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Essays on bargaining games
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Content
ESSAYS ON BARGAINING GAMES
by
Y usuf Ertas
A Dissertation Presented to the
FACUL TY OF THE USC GRADUA TE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(ECONOMICS)
December 2012
Copyright 2012 Y usuf Ertas
Acknowledgments
I would like to take this opportunity to thank first my advisor Prof. Simon Wilkie
who suggested the problem that comprises the first chapter of this dissertation. I
also thank Prof. Anne van den Nouweland for pointing out a mistake in one of the
crucial lemmas in the first paper. The seminar participants at USC and the Midwest
Economic Theory conference have provided valuable feedback and suggestions.
My committee members Prof. Harrison Cheng and Prof. Kim C. Border have
set apart time to listen to this research and have provided generous comments and
suggestions. All errors in this dissertation are solely mine.
ii
Table of Contents
Acknowledgments ii
List of Figures iv
Abstract v
Chapter 1 Characterization of a Dual to Nash Bargaining Solution with an
Application to Bankruptcy Problems 1
1.1 Introduction ...............................................................................................1
1.2 Definitions .................................................................................................6
1.3 Axiomatic Characterizations ...................................................................10
1.3.1 Duality with Nash Solution and Proportional Gain vs. Loss ….10
1.3.2 The Nash Solution Approach .....................................................12
1.4 A Simple Application to Bankruptcy Problems ......................................21
1.5 Conclusion...............................................................................................23
1.6 Appendix A: Missing Proofs in Main Text ........................................... 24
1.7 Appendix B: Continuity ..........................................................................29
Chapter 2 Optimism in Bargaining Games without Common Priors:
Further Results on Immediate Agreement 34
2.1 Introduction .............................................................................................34
2.2 Model and Preliminary Results ...............................................................40
2.3 Results .....................................................................................................45
2.4 Conclusion...............................................................................................48
2.5 Appendix: Missing Proofs in Main Text .............................................. 50
Bibliography 55
iii
List of Figures
1.1 Nash Solution ..........................................................................................13
1.2 Impossibility Result.................................................................................16
1.3 Delegation to Symmetry ..........................................................................17
1.4 Yu solution on hyperplanes normalized to −1 on y coordinate ...............20
1.5 Curves on which we set an arbitrary solution equal to the original
ICST solution ..........................................................................................21
1.6 Level Sets and Impossibility Result ........................................................33
iv
Abstract
This dissertation is centered around bargaining games. I study both the cooperative
and the non-cooperative approaches to the bargaining games. The two chapters of
my dissertation deal with each of these approaches.
Coming back to the bargaining problem; in the first chapter of this dissertation, I
study two related but separate questions on a cooperative bargaining problem. I will
give a brief historical background to explain the first question I tackle. Nash (1950)
characterized a solution for bargaining problems in terms of four simple axioms.
Later research showed that when the disagreement point in the analysis is replaced
with an infeasible ideal point, there is no solution that can be characterized with
the axioms that are counterparts to the ones used in Nash’s approach. One way to
deal with this problem is suggested by Conley, McLean, and Wilkie (forthcoming).
Instead of using independence from affine transformations, their characterization
of the Nash solution employs a property that guarantees proportional gains from
an endogenous disagreement point. Then, the same property is used to characterize
another solution when the endogenous disagreement point is replaced by an infeasi-
ble ideal point. That is, the corresponding property guarantees proportional losses
from this ideal point. I extend their characterization to weighted bargaining games
by a simple modification of their proportional loss property.
v
In the second part of the paper, I tweak Nash’s original proof to characterize
solutions that are independent under common scale transformations. This weaken-
ing of the independence from affine transformations is required since we know that
there is no solution that satisfies it along with other nice properties under the ideal
point approach. Then, instead of showing that the Nash solution can be obtained in
the entire domain from the solution of a symmetric set, I show that any solution that
satisfies independence under common scale transformations can be obtained from
the solutions of a certain class of hyperplanes.
In the second chapter of this dissertation, I look at the bargaining problem from
the non-cooperative side. I provide results that extend and complement Yildiz
(2003). Yildiz studies bargaining games without a common prior; that is, games
where players have subjective beliefs about their probabilities of getting selected
at each period. This departs from the original alternating offers games where such
selection procedures are strict. Assuming no learning and transferable utility, he
shows that as long as both players are optimistic for long enough, they will reach an
agreement immediately. Intuitively, players will reach an agreement immediately
if both of them are highly optimistic, since their expectations about rents in future
periods are so low that they settle immediately. Another key result from his work
is that players will also reach an immediate agreement if the drop in optimism from
one period to the next is within certain bounds. This second result does not require
high optimism.
