the production per day, then the technological options available for F irm 1 (the established firm ) on day t m ay be given by S x (t + 1) = S x (t) + a ffj (t) - d L(t) (4-1) S 1 (t) > ^ (t) * 0 o-j (t) < V (4-2) w here (t+1) is the sto ck on hand at the beginning of day (t+1). Equation (4-1) is a stock adjustm ent equation w here stock level is adjusted dynam ically fro m day to day. The f ir s t inequality of (4-2) says that the amount of stock ^ ( t ) the firm decides to allocate to the production p ro c e ss is non-negative and cannot be g re a te r than the available stock S^(t) on hand. The second inequality fu rth er co n strain s ttj(t) to be le ss than and equal to the absorption capacity of F irm l 's production capacity. F i r m l 's input absorption capacity V is tim e- invariant because the f ir m 's production capacity is assum ed to rem ain fixed during the entire gam e. Since this firm is the original monopoly 33 firm with sufficient production capacity to m eet the total local dem and to begin with, there obviously is no good reason why that capacity should be expanded. F u rth e rm o re , the possibility of reducing the production capacity is ruled out for the reaso n that th ere would be either no m a rk e t for the used facility or prohibitively high co sts of transaction, though there ex ists an apparent need for the reduction of production capacity with the prospect of sharing the m a rk e t with the new firm . On the other hand, F irm 2 (a new firm ) has the following technological options open to it on day t, ( t + 1) = S£ (t) + b ^ (t) - d2 (t) G (t + 1) = G (t) +Cd2 (t) (4-3) S2 (t) > o2 (t) > 0 £*2 (t) - PG (t) (4-4) where G (t) denotes accum ulated "good w ill" o r the firm e d -u p m ark e t share of F ir m 2 on day t. C is the average frac tio n of the new cu sto m ers who becom e perm anently attached to F ir m 2 's brand of products (a conversion facto r from sales to m arktet sh are. ) The constant production coefficient for F ir m 2 is b and (3G (t) is the input absorption capacity of F ir m 2 that is assum ed p roportio nal to G (t) on day t. The sam e in terp retatio n given to the equations of F ir m 1 34 applies to (4-3) and (4-4) of F irm 2 with the one exception that the second inequality of (4-4) dem ands a little different in terp retatio n ; production facility of F irm 2 is not assum ed fixed. R ath er it is assu m ed to in cre ase in proportion to the expanding m a rk e t sh a re with constant proportionality p. The second equation of (4-3) is fo r the adjustm ent of the m a rk e t share fro m day to day. It is im p o rtan t to observe that the stock allocation pro b lem for F irm 2 is slightly m ore com plicated than that of F ir m 1. W hereas F irm 1 has to decide on each day of its operation how m uch of its existing stock to allocate to the production for the p re s e n t m a rk e t consum ption and how m uch to stockpile against future dem ands, F ir m 2 m u st m ake the sam e decision taking into co n sid eratio n not only the future m a rk e t dem ands v e rsu s p re s e n t consum ption but also the effects of p resen t sales on its m ark et sh a re. In a way, then, F ir m 2 faces a difficult problem of three-W;ay allocation w herein the stock allocated to the production for the p resen t consum ption also helps expand the f ir m 's m a rk e t sh are. The d e sire of F ir m 2 to expand its m a rk e t share and its d e sire to obtain la r g e r p re s e n t revenue, how ever, conflict, for la rg e r sales revenue re q u ire s the firm to change a hig h er p rice, w hereas a low er p ric e is re q u ire d for rapid expansion of m a rk e t share. As a ty p ica l price com petition p resu m ab ly la s ts a few y ears, differential equations a re used as an approxim ation for the difference equations (4-1) and (4-3), which describe the lo n g -ru n d isc re te p ro c e ss e s under consideration. Use of continuous functions h as an advantage of being m athem atically tractable as will be seen la te r. 35 Continuous v ersio n s of (4-1) and (4-3) are sim ply given by S j' (t) = b «j (t) - d1 (t) (4 -la) S2 ' (t) = ba2 (t) - d2 (t) G' (t) = Cd2 (t) (4-3a) where S^' (t), S2 ' (t), and G' (t) a re the time derivatives of (t), S2 (t), and G (t), respectively. d2 (t) are yet to be defined. Since both firm s produce and m a rk e t essen tially hom ogeneous products, i.e . , slightly different but e asy to substitute for each other, a sim ple function, II, will be specified shortly with the p ro p erty that it divides the total m a rk e t dem and among com peting firm s on the basis of the relative p ric e s only. C onsider the two demand functions product, and Dq is the total m ark e t demand. Notice that the dem and function of each firm is in v ersely related to the price that the firm charges for its product. This choice of the dem and function is m ade Now in (4 -la) and (4-3a) the demand functions dj(t) and (4-5) w here P^(t), i = 1, 2 denotes the net price after costs of F ir m i 's 36 d elib erately here because we will be p rim a rily in te reste d in the effects of pricing policies on m ark e t behavior in the subsequent an alysis. A dm ittedly such a dem and function is an oversim plified a b strac tio n of the reality. Before we specify the function II, let us review b riefly the H otelling's lin e ar m arket, for it is rele v an t to the function to be specified. In H otelling's m arket, co n su m ers are assum ed to buy ice c re a m fro m the stand nearest them. It is, th erefore, c lear that the decisive consideration, as fa r as m a rk e t division between the two ice c re a m stands is concerned, is the im plicit tran sp o rtatio n cost or m o re generally transaction cost (travelling time converted to dollar cost) which the consum ers m u st b e a r. Now, suppose the p rice s charged fo r ice c re a m at the two stands are different in spite of the fact that the ice c re a m sold is hom ogeneous. Then presum ably the tran sp o rta tio n c o st plus price would be the new guiding consideration fo r the c o n su m e r's decision on which stand to patronize. In addition, if the tran sp o rtatio n oost were borne by the se lle r, which is the case fo r m o st products in the real world (price includes tran sp o rtatio n cost) then co n su m ers would be guided by the relativ e p ric e s o r the p rice difference only, and m ore people would buy from the s e lle r who c h arg es le s s . In general, this tendency becom es m ore pronounced when the products involved are hom ogeneous. With this introduction established, let us define o u r function II. Let x denote the price difference betw een the two firm s, 37 i .e . , p2(fc ) _ P j^ ) or P j(f c ) - P2(f c )* Then the function II is formally- defined as, w here r\ is the m axim um price difference that is m eaningful insofar as the determ ination of m arket division is concerned, i .e . , when the p rice difference reaches r|, one firm dom inates the e n tire m arket. It can be shown that the function II h as the following useful p ro p e rtie s, A - - n(ri) = 1 B -- n(0) =j C -- 0< n(x) < 1 for - T | < x < r| d -- n(-x) = i -n(x) e ~ n't*) = F -- n»(x) = -^ n '(x) = 0 (4-7) The value of the m a rk e t division function II in c r e a s e s continuously and m onotonically fro m 0 to 1 as the p ric e difference x tra v e rs e s the line fro m - r | to + t j . Thus it is c le a r that the function II d eterm in es the relative m ark et demand of the two firm s s tric tly on the b a sis of price difference. T herefore, in a sense, II is a g e n eraliza tio n of the H otelling's lin ear m ark e t. With this definition of the m ark e t-d iv isio n 38 function we can see that firm s have another stro n g 'p o licy variable — the price of their product. F irm s would now try to m axim ize their resp ectiv e utilities by m anipulating the two policy v a ria b le s available to them , i. e. , stock allocation decision Q^(t) and price policy P^(t), i = 1, 2 . In our m odel for price com petition, we assu m e that both firm s w ish to m axim ize their respective profits over the entire planning horizon. Let P q be the net valuation of m a rk e t sh are and P,p the net comm on price of the products of both firm s prevailing at the end of price com petition. Then F ir m 1 w ishes to m axim ize over the planning horizon t < t < T o U j = P t (‘s i (T) - S ^ t j l + fT Pj,(t) d1(P2 - P L) dt (4-8) f c o while F ir m 2 w ishes to m axim ize U 2 = + P G [ G <T > - G <to>] + f P2(t) d ^ P , - P 2 ) dt . (4-9) f c o The integral payoffs of the above utility functionals re p re se n t the cum ulative sum of undiscounted p resen t revenues, while the other term s re p re se n t valuation of the in cre ase in stocks and the in cre ase in the m a rk e t share at the end of the planning period. Although it is 39 straightforw ard to incorporate a tim e discount, we choose to ignore this for sim plicity. Because the initial stocks and the initial m ark e t sh a re a re not going to influence the decisions of firm s, they will be safely ignored in the subsequent analysis. The problem of m axim izing the utility functionals (4-8) and (4-9) subject to the differential equation co n strain ts (4 -la) and (4-3a) and the inequality constraints (4-2) and (4-4) is a n o n zero -su m differ ential game of the so rt discussed in C hapter III. If we denote by S the state vector (Sj, S^, G), the strategy of F ir m 1 is a pair of continuous functions <*j(t, S) and P j(t, S) defined on the E uclidean space (tQ, T) X (S |S j> 0, S£ 2 0, G 2:0) and taking values in a region of defined by the inequalities (4-2) and > 0, i. e . , F ir m 1 observes the state S(t) of the gam e on day t and chooses its strateg y <*j(t, ^) and P^(t, S) such that it m ay m axim ize the utility functional U j. In an analogous m an n er, F irm 2 plays its game rationally. Solution of the Game Before attem pting to solve the gam e, let us co n sid er what kind of solution we would like to have for the price com petition model. With the m a rk e t dem and inelastic, the two firm s can conceivably get together, charge a monopoly p rice, and split the profits between them. But instead of exploiting the inelasticity of the demand, firm s com pete rationally against each other to obtain a la rg e r sh are of the m a rk e t. (Adam Sm ith's "invisible hand" is om nipresent even in this im p e rfec t m ark e t!). T here can be m any rea so n s why firm s cannot 40 get together; the local law m ay prohibit any o v ert and co v ert collusion; firm s sim ply cannot tru st each other; or, m o st im portantly, it is the fundam ental hypothesis of our m odel that the entering firm intends to engage in a "price w ar" against the m onopoly firm not only to force it to share the m ark et but also to outm aneuver it. It is presum ably this kind of highly com petitive s p irit that p rev en ts the two firm s from getting together. With this ratio n alizatio n we seek to obtain a Nash equilibrium solution fo r the n o n z ero -su m differential gam e. The Nash equilibrium solution im p lies that each firm m anipulates its strateg y v ariab les, i. e. , o^(t), P^(t), i = 1, 2 on the basis of time and the system state (Sj(t), S2 (t), G(t)) ag ain st the b e st strategy of his opponent so as to m axim ize his aggregate profit over the entire plan ning period. F o rm ally we seek an eq u ilib riu m solution consisting of a strategy (P^ (t, S), (t, S) for F ir m 1 and a stra te g y (P2 (t, S) , <*2 (t, S)) fo r F irm 2 such that the inequalities U1 (P1’ «1' P2 * *2 *> * U1 <P 1*’ *1*’ P2 *’ "2 *) U 2 ( f / . o r / P 2 , a 2 ) < U 2 ( p / o r / . p / 0 / hold for all adm issible stra te g ie s (P^» 01 and (P2, ot^). Individual firm s, realizing that the sy stem state is influenced by their decisions as well as by those of their opponent, choose th eir optim al strategy for a given day of the gam e to re fle c t the m a rg in a l utility of various state v ariab les on that day. 41 Optimal S trategies We will obtain a com plete solution of the game by m eans of the solution algorithm discussed in Chapter III. F irst, the optim al stra te g ie s are obtained from the two H am iltonians by m eans of the Lagrangian m ethod of m axim ization. The two H am iltonians are obtained fro m the integrands of the integral payoffs (4-8) and (4-9), and the differential equation constraints ( 4 - la) and (4-3a), H 1 = P 1 Don 'P 2 - P l> + h [ a “l - Don ] + X 2 [b02 - Don ] + X3 [CDon ] Do [ P l - h + V C,l3 ] n P ( P 2 - P l> + ^ aafj + \ 2 ba2 + ( \ 3C - X^) Dq H2 = P 2 Don ( P l - P 2 > + - Don (P 2 - P l>] + 1 *2 [b «2 - Don ] + ^3 [CDon ] = Do [ P 2 + ^1 - H -2 + ^ 3 ] 1 1 (P 1 - P 2> + acxl + n2 b az - (4-10) 42 w here (X^, X2> Xg) and (p^, p2> ^ 3) are L agrangian m u ltip liers. The ❖ optim al stock decision 0% , i = 1, 2 is found as that value of adm issible Q^(t), i = 1, 2 which m axim izes the respective H am iltonian, i. e. , 0 if (x 2 (t) < 0 pG(t) if (x 2 (t) > 0 and S2 (t) >pG(t) S2 (t) if ji2 (t) > 0 and S2 (t) < pG(t) . (4-11) > J c We note in the above solutions that a^ (t), i = 1, 2 is dependent not only on time and the sy stem state (S^(t), S2 (t), G(t)) at that time, but also on the L agrangian m ultipliers. It is a little m ore com plicated to find the optim al price stra te g y fdr. reastmfe to be explained shortly. L et 0 if X^t) < 0 Y if Xj(t) > 0 and S j(t) > Y S^(t) if Xj(t) > 0 and S^(t) < Y h j — Xj - \ 2 + GXj h 2 s p2 - V-i ~ C^ 3 (4-12) Now, for fixed values of h^ and h 2, a pair of optim al * * stra te g ie s and P 2 m u st m axim ize sim ultaneously the following two equations in p^ and p2> R 1 = (P 1 " h l ) 1 1 (P2 ' P l } R 2 = (P2 - h 2 ) n ( P x - P 2 ) (4-13) in o rd e r to satisfy the req u ire m e n ts of the Nash equilibrium i.e., F irm 1 m ust m axim ize R j syith re s p e c t to its policy variable against the optim al stra te g y P 2 of its opponent, and F irm 2 m ust likew ise m axim ize R 2 with re s p e c t to P2 against P^ of F irm 1. Function II, through which the pricing stra te g ie s of the two firm s in teract, m akes difficult the d eterm in ation of P j (t), i = 1, 2. Since II is a function of both Pj and f?2, it is obvious that both utility functionals and the differential equations for state v ariab les are affected by the opponent's price stra te g y as well as by one's own. To solve for optim al price stra te g ie s, we consider (4-13) as a sub-gam e and seek a Nash equilibrium solution, i. e . , to find (P^ , P 2 ) such that (P x* - h p n (P2* - P j* ) > (Pj - h L) n (p 2 * - p ^ (P2* - h 2) n (p* - P 2 ' ) > (P2 - h 2 ) n ( p x* - p 2) (4-14) hold for any adm issible p air of (Pj» 44 Now, the f ir s t conditions for the equilibrium solution are V = 9P 7 = (p i - h i>n' (p 2 - p !) - n ^ - P ^ = 0 3R. 2 R. 2 “ 8P 2 (P2 - h2) n' (Pj - p 2) - n ( P 1 - p 2) = o (4-15) If we let P^ (P2* anc* P 2 (P^» ^ 2 ) denote the functions which satisfy the first equation and second equation of (4-15), respectively, then the existence of the unique equilibrium solution dem ands that these two functions have a unique in te rse c tio n point. By making use of the function II defined by (4-6) and the p ro p e rtie s of II given by (4-7), we will establish the existence of the unique intersection. F ir s t consider an equation n x (y, h ) = (x - h) n ' (y - x) - n (y - x) = o where or consider i r y_x i ,x - h , 2^ ' J 2? ^ = °' -n (4-16) 45 By carrying out the integration and re a rran g in g te rm s, we obtain x = (y + T ] + h) . (4-17) Now, by setting*'x = P^ ., y = P£ and h = we obtain from (4-17) the solution of the firs t equation of (4-15) P 1* = Y P 2 + Y (r] + h l ) • (4 -18a) S im ilarly we obtain the solution of the second equation of (4-15) by >Jc substituting x = P^, , y = P^ and h = : P 2* = 7 P 1 + r (n + h2 ) * (4 -18b) We now have to solve (4-18a) and (4— 18b) sim ultaneously for the two unknowns P^ and P^, to obtain the unique solution for the sub-gam e (4-13): P L* = | (ti + h x) + j fn + h 2 ) = p x* (hr h2 ) P 2* = y ( T 1 + h i) + y ( T l + h2) = P 2 > ! C (hr h 2 ) . (4-19) It rem ains to show that P^ and P 2 obtained above are positive, for they are prices, and that P j and P 2 jointly attain the absolute m axim a of both R^ and R2 (4-13). F i r s t we show that indeed 46 'n P j and T?^ are positive. To this end we hypothesize that both h^ and h 2 of (4-12) are non-negative re a l num bers, i.e., h. = + C x, > 0 x2 = ^2 ~ ^1 " Cfi3 “ 0 ' (4-20) The non-negativity of h^ and h^ will be justified in term s of the econom ics of the p rice com petition m odel. In the model, the L agrangian m ultipliers (X. ^ \ ^ ) a re the subjective m arg inal utilities of state v ariab les for F irm 1, i.e., 'OV1 9V1 9V1 9 S 1 * a s 2 * 9G and the Lagrangian m u ltip liers (p.^, p 2, ^ 3 ) a re t^ le subjective 2 m arg in al values of state v a ria b le s for F irm 2, i. e . , (9V /9S^, 9V /9S2> 9V /9G) w here V1, i = 1, 2 denotes the value function of the game fo r F irm i. We have, therefore, h , = V Fg + CV* * 0 ? ? 2 Vg - Vg - CVq > 0 2 X (4-20a) f 47 or VS X + CVG * VS2 V? > V? + CV?, . (4-21) 2 1 (4-21) im plies that the subjective m arg in al value of each fir m 's stock is at le a st as g re a t as that of its opponent's stock after the p ro p er account is taken for the m arg in al value of F irm l 's accum ulated "good will. " In other w ords, the subjective m arg in al valuation of the state v ariable(s) for each fir m is at le a s t as g re a t as that of the state variable(s) belonging to the opponent. It will be shown shortly that an additional hypothesis is needed concerning P^ and h^ before we can prove that ', i = 1, 2 is indeed a positive quantity, i. e . , P. (t) > h . ( t |, i = 1, 2 . (4-22) Now, the im plications of the two hypotheses become clear: to m axim ize its utility function, F ir m i m u st charge for its product a p rice P^ at le a s t as high as the ex cess of the m arginal value of its own stock over that of its opponents stock after p ro p er adjustm ent is allowed for the m arg in al value of "good w ill" V^.. Intuitively this is a reasonable, rational behavior expected of each com petitor in this model, for each firm has a choice of e ith er to sell now for present 48 revenue or stockpile for future dem ands. This optim al pricing under im p erfect price com petition is in sharp co n trast with the optim al pricing under pure com petition. As we know, under pure com petition e v ery com petitor em ploys the sam e pricing policy, i . e . , price is always equal to the m arg in al value of the product. Now, we a re read y to prove under the two assum ptions cited above, P., , i = 1, 2 is indeed positive. Consider the n e ce ssa ry condition for the optim al solution (P^ , P^ ) given by (4-16). If x ^ 0, then the fir s t te rm of the equation (4-16) is negative, i. e. , (x - h)IT < 0, because h £ 0 and II1 > 0, and, on the other hand, the second term of (4-16) is non-negative becausell > 0. It is, therefore, c le a r th atn (y , h) 4 0. Thus x < 0 fails to satisfy (4-16). It follows, therefore, that the n e c e ssa ry condition (4-16) req u ire s that x > 0, s |c # i . e . , by setting x = P j or P 2 we can see that both P^ and P 2 m u st be positive to satisfy the n e c e s s a ry conditions (4-15). Next, we prove the sufficiency of the equilibrium solution ( P ^ , P 2 ). C onsider the subgam e (4-13), i . e . , R = (x - a) n (y - x) w here II is defined by (4-6). If we exam ine the behavior of the function R (see F igure 4-1) R i L Figure 4-1;Beha*iorofc Function R j 49 as a function of x, given a and y, we note that R takes positive or negative value, depending upon the value of x relative to the given non-negative constant a, i. e . , R < 0 if x < a R > 0 if x > a . Also, for given y, we have lim II(y - x) = n(-n) = 0 by (4-7). x -» C O Thus, lim R = lim (x - a) n (y - x) = 0 . x -* co x-*y + r| It follows, therefore, in Figure 4-1 that there m ust be a point along the x -a x is at which R attains an absolute m axim um . T hat m u st be the point w here R' = 0. Now by setting x = Fj, a = h^, and y = F^ we see that R is identical with R^ of (4-13). T herefore R' m u st be identical with R '^ of (4-15). Thus it follows that the value of P j at > ! c which R 'j = 0 m u st be i. e . , the optim al p rice strateg y we obtained before for F irm 1. S im ilarly we obtain R = R£ and R' = R2 f if we substitute x = P^, a = and y = P j in R. It follows, therefore, that ❖ the value of |?2 which m ax im izes R£ m ust be P 2 , i. e . , the optim al price strategy of F ir m 2. This com pletes the proof that optim al 50 p ric e s and T ?2 a re positive, and that the pair of (P j , P ^ ) is indeed the Nash equilibrium solution of the subgame (4-13). equations of the game, let us study other p ro p erties of the optim al price stra te g ie s. It is interesting to observe in (4-19) that each firm gives m o re weight to his own valuation of the state v a ria b le s than that of his opponent in the determ ination of its optim al p rice. Since the p re se n t game is a d eterm in istic one with p erfect inform ation, the other f ir m 's valuation of stocks and m ark e t share is assum ed to be known. The im plication of im p erfect inform ation or uncertainty in the differential gam e is the subject of Chapter VI, w here stochastic differential gam es are discussed. To facilitate the analysis of the effect of price policies in the subsequent analysis, let us exam ine the behavior of P^ and P 2 ' as functions of tim e. We note f ir s t that both P j ' and P 2 ‘ are the functions of h^ and h 2> and h^ and h,, are in turn the functions of tim e. F ro m the ex pressio ns for optim al p rice stra te g ie s (4-19) we have, Before we proceed to the solution of H am ilton-Jacobi " 3 (h2 . " h l ) 51 # * It follows, therefore, that (P ^ ) in c re a se s m onotoni- cally w ithout bound if and only if (l^ - h^) does likew ise as a function of tim e. We re c a ll that the strateg ies given by (4-11) and (4-19) are optim al in the sense that they jointly m axim ize the H am iltonians (4-10). The optim al stra te g ie s are, however, not yet determ inate, for they a re not only functions of known variables, e .g ., time t and state v a ria b le s (Sj(t), (t), G(t)) on day t, but at the sam e tim e they a re also the functions of the now unknown shadow p rices (\^, \ ^ ) and (p^, p^, Inasm uch as we are dealing with the dynamic system , we know that both the optim al stra te g ie s and shadow p rices are the functions of tim e, that is, these variable-s do change, with tim e . I t i s , th e re fo re ,. n e c e ssa ry to determ ine the optim al values of tim e-varying shadow p ric e s on each day before the optim al strategy for that day can be d eterm ined. F u rth e rm o re we re c a ll fro m Chapter III that the com plete behavior over time of the optim al stra te g ie s m u st be known before one can evaluate the effects of the optim al stra te g ie s on the tim e paths of both the state v ariables and the value functions. As will be shown la te r, knowledge on the optim al tra je c to rie s of state v a r i ables is n e c e ssa ry to determ ine com pletely the switching optim al s tra te g ie s a-/(t), i = 1, 2 given by (4-11). Thus there exists a s o r t of feedback loop between stra te g ie s and state variables. Solution of H am ilton-Jacobi Equations We now proceed to obtain optim al shadow p ric e s. The optim al values of shadow p ric e s are obtained fro m the solution of the 52 H am ilton-Jacobi equations. We re c a ll that the H am ilton-Jacobi equations are the f ir s t- o r d e r partial differential equations in unknown 1 2 value functions V and Y . The H am ilton-Jacobi equations are solved re tro g re s s iv e ly by the dynamic program m ing technique, starting fro m the term in a l stage of the game. 1 — 2 — C onsider the value functions V (t, S) and V (t, S) for F irm s 1 and 2 resp ectiv ely , w here = (S^(t), S^ft), G(t)) . They m ust satisfy the H am ilton-Jacobi equations, as well as the initial conditions, V 1(T, S) = P T S ^ T ) V2 (T, S) = P T S2 (T) + P G G(T) . (4-23) The initial conditions of the value functions a re dictated by the fo rm of utility functionals (4-8) and (4-9). Since a p a rtic u la r solution of the H am ilton-Jacobi equations and, th erefore, a p a rtic u la r solution of the gam e, depends on the initial conditions of the value functions, it is im portant fro m the application point of view that the conditions at the term in a l stage of the gam e be com pletely known to both p a rtie s, i. e . , in p a rtic u la r the com m on p rice P,p of stocks expected to p revail at the end of the game and the term inal valuation P q of F irm 2 's m a rk e t sh are. F ro m the planning viewpoint P^, m ay be taken to be equal to the p re -g a m e p ric e level, p a rticu la rly fro m the viewpoint of the 53 entering firm . The new firm m ay also be able to estim ate P ^ in the light of the m a rk e t sh are G (T) it plans to attain by the end of the planning period. The H am ilton-Jacobi equations are a rriv e d at by f ir s t in sertin g the optim al strateg y p airs (o^ , P^ ) and (a^ , P ^ ) of F irm s 1 and 2, respectively, into the H am iltonians (4-10), adding new 1 2 te rm s and V^, and then equating the resulting e x p re ssio n s to zero, i. e . , y l + Do ( P ^ - ^ ) n ( P 2* - p ^ ) i * i , * * VSX aflrl + VS2 2 K - v 0 + i c v x - v ; ) d q = o Vt + Do ( P 2* - h 2) n (P i * - P 2^ + a Bl* + b c 2 * - D V ^ = 0 (4-24) 1 2 The new term s V ^. and V2 are defined as A y 1 (t, S) a n d A y 2 (t, S ), respectively. 54 1 2 Vj. and V re p re se n t the integ ral payoffs of F irm s 1 and 2. Now, instead of d irectly solving the system of p artial differential equations (4-24), the sy stem is f ir s t reduced to a sy stem of o rd in ary differen tial equations by m eans of a reasonable tria l solution. The advantage of this m ethod of solution is obvious, as p artial differential equations a re h a rd e r to solve. C onsider the tria l solution, V l (T) = K ^ t ) S j + ^ (T ) V2(T) = p.2(T) S2 + h 3(t)G + 4 * 2( t ) (4-25) w here t denotes the re tro g re s siv e time v ariab le that is defined by t = T -t, and X^(t), M - 3 (t )» and ^ ( t ) a re functions to be determ ined. When the above tria l solution (4-25) is com pared with the utility functionals (4-8) and (4-9), we notice that the fo rm of the tria l solution is identical with that of the utility functionals if we m atch \ j ( t ) and ij,^(T) with P ^ , ^ ( t ) with P q , 4^ ( t) with the in te g ral payoff of F ir m 1, and finally ^ ( t ) with the in teg ral payoff of F ir m 2. Since the trial value functions (4-25) account for all the payoff te rm s of the utility functionals, they seem to be reasonable. Now if (4-25) is substituted into the H am ilton-Jacobi equations, then the resulting system of o rd in ary differential equations m ay be solved for the unknowns of the tria l solution. If we su ccessfu lly do that, we obtain the optim al shadow p rices as well as optim al value functions, as \ ^ ( t ), [ jl^ (t ) and (^ (t) a re the shadow p rices of in te re s t. H ere 55 optionality m eans that these unknown functions are determ ined under the optim al play by both p arties, as the H am ilton-Jacobi equations already include the optim al stra te g ie s of both firm s. Using the value functions defined by (4-25) we have, V X ^r), Vg = 0 , = 0, v ; = + v < x ) \ - - 0> v k - ^ (T)- v ° = ^ ,T)- = tV < T )S 2 + |X 3 '(t)G + + 2 '(t). Hence, by (4-20a) we also have h l = VSj - VS2 + CVG = V T> h2 = VS " VS ' CVG = ^ 2 (t) “ C |JL 3 (t) • (4-25a) 2 1 Substituting these values into the H am ilton-Jacobi equations (4-24) and bringing V*, i = 1, 2 to the right side of the respective equations, we have Do [P i* (T) " Xl (T>]n (P 2 *(t) " P l > » + X j ( t ) a o ' 1 = ^ ' ( t J S j + 4*j 1 ( r ) 56 Do [P 2*(T) " ^ 2 (t) + ^ 3 ^ ) ] n (P i*(T) “ P 2> !< (T )) + h 2 (t ) bcr2 ' = (jl2 ' ( t ) S 2 + |o.3 '( t ) G + ^ 2 ' ( t ) . That -V* = y j , i = 1, 2 is used to obtain (4-26) fro m (4-24). The H am ilton-Jacobi equations a re e x p re ssed in te rm s of the re tro g re ssiv e tim e variable because they have to be solved backw ards starting fro m the term in al stage. The solution of (4-26) req u ire s that for each j j c possible com bination of the optim al stra te g ie s (o^ , a 2 ) given by (4-11) v l / V * V * y A - - Substitute a chosen p air of (a^ , a2 ) into (4-26), B - - Regroup te rm s such a way that all the term s containing the sam e state v ariab le are brought together, C - - Set all the coefficients of state v ariab les equal to z ero to solve for X^(t), D - - Equate 4 * ^ 1 (t) and ^ ' ( t ) with the respective rem aining term s containing { ^ | to solve fo r 4j(t), i = 1, 2. We re c a ll h ere that 4^(t) and ^("O account for the in teg ral payoffs of ‘ F irm s 1 and 2, resp ectiv ely . The solution re q u ire m e n ts suggest a sim ple procedure to solve (4-26) fo r the unknowns Xj(t), p 2 (t), p 3(t), and 4.(t). A fter substi- > ! < $ tuting the values of and a2 given by (4-11), equate the coefficients of Sj, S2, and G on the right side of (4-26) with the coefficients of corresponding v a ria b le s on the left side to solve for \^(t), jjl2 (t), and >;< P3 (t) fo r each p a ir of (o-j , a 2 ). Then ^ ' ( t ) and ^ ( t ) can sim ply be obtained by q u adrature. 57 Out of the total nine possible com binations of the optimal strateg ies (o-j ', ) given by (4-11), only four n o n -triv ial p airs of strateg ies are co n sid ered to determ ine the optim al values of shadow p ric e s (X .^, ^ 3)! since shadow p ric e s m ust be non-negative to be meaningful, the values of , i = 1, 2 corresponding to the negative shadow p ric e s a re elim inated fro m fu rth e r consideration. F irm l 's O ptim al Shadow P ric e s If F ir m l 's stock condition is Sj 2Y, then the optim al stock xl> strateg y is given by = Y in accordance with the solution of the H am iltonian (4-11). In this case we have only on the right side of the f ir s t equation of (4-26). It follows, therefore, that o ^(t ) m ust obey the equation, Xj '(t ) = 0 according to the solution procedure c. If, on the other hand, the stock condition is S^< Y, then in the f ir s t equation of (4-26). T herefore, in this case X j (t ) m ust satisfy \ j '( t) = aX.j(T). Solving the two differential equations we have Xj (t) = P t if Sj >V X 2 (t) = P Te aT if S 1 < Y (4-27) In solving the differential equations, use is m ade of the initial condition of X^(t ) given by (4-23), i . e . , X1(0) = \(T ) = P T 58 F irm 2's Optimal Shadow P ric e s Case 1: < (3G If the c u rre n t stock level of F ir m 2 is insufficient to operate production at full capacity, then the optim al stock strategy m ust be ^ (4 -H ) so that the differential equations for (a2'(t) and P3 (t) in this case are P2 '(t) = bix2 (r) P3 '(r) = 0 The solution of the equations is given by / \ -o b T (j.2 (t) = P Te M-o(t) = Pr . (4-28) Case II: S2 > (3G If the stock condition is such that the above status holds, ❖ then «2 = pG by (4-11). p 2 (t) and ^ ( t ) m u st *n ^ i s case satisfy the equations p 2 ' ( t ) = 0 jjl3 '(t) = bpf x2 (T) 59 the solution of which is given by \x3 (r) = P G + b(3PT T . (4-29) Now, ^ ( t ) and are given by quadrature, V T) = D o / T [P i :<((r) " Xl (<r)] 11 [ P 2 * (<r) " P i * (<r>] dcr f ^ P 2 (or) - (x 2 (<r) + C|x3(cr)J n [ P j ' (or) - P 2 " (cr)] dcr. The req u ired integrations can be c a rrie d out over any tim e in terv al 0 < t< A, w here A < tQ if |p^ (cr), i = 1, 2 | , \^(cr) and j (jl^(ct), jf = 2, 3 | a re known on that interval. However, the knowledge of 4^(t )» i = 1 » ^ is not n e c e ssa ry for the Nash eq uilib rium solution of the gam e, for both P^ and Q\’, i = 1, 2 can be determ ined absolutely by shadow p ric e s only. E q uilib rium Switching S trategies and T ra je c to rie s of State V ariables We re c a ll that the optim al stra te g ie s of F irm s 1 and 2 can be d eterm in ate only if the Lagrangian m u ltip lie rs o r shadow p ric e s are known. Since we have now resolved the H am ilton-Jacobi sy stem of p a rtia l differential equations for the unknown shadow p rices, the 60 optim al switching stra te g ie s can be determ ined by substituting the optim al values of the shadow p rice s in the e x p ressio n s for a-. (t) and P / ( t ) , i = 1, 2 (4-11) and (4-19) respectively. The optim al stra te g ie s as given by (4-11) and (4-19) a re called switching stra te g ie s, fo r the stra te g ie s do change as shadow p ric e s and state v a ria b le s change with tim e, i . e . , the optim al stra te g ie s reflect both the c u rre n t m arginal values and status of state variab les. While the determ ination of optim al pricing stra te g y involves a sim ple substitution of optim al shadow p rices in (4-19), knowledge on the c u rre n t status of state variables as well as optim al shadow p rice s a re req u ired to determ ine the optim al stock allocation decision, > ! < (t), i = 1, 2. But the c u rre n t status of the state v a ria b le s reflects in turn the re su lts of the optim al stra te g ie s of both firm s applied to the status of state v a ria b le s on the previous stage. The optim al tra je c to rie s of the state v a ria b le s a re d e te r m ined by substituting the known optim al stra te g ie s including pricing policies into the sy stem dynamic equations (4 -la) and (4-3a). It will be shown shortly that the tra je c to rie s are determ ined re tro g re ssiv e ly , i. e . , starting at the term inal stage t = T and working backw ards to the beginning of the game. This so -called dynamic program m ing m ethod is n e c e ssa ry because the n e c e ssa ry shadow p rice s fo r the d e te rm in ations of the optim al stra te g ie s and the values of state v a ria b le s are initially known only at t = T by assum ption. Due p rim a rily to the fact that there is only one state v ariab le S^(t) involved in the determ ination of the optim al stock allocation decision (t), the com plete resolution of the optim al 61 stra te g ie s for F ir m 1 is m uch sim p ler than that of F irm 2. We will, therefore, take up this case firs t. We will f ir s t exam ine the direction of change in S^(t) at t = T. F ro m (4 -la) and (4-11), the app ropriate dynam ic sy stem equation for F ir m 1 under optim al play m ay be ex p re ssed in term s of the re tro g re s siv e time variable and S j'fr ) = ■aofj (t) + d j (t) w here ^ ( t ) y if S j (t ) > V S j( t ) if S j t i X V , d i (T) = Do - d 2 (T) (4-30) L et us now re c a ll the assum ption that F irm 1 has enough production capacity to m ee t the total m ark e t dem and Dq, i. e . , D < a \ o (4-31) 62 w here aV is the full-capacity production rate of F irm 1. By making use of (4-31) we have for (4-30) at t = 0, i . e . , at the term inal stage of the game S x'(0) = Dq - d2*(0) - aY . < 0 if S^O) = Y Sx' (0) = Dq - d2*(0)>0 if S^O) = 0 . (4-32) The two extrem e conditions chosen above for Sj at the end of the optim al game, i . e . , F ir m l 's stock position at t = 0, are dictated by the econom ics of o ur p rice com petition m odel. At the end of the com petition F i r m 1 obviously does not need its stock level g re a te r than what it needs to operate the produc tion facility at full capacity. In fact, it is m o st likely that F ir m 1 is fo rced to sh are the m a rk e t with its new com p etito r under optim al play by the end of the com petition. This m ean s that F ir m l 's optim al stock level would in all probability be le s s than what it needs to m aintain full-capacity production (S^ (0) < Y). On the other hand, the optim al stock level of F irm 1 a t the end of com petition m u st be g r e a te r than zero (S j'(0) >0), for without stock on hand, i . e . , with out n e c e s s a ry input to production, the f irm would no longer rem ain in b u sin ess beyond the end of the gam e. In fact, it can never be optim al fo r e ith e r firm , against any stra te g y of the other, to allow its stock to be exhausted at any time during the com petition, as firm s without any am ount of stock have to go out of b u sin e ss. Thus rational play by 63 both p arties im plies that S^(t) > 0 fo r all tQ S t ^ T if S^(t) was positive at t = tQ. Inform ation m ade available by (4-32) allows us to draw optim al tra je c to rie s of F ir m 1. In F igure 4-2 the curves S j (t ) and jSj (t ) starting fro m Sj(0) =Y and Sj(0) = 0, respectively, establish the boundaries of optim al tra je c to rie s for F ir m 1. The term in al stock level re q u ire d fo r optim al play by F irm 1 has an in terestin g im plication for the f ir m 's optim al pricing strategy; in F igure 4-2 if the gam e s ta rts anywhere below the curve S j (t ), F irm 1 m u st charge a h ig h er price to delay the exhaustion of its stock and consequent b u sin e ss failu re; if, on the other hand, the game sta rts anywhere between the line Sj = V and the curve S j ( t ) ( the firm should low er the p rice to avoid a costly excessive stockpile. T herefore, all possible tra je c to rie s lying outside the a re a bounded by i T - RETROGRESSIVE TIME VARIABLE S, - FIRM VlSTOCK ON HAND 7 " MAXIMUM INPUT ABSORPTION CAPACITY OF FIRM 1’« PRODUCTION FACILITY “ 7 ALL OPTIMAL TRAJECTORIES MUST LIE BETWEEN THE CURVE8T, IT) ANDJ5, IT). A PARTICULAR TRAJECTORY DEPENDENT UPON THE INITIAL CONDITIONS OF FIRM 1 AT THE START OF THE PRICE COMPETITION. ARROWS INDICATE THE INITIAL DIRECTION OF CHANGE IN STOCK S, AT THE TERMINAL STAGE OF THE GAME. Figure 4-2. Optimal Trajectories of Firm 1 64 the curves and are nonoptim al fo r F irm 1. A particular tra je c to ry under optim al play depends on the condition of Sj at the beginning of the game. In all c a se s the optim al trajec to rie s m ust end between = 0 and = V to be m eaningful. It is interesting to o b serv e in the case of F ir m 1 that the >; < stock allocation strategy a^ (t) plays a ra th e r insignificant role insofar as the rational behavior of the firm is concerned. In fact it is dom inated by the pricing stra te g y P^(t) m o st of the tim e because F irm 1 m ust steer its stock lev el all the tim e during the gam e by adopting optimal pricing policy so that it m ay achieve the d esired stock level, i. e . , 0 < Sj(0) < Y, at the end of the game. The determ ination of optim al stra te g ie s for F irm 2 is a little com plicated because two state v a ria b le s a re involved in the determ ining of the optim al stock allocation. We observe in (4-11) • ' I ' that the optim al strategy (t) depends on the relative amount of available stock (t) with re s p e c t to the m axim um production capacity (3G(t), i . e . , S£(t) > pG(t) or $2 (t)< pG(t). This m eans that ignoring for the time being the influence of shadow p rice ^ ( t ) , which happens to be insignificant in this case, F ir m 2 m u st switch its stra te g ie s in re a l tim e depending upon w hether its available stock on that p a rticu la r day is g re a te r than what its new production capacity on that day can ahsorb. In this case the production capacity is a function of the state variable G(t), which generally changes with tim e. Because of this, the tra je c to rie s of G(t) as well as S^(t) play an im portant role in the determ ination of (t). This fac t constitutes a com plicating factor for the case of F irm 2. In the sequel, we will study in detail the 65 optim al tra je c to rie s of F ir m 2 's stock and m ark et sh are, i. e. , S2 (t) and G(t), in two different regions, i . e . , (t) >pG(t) and S^(t) < pG(t). If the tra je c to rie s c ro s s over fro m one region to the other any time during the game, we know that the optim al strategy (t) m u st be switched fro m one value to the other given in (4-11). The optim al tra je c to rie s near t = 0, i . e . , near the end of the gam e, are determ ined sim ply by substituting the optim al stra te g ie s given by (4-11) in the dynamic system equation (4-3a). This is so because, as we recall, the initial conditions are assum ed to be known. That is, we know exactly the c u rre n t status of state v ariab les as well as ^ ( t ) at T = i. e . , F ir m 2 finds itself either in 3^(0) = : pG(0) or in (0) < PG(0). It follows, therefore, that we know exactly what our optim al stock stra te g y a^ (0) should be by (4-11). If t is large, how ever, i . e . , away fro m the initial known conditions, we no longer can a sc e rta in [^ (t), S2 (t), and G(t), for they are functions of time t , and fu rth e rm o re S2 (t) and G(t) are influenced by pricing stra te g ie s * * P j ( t ), i = 1, 2 of both firm s . Since P^ (t ), i = 1, 2 depends on shadow p rices of state v ariab les, we will see shortly how the optim al tra je c to rie s of S2(t) and G (t) for Ihrge values of t a re affected by shadow p rice s. Region 1: S2 < PG If the initial stock condition is such that the above relatio n - > ! < ship holds, we have (t) = S2 (t) b y (4-11). 