By relaxing the transferable utility assumption, I study the effects of risk aver-
sion on bargaining games without common priors. I find that it takes a higher level
of optimism (as compared with transferable utility) to reach an immediate agree-
ment as the agents get more risk averse. Intuitively, risk aversion means higher rents
vi
in future periods, which will make immediate agreement a less attractive option.
This is in line with Yildiz’s findings. Also, I show that the bound required for
immediate agreement as mentioned in the above paragraph is strictly narrower than
the transferable utility case. In fact, I am currently trying to show that immediate
agreement is only reached at constant levels of optimism as the agents get more and
more risk averse.
vii
Chapter 1
Characterization of a Dual to Nash
Bargaining Solution with an
Application to Bankruptcy Problems
1.1 Introduction
Nash (1950) introduced a solution concept for bargaining games which had a sim-
ple characterization in terms of pareto optimality, symmetry, independence of irrel-
evant alternatives, and independence under affine transformations. This solution
and many of the other solutions considered in the literature takes a disagreement
point inside the set with the interpretation that this is the point where the negoti-
ations break down. Extending the Nash’s approach to problems with claims has
been problematic. If we represent the maximal point of each of the coordinates
that lie within the bargaining set as each agent’s claim, we have a new approach of
looking at the bargaining problem. These type of problems have been studied by
Chun (1988), Chun and Thomspon (1992), Bossert (1992), (1993). We call these
points utopia points throughout the paper. The Euclidean Yu solution is the distance
minimizer from the utopia point to the bargaining set.
Replacing the disagreement point with the utopia point as a reference point
leads us to a different interpretation of the bargaining set. Multi-objective decision
1
making is one such possible explanation. Consider several alternative methods of
producing an item. Each of these methods differ in their levels of cost, time to
completion, need for new personnel etc. These multi objectives then can be mapped
onto a convex set to find the best linear combination among the objectives that is
optimal. Such an approach would treat the optimal solution as shared losses from
the utopia point, which would be considered as the maximum attainable levels in
each objective. We also provide an interpretation along the lines of bargaining
with claims literature. A bankruptcy problem is defined as claims of the bargaining
parties and a resource available that is in general less than the sum of the claims.
Then, interpreting the claims as the coordinates of the utopia point, we can pick
a solution from the linear combination of the claims using our solution and then
distribute the resource in a proportional manner to the solution. We define our
solution to the bankruptcy problem mathematically in section 5.
Unfortunately, there is no straightforward way to approach to characterize the
new solution with a different reference point (the utopia point). That is, there is
no simple modification of the four axioms that will yield a new solution for these
class of games. In fact, there is no solution that satisfies pareto optimality, sym-
metry, independence of irrelevant alternatives with utopia points, and independence
under affine transformations. Therefore, one has to adapt a different approach rather
than simply trying to modify the Nash axioms. This paper considers a solution
that is a dual to the Nash solution in the following sense. The Nash solution pro-
vides proportional gains with respect to the disagreement point and EY solution
provides proportional losses with respect to the disagreement point. Such a dual
characterization for equal weight bargaining games is given by Conley, McLean,
Wilkie (forthcoming). This paper completes the characterization by extending it to
2
weighted bargaining games. We present this characterization first, as it is simpler to
state and illustrative of the duality between the Nash solution and the Yu solution.
The failure of EY solution to satisfy independence under affine transformations
makes it characterization in a similar way to Nash much more difficult. Nash
showed that if an arbitrary solution satisfies the axioms given above and is equal
to the Nash solution at a single point
1
, then it is equal to the Nash solution on the
entire real domain. Our approach is very similar to this; however, we show that if
an arbitrary solution is equal to the Yu solution on a curve (manifold inR
n
) in the
plane, then it is equal to the Yu solution in the entire domain. An interesting obser-
vation is that such an argument can be used to characterize any solution invariant
under a common scale transformation including the entire class of Yu solutions.
Another interesting observation that arises out of this analysis is that in the equal
weights case, for a class of asymmetric bargaining sets the asymmetric problem
can be converted to a symmetric problem. A similar situation holds for weighted
bargaining games. This time, the solution of a normalized bargaining problem can
be obtained from the solution of a canonical asymmetric bargaining set. All of these
notions will be made precise in the following sections.
There is a particular reason why Euclidean distance is special for our character-
ization. Euclidean distance allows us to delegate normalized asymmetric problems
(this kind of problems will be explained later in the paper)to symmetric problems
in a one to one fashion. As we will see, the slope of the tangent at any point x is
equal to−x when appropriately normalized. This fact will be useful in a number of
1
Namely the vector with all coordinates equal to 1.