66 By substituting the optim al strateg y into the dynamic sy stem equation (4-32a)we obtain the dynamic equations under optim al play S2 '(t) = Do n [ P j V ) - p 2 *(t)] - bS2 (T) G '(t) = -C D o n [ p ]L V(T) - P 2 * ( t) ] < 0 (4-32a) w here P j ( t ) and P 2 ‘(t) a re given by (4-19), and all v a ria b le s a re e x p re sse d as functions of the re tro g re s siv e time variable t . We note that the sign of S2 '(t) depends on the relative size of F ir m 2 's demand with re s p e c t to the stock on hand, i. e . , S2 '(t) > 0 if D n * ( T ) > bS2 (r) S2 '(t) = 0 if DqII*(t) = bS2 (x) S2 '(t) < 0 if D n*(T ) < bS2 (T) . (4-33) T hese initial tendencies of S2(t) and G(r) n ear t = 0 are shown in the(S2> G)-plane in F igure 4-3. When S2'(t) > 0, there is an initial tendency away fro m the region S2 < pG. This is shown in F igure 4-3 by having the subregion S2 '(t) > 0 draw n c lo se r to the boundary line S2 = pG. To determ ine the behavior of S2(t) and G (t) for la rg e values of t , we m ust firs t study the behavior of shadow ❖ p rice p2 (T) anc* Pr i cin£ policies P^ ( t ) , i = 1, 2 under optim al play, for, depending on their behavior, the tra je c to rie s m ay c ro s s fro m 67 Sj <pG si > 0 o'<o S j'< 0 G * < 0 G • S*< 0 Figure 4-3. Initial Tendecies of S2 (r) and G (r) Near r = 0 one region to another, thus requiring a switch in the stra te g y ( t ) . Now, by the solution of the H am ilton-Jacobi equations (4-28), we know that *s an in creasin g function of t in Region 1 since both P,p and b a re positive. It m eans that once we have H -2 (T) ® ^or some 0 < t < A, (T) will rem a in positive for all t > A. In other w ords, once in Region 1 it is not possible to c ro s s over to the third region w here (T) = 0 [see (4-11)]. The econom ic im plications of being an in creasin g function of t are presum ably that in Region 1, where stock is relativ ely sc a rc e in com parison with production capacity, the shadow p ric e of S£ is p a rtic u la rly high in the e a rly stages of the game, and then tends to dim inish as the game p ro g re s s e s toward the end, for F ir m 2 is a new firm eyeing an ev er la rg e r share 68 of the m ark et. S2 is p a rtic u la rly in g reat need in the e arly days of its existence if S2 < pG. As the firm gets established, how ever, the demand for la rg e r stock becom es le ss urgent, fo r the firm can by this time afford to in cre ase its stock relative to production capacity by m anipulating price strategy. Next, consider the behavior over time of (h2 - h^) in this region. We have h2 " h l = ^2 ^ " CI jl3(t ) " = P T (ebT - eaT) - C P G = k 1(ebT - e aT) - k2 by (4-25a), (4-27) and (4-28) (4-34) where and k^ a re positive constants, and b -a > 0 by assum ption, i. e . , the rela tiv e production efficiency of thernew firm is assum ed g re a te r. Hence (4-34) im plies that (h^ - h j) is an increasing function of t . Now, by the growth pro p erty of (P2 - P j ), it is c le a r that ( P ^ - P^ ‘) m u st in c re a se m onotonically as (h^ - h^) in c re a se s ❖ m onotonically with t . This in turn im plies that DqH ( t ) m ust sje > J c d ecrease m onotonically to zero as P 2 ( t ) - P j ( t ) in c re a s e s with t . When this behavior of DqI1 is considered in conjunction with (4-32), 69 w e ob tain the fo llo w in g in te r e s t in g r e s u lt s on the b e h a v io r of S 2 (t ) and G ( t ): A - - The slope of the curve S2 (t), as a function of t, tends to become negative, and this tendency becom es m ore pronounced as t in c re a se s. B - - The negative slope of G(t) d e c re a se s as t in c re a s e s, the slope approaching zero eventually. If we take the second derivative of ,S2(t) by differentiating again S2 '(t) of (4-32) with re sp e c t to t we get S2 "(t) = DoII' ( P j V ) - P 2 V ) ) - bS2 '(r), which im plies that S2 "(t) = DoT I' (t) < 0 for all those t for which S2 '(t) = 0. (Note that n'(T) < 0 because II'(t) is a decreasin g function of t in the region. ) Thus we obtain the third im p o rtan t re su lt on the tra je c to rie s of S2 (t) and G(t) in Region 1, i. e . , C - - S2 (t) m ay have one local m axim um , but no lo cal m inim a in the region S2 < pG. These re su lts are shown in F igu re 4-4. It is in terestin g to observe in the fo rw ard -tim e sense, that under optim al play S2(t) in c re a s e s v ery rapidly in the e a rly stages of the gam e, and then the growth of the stock tends to level off as the game approaches the end. G(t), on the. other hand, in c re a se s at a m od erate rate in the beginning when S2 (t) is in creasin g rapidly, and then the growth of G(t) picks up when the growth of S2 (t) levels off toward the end of the gam e. This 70 behavior of (t) and G(t) appears to be rational in the region (t) < pG(t). The contrasting behavior of (t) and G(t) p resu m ab ly reflects the fundam ental fact that the m a rk e t demand which tends to reduce ( t ) is the v e ry factor that tends to expand G(t). F igure 4 -4 shows m any possible optim al trajec to rie s of S2(t) and G (t) in the r e t r o g re ssiv e tim e t. A p a rtic u la r trajec to ry of S2 (t) or G (t) depends obviously on the initial conditions at = 0. and G(r) m ay be continuous a c ro ss S2 (t) = PG(t). In p a rtic u la r, those tra je c to rie s originating sufficiently near S2 = PG in the su b-region 8 2 ' > 0 m u st c ro s s over the line into the region S2 > pG. It follows, th ere fo re , that in the forw ard -tim e sense a game that sta rte d out in As both S2(t) and G(t) are continuous functions of t, S2(t) G ✓ / --am — ► T S j, - 0 0 Figure 4-4.1 OptimalTrajectories of S2<(r):and G (r) in the Region S2 <( $ G 71 the region S£ > PG could end in the other region < PG. F u rth e r m ore, the re s u lt 3 im plies that among those tra je c to rie s that have their beginning in the subregion 8 2 ' > 0 there m u st be a trajec to ry which strik e s the plane Sg = PG at a point where $2 ' = 0 and then bounces back into S2 < PG. Suppose that the game s ta rts at such point of tangency on the plane S2 = PG. Then it m akes no difference for F ir m 2 w hether it moves into S2 < PG or into S2 >pG under optim al play, for the payoff of the gam e fo r F irm 2 is the sam e in eith er case. We re c a ll that the payoff function depends on shadow p ric e s and that the optim al shadow p ric e s are different depending on w hether the game is played in the region S2 < PG or in the region S2 > PG. It follows, th erefo re, that the payoff of the gam e rem a in s the sam e as long as the gam e is played on S2 = PG. In this case the game m ay end in e ith e r region. The economic im plications of the optim al pricing strategy of F ir m 2 m ay be deduced fro m the growth p ro p erty of P 2 - P j . Should F ir m 2 find itse lf in the situation S2 < PG, it musjt; try in the beginning to build up the sc a rc e stock S2 by raising the p rice at the expense of ignoring m ark et expansion G. But as the gam e p ro g re sse s, the firm can m anage to build up stock to a sufficiently high level, for its production efficiency is higher than that of its com petitor. C onse quently the firm can afford to low er the price to allow its m ark et to expand as the game p ro g re sse s. Region 2: S2 > BG In this region of abundant stock, the optim al stock policy m u st allocate as m uch stock as the production capacity can take in > ! < o rd e r to produce at full-capacity level, i . e . , = PG by (4-11). Recalling that stock has three uses— c u rre n t revenue generation, expansion of m a rk e t through sale, and stockpile for future dem ands — it is easy to understand intuitively why F ir m 2 should operate its production at full-capacity level if the stock condition allows it to do so. The dynamic equations under optim al play in Region 2 are given by substituting the optim al stock and p rice stra te g ie s into (4-3a), i. e . , S2 '(t) = D J I ^ P j V ) - P 2 *( t) ] " b(3G(T) G '(r) = -C D Q n [ P j * ( t ) - P2V)] < 0 (4- 35) w here the sign of ' (T) is determ ined by r> 0 if D II*(t ) > bpG(T) S »(t ) = < = 0 if D II* (t) = bpG(T) u O 0 if DoII*(t ) < bpG(T) (4-36) 73 The initial directions of change in G(t) and S2(t) are shown in F igure 4-3. 0 constitutes the subregion of transition in Region 2. Now, for large values of t , we m u st exam ine the behavior of shadow p rice s to determ ine the optim al tra je c to rie s. Before we do that let us study the effects of a com plicating facto r G(t) on the right- hand side of the fir s t dynamic equation (4-3 5). Unlike the dynamic equations (4-32) of Region 1, S2 '(t) is a function of G(t) in (4-35) and accordingly re q u ire s m ore careful exam ination of the behavior of S 2 (t) and G (t), as both of these m ay change with tim e. We will solve, therefore, the sy stem of differential equations (4-35) f ir s t to e sta b lish the functional relationship between S 2 (t) and G (t), which m ust be known fo r the tra je c to rie s of S2 (t) and G(t) to be determ ined. To this end, we f ir s t note that the solution of (4-35) m u st lie som ew hete betw een the solutions of the following two d erived sy stem s, for the term s containing DqII (t) in (4-35) m u st satisfy the basic p ro p e rty of the function II, i. e. , 0 < D II (t ) < D : o o (4-37) (4-38) [S2 '(t) = -bpG(r) hom ogeneous system G 1(t) = 0 inhomogeneous system [S2'(t) = Do - b(3G(r) G '(t ) = - C D o 74 A ccordingly these two sy stem s are solved f ir s t in the sequel, and then the solution of (4-35) is deduced fro m the solutions of the above two sy stem s. The solution to the hom ogeneous sy stem (4-37) is obtained f ir s t by solving the second differential equation to get G(t) = G(0) . (4-39a) Then, by sustituting G(t) in S^'(r) we get S2 '(t) = -b(3G(0) that can then be solved to obtain S2 (r) = -b(3G(0)t + S2 (0) . (4-39b) The solution of the inhom ogeneous sy ste m (4-38) m ay be obtained in a sim ila r m anner, G(t ) = - C D q t + G(0) S2 ,(t) = Do " b P [ - CDoT + G(0)] = fo - bpG(0)l + bpCDQ T . (4-40a) 75 Thus, S 2 (r ) = -ib (5 C D o T2 + [ d o - bpG (O )] t + S 2 (0) . ( 4 -4 0 b ) G (t) and S2 (t) a re plotted in F ig u re 4-5 as functions of t . F o r the sake of c la rity only a few optim al tra je c to rie s a re shown, each of which has its origin at a given initial condition (S2°, G°) in Region 2. It should be kept in mind, how ever, that there a re an infinite num ber of possible initial conditions in Region 2, and accordingly there are an infinite num ber of possible solution cu rv es like the ones shown in F igure 4-5. The solution of (4-37) gives a ho rizontal tra je c to ry that runs into Region 1 as t in c re a se s, starting at an initial point (S2°, G°) in Region 2. In this case only S 2 ( t ) d e c re a se s as t in c re a se s while . G ARROW INDICATES THE DIRECTION OF MOVEMENT AS T INCREASES. <S°G°) DENOTES THE INITIAL CONDITION OF STATE VARIABLES AT T - 0. SOLUTION OF HOMOGENEOUS SYSTEM SOLUTION OF INHOMOGENEOUS . SYSTEM Figure 4-5. Optimal Trajectories of Firm 2 in the Region S2 > |3 G 76 G ( t ) rem ains the sam e at its initial value G°. In the case of the inhomogeneous system , on the other hand, we can have m any possible tra je c to rie s for a given initial point in Region 2. To see this, consider S2 '(t) = Jdq - b(3G(0)J + b(3CDQ T fro m (4-40). If Dq > b(3G(0), i. e . , the total m a rk e t demand is g re a te r than the m axim um production ra te of F ir m 2 at t = 0, then 82' ( t ) is positive and in c re a se s linearly with t . If Dq < bpG(O), on the other hand, S2 '(t) is negative initially but becom es positive and in c re a s e s lin e arly with t after the point bpCDqt ^bpG(O) - Dq is reached. Since in all likelihood the m a rk e t would be sh ared with the monopoly firm (0 < DQII (t ) < D ) throughout the gam e, we know that the optim al tra je c to rie s of the original sy ste m (4-35) would behave m o re like those of the inhomogeneous sy ste m except that S2(t) of the sy ste m (4-35) would be a parabola that is fla tte r than Sj>(t ) 2 inhomogeneous sy stem (4-40b), for the coefficient of t - te r m is sm a ller in case of (4-35). (We know fro m the e le m e n tary geom etry 2 that the size of the coefficient of t - te r m d e te rm in e s the flatn ess of the parabola. ) If, on the other hand, E H 1 ' is so sm all that it is c lo s e r to zero, then the tra je c to rie s of (4-35) would be m o re like those of the hom ogeneous system . Thus, one can deduce fro m the above d isc u s sion an interesting im plication about the nature of the tra je c to rie s of F ir m 2 in the region Sg > pG: The sm a lle r the m a rk e t dem and of F ir m 2, the g re a te r the chance that the tra je c to rie s of F ir m 2 s ta rt 77 in Region 1 (S£-< PG), c ro ss over the line S£ = pG, and then move into Region 2 as the game p ro g re s se s. If, on the other hand, F ir m 2's m a rk e t dem and w ere always substantial, then the tra je c to rie s would rem ain in Region 2 during the entire gam e. In this case the gam e does, however, end with a la rg e r m a rk e t share and a sm a lle r stock on handl It is im portant to observe that the above effects of relative m a rk e t demand for F ir m 2 depend on the pricing stra te g ie s of F irm s 1 and 2. L et us exam ine m ore closely the behavior of optim al p rice stra te g ie s over time in this region. H ere we have h2 " h l = ^2 ^ ' C^3<t > " = P T - C |P G + bpP T r j - P Te aT by (4-27) and (4-29) , and (h^ - h^) is, therefore, a decreasing function of t. Since it 5 |< > ! < d e c re a se s m onotonically without bound as t in c re a se s, does likew ise by the growth p ro p erty of - P j . T his m eans that D II (t ) m ust in cre ase to D as t in c re a se s. The econom ic im plica- o o c . tion of all this is that F ir m 2 m ust lower its price substantially so that it m ay capture a large share of the m ark e t dem and in the e a rly stag es of the game if its stock condition is given by > pG. In other w ords, F ir m 2 m ust concentrate on expanding its m a rk e t share by selling m ore in the beginning, and then as the game p ro g re s s e s and its m a rk e t expands, F irm 2 m ust gradually level off the reduction of stock tow ard the end of the game. These h e u ristic re su lts on the behavior of shadow p ric e s are substantiated when we consider, in ❖ conjunction with the dynamic sy stem (4-35), that D q I I ( t ) is an increasing function of t , i . e . , G '(t) is negative and the negative slope in c re a s e s with t , while 82 ' ( t ) is positive and its positive slope in c re a s e s with t . This concludes our solution p ro c e ss . We are now in a position to know the entire subsequent behavior of both firm s under optim al play once the game s ta rts at any a rb itr a r y initial point (t , S j 0, S2°, G°). CHAPTER V INTER-REGIONAL TRADE Introduction i T rade among the different regions of the sam e country in a h a rro w sense, and of the w orld in a wide sense, was com m on even in the days of Adam Smith. But it was R icardo who f ir s t studied analyti- j ; ! cally the international trade that has its base on the econom ic special- j ization in the sense of Adam Smith. The c la s s ic a l international trade j 1 I theory of R icardo m akes certain assum ptions of which two im portant ones to us h e re are (1) the specialization of a region o r a nation is derived p rim a rily fro m the natural endowment of sc a rc e re so u rc e s, j i i j I ;and (2) som e re so u rc e s a re perfectly m obile within a national boundary ! i : i but im m obile a cro ss cou ntries. In p a rticu la r, labor, an im p ortant | jresource, was considered im m obile a c ro ss national boundaries and p erfectly m obile within a given country. A gainst the background of this p recep t of the c la ssic a l international trad e theory we see today, jat lea st in p a rts of our m odern-day world, ra th e r c o n tra ry economic phenom ena, e .g ., relatively fre e m ovem ent of labor a c ro ss countries I in the E uropean com m on m arket, and stran g e im m obility of labor in J A ppalachian and other d epressed regions within the United States. j 79 The traditional economic theory of in ternatio nal trade has | also failed to recognize technology or technical know-how as a sc arc e econom ic re so u rc e in which a nation m ay specialize for trade with other nations. It is, however, com m on knowledge today that there a re at lea st two g rea t economic pow ers, Japan and W est G erm any, whose I econom ies thrive on the specialization in advanced technologies. In fact, in the case of Japan, this sm all country with a trem endous popu- ! i | lation p re s su re , with little or no re so u rc e s to speak of other than i j j labor, has successfully rebuilt its economy, devastated during World ! I W ar II, p rim a rily by engaging in a g g ressiv e intern ation al trad e that I ; i had its base in high technology. i j j Tapping such sc arc e re s o u rc e s as technology and cheap j 1 ! labor for trad e in this m anner is no longer lim ited to within a given j : I jcountry. Advanced econom ies such as that of the United States that isuffer fro m either the short supply of or high co st of labor are today i I jreaching to every co rn e r of the w orld for cheap and abundant labor i I jsources. They a re doing so because it is not only n e c e ssa ry but j alm ost im p erative that advanced countries se c u re for them selves jcheap labor sources outside of their national boundaries in o rd er to ! survive in the ever-intensifying in ternational trad e com petition. To cite a few specific cases, F airch ild C am era and M otorola, two US g ia n ts in the field of sem iconductor devices, alread y have fully opera- | jtional plants in South K orea to utilize cheap but w ell-educated indige nous labor there. In addition, they also have plants in F o rm o sa and Kong. G eneral M otors, another US giant, is planning to set up aj i I la rg e auto assem bly factory in South K orea. What is interesting to j fHong ! 81 observe h e re is that these com panies a re in Korea, for instance, not to exploit the local m arket, nor as a re s u lt of the K orean governm ent's efforts to build up im port substitution in d u strie s in Korea, but m erely to utilize cheap labor. In fact, v ery little of their product, if any, is consum ed locally, but ra th e r is e ith er brought back to the United States for dom estic consum ption or exported to third countries. D irect foreign investm ents of the s o rt discu ssed above are certainly not the only driving fo rce that one can observe in rapidly | ; (developing countries today. To take the K orean experience again as an exam ple, South K orea im ports a v a rie ty of sem i-finished or intermedi-i ate goods fro m Japan, e .g ., synthetic fib ers for textiles, electronic ! i ; p a rts and com ponents for TV and radio, e t c ., m akes final o r finished : i j (products using dom estic in d u strie s, and exports them to a third (market such as the United S tates. The developm ent of im port sub stitution in d u stries adds fu rth er im petus to the econom ic developm ent ; jof less developed countries. Im p o rt substitution industries a re j | | developed as joint ventures with e ith er Japanese or A m erican firm s in ; i I jthe case of K orea. All three of these fo rm s of "trad e" betw een I advanced econom ies and developing econom ies have one thing in com - jmon, i . e . , the advanced countries "export" technology including j technical know-how and capital, while developing countries "export" (cheap labor. Thus, though labor is indeed quite im m obile a c ro ss jcountries in A sia in the physical sense, this im po rtant facto r of jproduction has turned out to be v e ry m obile indeed a c ro ss national boundaries, thanks to m odern-day tran sp o rtatio n and com m unications, i in the sense that it is indirectly "exported" through lab o r-in ten siv e products to such advanced nations as the United States. Sim ilarly, to the extent that le ss developed countries le a rn the new technology through th eir "tra d e " with the advanced nations, the sc a rc e re s o u rc e of technology is "exported" to developing countries fro m advanced j econom ies. In the sequel we w ill consider this unconventional trade involving labor and technology as "exportable" com m odities. We w ill be p rim a rily concerned with an in te r-re g io n a l trade in a given country.! N eedless to say, how ever, the re su lts of our analysis apply ju st as w ell to the in tern atio n al trade. We m ust, however, observe one dif ference, as we w ill be in te re ste d in a P areto -o p tim al solution later, I that it is m uch m o re difficult for nations to cooperate than for regions j | ; of the sam e country. We chose to m odel and study the unconventional J trad e because such trad e seem s to play an im portant role in the ! l developm ent of le ss developed regions and countries today. In som e j I ! sen se J. S. M ills' "vent for su rp lu s" aspect v is -a -v is R icard ian ! j "com p arative cost" doctrine of the c la ssic a l theory of international l I trad e (Myint 1958) seem s to apply to the kind of trade under c o n sid e ra tion. It is the surp lus of labor in underdeveloped regions and the j"surplus of technology" of advanced regions that seem to prom ote both the developm ent of the le ss developed regions and the growth of I I ^advanced regions. j It is quite conceivable and understandable, p a rticu la rly in i j the light of seem ingly irra tio n a l im m obility of labor in som e a re a s , j I i | th at the underdeveloped regions, once given an initial "shot in the a rm '| by such trade, m ight gain a badly needed m om entum and then graduallyj 83 take off on a sm ooth developm ent path through H irschm an-ty pe "linkage effects" (H irschm an 1958) or the sp read effects of M yrdal (1957). H irsc h m an linkage effects p articu larly would bring about the gradual in d u strializatio n that w orks its way backw ard fro m the "final touches" j stage to local production of interm ed iate, and finally to the local p ro duction of basic, in d u strial m a te ria ls . This kind of developm ent p a ttern is notable p a rticu la rly on today's international scene, e .g ., the s o rt of trade under consideration has certainly m uch to do with the p re se n t econom ic developm ent in South Korea, F o rm o sa , and to a le s s e r extent Mexico. Some southern regions of the United States appear : lately to be benefitting fro m such trade patterns too. The econom ic developm ent p attern that we are considering h e re is obviously quite different fro m that of N u rk se's balanced growth; j (N urkse 1953), for exam ple. The basic idea of the balanced growth th e sis is that the le ss developed countries or regions can achieve much; ; j 'desired econom ic developm ent only by developing a whole line of or isubstantial range of in d u stries sim ultaneously so that m axim um | advantage can be taken of the com plem entarity among different indus tr ie s to sustain the growth. In spite of the seem ingly im possible | p ro b lem of raising larg e sum s of n e ce ssa ry capital on the p a rt of the developing countries (capital is known to be the s c a rc e s t re s o u rc e of jail in m o st of the developing countries) there still seem s to exist today !a p e rs iste n t tendency, at lea st on the p art of the governm ents involved, ito bo rro w larg e sum s of capital fro m foreign so u rces and invest in | i v a rio u s in d u stries sim ultaneously, often including such basic indus- ! trie s as steel. Thus what we observe today in m o st of the less 84 | (developed countries appears to re s u lt fro m a m ixture of these two kinds of developm ent. R egardless of the pros and cons of these two stra te g ie s of economic development, the trad e p attern that we will study next undoubtedly m e rits the attention of governm ent officials, businessm en, and econom ists alike of both advanced and less developed countries, for it seem s to work quite w ell in many p arts of the world today. F orm ulation of Model F o r the sake of sim plicity, we consider a sim ple case of ; i | fwo regions within a given country trading with each other two com- I ; m odities, i . e . , technology and labor on the basis of resp ectiv e j specialization. Suppose that Region 1, an underdeveloped region, im p o rts sem i-finished products of high technological content from | f Region 2, an advanced region, at a ra te o(t) on day t, and Region 2 j im p o rts finished products from Region 1 at a rate (3(t) on day t. If we i ! | jdenote by X(t) the quantity of finished products that Region h as on hand ; jon day t, y^(t) the quantity of finished products Region 2 has on hand on iday t, and y ,(t) the quantity of se m i-fin ish ed products Region 2 has on I ' hand on day t, then the dynam ics of trad e betw een the two regions m ay j be d escrib ed by the differential equations { j ! X'(t) = a«(t) - p(t) - C x(t) ! | y'(t) = p(t) - C2(t) I y ^ t) = b(t) - «(t) (5-1) | 85 Where jc^(t), i=l, 2 J denote the ra te s of consum ption for Regions 1 and 2, a constant production coefficient for Region 1, and b(t) a tim e- i varying rate of production of Region 2. ' A variable production ra te is assu m e d for the advanced region on account of high technological change, while the less devel oped region is assum ed to have a fixed input-output coefficient to account for relativ e low technological change. We assum e that all ; {state v ariab les X(t), y^(t) and y2(fc ) are always non-negative for {obvious reasons, and that both consum ption ra te s and im port ra te s are: {assumed to be constrained by inequalities O C. (t) < C1 O < ff(t) < a 1' ' m ax m ax ° - C2( t ) S C m ax O s |J(t> s Pm ax (5-2) 2 w here C m ax denotes the m ax im um r a t e1 of consum ption possible by {the population of Region i (the gam e period is such that the effects of j ! I population growth m ay be ignored), a the m axim um rate of im port i rxicix i possible for Region 1, and Pm ax the m axim um possible rate of im port j j for Region 2. We assum e that the constrained consum ption ra te s and { im p o rt ra te s are policy v a ria b le s in the sense that regions m ay m anipulate them to prom ote the w elfare of resp ectiv e regions. The lim itations im posed on im p o rt ra te s a re presum ably due to such i i {physical considerations as road capacity, the num ber of trains and Itrucks available for the trad e, etc. In the case of international trade, j i {imports a re often re s tric te d by such non-physical facto rs as quota, {import duties, and other a d m in istrativ e re d tape. j ’ .........................................................................................86 .... Corresponding to the flow of goods between the two regions, consider the flow of m oney betw een these two regions, i. e . , the balance of payments of trad e R'(t) = P ^ t ) p(t) - P2(t) a(t) (5-3) where R(t) denotes the quantity of money that Region 1 has in its pos session on day t as a re s u lt of trad e with Region 2 (the trad e revenue of Region 1 on day t), P^(t) the p rice Region 1 charges fo r its finished- goods on day t, and P2(t) the p ric e Region 2 charges for its se m i finished products. In the c ase of international trade, the balance of j j payments might be m e a su re d e ith er in US d ollars or in gold. Since we j | j a re considering a sim ple tw o-regio n trade, another equation for the j i i balance of payments for R egion 2 is not n ece ssa ry , for in this sim ple i ! pase what one region gains is a loss to the other region insofar as the | balance of payments is co ncerned. We note h e re that with (5-3) each j region acquires one m o re policy variable, i . e . , P.(t), i = 1, 2. ; i i | We assu m e now that Regions 1 and 2 try to m axim ize the utility functionals and U^, resp ectiv ely , by m anipulating all the i I i i policy variables available to them , i ! j v p I V = h x( r ( T ) ) + h-2 (x(T )) + J C j(t)dt ! to ! I | U2 = Vj (r( T )) + v2 (y2(T)) + J C2(t)dt (5-4) ; ! to i where the utility functions [x^( •), i=l, 2, and •) a re increasing func tions of resp ectiv e argum ents, while v^( •) is a decreasing function of R(T), Two im portant assum ptions a re m ade in the form ulation of the utility functionals: (1) consum ption utility is additive over tim e and is proportional to the absolute ra te of consumption, and (2) both regions j 1 I have basically the sam e attitude tow ard money, i . e . , the utility of ; i money is com parable in both regions in the sense of Von Neum ann and : M orgenstern. To be m o re re a listic , different weights m ight be | assigned to the different te rm s of the utility functionals. F o r the sake ! of sim plicity, how ever, we will use unweighted functionals. ! The problem of m axim izing the utility functionals (5-4) ! | I ; i subject to the differential equation constraints (5-1) a n d (5-3) and the j inequality constraint (5-2) is c learly a non zero -su m differential gam e | of the so rt discussed in C hapter III. This problem may, therefore, be! : i ! I solved for a Nash equilibrium solution in a m anner quite analogous to : i the price com petition m odel of the preceding chapter. Instead of ! ! j solving the problem , how ever, we will study qualitatively the nature ; ; i : i :of this model, assum ing the existence of an equilibrium solution. We j j ' | will do this, keeping in m ind that our ultim ate purpose h e re is to I | i |determ ine the conditions under which the Nash equilibrium solution of i 1 jthe above problem is also P a reto -o p tim a l. Our in te re st in the P a re to - ! joptimal solution derives fro m the fact that there are m any re a l-w o rld situations w here regions can benefit fro m close cooperation among them . F ro m the planning viewpoint, this kind of cooperation poses a 1 i ;real challenge to fre e econom ies, and it m ight very well be a practicalj n ecessity in planned econom ies. j i 88 i i | We observe in (5-1) that there is no com pelling reason why Region 2 has to cooperate with Region 1 by adjusting its production rate to the input req u irem en ts of Region I's industry as long as Region 2 can sell its sem ifinished products to other regions. Indeed, Region 2 m ay even w ish to stockpile its products against future dem ands instead] of supplying Region 1 with what it needs so badly to keep its production: going. In this case it is obvious that the Nash equilibrium solution cannot be P a reto -o p tim a l. If, on the other hand, Region 2 has no other cu sto m e rs besides Region 1 for its products, it is b e tter off to cooper-' ate with Region 1 out of its own in te re s t by adjusting the output as perfectly as possible to the need of Region 1. In this happy situation i we m ight have a p erfect coordination betw een the two regions, promot-! i i ; f ing the w elfare of all p arties concerned though each party p ursues only j its own in te re st. It, therefore, follows in this case that a Nash equi- 1 ! | lib riu m solution m ight very well be P a reto -o p tim a l as well. j Modified Model L et us assu m e that b(t) = ^(t), t < t < T, for the case of per fect cooperation. F o r the sake of sim plicity and also to elim inate in I (5-4) the utility te rm s that a re dependent on the term in a l stocks, we also assu m e that Region 1 consum es all its products left over after i l export and that Region 2 consum es all finished products it-im ports I I |from Region 1. The la tte r assum ption m ay not be n e c e ssa ry for the j ! t N ash equilibrium solution of our game to be P a reto -o p tim a l. With I I j these assum ptions, the dynamic system (5-1) becom es ! 89 CJ(t) = a«(t) - (3(t) ' C ^ (t)= p (t) (5 -la); If we assum e that v ^ R (t)j = -k|o,j^R(T)^ in (5-4) w here k is a positive j constant, then we obtain a sim p ler e x p re ssio n for the utility functionals T U x = h (r (T)) + / (t)dt T t o •T U2 = -k|jL(R(T)) + f C2(t)dt. (5-4a)! to i ;The assum ption v ^ R (T )^ = -k|o,j^R(T)j seem s to be reasonable, for R(T) is the trade revenue with which Region 1 ends up at the conclusion! of the gam e, and both regions a re assu m ed to have basically the sam e j attitude toward m oney. In particu lar, if k = 1 it im plies that both ! j : jregions have an identical utility of m oney. In this m odified gam e we ! | a r e now in te reste d in the behavior of the consum ption utility and the jbalance of paym ents only. Solution of the Modified Game Our m odified problem now is to m axim ize utility functionals (5-4a) subject to the differential equation co n strain ts ( 5 - la) and (5-3) 'and the inequality constraint (5-2). L et us solve the problem for a i iNash equilibrium solution. To this end, we f ir s t obtain two appro- i p riate H am iltonians H 1 = C 1 + Ki ( aa(t) - P(fc>) + X2p(fc) + X3 ( p i (t)p(t) - P2(t)a(t)) = Cj + (& \1 - P 2\ 3 )a + (x2 - x x + P ^ 3)P H 2 = C 2 + u ^ a a f t ) - P ( t ) j + 0J2 P (t) + w3 ( p i (t)P(t) - P2(t)tt(t)) ! = C2 + (acoj - P2co 2)ff + (co2 - + PjCo^JP (5-5) ! i j ! ! t j i I for Regions 1 and 2, respectively. Region 1 trie s to m axim ize Hj with •respect to its strategy variables ^a(t), P^(t)^, and sim ila rly Region 2 j |tries to m axim ize H2 with re sp e c t to ^P(t), P2(t)j. (X^, X 2> X^) and | ,|(coj, co 2, c u >2) a r e L agrangian m u ltip lie rs fo r Regions 1 and 2, jrespectively. It is now apparent fro m (5-5) that the optim al stra te g ie s which m axim ize the respective H am iltonians a re given by « * (t) = P*(t) = a if aX, - P0X _ > 0 m ax 1 2 3 /r M 0 if aXj - P2X3 < 0 p “o; Pmax lf "2 - “ 1 + P l “ 3 > 0 0 if u 2 - U j + P j m , < 0 ' " j ...................'..'... " ........................................................' ................... 9 1 ; ! ; Notice in (5-5) that the im p ort strateg y v ariab les have no influence on the resp ective H am iltonians if the coefficients of the v a ria b le s are zero. This constitutes an un interesting triv ia l case, l which we choose to ignore in the subsequent analysis. The optim al ; im po rt stra te g ie s a and (3 given by (5-6) and (5-7) m ay be given an i in terestin g econom ic interp retation ; according to the optim al strategy, ; ! Region 1 should im po rt the sem i-finished products fro m Region 2 at a ! m axim um rate only if the price that Region 2 charges for its goods is j less than Region l 's shadow price of its derived consum ption, d is counted by the shadow price of the trade revenue, i . e . , aX^/X^ > 1 ?2» and im p o rt none otherw ise, i . e . , aX^/X^ < Pg* * i > ! < i One can draw sim ila r im plications fro m (3 (t), i. e . , Region ! i i j i2 should im port the finished goods at a m axim um ra te as long as the j p ric e of finished goods is less than the excess of the shadow p rice of j j . j its own consum ption over the shadow price of Region l 's consum ption, j | j jafter p ro p erly discounted by the shadow price of Region l 's trade j jrevenue, i. e . , > P j (Note that co^ < 0 because is Region 2's shadow p rice of the m oney that Region 1 p o sse sse s as a re s u lt of trade), and im p o rt none if the price is so high t h a t ^ - co^/cjO ^ < Pj» We note h e re that the demand is not elastic with re s p e c t to price because, for instance, Region 1 buys at the m axim um ra te < * m ax jfor a range of p rices, i. e . , 0 < < aX^/X^ though the decision to ! im port is related to the price that the ex p o rter ch arg es. By assu m p tion, im p o rt is in fact re s tric te d only by non-price facto rs such as the ! i available capacity of transpo rtation. L et us proceed now to the solution of the H am iltonians for ❖ > ! < the optim al price strateg ies, i . e . , P j(t) and P2(t). In o rd er to m ax im ize H j and H2 of (5-5) with re s p e c t to P^(t) and P2(t)» respectively, i we m u st satisfy the following: j ) ' ! m a x I M P ) = m a x ( \„ - \ 1 + P - A - J p | ; p ± p ^ r 1 J 1 1 j m ax n_(a:) = m ax (aco. - P-coAtt. (5-8) I P» ^ P0 1 1 C , 6* We observe above that the function 11^((3) that is to be m ax- j im ized with re s p e c t to P^ by Region 1 contains a strateg y variable (3 ; that belongs to Region 2. S im ilarly the function n^ff) of Region 2 bontains a strateg y variable a that belongs to Region 1. This m eans j ' i that each region m ust try to m axim ize its resp ectiv e function by ; i | m anipulating its p rice strateg y against the best im p o rt strategy of its j opponent. This is precisely what a Nash equilibrium solution req u ires. j | L et | n ^ p * ) '= ( \ 2 - | n 2 ( o r ) = ( a w j - P 2 c o 3 ) q - ‘ | i I I I ! % # w here a and P a re the optim al im p o rt stra te g ie s given by (5-6) and (5-7), respectively. It is c lear that II (p ) is an increasing function of j 1 K on ! ■ C O -i “ j o 0. ; as (3 = 0 for such high price. Hence it follows that the best pricing ! s j c strateg y of Region 1 against the best im port strategy p of Region 2 is ! to charge the m axim um possible p rice which would still allow Region 2j to im port at m axim um ra te . Since Region 2 would im p o rt at m axim um | irate as long as < (coj - co^/cog, it follows that Region 1 m ust charge a| price as close to^jj - w^cjg as possible, i . e . , ‘ =^tOj - co^cog. j It is not an optim al strateg y for Region 1 to charge a pro- | hibitive price unless the shadow price of its own consum ption is so | I high that Region 1 w ishes to consum e all its finished products. II_(<* ) j ! > J e is also an increasing function of P - on 0 < P_ < aX.1/ \ of for a - a ° 2 2 1 3 max ! | j there by (5-6) and < 0. ^ ^ ^or ^2 > Thus it j > ! < ! follows that P^ = aX^/X^, for this p rice guarantees a m axim um sale I at a m axim um p rice for Region 2. We obtained the above resu lts under I the assum ption that X ^ > 0 an(i W g < 0. We know fro m the Ham iltonians ! | |(5-5) that X^ is the shadow p rice of Region 1 for its trade revenue, j I i. e . , the amount of m oney that Region 1 has in its p o ssessio n as a financial rew ard of trading with Region 2. It is obvious, therefore, i that X, > 0, for X_ cannot be negative or zero as long as Region 1 has a j 3 3 positive attitude toward m oney, cjg is, on the other hand, the shadow S price of Region 2 for the trad e revenue of its opponent, and therefore m ust be negative, for Region l 's financial gain is n e ce ssa rily a finan- ! cial loss to Region 2 since th ere is no third party involved in the trade.! ! 94 ❖ # We note h e re that the optim al strateg y p airs (a , ) and 5 } : i(P , P2 ) are determ in ate up to L agrangian m u ltip liers or shadow p ric e s. We need, th erefo re, to determ ine the optim al shadow p rices for the com plete solution of the gam e. The shadow p rice s under Optimal play are obtained by solving the H am ilton-Jacobi equations, Vt1 + ™Pl l Cl + (aVC1 - P2*VR)“ + ( v c 2 ' v c x + p i v r ) p |C 2 + K - P 2V r ) » * 2 . m a x t + p , p 2 . - . - j + (v c - v c , + p i* v r ) H =0 (5- 9) w here ^ri 9V1 • , , t ~ 8t * 1 ’ V1 = jj l (R ) + V1 (t, Cv C2) V2 = -kji(R) + V2(t, C j, C2) We re c a ll that the H am ilton-Jacobi equations a re derived from the H am iltonians (5-5) by substituting the optim al values for the strategy v a ria b le s. A lso the L agrangian m u ltip liers (X.^, \ 2> \^ ) and '(coj, a > 2, co^) a re rep laced by the equivalent m arg in al values of the 95 state v ariab les (C^, C^t R), i . e . , '9Vl 9 V1 9V_\ . /aV2 9V2 9V g c ^ a c 2’ 9Ry ana laCj* a c 2’ 9R/* respectively. The value functions | V*, i = 1, 2 j m u st satisfy the Hamilton- Jacobi equations (5-9) and the initial conditions v V , C r C2, R) = [x(R) V2(T, Cj, C2, R) = -k[x(R). (5-10)1 Once the optim al shadow p ric e s a re determ ined, then the com plete j solution of the gam e, i . e . , the Nash equilibrium stra te g ie s, is obtained by substituting the optim al shadow p rices in the ex p ressions for the optim al stra te g ie s given above. | If we substitute the expressions of the optim al price s tra te - ; * $ gies P j and P2 into the H am ilton-Jacobi equations (5-9) and re a rra n g e the te rm s, making use of the fact that the optim al im port i I stra te g ie s and p rice stra te g ie s a re determ ined sim ultaneously, we i obtain V2 - c * c 1 1 * 1 7 1 * v t + c i + v c p ‘ v c p + " ^ k p = 0 1 V ^ + C2 + a V2 / I m aV ^ a* = 0 (5- 11); = -k V ^ because v^(R) = ' I 1 * -kp,j(R) by assum ption, a and P of (5-11) a re given by a = r = a if aV * + kaV* > 0 m ax C. C. 0 otherw ise v 2 v 2 i i c r 2 6 if - v J ; - — ---- > o rm ax C , C1 k 0 «. • otherw ise. Now, if we m ultiply the firs t equation of (5-11) through by k, J add the resu lta n t equation to the second equation, and re a rra n g e terms,! | we obtain (mV* + V*) + (kC1 + C 2) + (ka* - p*)(kV^ + ) 1 2 + p*(kvJl + V* ) = 0 . C1 2 (5-13) jWe w ill show that (5-13) is the H am ilton-Jacobi equation which the i freighted sum of the value functions of the two regions m u st satisfy. | 97 | Let us define a new value function W as a weighted sum of the value functions of Regions 1 and 2: W = kY1 + V2. (5-14) The value function W m ust satisfy its H am ilton-Jacobi equation ; Wf c + (kCx + C2) + (act* - p*) Wc + P* Wc =0 (5-15) 1 2 ❖ s j e ' that can be shown to be identical with (5-13), w here a and (3 a re i i | given by ❖ a = p* O ' if aW „ > 0 m ax C j j 0 otherw ise Pmax if WC > WC ' 2 1 (5-16) 0 otherw ise j iNotice that (5-16) is also identical with (5-12) by the definition of the i value function W (5-14), Thus the value function W is indeed a weighted j isum of the value functions of Regions 1 and 2 that m u st satisfy the jHam ilton-Jacobi equation (5-13). Solution of C ooperative Game j j Now suppose that a higher governm ent having ju risdiction I over both regions decides to m axim ize the weighted sum of consum ption' | 98........ utility functionals of Regions 1 and 2, internalizing the balance of payment of trade betw een the two regions, i. e . , to m axim ize the j aggregate consum ption utility functional m a x a, (3 f T = J t + c2(t) I at (5-17). subject to Cj(t) = a«(t) - P(t) C ^ t) = (3(t) 0 ^ <y(t) < a , 0 S p(t) < p ' ' m ax’ ' r m ax jThen, the H am iltonian fo r this com pletely cooperative gam e is | H = (kCj + C2) + P l (ao - p) + P2 p = kC j + + Pjao-+ (P2 - Pj)P • (5-19] It is c lear fro m the H am iltonian that the optim al im port strateg ies, a" and P , a re given by * (t) = P (t) = a if ap, (t) > 0 m ax r l ' ’ 0 otherw ise Pm ax if P2(t) > Pl (t) I 0 otherw ise . < 5- 2°> ! 9 9 Now the com plete solution of a and p req u ire s the determ ination of shadow p rice s under optim al play. The optim al shadow p rices a re ;obtained by solving the H am ilton-Jacobi equation of the game. To this ;end, let us define a value function W' = w°(t, c v c 2). Then, the value function W° m u st satisfy the Hamilton- Jacobi equation and the initial condition Wt ° + k C x + C2 + ( a / - P*)W£ + p*W£ = 0 1 2 W°(T, C r C2) = 0 (5-21) I w here * a = p* = a if aW° > 0 m ax C j 0 otherw ise j p if W° > W “ I 2 C 1 (5-22) 0 otherw ise. ! Both the fo rm and the initial condition of the value function W° are dictated by the utility functional (5-17). The optim al im p o rt stra te g ie s a and p of the cooperative gam e have an interesting i (economic im plication: Region 1 im po rts the in term ed iate products from Region 2 at m axim um ra te Q 'm ax as long as the shadow p rice of Region l 's consum ption is positive. Then, the finished goods I .............................. *.................. 1 0 0 ............ produced by Region 1 a re divided between the two regions in accordance with the relativ e shadow p rices of consum ption in Regions 1 and 2. jSuch im p o rt policies of the two regions se em to be reasonable, p a r ticu larly in the light of the fact that the goal of the higher governm ent is to m axim ize the combined consum ption utility of both regions. I W e observe that the H am ilton-Jacobi equation (5-15) of the ! I 'original com petitive game is identical with that of the com pletely j i ; Cooperative game (5-21). Since the H am ilton-Jacobi equation of a gam e m u st be satisfied by the value function of that gam e under j optim al play by all p a rties, it follows that the Nash equilibrium s tr a t- : egies of the com petitive gam e a re identical with the P a reto -o p tim a l j (strategies of the com pletely cooperative gam e. Recalling that the j | I iH am ilton-Jacobi equation (5-15) is to be satisfied by the weighted sum ! I ’ I of the individual value functions of the two regions, we know that the I ! i I Nash equilibrium solution of the com petitive gam e is P a re to-optim al j i | jin the sen se that it m axim izes the weighted sum of the utility func- | i j tionals of both regions. j In our com petitive game, the fact that the two regions have opposing m onetary objectives would not prevent them fro m employing [ P a re to -o p tim a l strateg ies so long as both regions have the sam e basic attitude tow ard money, for their m a rk e t c ircu m stan ces a re such that lit would pay fo r both regions to cooperate and then to divide the fru its | |of cooperation betw een them on m utually acceptable te rm s. We re c a ll i 'that two assum ptions p articu larly im p o rtan t to obtain the P a re to - j I ■ j 'optim al solution for our tw o-region trad e gam e w ere (1) both regions ! ! 101 ' ” | have com parable m onetary utility, i . e . , Vj(R) = -k|j,j(R), and (2) Region 2 adjusts perfectly its production of interm ediate goods to the input req u irem en ts of Region l 's industries, i.e ., b(t) = a(t). The ; fir s t assum ption seem s to be quite reasonable, for different people do | in general have com parable attitudes toward money, p a rticu la rly those i living in the sam e country. The second assum ption, on the other hand,; im poses ra th e r stringen t req u irem en ts, because except for very r a r e ' C ircum stances, in d u strie s in a region usually enjoy m a rk e ts extending ; to sev eral regions. In other w ords, one would be h a rd put to find a rea l-w o rld situation in f re e -e n te rp ris e econom ies today w here regions ; : i depend exclusively on each other in the sense we are discussing h e re , j ; j Accordingly, there usually exists no com pelling reaso n why one regionj ; | has to cooperate with another region in the sense of P a re to optim um , j The com plete cooperation n e ce ssa ry to achieve P areto optim um m ay com e about in the fre e -e n te rp ris e system only when a ! higher authority com pels subordinate regions to do so. The re su lts j we obtained in this chapter a re p a rticu la rly in terestin g fro m the I | jplanning viewpoint, for they im ply that a higher governm ent should j i i jcreate a situation that tends to induce voluntary cooperation among jregions ra th e r than co erce them to cooperate. C oerced cooperation is [difficult to ad m in ister as there always exists a tem ptation to cheat the I isystem. Induced cooperation, on the other hand, has an additional I 'advantage of keeping com petitive sp irits alive that tend to prom ote the |economic efficiency. j We re c a ll fro m C hapter II that the Nash equilibrium solution; | i is not an optim al solution in m any game situations. Thus, the Nash 102 eq uilib rium solution of our trade gam e being P a re to -o p tim a l is re a ss u rin g , for it m eans that the particu lar trade a rra n g em e n t is not only beneficial to all p arties concerned in the sen se of such fre e -tra d e proponents as Gottfried H ab erler and H arry G. Johnson, but is also optim al in the sense that one region cannot im prove its lot without hurting the other party. We have shown by m eans of a sim ple exam ple that under certain suitable conditions a trad e can m axim ize the aggregate w elfare of all p arties involved in the sense of P a re to - optim um . We have also seen that the traditional g am e -th e o retic concept of P a reto -o p tim ality extends readily to the n o n zero -su m differen tial gam e. CHAPTER VI UNCERTAINTY IN D IFFEREN TIA L GAMES In the two previous chapters we studied the m odels of dynamic com petition in the im p e rfec t m a rk e t by m eans of the nonzero- sum differential gam e. In such gam es all participating p lay ers are assum ed to p o ssess com plete knowledge on all aspects of the game at any time during the game including the state of the m ark et, i . e . , each com petitor knows p rec ise ly w here he stands in the m ark e t with re sp e c t to his opponents' positions throughout the gam e. Only unknown to a player in such gam es of p e rfe ct inform ation are the stra te g ie s of his opponents, as individual p lay e rs a re assum ed to choose their respective stra te g ie s sim ultaneously. The assum ption of p e rfe c t inform ation on the p a rt of all the p lay ers is obviously an oversim plification of m any re a l-w o rld situ a tions. As we observe in our daily life, it is quite com m on for b u sin e ss m en to make im portan t busin ess decisions in the face of incom plete inform ation not only about their opponents' decision v ariab les, e .g ., relative m a rk e t share, inventory level, costs, e tc ., but also about their own as well. An o rd in ary citizen also has to face the problem of decision under uncertainty in som e c irc u m sta n ce s. F o r instance, people buy stocks am id a g re a t deal of uncertainty. The Cobweb theorem (Bean, 1929) and the hog cycle deal with the m a rk e t situation w herein the individual fa rm e rs do not know the behavior p attern s of 103 r ................................. 104 ■ j I their opponents. One can cite num erous situations in one's own i | experience w herein individuals have to m ake decisions on the b asis of j j j the best but incom plete inform ation available. D espite all this, m any i th eo ries of econom ics have been based on the assum ption of perfect ; I 'information. The th eo ries of individual demand and the behavior of jthe individual firm as they a re p resented in textbooks and econom ic j lite ra tu re a re m ostly theories of "ratio nal behavior under p e rfe ct | i inform ation. " It is assum ed that the individual always knows fairly | | ; well what he wants, what alternatives a re available to him , and what jth e m a r k e t s i t u a t i o n i s . ! ! i | F. H. Knight (1933) m ade the distinction between ris k and j i : uncertainty. The Knightian uncertainty m ay be broken down into two I jcategories— b eh av io ristic uncertainty and ignorance. The f ir s t cate- Igory re fe rs to a state in which an individual is com pletely inform ed of j | ; his own environm ent (the rules of the game) but does not know what ! behavior to expect fro m his opponents. The differential gam e of I p e rfe ct inform ation deals with this kind of situation. The other Knightian uncertain ty re fe rs to a situation w here an individual does not know all the ru les of the gam e, that is, he does not know the state of the m ark et. The differential gam e of im p erfect inform ation deals jW ith t h i s k i n d o f i g n o r a n c e . I T h ere have been m any m athem atical and sta tistic a l studies on the problem of behavior under uncertainty. The lite ra tu re on gam es against nature and sta tistic a l decision-m aking deals with the uncertain ty caused by chance phenomena. The m axim in principle, j ithe m inim al re g re t principle (Savage, 1954), the B ayesian principle j ! 105 ;(Wald, 1950), and the Hurwicz principle (Hurwicz, 1951) a re some of the well-known m ethods of solving gam es against nature. Von Neumann and M o rgenstern (1944) applied the concept of m ath em a tica l expecta tion to their gam es of stra te g ie s w here two o r m o re p lay ers are involved. Each player m ay random ize his choice of stra te g ie s in such a m anner that the expected value of his payoff m ay be m axim ized. The traditional gam e theory of Von Neum ann and Morgen-i Istern is concerned with the static gam es, and as such it fails to i | account for the learning aspects of the gam e. In a gam e of long d u ra- ' i I ; |tion, players can in general le a rn about the nature of unknowns as the j jgame p ro g re sse s and accordingly tend to take advantage of this newly igained knowledge for the subsequent game. The adequate analysis of ! isuch gam es, therefore, calls for a m ultistage d ecisio n approach wherej- ! . | |by a sufficient provision m ay be m ade for the learning p ro c e ss. The i i 1 . itheory of stochastic differential gam es offers a m ethod of handling j 1 jsuch m ultistag e decision problem s. i # j | In the game of im perfect inform ation, the u n certain ty m ay I | j |in g e n eral p ertain to any one of many variables of the gam e, e. g . , j j state of the system , system p a ra m eters, se t of allow able decisions, | jeffects of these decisions, c rite ria by which to evaluate the strategy employed, etc. In the context of the differential gam e theory, however, | jthe uncertain ty pertains only to the state of the m a rk e t or sy stem unde? I 1 i ! Iconsideration, not the m ark e t p ro cess itself nor sy ste m p a ra m e te rs j jnor any other variable of the game. In a stochastic d ifferen tial game, j iplayers have available only noisy observations of the state of the sy s- j |tem, and have to select strateg ies based on the lim ited inform ation. j j 106 ' Under such circu m stan ces, a ration al player would m ake his decision on the basis of the "best estim ate" of the sy stem state that he could obtain fro m both past and c u rre n t observatio ns. We re c a ll that in the d ifferential game both the state and the payoffs of the gam e are influenced by the stra te g ie s that individual players employ. Now, since players have to base their resp ectiv e stra te g ie s on the estim ates! of the state under uncertainty, it is intuitively c le a r that incom plete I inform ation significantly affects the e n tire co urse as well as the pay- | offs of the game. i It is hardly n e ce ssa ry to em phasize that the state of informaj- tion m ay be a determ ining factor in m any m ark e t situations. Even the Cournot duopolists1 behavior is not independent of their inform ation j state. The m odern-day corporation o r firm operates in a w elter of j 1 uncertainty and half-knowledge under m a rk e t situations that som e- | ; 1 j j jtimes m ake special inform ation its m o st valuable asset. It is im p o r- : I • |tant to observe in the stochastic gam e that not all the players know the i | ;system to the sam e degree, i. e. , the m a rk e t visibility of an individualj f ir m frequently depends on the relativ e strength of the firm in the i ! I I I jindustry. j | Because an enorm ous sum of m oney is n e ce ssa ry to gather, I jstore, p ro cess, and com m unicate data, the inform ation available to a i i f ir m often depends on the relative a s s e t position of that firm in the iindustry. This unequal knowledge about the m a rk e t c re a te s a tem pta- ! jtion for stro nger and m o re knowledgeable firm s to exploit to their | ; j advantage the weak and relatively ignoran t firm s. Indeed it will la te r | i i be shown analytically that under c e rta in conditions players do take advantage of the ignorance of their opponents. The m odern K alm an filtering theory has played a significant role in the developm ent of both the stochastic control theory and the stochastic differential gam e theory. Kalman and Bucy (1961) extended the now c la ssic W iener-K olm ogorov theory of filtering to include non- , statio nary tim e s e rie s . The Kalm an filtering theory thus helped to develop f ir s t the stochastic control theory and then the stochastic dif- ; i feren tial gam e by providing a m ethod of obtaining dynam ically a "b est : estim ate" of the state of a linear sy stem in the le a st-m e a n -s q u a re - ;error sense fro m the available nonstationary tim e se rie s of o b serv a- ; j tions. Since the K alm an filtering theory was f ir s t published in the ! ] ! e a rly 1960's, m any papers have been w ritten on both stochastic control; ! problem s and stochastic differential gam es. Work done to date on the j stochastic differential gam e is, however, exclusively on the lin e ar ; i stochastic differential gam es of z e ro -su m variety. As far as the j | ! author knows, th ere has been no work done at all on n o n zero -su m j i | Stochastic gam es. | I | In the stochastic optim al control problem with im p e rfec t j i ! i i jstate inform ation it has been proven that the certainty-equivalence ^principle holds under the assum ption that the system under study is i lin e a r, and output m e a su re m e n ts a re corrup ted by additive white n o ises. In other w ords, under these assum ptions the optim al control |for the stochastic control problem m ay be obtained by first, getting [the best estim ate of the lin ear system in the le a s t- m e a n -s q u a re -e r ro r j ! i sense fro m available noisy m easu rem en ts, and then using this b est i ; j - estim ate in place of the unknown state in the optim al solution of the J | 108 I corresponding d eterm in istic control problem . It is im portant to recognize that this handy way of solving the difficult stochastic problem j becam e available only after the advent of the K alm an filtering theory. 1 Now, it naturally com es to m ind w hether this convenient m ethod of solving stochastic control problem s can be extended to z ero - sum stoch astic differential gam es, which m ay be reg a rd ed as two- sided control problem s. Unfortunately the very sam e se t of assu m p tions ap p ears to be not sufficient for the optim al stra te g ie s of the stochastic differential game to be given by the certainty-equivalence principle (Rhodes and L uenberger 1969). Indeed, even under these | re s tric tiv e assum ptions, a solution to the stochastic differential game se e m s to req u ire in general..an infinite-dim ensional state estim ato r, i „ ; i. e . , the p reserv a tio n and use in the determ in atio n of optim al s tra te - ; 'gies at each tim e of the en tire output m e a su re m e n ts up to that tim e. j i jThis g en eral stochastic problem rem ain s unsolved as fa r as the author; i | know s. j ! I Rhodes and L uenberger (1969) and Behn and Ho (1968) ; i ! 'studied, how ever, a special class of z e ro -s u m lin e ar stochastic j i ^differential gam es and obtained very in terestin g c lo s e -fo rm optimal I jsolutions. The authors had to m ake n e c e ssa ry additional assum ptions j t I j I !to alleviate the above-m entioned "clo su re problem " of general j stochastic differential gam es. The optim al stra te g y obtained for each player in these special cases was then shown to be closely related to j that which would be obtained by applying the certainty-eq uiv alence j j i ; i prin cip le. F u rth e rm o re , the various te rm s in the resulting optim al ipayoff function w ere shown to be readily assignable to the appropriate | |.................. "............... . 109........... 1 sources, such as the optim al payoff of the corresponding determ inistic game, the effect of estim ation e rr o r, and the effect of m easu rem en t i i e r r o r . j i i Some interesting z e ro -su m lin e ar stochastic differential , gam es will be review ed next in som e detail. Then an im perfect m a rk e t of in te re s t will be c a st as a n o n z ero -su m lin e ar stochastic differential game with a discussion of a prom ising m ethod of solving the game. ; " ■ i Sufficient Conditions for Solution of G eneral Stochastic D ifferential Game — ! ' " ! Rhodes and L uenberger (1969) investigated the sufficiency j ! i | i conditions for the solution of the z e ro -s u m stochastic differential game;. iThe re su lts of their investigation w ere su m m arized in their theorem . i j iBy applying the theorem , they w ere able to obtain analytic solutions j ! i 'for se v era l interestin g special c ases of the z e ro -su m linear stochastic! i j i _ i 'differential gam e. In the following, all differential gam es a re assum ed i to be m ulti-dim ensional unless otherw ise specified, i. e . , th ere a re j j m any state variables and m any stra te g y v ariab les in the game, and these variables and their coefficients a re e x p re ssed in vectors and m a tric e s of appropriate dim ensions. i Consider a general dynam ical sy ste m described by the |vector differential equation i j i j i i ! X = f(x, [x(t), v(t), t) (6-1) i w here jju(t) and v(t) a re strateg y vectors at tim e t of P la y ers 1 and 2, respectively, and X(t) is the state vector, f is a sufficiently sm ooth function in its v ariab les. L et us assum e that P lay ers 1 and 2 both have available only noisy observations of the state of the sy ste m in the form ' of, y 2(t) = h ^ x f t) , (^(t)) y 2(t) = h 2(x(t), w2(t)) (6-2) | w here Jy^(t), i = 1, 2 J a re observed system state v ecto rs, and coj(t) ;and co2(t) a re random disturbance vectors whose sta tistics a re assum ed known to both p layers. Functions h^ and h^ are sm ooth functions. Thei iinitial state X(t ) is assu m ed by each player to have a p rescrib ed d is- j o ; tribution, and each player is aw are of the o th er's assum ptions on the :a p rio ri distribution of X(tQ). Both players assum e that both and c i> 2 j have the sam e distribution function. | i L et us also assu m e that c rite rio n functional is specified by, ! | ' ! i 1 1 ! j | ! ! /*T j | j(jx, v) = J g(X, jjl, V , t)dt + L(x(t)) (6-3) I t i O w here the integrand payoff function g and term inal payoff function L a re continuous functions in all v a ria b le s. Let Y.(t), i = 1, 2 denote ! 1 jthe se t of observed sy ste m state over the tim e in terv al (tQ, t) i . e . , ! I l l | Now, the problem of this general, z e ro -su m stochastic differential ! gam e is to find two decision functions, fx°(Y^(t), t) and v°(Y2(t), t) such ; i that for any adm issible stra te g ie s |x(Yj(t), t) and v(Y2 (t), t) there holdsj for all Te(t0» T) E(j(p.°, v° ) | V 1(t )) < E ( j ( n , v° ) | y i (t )) ! E ( j (h -°, v) j Y2(t)) < E (jCfx0, v° ) J y 2(t )) (6-5)| i Notice that P layer 1 with strateg y ( jl is assu m ed to be the | m in im iz er and P layer 2 with strategy v to be the m ax im iz er. Both | jplayers condition the expectation of the payoff on their resp ectiv e set j |Of noisy observations including their knowledge on the a p rio ri d is tri- j jbution of the initial state. J ; ! i T heorem ! ! | A sufficient condition for the existence of the optim al j isolution, u.° and v° of the above stochastic differential game is that i I jthere ex ist two value functions Vj(X, z, t) and z» ^ differen- j jtiable in each variable, which together with p0 and v satisfy for all j ! i |te(t , T) the following five conditions: ; i ° j | A - - m in E js^ X ft), z(t), (j., v° (Y 2(t), t), tJ / Y x(t) | = 0 j | M > j ! B - - E jSj [x(t), z(t), (i.° ( y J (t), t), v °(Y z (t), t), t ] / Y 1(t)j = 0 j ! | C - - m ax e )s 2 [x(t), z(t), p.° (Y j(t), t), v, t ] / Y 2 (t)|.= 0 | 1 12 D - E | s 2 |x(t), z(t), ( j l ° ( v ^ t) , t), v° (Y 2(t), t)]/Y2(t)| = 0 E — V.(X, z, T) = L(X), i = 1, 2 (6-6) [where S.(X, z, p., v, t) = Vi t (X, z, t) + (X, z, t) f (X, |x, v, t) + V.z (X, z, t) Y (X, z, p., v, t) + g(X, | j l, v, t) 0V. V. = i = 1 2 . lm 8m * * z = y ^X(t), z,p(t), v(t), t^ is a differential equation that z(t), an estim ate 1 ! 'of the sta te X(t), m ust satisfy. It is in terestin g to observe that the fivej [conditions of the above theorem re p re se n t a stochastic versio n of the I 'H am ilton-Jacobi partial differential equations and the initial conditions! I _ I ithat the value function of the d eterm in istic differential gam e m ust j satisfy. A lso notice that th ere a re two value functions and two j i H am ilton-Jacobi equations although the gam e is of z e ro -s u m v a rie ty | (there is only one H am ilton-Jacobi equation for a d eterm in istic z ero - j sum gam e). This is so because each player conditions the single j H am ilton-Jacobi equation on his own set of observations. [See (Rhodes! i i | and L uenberger 1969) for the proof of the theorem ). j With a slight m odification of the above th eo rem of the z ero - S isum gam e and a little different in terpretation, a sufficiency theorem j , i •may be obtained for general n o nzero-su m stochastic differential games!. We consider a sim ple tw o-player game to state the theorem . We f ir s t note that the dynamic sy stem equation (6-1) and the definition of observed sy stem outputs (6-2) rem ain the sam e for the n o n zero -su m differential gam e. Unlike z e ro -su m gam es, how ever, th ere a re two c rite rio n functionals, X J i (|ji, v) = S %<*• H . - t)dt + L.(x(T).), i = 1, 2 t o in the case of a nonzero-sum , tw o-player gam e. The problem of a n o n zero -su m stochastic differential game is to find two decision functions, p°^Yj(t), t^ and v0 (Y2(t), tj such that for any adm issible stra te g ie s p^Y^(t), t^ and vjjV^t), t) there holds for all Te(t0» T), E ( j 1(|J L , v°) | Y x(t)) s E j j j ^ 0, v ° ) | Y x(t) j j e (j 2(p.°, v) |Y 2(t)) S e (j 2(^°, v ° ) |Y 2(t)) (6-7) ■ i ! ’ | In other w ords, we seek a Nash equilibrium solution w here each player! ! j trie s to m axim ize his expected utility functional against the optim al j i j stra te g ie s of his opponents by manipulating his own strateg y . With j ! | this new definition of the solution of the non zero -su m gam e, the sam e theorem given above for the z e ro -su m gam es (6-6) holds true for the i i n o n zero -su m stochastic differential gam es except that in the condition t j A of the theorem , taking m inim um with re sp e c t to p m u st be read as | j I taking m axim um with re sp e c t p. In addition, the condition E of the j ; # ! th eo rem m ust read as V^(X, z, t) = Li^(X), i = 1, 2, reflecting that therq jare two distinct initial conditions for the two value functions one for 1 1 4 each player. In the case of z e ro -su m stoch astic gam es there exists only one c rite rio n function, which one player trie s to m axim ize while the other player trie s to m inim ize to a rriv e at the saddle-point solu tion. One obtains, however, two different value functions owing to the fact that there a re two different estim ates of the sy ste m state, one for I each player. In the case of the n o n zero -su m stochastic game, on the other hand, there a re also two value functions involved, one for each player, but unlike the z ero -su m case, the two value functions r e p r e sent in this time the two different c rite rio n functionals. Although the theorem holds true for sufficiently g en eral ■ -stochastic differential gam es, it appears possible to find value func tions and corresponding optim al stra te g ie s which together satisfy the hypotheses of the theorem (6-6) only in a special c la ss of differential g am es which is ch aracterized by linear system , quadratic c rite rio n j jfunctional and lin ear output functions co rru p ted by additive white j I I inoises. | j ; | j i ! I C lass of L in ear Stochastic j I j | D ifferential Gam es of Z ero -S u m V ariety j | I Before examining lin ear stochastic differen tial gam es, it j | seem s appro priate at this point to review the K alm an filtering theory (1961) to appreciate why only the class of lin ear stochastic differential j i gam es m ee ts the hypothesis of the sufficiency theorem . During the j jlate 1950's and early 1960's, Kalm an and Bucy g eneralized the c la ssic | W iener-K alm ogorov theory of filtering to include nonstationary 115 ! 'dynamical sy stem s by m eans of the "sta te -tra n sitio n " m ethod of d escribing dynam ical system s (a dynamic program m ing technique), and the lin e ar filtering reg ard ed as orthogonal projection in H ilbert space. The Kalman filte r generates a "best linear estim ate " of the lin ear sy stem in re a l tim e in the m in im a l-m e a n -s q u a re -e rro r sense, provided the observed outputs of the linear system a re corrupted by 1 additive white noises. If, m oreover, the white noises a re G aussian, then in addition to the "best estim ate" being linear in the observed outputs, the sam e estim ate is "best" not only in the m in im al-m ea n - isq u a re -e rro r sense but also for a wide variety of o ther loss functions, j : i In the following, capital le tte rs denote m a tric e s and sm all le tte r s v e cto rs. F o r concreteness, consider the following linear | dynam ical system , j X(t) = F(t)X(t) + G(tV(t) (6-8) j I | j ' j Jwith noisy observed outputs given by j ; t I I I | y(t) = H(t)X(t) + w(t) (6-9) i I i I I j w here o j (t) is assum ed to be a random p ro cess with zero m ean and ! covarience m a trix given by C o v L (t), o j(t) ) = W(t)6(t - t) w here 6 is the! i D irac delta function. It is also assum ed that observed output s e rie s j ljy(t) J s ta rts at some fixed beginning tim e tQ, at which tim e j C ov|x(to ), X(tQ)J is known. Under these assum ptions, the optim al j : A | estim ation problem is to find a lin ear estim ate X(t) given past j 116 A observations J y(t), te(tQ , t) j such that E || X(t) - X(t) || ^ = m inim um , w here I I X denotes the n orm of the vector X. The solution to this dynam ic estim ation problem is given by the K alm an filtering theory as A follows. The optim al lin e ar estim ate X(t) is generated for all t > t by the following lin ear dynam ical sy stem known as the K alm an F ilter: X(t) = F(t)X(t) + K(t)y(t) (6-10) w here y(t) = y(t) - H(t)X(t). (The optim al weight function K(t) of the Kalm an filte r is given by, K(t) = M I t j H V j w V ) (6- 11) w here the prim e denotes the transp osed m atrix . The covariance m a trix of the e r r o r s of the b e st estim ate, M(t) = Covjx(t), X(t)J is the solution of the m a trix differential equation known as the variance equation, M(t) = F(t)M(t) + M (t)F'(t) - M(t)H,(t)W "1(t)H(t)M(t) I I M(to ) = Cov[x(tQ), X(tQ) ] s Mq • (6-12) I 117.....'........! The optim al e r r o r X(t) is generated by the sam e dynam ical system as k t), i.e ., X(t) = F(t)X(t) + G(t)p,(t) - K (t)[w (t) + H(t)X(t)| (6 -12a) w here : x(t) = x(t) - x ( t ) . ; Now, to initiate the feedback loop of generating the best ■ I E stim ate of the state of the lin ear sy ste m in re a l tim e by m eans of the 1 above schem e, it is only n e c e ssa ry to specify the initial state of the Optimal filter. L et it be given by X(to ) = 0. Then one can compute j |M(t) by the nonlinear differential equation (6-12) for any t > tQ , for M(tQ) is known at the starting point tQ, and F, H, and W a re all known j m a tric e s. The optim al weight function K(t) is obtained by Equation : (6-11) for tim e t > t , that can then be substituted in the Kalm an filte r j i . ° ! j(6-10) to get the b est estim ate of the lin e ar sy ste m state at t. In this I i m anner, the Kalm an filte r generates in re a l tim e the le a st-m e a n - j s q u a re -e rro r, lin ear estim ate of the state of the lin ear sy stem by | ! ^ making observation y(t), form ing the e r r o r of the estim ate y(t), and i then feeding the e r r o r forw ard with the optim al weight K(t). With the j i Kalman filter, it is now possible to investigate stochastic differential j I i gam es involving a linear system . | 118 Consider a lin ear stochastic differential gam e of z e ro -su m variety, w here the lin ear dynam ical sy ste m is ruled by X(t) = F(t)X(t) + Gj (t)p,(t) + G2(t)v(t) X (‘0) = x„. (6-13) L inear output functions a re given by y^(t) = Hj(t)X(t) + Wj(t) for P lay er 1 y 2(t) = H2(t)X(t) + u 2(t) f ° r P lay er 2 (6-14) w here co, and to, a re independent, z e ro -m e a n G aussian p ro c e sse s with '1 2 covariances, Cov Cov Cov coj (t), co^t) a > 2(t), u2(t) Wi(t), co2 (t ) = W j(t)6(t - T) = W2(t)6(t - r) = 0 (6-15) The initial state X(t ) is assum ed to be G aussian with o E (x (tQ)) = XQ and Cov[x(tQ), X(tQ)] = Mq, and is unco rrelated with coj(t) and co2(t) for all t. Suppose that P lay er lj trie s to m inim ize while player 2 trie s to m axim ize the following quadratic c rite rio n functional ..............................1 1 9 ... .. / ' r^ i i i i , v) = J (fi. Q lHL + V Q2 v)dt + X (T)C CX(T). (6-16) t o With these specifications of the dynam ical system and the c rite rio n functional, the problem of the lin ear stochastic differential game is the sam e as that given for the general stochastic differential gam e considered in Section 1. The sufficiency theorem given in that section is, therefore, applicable to the problem on hand. Indeed Rhodes and L uenberger (1969) obtained solutions to a num ber of special c ases of this problem by applying the sufficiency theorem . Several special cases m ay be derived by assum ing the different degrees of perfection in the available inform ation for p lay ers. Only two special cases of the lin e a r stochastic differential gam e are selected for detailed review because of their obvious relevance to econom ic applications; In Case 1, P lay er 1 is assu m ed to have perfect inform ation about the 'state of the system , while P la y er 2 is assum ed to have available only noisy observations. Both players a re assum ed to have available only I i Inoise-corrupted observations on the state of the system in Case 2. | To facilitate the evaluation of the effects of uncertainty, the j | > [solution of the d eterm in istic lin e a r differential gam e corresponding to j I the stochastic game (6-13) to (6-16) is given in the following. I ! | | The optim al stra te g ie s for a z e ro -su m linear differential j I gam e of perfect inform ation a re given by p.°(t) = -Q j" 1(t)G1'(t)P(t)X(t) v°(t) = -Q 2" 1(t)G2'(t)P(t)X(t) (6- 17b) 120 w here the square m a trix P(t) is the sym m etric non-negative definite solution to the m a trix Riccati equation, P + PF(t) + F*(t)P - p [ G 1(t)Q j1(t)Gj(t) + G2(t)Q21(t)G 2(t)]p= 0 (6-18) with boundary condition P(T) = C'C. The optim al payoff of the z e ro -su m linear differential gam e ’ of p erfect inform ation is given by J(\x°, v°) = X'(to)P(to )X(to) . (6-17a) Case 1: P la y e r 1 has perfect inform ation, while P lay er 2 has only noisy observations of the sy stem state. To avoid the closure problem of stochastic differential Igames we need to m ake the additional assum ption that P la y er 1 with j | 1 p erfect state inform ation is in such a strong and advantageous position; | jthat he also knows exactly his opponent's state estim ato r. Case 1 is a | ispecial case of the lin ear stochastic differential gam e defined by (6-13)| ito (6-16) w here Hj(t)=I, unit m atrix , and co^(t)=0 because P la y er 1 enjoys p e rfe ct state inform ation. It can be shown that the following j value functions and V2, and the following optim al stra te g ie s |x° and j y ° satisfy the hypotheses of the sufficiency theorem (6-6) i. e . , (jl° and v° a re the optim al solution of the Case 1 problem : 121 v x = (x(t), x 2(t), t) = v 2(x(t), x 2(t), t) |j(n, v °)|[x(t), x 2 (t,]| |j(n°» v°)| [x(t), X2(t)]| = m in E = E X'(t)P(t)X(t) + X^(t)N(t)X2(t) •T + t r f N(S)M(S)H2(S)W2 1(S)H2(S)M(S)ds (6 -19k) t o w here t denotes the tra c e of a square m atrix, and x 2 = X(t) - x 2 . H°(t) = -QjVjGjWjpOOXtt) + N(t)X2(t)J v°(t) = - Q ^ W G ^ t l P ^ y t ) (6 -19b) Jwhere the sy m m etric m a trix N(t) satisfies the H am ilton-Jacobi p a rtia l j i . ! i ! differen tial equation, ; i ! | j A + NF(t) + F'(t)N | ! j ! + P W t G O O Q j ^ t j G j V ) + G 2 ( t ) Q 2 1 ( t ) G 2 ( t ) ] P ( t ) j I - (P(t) + N) G^(t)Q~ 1 (t)Gj(t)(p(t) + N) - N M (t)H '(t)¥ ‘ 1(t)H2(t) - H^(t)W“ 1(t)H2(t)M(t)N = 0 122 with initial condition N(T) = 0. M(t) = E |x 2(t)X2(t) | Y2(t)| is the solution of the variance equation sim ila r to that asso ciated with the K alm an filter. A In this gam e P lay er 2 obtains his b est e stim ate X2(t) by m eans of the K alm an filter. By substituting the optim al stra te g ie s (6 -19b) into the lin e ar sy stem (6-13), we f ir s t obtain the lin ear sy stem under optim al play X = (F - G ^ ^ P - G ^ ^ G ^ P J X + ( G ^ ^ G ^ P - GjQ^ 1g '1N)X2. In the following let us call this sy stem the optim ized linear sy stem . The Kalm an filte r based on the optim ized dynam ical sy stem A g en erates the b e st estim ate X2(t), i . e . , I j X2 = (F - G jQ j X G j P - G ^ ^ G ^ P ) ^ + MH2W2 1(y2 - H ^ ) ! A - - with initial condition X -,(fc ) = X w here X is an a p rio ri estim ate, i 2 o o o c i [Similarly the optim al estim ation e r r o r m ay be generated in re a l tim e j ! i ; I Sby the K alm an filte r. * I * \ : i f I I The solution of this problem m ay be understood heuristicallyj. [Not knowing the true state, P lay er 2 can only base his decision on the j best estim ate of the state he can obtain from available observations. Thus, it is best for him to use the sam e optim al stra te g y as he would ; 123 employ in the corresponding d eterm in istic gam e except that it is now applied to the best estim ate of the unknown state. Indeed if we com p are (6 -17b) with (6 -19b), it is apparent that the certainty-eq uiv alence ; principle applies exactly in the case of the player with im p erfect inform ation. The optim al strategy for P lay er 1 with p erfect in fo rm a tion is found to be the sum of two te rm s— a te rm which is identical with the optim al strateg y in the corresponding d eterm in istic gam e, and another te rm to account for the exploitation by P lay er 1 of the opponent's e r r o r of estim ation. It is in terestin g to observe fro m (6 -19b) that the optim al strateg y of P lay er 1 for the stochastic game reduces to that of the corresponding d eterm in istic gam e if his Opponent's estim ate of the state is perfect. This m eans that P la y er 1 m ay take advantage of the o th e r's ignorance as long as h is opponent's i e stim ato r is not perfect. The optim al payoff function at tim e t given by (6 -19a) has I I th re e te rm s . The f ir s t te rm correspo nds exactly to the optim al payoff of the corresponding d eterm in istic gam e. The second term , which is ; ^quadratic in P lay er 2's estim ation e r r o r , accounts for the reduction | jthat occu rs in the payoff of the gam e as a re s u lt of the inability of jpiayer 2 to estim ate the state exactly. The la s t te rm of the value j I I function accounts for the reduction in the payoff owing to the o b se rv a- i ! I ition e r r o r s by P lay er 2. Both the second and third te rm s a re j 1 i [negative because N(t), the solution of the H am ilton-Jacobi equation, is I a negative-definite m atrix while other m a tric e s involved a re positive- ; definite. 124 Case 1 m ay be used to m odel an im p e rfec t m a rk e t organiza tion in which the lead er of the industry with its enorm ous financial and other re so u rc e s and sem im onopolistic power enjoys a p erfect visibility; of the m arket, while other firm s suffer fro m im p erfect inform ation about the m a rk e t either due to the lack of re s o u rc e s or due to being deliberately kept in "d ark n ess" by the powerful lead er firm . F irm s ' re so u rc e s have m uch to do with the m a rk e t visibility of that firm , for a very expensive and extensive m arketing re s e a rc h is frequently n e c e ssa ry in today's economy to lea rn enough about the m ark et, and sm all firm s often do not have the n e c e ssa ry m eans. Case 2: Both players have only noisy observations on the state of the system . This case re p re se n ts the original z e ro -s u m lin e ar stochastic differential gam e as form ulated at the beginning of the p re se n t section, j i I Since the solution of this g eneral problem suffers fro m the "closure j I ■ problem ", we con strain our players to the use of e stim a to rs of finite j i | dim ension. In p a rticu la r, each player is allowed an n-dim ensional lin e ar dynamic system to generate the estim ate of the sy stem state, j I i e ! c • 9 ! i | I Z. = A.(t)Z. + B .(t)[y.(t) - H .(t)Z.] + D.(t)6.(t), i = 1, 2 (6-20)1 I j w here | ; i 6 j(t) = |a.(t) and 62(t) = v-(t). Inputs to the above estim ato r at any given tim e t a re re s tric te d to the newly observed system output y^(t) at t and the stra te g y 6^(t) that is applied by the players to the m ain dynamic sy ste m of the game at tim e t. The coefficient m a tric e s A., (t), B^(t), and D .. (t) and the initial con dition Zj,(tQ) a-re unknown and are to be determ ined by P la y e r i along with his optim al strateg y 6^(t) = 6^[Z^(t), t ] at each tim e t. The linear dynam ic sy stem (6-20) has interesting p ro p erties; it is M arkovian in the sense that the c u rre n t estim ate depends only on the previous estim ate ra th e r than on the entire se t of past observations; the system also re p re se n ts a learning p ro cess by which the e r r o r of the previous estim ate is taken into consideration in computing the c u rre n t estim ate. The learning p ro cess is sim ila r in c h a ra c te r to W ald's sequential analysis and B aysian m ethod (Wald 1950). As we no longer req u ire each player to p re s e rv e the entire observed output up to tim e t, players need to condition their expecta tion of the payoff only on Z^(t) and Z^(tQ) instead of Y^(t). L et |Z^(t) = [Z^(t), Z^(tQ)]. Then the problem of Case 2 is to find (if they exist) the m a tric e s A°(t), B°(t), and D?(t), i = 1, 2, the initial condi tions Z °(tQ), i = 1, 2, and the stra te g ie s (x°(Zj(t), t) and v°(Z 2(t), t) ithat a re optim al in the sense that given the optim al e stim a to r-stra te g y i fo r P lay er 2 there holds for all tE(tQ, T) 126 and given the optim al e s tim a to r-s tra te g y for P la y er 1 there holds for all te(tQ , T) E [J(h °, v ) Z ? (t)] S E tK fj.0, v°) Z (t)]. ( 6-21) The solution of the above lin e ar stochastic differential game with im p erfect state inform ation for both players can be shown to be .-1 H - (t) = - Q 1i (t)G1(t)P(t)Z 1(t) v°(t) = - Q " 1(t)G2(t)P(t)Z2(t) (6- 22) w here the optim al estim ates a re g enerated by the linear dynamic system , Z x(t) = F (t)Z x(t) + G 1(t)|j,(t) + G2(t)E |v °(t)| Z 1(t)| + M n (t)H j(t)W j1(t)[y 1(t) - H ^ t j z j |with the initial condition Z®(tQ) = XQ , Z 2(t) = F (t)Z 2(t) + G2(t)v(t) z 2(t) + Gx(t)E ((j.°(t) + M22(t)H2 (t)w2 1(t)[y 2(t) - H2(t)z2] (6-23) 127 with initial condition Z~(t ) = X . I L Q O Since both players have noisy observations only, the e rr o r covariance m atrix M(t) and the noise covariance m a trix W(t) a re defined as M(t) = E |m (t)m ' (t) w here the extended e r r o r vector is given by m ( t) = (x'(t), X'(t) - Zj[(t), X'(t) - Z ^ t)), and W (t) = E |w (t)w '(t)J w here the noise vector is w(t) = (w ^ t), w 2(t)^. |The sy m m etric nonnegative definite m a tric e s M j j and M22 are the i diagonal components of the com posite m atrix M(t). The m a tric e s I iW,(t) and W~(t) a re sim ila rly the diagonal components of the com posite i Imatrix W(t) defined above. i ! | Com paring the optim al stra te g ie s (6-22) with those given for ! jthe d eterm in istic gam e of p erfect inform ation ( 6 -17b), it is c le ar that the solution (6-22) obeys the certainty-eq uiv alence principle in the sense that the state e stim ate s Z^(t) and Z 2(t) replace the now unknown state X(t) in the solution of the d e te rm in istic gam e to give the solution of the stochastic game. 128 It can be shown that the e stim ate s J Z^(t), i = 1, 2| obtained by (6-23) are the fam ilia r b est lin e ar estim ates in the sense of the le a s t-m e a n -s q u a re -e rro r. They a re also unbiased estim ates of X(t), i. e ., given the optim al e s tim a to r-s tra te g y of P layer 2 E |x(t) | Z x (t) j = Z x (t) , and given the optim al e stim a to r-stra te g y of P lay er 1 E |x (t) | Z 2 (t) | = Z 2 (t) The optim al e stim a to rs (6-23) appear to be intuitively reasonable in the light of the Kalm an filtering theory. Indeed they can be obtained by applying the K alm an filtering theory to the optim ized linear dynamic sy ste m that is obtained by substituting the optim al stra te g ie s (6-22) I into the linear sy ste m (6-13). In com parison to the optim al estim ato r j ! I (6 -19c) of Case 1, w here only one of the players had im perfect state Inform ation, the optim al e stim a to rs of Case 2 have the sam e form jexcept that the Case 2 e stim a to rs have E(v°(t) | Z^(t)) for P lay er 1 and E(u.°(t) Z 7(t)) for P la y er 2 in place of v°(t) and |i°(t), respectively. ! • ^ j The optim al payoff function for Case 2 can be shown as | E (J (,j,0 , v ° ) | z . ( t ) | = E j x ' ( t ) P ( t ) X ( t ) | Z . ( t ) | + b ( t ) , i = 1, 2 w here the value function V(t) = X1 (t)P(t)X(t) + b(t) and the optim al stra te g ie s (6-22) satisfy the sufficiency conditions (6-6) for the 129.......... existence of the solution to the stochastic differential gam e. If we com pare the optim al payoff function with the optim al payoff function of the determ inistic problem (6 -17a), we notice that the f ir s t te rm of the value function for the stochastic problem is identical with the optim al value function of the corresponding d eterm in istic problem , and the second te rm b(t) m u st th ere fo re account for both the effects of estim ation e r r o r and observation e r r o r s , i. e . , b(t) = tr + P(T)G 2 (T)Q21(T)G2(T)P(T)M 2 2 (r)dT A Stochastic Im p erfect M arket P ro c e ss In this section a sim ple tw o-person non zero -su m lin ear stochastic differential gam e is form ulated to m odel an im p erfect com petition situation of in te re st. Case 1 m odel of the previous section I I is adopted for the n o n zero -su m gam e under consideration, for it is m o re interesting fro m the econom ic application point of view. We will | ;consider a situation w here the lea d er or dominant firm in a given I industry, with perfect inform ation on the m arket, trie s to take j advantage of any m istak e that its w eaker com petitor m ight m ake owing j to ignorance. The im p erfect com petitive situation that we like to m odel h e re pertains to m odern-day in d u strie s w here firm s com pete for la rg e r m a rk e t sh a re s for th eir products p rim a rily by m eans of advertisem ent^ 130 . packaging, and other m arketing p ractices. These industries a re ch ara c te riz e d by the fact that the products of individual firm s a re essen tially hom ogeneous except for different brand nam es, different packaging, e t c . , and that individual firm s m ay be able to extend th eir sh a res of the m a rk e t by e ith er "stealing" cu sto m ers from com petitors or by bringing in new c u sto m e rs fro m outside the industry, i . e . , through both in tra -m a rk e t and in te r-m a rk e t expansion efforts. New cu sto m e rs m ay be brought into the m a rk e t of this industry fro m out side eith er through the physical extension of geographical m a rk e t 'sphere by absorbing additional tran sp o rtatio n costs or by sim ply enticing "n o n -cu sto m e rs" to try this in d u stry 's products. Typically these in d u stries a re also c h a ra c te riz e d by the fact that the m arketing expenses account for the bulk of the total cost w here m arketing costs include tran sp o rta tio n cost. Also the technological content of the in d u stry 's products is relativ ely low, so that com petitive equilibrium | is not likely to be u p set easily by technological innovation once that ^equilibrium is established. j | i i Although individual firm s can theoretically extend th eir ! I i : | jm arket indefinitely by in cu rrin g la rg e r m arketing expenses, it is | j j obvious that the law of dim inishing re tu rn s will set in som ew here j along the expansion path. Thus individual firm s would like to m axi- jmize their rela tiv e sh a re s of the m ark et, and at the sam e tim e m in im ize m arketing expenses. j ' L et y(t) = (y j(t), y 2(t))' be the state vector w here | i I y^(t), i = 1, 2 denotes the m onetary value of the m ark et sh are for F ir m ii at tim e t. We assu m e that the state vector is governed by a linear dynamic system in strateg y v ariab les jx(t) and v(t), w here jj.(t) and v(t) denote the dollar values of the m arketing expenses of F irm s 1 and 2, I respectively, i. e . , y(t) = a(t)|a,(t) + b(t) v(t) (6-24); With initial condition y(tQ) = y , w here a(t) and b(t) are tw o-dim ensional vectors. Specifically, y ^ t ) = a 1(t)[ x(t) - b 2 (t)v(t) j y 2(t) = - a 2(t)(j.(t) + b 2(t) v(t). In other w ords, the tim e rate of change in the m ark et sh are ! for a firm is d irectly proportional to the m arketing expense that the < J [firm incurs in its drive for m a rk e t expansion, but inversely propor- I ! jtional to the m arketing expense of its com petitor. In accordance with j Case 1 of the previous section, we also assum e that F ir m 1 enjoys perfect state inform ation including the knowledge on the state e stim ato r of its com petitor, and F ir m 2 m akes noisy observations on the state in f I Ithe form of, ! ! i ! i I Z(t) = H(t)y(t) + w(t) | ! or Z 1(t) = h jf tjy ^ t) + W j ( t ) j ; ! | z 2 (t) = h 2(t)y2(t) + w2(t) (6-25)' w here the noise vector w(t) is assum ed to be G aussian white noise with E (w(t)) = Cov^w(t), w (t)^ = 6(t - T)W(t). F irm 1, with p erfect inform ation, is assum ed to m axim ize ' the expected value of its payoff functional, X e Jj ^ h, v ) |(y , Z)| = E ^ y ^ T ) - / p2(t)dt] (6-26a) f c o ! I : iwhile F irm 2, with im p e rfec t inform ation, is assum ed to m axim ize the; i expected value of its utility functional, j X j e [ j 2(h, v) z ] = E[cv2y 2(T) - p2 f v2(t)dt] (6-26b) to i ! \ \ \ jwhere (3^, i = 1, 2 j a re the weights of relative im portance to be j assigned to the term in a l m a rk e t sh are and the cum ulative m arketing ! • I jexpenses in cu rre d during the gam e for the attainm ent of that m a rk e t | i sh a re . | | The stochastic problem of m axim izing the two utility func- j |tionals (6-26) subject to the lin e ar dynamic system (6-24), with F ir m 1 j I having perfect knowledge and F irm 2 making noisy observations of the j !form given in (6-25) falls in the class of n o nzero-sum linear stochastic differential gam es. Hence the p re se n t n o nzero-su m gam e m ay be solved for the Nash equilibrium solution. The problem is, therefore, to find two optim al decision functions |x°(y(t), Z(t), t), and v>°(Z(t), t) such that for any adm issible stra te g ie s fj,(y(t), Z(t), t) and v(Z(t), t) there holds for all te(tQ , T) v ° ) | (y(t), Z(t)j < E ^ ^ f x 0, V°) (y(t), Z (t)| e [j 2(|x°, v)|z(t)] < e [j 2(p°, w°)|z(t)]. (6-27) To facilitate the solution of the stochastic gam e, we assum e ; that the stra te g y v ariab les a re lin ear functions of the state vector and j the estim ate of the state vector, i. e . , H-(t) = c(t)y(t) + s(t)y(t) v(t) = d(t)y (t) (6-28)1 ;where y(t) = y (t) - y(t), and y(t) is the b est estim ate of the state obtain-| ! . \ able fro m ( z (t ), Te(tQ , t)j. | | The tw o-dim ensional row vectors c(t), s(t), and d(t) a re j unknown weights to be determ ined. As pointed out elsew here, it is j jnecessary to a ssu m e for the alleviation of the closure problem of the i general stochastic gam e that F irm 1 is in a strong advantageous position to know the state estim ate of its opponent y(t) as well as the i state y(t). 1 A ccording to the specified form of strateg ies (6-28), the expense of F ir m 1, |i.(t) for m a rk e t expansion depends not only on the r ............................................................................... 1 3 4 .-"i c u rre n t relative m a rk e t sh a re s of both firm s but also on its opponent's; e r r o r in the estim ation of the relative m ark e t sh a res. We a ssu m e that F irm 1 w ill try to take advantage of any m istake that F irm 2 m akes in this connection. F ir m 2, on the other hand, bases its m arketing cam paign decision en tirely upon its own estim ate of the c u rre n t relative m a rk e t sh a re s . Now, substituting (6-28) into (6-26) and (6-24) a n d u sin g / ' y(t) = y(t) - y(t) to elim inate y(t), we obtain for the expected values of utility functionals of F irm s 1 and 2, v i = E “ i y(T) X A 'h f [ | y(t) C'C + y 's 'c y + y 'c 's y + y ^ s 's Idt V2 = E |«2 y(T) B - (3 . r T / I I y(t) - y(t) d'd dt (6-29 jthat a re to be m axim ized subject to the lin ear dynamic constraint, y(t) = a(t)c(t) + b(t)d(t) y(t) + [ a(t)s(t) - b(t)d(t)]y(t) (6-30) w here A = 10 00 and B = 00 01 J * 135 F irm 2 can obtain its b est estim ate of the state by applying the K alm an filte r on the dynam ic equation (6-30), i . e . , y(t) = [ac + bdjy(t) + M(t)H'(t)W" 1(t) [z(t) - H(t)y(t) y(tD) = yG <6-31) where the covariance of the e r r o r of the best estim ate is defined by, M ( t ) = E [y(t)y' (t)J = E [ y ( t ) y ' (t) ] ;M(t) satisfies the v arian ce equation, ; M(t) = [ac + asjM (t) + M(t) [ac + a sj ' - M(t)H' (t)W '1 (t)H(t)M(t) (6-32)| with initial condition M(tQ) = Mq . N o w let Y(t) = E [y(t)y'(t)J. | j 1 | Then the utility functionals (6-29) can be rew ritten in term s j i i | I |of Y(t) and M(t), i . e . , j i ■ i | ! /•T V j = tr [o^YfTJA - J | Y(t)c‘c + M (t)(s'c + c 's + s 's ) jd t] t o .T V2 = tr [cv2Y(T)B - p2 f | Y(t)d'd - M (t)d'd|dt] (6-33)] t _ I 136 w here Y(t) is governed by the dynamic system , Y(t) = - ^ E [y(t)y'(t)] = E [y(t)y'(t) + y(t)y'(t)] = acY(t) + Y (t)c'a' t ! + asM(t) + M (t)s'a' + bd[Y(t) - M (t)| ‘ j + |Y(t) - M (t)]d'b' . (6-34) i ; The stochastic differential gam e (6-29) to (6-32) has now ! been successfully tra n sfo rm e d into a d eterm in istic one (6-32) to (6-34); I ; jby use of the expected value operato r, w here Y(t) and M(t) a re the new; I j | | Istate v ariab les, c(t) and s(t) the new strateg y variables of F irm 1, andj |d(t) the new strateg y v ariab le of F irm 2. The transform ed d e te rm in - i i istic problem is to m axim ize the d eterm in istic utility functionals (6-33) subject to the two new dynam ic system equations (6-32) and (6-34). ! Now we have a fa m ilia r n o n zero -su m d eterm in istic differential gam e jwherein each player trie s to m axim ize his own utility functional j I 1 against the optim al stra te g y of his opponent, by m anipulating his own j strategy in an optim al m ann er. The m ethod of solution given in i ^Chapter III may, therefo re, be employed to obtain a Nash equilibrium 137 solution. F irs t, two H am iltonians a re obtained from the utility functionals (6-33) and sy ste m dynam ics (6-32) and (6-34) with two p airs of L agrangian m u ltip lier m a tric e s (A^, and (r^, T^), i. e . , H j = tr |p j[ Y (t)c'c + M (t)(s'c + c 's + s 's ) J + A^Y + A 2^ ] H2 = tr [p2 | Y(t)d'd " M(t)d'd | + r i Y + r 2 ^ ] * F ro m these H am iltonians the H -m axim al strateg ies for P la y ers 1 and 2 can be obtained by taking the f ir s t partial derivatives of the resp ectiv e H am iltonians with re sp e c t to the respective stra te g y v ariab les and equating them to zero. The H -m axim al stra te g ie s are the values of the stra te g ie s that specifically m axim ize the resp ectiv e H am iltonians. They a re usually functions of the state vector and m ultipliers.- Now it is only n e c e ssa ry to find the values of m u ltip lie rs : for the com plete solution of the tran sfo rm e d d eterm in istic gam e. This : can be done by solving the H am ilton-Jacobi equations. Once the solution of the tra n sfo rm e d d e te rm in istic problem is known, i. e . , the j optim al values of c(t), s(t), and d(t) a re known, the solution of the 1 s original stochastic gam e can be obtained by m e re ly substituting these values into the definition of strateg y v ariables (6-28). We have fo rm u lated an interesting im p erfect com petitive i m ark e t as a n o n z ero -su m lin ear stochastic differential game, and i : 1 shown a m ethod of attack to obtain a Nash equilibrium solution. The economic rationalization of the quadratic utility functions (6-26) is yet ! to be given. It is noted in (6-26) that both the utility of the term in al m a rk e t share, y^(T), i = 1, 2 and the disutility of the m arketing expenses jj,(t) and v(t) are quadratic in the respective v ariab les. This m eans that the utility of a sm all m a rk e t share is relatively sm all, while the la r g e r m a rk e t sh are enjoys m ore than proportional utility, i . e . , the utility of m a rk e t sh are in c re a se s m ore than proportionally as m ark et sh a re in c re a s e s. This m ight be so presum able because the term in al m a rk e t sh are m ight have an overwhelm ing influence on the stability, profitability, and future growth of the company. Likew ise the disutility of m arketing expenses m ay in crease m o re than p ro p o r tionally to the in c re a se in the expenses, for severe penalty of a wrong investm ent decision m ight be justified in the case of such sc a rc e re so u rc e as capital, i. e ., the fund could be used for other m o re lu crativ e investm ent. C H A P T E R VII SUMMARY AND CONCLUSIONS In this chapter some im portant aspects of the la s t six chapters are review ed, and suggestions a re m ade fo r fu rth e r work. Sum m ary of the D issertation With em phasis on their analytic capabilities, the h isto ric a l developm ent of the various theories of im p e rfec t m a rk e ts was review ed in the f ir s t chapter. A. C ournot's repognition of the strateg ic interdependence of com petitors was noted as his m ajo r contribution to the theory of im p erfect m a rk e t organizations. C ournot's theory of oligopoly suffers, how ever, fro m the so -called reaction curve hypothesis, i . e . , com petitors do not m ake their m oves sim ultaneously but ra th e r re a c t to the m oves that the other com petitors m ake. The gam e theory of Von Neumann and M orgen stern avoids this pro b lem but fails to account for the dynamic aspects of the econom ic gam es. The dynamic generalization of C ournot's oligopoly theory by T intner and N ied erco rn is an im portant step toward the developm ent of m ore re a lis tic oligopoly theory. It w as pointed out that their w ork also suffers fro m the sam e hypothesis on the behavior of co m petitors as does the Cournot oligopoly theory. 139 140 The recently developed theory of n on zero-su m , n -p lay er differential gam es provides the n e c e ssa ry technique with which re a lis tic im perfect m a rk e t p ro c e sse s can be analyzed to an extent hereto fo re im possible. The differential gam e theory takes full account of the interactions of com petitors' stra te g ie s over tim e, thus providing invaluable insights into the behavior of com peting firm s over tim e. It was the purpose of this d isse rtatio n to introduce this im p o rtan t and pow erful theory into the field of econom ics by w ay of m odelling and solving a few sim plified yet rea listic im p e rfec t m ark e ts. The differences between z e ro -s u m and n o n zero -su m gam es w ere review ed in Chapter II by m eans of sim ple b im a trix gam es. V arious theories of solution for the n o n z e ro -su m gam es w ere review ed. It was noted that the Nash theory of equilibrium solution for non-cooperative gam es and Von Neum ann and M o rg en stern 's theory of P a re to-optim al solution for cooperative gam es, a re p a rticu la rly relev an t to the study of im perfect m a rk e ts. The N ash equilibrium solution and P a re to-optim al solution w ere fo rm a lly defined for the g eneral n o n zero -su m n-p layer differential gam e. The Nash equilibrium solution gives a player the best stra te g y he can use against the best stra te g ie s of the other players over tim e. It is an equilibrium solu tion in the sense that no player can gain by deviating fro m his Nash stra te g y as long as the other players use their N ash stra te g ie s. These two highly desirable p ro p erties m ake the Nash equilibrium solution the best choice fo r m ost of the econom ic gam es of in te re s t. The P a re to - optim al solution gives, on the other hand, a se t of optim al stra te g ie s for all the play ers that m axim izes the joint utility. It is an optim al 141 solution in the sense that the m axim um joint utility is divided among p lay ers in a m anner that guarantees each p lay er at le a st as much as he can guarantee him self. In Chapter III the nonzero-sum , n -p lay e r differential game was form ally form ulated. The feedback relationship between the state of the game and the stra te g ie s of p articip an ts, and the dependence of a given p lay e r's payoff on the stra te g ie s of the other players, w ere stre sse d . In the n o n zero -su m differential game, each firm trie s to m axim ize his objective c rite rio n over the entire period of the game or over his planning horizon by m anipulating his strateg y or policy v a ri ables. The algorithm to obtain a Nash eq u ilib riu m solution for the no n zero -su m differential game was outlined. It was pointed out that the P a re to-optim al solution of a cooperative differential game, i . e . , single coalition differential game can be obtained by m eans of the well-known solution algorithm of the optim al control theory. The analytic relationship between the N ash equilibrium and P a reto -o p tim a l solutions was shown when the Nash solution is P a re to -o p tim a l under suitable game conditions. An interesting m odel of p rice com petition in an im perfect consum er product m ark et was developed in C hapter IV. F o r the sake of sim plicity, a sim ple tw o -firm gam e was form ulated, and a complete Nash equilibrium solution was given with the in te rp retatio n of economic im plications. A few assum ptions are m ade in the m odel concerning the nature of m a rk e t and firm s, i. e . , (1) the total local m ark e t demand for the consum er product is tim e -in v a rie n t for the perio d of game, 142 and the demand for the product is inelastic with resp ec t to price, (2) the m a rk e t is initially enjoyed by o n e -firm monopoly that has enough production capacity to m eet the total demand, (3) both the established firm and the new firm have the same Von Neumann-type production technology, i . e . , the output of the production p ro ce ss is the essen tial input to the production of the sam e product, and (4) the new firm can expand its production facility without difficulty if and when justified by its share of the m ark e t. The com petitive situation under consideration is assum ed to be brought about by the new f ir m 's attem pt to b reak into the m onopolist's m a rk e t by offering essen tially the sam e product at substantially low er price. In this game we assum e that both firm s wish to m axim ize their respective profits over the entire planning horizon w here the payoff of each firm co n sists of the value of the term in al m ark e t share, the value of the term in al stockpile of goods for future demands, and the cum ulative sum of the p resen t revenues over the entire period of gam e. Each tries to m axim ize its own payoff by m anipulating two strateg y variables, i . e . , the price it ch arg es for its product and the stock allocation decision. Each firm faces a difficult stock allocation problem , i. e. , to decide on each day of operation how much of the existing stock to allocate to the production for the p resen t m ark et consum ption and how m uch to stockpile against future dem ands, taking into consideration the effects of c u rre n t sales on m ark et share and p re se n t revenue. Since a firm 's d e sire to expand its m ark et and its d esire to obtain la rg e r p re se n t revenue conflict (larg er sales revenue 143 req u ire s the firm to charge a higher price, while the rapid expansion of m a rk e t takes a low er price), the selection of a right set of s tr a t egies at a right tim e poses a difficult problem for the firm s. We realize in our m odel that the two firm s can conceivably get together and reap a monopoly profit by exploiting the inelasticity of the demand. It is, how ever, the fundam ental hypothesis of our model that firm s com pete rationally against each other to obtain la rg e r sh a re s of the m a rk e t. The com plete Nash equilibrium solution given in Chapter IV shows c le a rly how each firm shifts its policy fro m time to tim e to refle ct the shadow prices of the changing inventory and m ark e t sh are in o rd e r to m axim ize its aggregate payoff over tim e. It also shows the tra je c to rie s or time profiles of the inventory and m a rk e t sh are as both firm s em ploy the respective optim al policies. In short, the solution d eterm in es the behavior of the two competing firm s over tim e under optim al play. Although one m ay be h ard put to find in the re a l world an im perfectly com petitive m a rk e t fitting exactly the description of the p re se n t m odel, sim ila r p rice com petition situations are ra th e r com mon in today's m ark e t place. In the m a rk e t of such daily consum er goods as beer, packaged food, e tc ., new firm s do seem to enter and successfully estab lish them selves by cutting p rice in a m ark et v irtu ally dominated by a few nam e-b ran d firm s. Typically in the e arly phase of such com petition the price of the new f ir m 's goods is d rastic ally reduced to get them introduced into the m ark et. As the new product is accepted, and once the new f ir m 's m ark e t share reaches a c e rta in com fortable level, the price is then raised to the 144 sam e level as that of the established firm s. The lo st revenue during the price cam paign is p resu m ab ly considered by the new firm as an im portant and n e c e ssa ry investm ent for the acquisition of a m ark e t for the firm 's products. An unconventional, but quite rea listic , tw o-region trade m odel was developed in C hapter V by m eans of the n o n zero-sum differential game. We con sidered a sim ple situation w herein two regions trade with each o th er two "com m odities", i . e . , technology and cheap labor. The developed region indirectly exports technology to the developing region through the sem ifinished goods of high technical content, while a developing region indirectly exports cheap labor to the developed region through the labor-intensive finished goods. This kind of trade is unconventional in that the two regions do not trade such conventional item s as finished goods and raw m a te ria l on the basis of com parative advantage, but ra th e r they trade the scarce "reso u rce s, " i . e . , cheap labor and technology. In some sense, then, J. S. M ill's "vent for su rp lu s" doctrine v is-a -v is R icardian com parative c o st doctrine of the cla ssic al theory of in te r national trade seem s to apply h e re . It is the surplus of labor in the underdeveloped region (presum ably due to the im m obility of labor) and the "surplus of technology" in the advanced region that fo rm the basis for trade between the two regions. The kind of trade under consideration here se em s to be p a rticu la rly relevant to some develop ing nations of the w orld today. The im portant im pacts of such trade on the econom ic developm ent of the underdeveloped regions o r nations were stre sse d . 145 In the in te r-re g io n a l trade game we assum e that each region trie s to m axim ize its utility c rite rio n by m anipulating three policy v a ria b le s, i . e . , the price it ch arg es for the goods it exports to the o th er region, the rate at which it im p o rts goods fro m the other region, and the dom estic consum ption rate . The payoff c rite rio n for each region con sists of the utility of the term in al balance of paym ent, the utility of the term in al stock of goods, and the utility of cum ulative consum ption over the entire period of gam e. A m ethod of obtaining a Nash equ ilibrium solution for the trade gam e was outlined. Then the original gam e was slightly m odified to show that the Nash equilibrium solution of the m odified game is P a re to-optim al. E ssen tially the following two assum ptions a re m ade for this purpose: (1) both regions have a com parable utility of money, and (2) the advanced region adjusts perfectly its production of in term ed iate goods to the input req u ire m e n ts of the in d u stries of the developing region. Under these assum ptions it was shown that the Nash equilibrium solution of our in te r-re g io n a l trade gam e is P a reto -o p tim a l, We note h e re that the second assum ption im poses a ra th e r strin g en t requ irem ent, fo r it re q u ire s that the regions depend exclusively on each other, i. e, p e rfe ct com plem entarity of in d u stries in the two regions. In the re a l world, how ever, there usually ex ists no such com pelling reaso n for one region to cooperate with another region, p a rtic u la rly in the sense of P a re to-optim um . The com plete co o p er ation n e c e ssa ry to achieve P a re to-optim um would in all likelihood come about in the fre e -e n te rp ris e system only when a high er authority com pels the subordinate governm ents to do so. The re su lts obtained 146 in Chapter V are p a rticu la rly in terestin g fro m the pegional planning viewpoint. The resu lts im ply that a higher governm ent should create a situation that tends to induce voluntary cooperation among regions ra th e r than coerce them to cooperate. C oerced cooperation is difficult to enforce as there always e x ists a tem ptation to cheat the system . Induced cooperation, on the other hand, has the additional advantage of keeping com petitive sp irits alive that tend to prom ote efficiency. In Chapter VI the effects of uncertainty on the optim al strateg y and the value of the d ifferential game w ere investigated. The Knightian classification of uncertainty was review ed with re sp e c t to its application to the differential gam e. In the stochastic differential game, the problem of uncertainty a ris e s fro m the p la y e rs' ignorance about the state of the game, not fro m the chance phenomena. Thus the Knightian "ignorance" is relevant. The Knightian behavioristic uncertainty is, on the other hand, not treated as uncertainty in the differential gam e. R ather, each p lay e r is assum ed to make his move without any knowledge of the m oves that h is opponents make. To date, v e ry little w ork has been done on the theory of stochastic differential gam es. A few p a p ers w ere reviewed in detail that have recently appeared in jo u rn a ls of engineering. The a rticle s a re all on the linear stochastic differential game of z e ro -su m variety. The authors prove that the certainty-equ ivalence principle holds fo r a c lass of lin e ar z ero -su m stochastic d ifferential gam es, i . e . , the solution of the stochastic game m ay be obtained by f ir s t getting the best estim ate of the state of the lin e a r sy stem in the sense of the 147 Kalman filter, and then using the b e st estim ate in place of the unknown state in the optimal solution of the corresponding d eterm inistic game. In this case the various te rm s of the payoff function are assignable to the appropriate contributing so u rces, i . e . , the effect of estim ation e r r o r , the effect of m e a su re m e n t e r r o r , and the optim al payoff of the corresponding d eterm in istic gam e. The Rhodes and L u en b erg er's sufficiency theorem on the existence of the optim al solution to the general z ero -su m stochastic differential gam e was extended to n o nzero-sum stochastic g am es. In m any m a rk e t situations the am ount of inform ation available to a firm is a determ ining factor. The m odern-day co rp o ration operates in a w elter of uncertainty and half-knowledge under m a rk e t situations that so m etim es m ake special inform ation its m ost valuable asset. Because an enorm ous sum of money is n e c e ssa ry to gather, store, p ro cess, and com m unicate data, the inform ation available to a firm often depends on the relative financial position of the firm in the industry. The effects of such unequal inform ation on the solution of the lin e ar stochastic differential gam es were exam ined. A tw o-player n o n z e ro -su m linear stochastic differential game was used to m odel an in te restin g im perfect m ark e t situation. The m ark e t situation p e rta in s to m o d ern -d ay industries where firm s compete for a la rg e r m a rk e t sh are by m eans of advertisem ent, packaging, and other m ark eting p ra c tic e s. V arious ways of expanding the m a rk e t share by in cu rrin g additional m arketing expense w ere discussed. Although firm s can theoretically extend their m ark et indefinitely by incurring la r g e r m arketing expense, it is obvious that 148 the law of dim inishing re tu rn s w ill se t in som ew hete along the expansion path. It is, therefore, assum ed that firm s try to m axim ize their relative sh a re s of the m ark et, and at the same time try to m inim ize the m arketing expenses. In this gam e the decision on the m arketing expense is considered as a strateg y variable that each firm can m anipulate to m axim ize its utility c rite rio n . We also assum e that F irm 1, a dominant firm , enjoys p erfect inform ation, while F irm 2 has available only im p erfect inform ation about the m ark e t. Each firm tries, therefore, to m axim ize the expected value of its respective utility criterio n . A prom ising m ethod of solution for this stochastic game was outlined. It is based on transfo rm ing the stochastic gam e to the corresponding d eterm in istic gam e by use of the expected value operator. Once the solution of the tran sfo rm ed d eterm in istic game is known, the solution of the original stochastic game is obtained by sim ple substitution. In the lin e ar differential gam e, the objective functional is quadratic, and the dynam ic sy stem governing the state of the game is lin ear. It follows, th erefo re, that the quadratic utility functional of our game m u st be justified. Both the utility of the term in al m a rk e t share and the disutility of the m arketing expense are assum ed to be quadratic in the respective v a ria b le s. This m eans that a la rg e r m ark e t sh are enjoys m o re than proportional utility. This m ight be so because the term inal m a rk e t share has p resum ably overw helm ing influence on the stability, profitability, and future growth of the 149 company. Likew ise the disutility of m arketing expense in cre ases m ore than proportionally as the expense in c re a se s because severe penalty of a wrong investm ent decision m ight justifiably be called for in the case of such sc a rc e reso u rce as capital. Suggestions for F u rth e r Work T hree m ajo r a re a s are identified fo r fu tth e r work, i . e . , (1) the extension of the p re s e n t capabilities of the differential game theory, (2) the developm ent of m ore com plicated and re a listic differential gam e m odels for im p erfect m ark e ts, and (3) em p irical studies such as ex p erim en tal gam es and com puter sim ulation. As pointed out in Chapter VI, the p re se n t theory of stochastic differential gam es is applicable only to the gam es involving lin ear system s, i . e . , the gam es having quadratic utility functional(s) and a linear sy ste m that governs the state of the game. This lim ita tion e x ists because no m ethod available can be used to estim ate in rea l tim e the state of no n -lin ear sy stem . The Kalman filte r provides the n e c e ssa ry m ethod of estim ation fo r the lin e ar sytetem. The extension of the Kalman filtering theory to non-linear sy stem s would aid fu rth e r developm ents in this a re a. E xcept for the sm all s ta rt m ade in this p resen t resea rch , no w ork has been done on the n o n zero -su m stochastic differential gam es. Since the n o nzero -gum gam es a re m ore applicable to and stochastic gam es a re m ore re a listic for the analysis of im perfect m ark e ts, m o re w ork is needed on the n o n zero -su m stochastic differential gam es. 150 An obvious extension of the p resen t re s e a rc h would be to develop m ore com plicated and re a listic m odels of im perfect m ark ets. Only by so doing will the g re a t potentiality and capabilities of the differential game theory be unfolded. F o r instance, the differential game m odel of a re a lis tic m u lti-nation trade that includes the trade re s tric tio n s and international balance of payments would be interesting and ex trem ely rele v an t to the c u rre n t issu e s in international trade. Again, any attem pt to solve the re a listic sh areh o ld er's utility m axim ization problem as defined in Chapter I would be welcom e. Finally, m uch m o re work is needed in the a re a of experim ental gam es and com puter sim ulation gam es to verify against and supplem ent the differential game theory. It is the a u th o r's belief that in due tim e the theory of differential gam es will be to the theory of im p erfect m ark et organ iza tions what the m o d ern optim al control theory already is to the developm ent of m acro -eco n o m ic m odels. BIBLIOGRAPHY Baumol, W. J . , B usiness Behavior, Value and G row th. New York: M acm illan, 1959. Bean, Louis, H. , "The F a r m e r 's Response to P ric e , " The Journal of F a rm E conom ics, XI (July 1929), pp. 368-385. Behn, R. D ., and Ho, Y. C. , "On a C lass of L inear Stochastic D ifferential G am es, " IE EE T ran s. Autom atic C ontrol. Vol. A C -13, No. 03, (June, 1968). Bellm an, R . , Dynamic P ro g ra m m in g . Princeton, N. J. : P rin ceto n U niversity P r e s s , 1957. B ertrand, J . , "T heorie m athem atique de la ric h e sse sociale" (review), Journal des Savants (P a ris: Septem ber 1883), pp. 499-508. Case, J. H . , "T ow ards a T heory of Many P la y e r D ifferential G am es, " SIAM J. C ontrol, Vol. 7, No. 2, (May 1969).. C ham berlain, E dw ard H. , The T heory of Monopolistic C om petition. C am bridge: H arvard U niversity P r e s s , 6th e d ., 1950. Cournot, A ugustin A ., R e se arch e s into the M athem atical P rin c ip les of the T heory of W ealth. New York: M acm illan, 1897, pp. 79-80, p. 84. Edgeworth, F. Y. , M athem atical P s y c h ic s. London: C. Kegan Paul, 1881, pp. 20-25, pp. 18-19. F e lln e r, W illiam , Com petition Among the F ew . New York: Knopf, 1949, p. 51, p. 53, C hapter 7, pp. 169-174. H irschm an, A. O ., The S trategy of Econom ic Developm ent. New Haven: Yale U niversity P r e s s , 1958. Hotelling, H. , "Stability in Competition, " The Economic Jo u rn al, XXXXI (M arch 1929), p. 41. > H urw icz, L . , "Some Specification P ro b le m s and Applications to E co nom etrics Models, " (abstract) E conom itrica, XIX (July 1951), pp. 343-344. Isaacs, R. D ifferential G am es. New York: John Wiley and Sons, 1965. 151 152 Isa rd , W alter, G eneral Theory Social, P olitical, Econom ic and R egional. Cam bridge, M ass. : The M. I. T. P r .ss, 1969. Kalman, R. E. and Bucy, R. S., "New R esults in L in ear F ilterin g and P re d ictio n Theory, " T ran s. ASME I. Basic E n g rg . , Ser. D. Vol. 83, M arch 1961. Knight, F. H. , Risk, U ncertainty and P ro fit. London: London School R eprints of Social W orks, No. 16, 1933. Luce, R. D ., and Raiffa, H ., G am es and D ecisions. New York: John W iley and Sons, 1958. Myint Hla, "The 'C la ssic al T heory' of International T rade and the U nderdeveloped C ountries, " The Econom ic Jo u rn a l, (June, 1958). M yrdal, Gunnar, Econom ic T heory and Underdeveloped R egions. London: G erald Duckworth & C o., L td ., 1957. Nash, J. F . , J r . , "The Bargaining P roblem , " E conom itrica, XVIII (April 1950), pp. 155-162. Nash, J. F . , J r . , "T w o -P erso n Cooperative G am es, " E co n o m itrica, XXI (January 1953), pp. 128-140. N urkse, R agnar, P ro b le m s of Capital F o rm atio n in U nderdeveloped C o u n tries. Oxford: B asil Blackwell S. Mott, Ltd. , 1953. P a re to , V ., Manual d'econom ie politque. P a r i s : G irard, 2nd e d ., 1921, C hapter VI, Appendix. Rhodes, I. B . , and L uenberger, D. B ., "D ifferential G am es With Im p erfect State Inform ation, " IE E E T ran s. A utom atic C ontrol, Vol. A C -14, No. 1 (F ebru ary 1969). Rhodes, I. B . , and L uenberger, D. B ., "Stochastic D ifferential G am es With C onstrained State E stim a to rs, " IE EE T r a n s . Autom atic Control, Vol. A C -14, No. 5, (October, 1969). Robinson, Joan. The Econom ics of Im p erfect Com petition. London: M acm illan, 1950, p. 21. Savage, L. J. , The Foundations of S ta tistic s. New York: Wiley, 1954, C hapter II, p. 30. Shubik, M artin. S trategy and M arket S tru c tu re . New York: John Wiley & Sons, 1959. 153 S ta rr, A. W. and Ho. Y. C ., "F u rth e r P ro p e rtie s of N onzero-Sum D ifferential Gam es, " Journal of Optim ization Theory and A pplications, Vol. 3, No. 4, (1969). S ta rr, A. W. and Ho. Y. C ., "N onzero-Sum D ifferential G am es, " Journal of O ptim ization T heory and Applications, Vol. 3, No. 3, (1969). Sweezy, Paul, "Demand Under Conditions of Oligopoly, " Journal of P o litical Econom y, XLVII (1939), pp. 568-573. T intner, G. , and N iedercorn, J. H. , "Cournot Oligopoly R econsidered, " To Be Published. Von Neumann, J. and M orgenstern, O ., Theory of G am es and Econom ic Behavior. P rinceton: Princeton U niversity P r e s s , 1944. Wald, A braham , S tatistical Decision Functions. New York: John Wiley & Sons, 1950.

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Creator Chung, George Sae-Ho (author)

Core Title Differential Game Theory Approach To Modeling Dynamic Imperfect Market Processes

Degree Doctor of Philosophy

Degree Program Economics

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag Economics, theory,OAI-PMH Harvest

Language English

Contributor Digitized by ProQuest (provenance)

Advisor Tintner, Gerhard (committee chair), Niedercorn, John H. (committee member), Pounders, Cedric J. (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c18-497822

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