3
observations. This is not true for other Yu solutions for which 1 < p <∞, where
general Yu solution is the following minimization problem:
min
x∈B
n
X
i=1
(u
i
−x
i
)
p
1/p
where u = (u
1
,u
2
,··· ,u
n
) is the vector such that u
i
= max{x
i
|x ∈ B} ∀i ∈
{1,··· ,n}. Throughout this paper, we denote the classes of compact and convex
sets by B and a generic compact and convex set by B. The casep = 2 is our main
focus, and we will have a chance to discuss the case ofp =∞. This case is closely
related to Chun (1988).
As we have indicated in the opening, the main difficulty with the characteriza-
tion of this solution is that it is not invariant under affine transformations. However,
as we will see the Euclidean Yu solution, or for that matter any Yu solution with
1 < p <∞ satisfies what we call the independence under common scale changes.
This terminology is due to Roth (1979). Nash (1950)’s main idea was to transform
the compact and convex set in such a way that its disagreement point is normalized
to 0 and the solution is transformed to match the solution picked by the solution of
a symmetric bargaining set. However, independence under affine transformations
is a really strong characteristic for a solution to possess. For example, Kalai (1977)
and Kalai and Smorodinsky (1975) show that in order to characterize proportional
solutions, continuity is necessary. We are not aware of any characterization of the
proportional solutions without continuity, however neither of our solutions use con-
tinuity. In fact, we will show that neither of the usual axioms other than the ones
we introduce in this paper imply or are implied by continuity. We take up this and
other related issues in an appendix at the end of this paper. Since independence
4
under common scale changes is a weaker condition than independence under affine
transformations, the additional assumptions required to characterize our solution is
stronger.
In section 2, we describe the definitions that will be used in the paper. In section
3, we state and prove our characterization results. Our first characterization is a
strictly axiomatic one and is based on the analysis of CMW (forthcoming), and
CMW (2008). Our second characterization is actually a procedure to establish the
solution on a class of sets first, then extend it to all sets inB in spirit of Nash (1950).
In section 4, we introduce a new solution to the bankruptcy problems. We conclude
in section 5. Appendix A contains the proofs of the results omitted in the text, while
Appendix B comprises of a discussion of continuity of solutions with respect to the
Hausdorff metric topology.
5
1.2 Definitions
In this section, we state the axioms that will be used throughout the paper. Most of
them are standard in the literature. Let N = {1,··· ,n} be an index sets rep-
resenting the players. We only consider games which are represented as com-
pact and convex subsets of R
n
in the paper until Appendix B. This class of
sets is denoted B. We refer to an element of this collection by B. The utopia
point and the nadir point of a set is u
i
= max{x
i
|x ∈ B} ∀i ∈ {1,··· ,n},
m
i
= min{x
i
|x ∈ B} ∀i ∈ {1,··· ,n} respectively. The strong Pareto set of
a set B is denoted as P(B) = {x ∈ B|y ∈ B withy ≥ x} and the weak Pareto
set is defined by WP(B) = {x ∈ B|y ∈ B withy ≫ x}. Here; y ≫ x means
y
i
> x
i
∀i, andy≥ x meansy
i
≥ x
i
∀i.Also,y > x means thaty
i
≥ x
i
with strict
inequality for at least onei.
The convex hull of a set inR
n
is the minimal convex set that contains it and is
denoted by
con(S) =
x∈R
n
|x =
P
m
i=1
λ
i
y
i
where
P
m
i=1
λ
i
= 1,λ
i
≥ 0∀i, andy
i
∈
S∀i
Finally, we define affine and common scale transformations and permutation
operators. For anyb,x∈R
n
, define the vectorb∗x so that(b∗x)
i
=b
i
x
i
. Also, let
b∗B ={b∗x|x∈B} for a compact and convex B. A positive affine transformation
is a mapf :R→R, such thatf(x) = (a∗x)+b for somea∈R
n
++
andb∈R
n
. A
common scale transformation is an affine transformation for whicha
i
=a
j
∀i6=j.
As a convention, we define the setf(S) ={y∈R
n
|y = f(x),x∈ S}. e
i
denotes
the vector whose i
th
element is 1 and all other elements are 0. Let1 denote the
6
vector whose elements are all1s. We use the following convention for reciprocal of
a strictly positive vector inR
n
: b
−1
≡
1
b
1
,··· ,
1
bn
.
A permutation operator, π, is a bijection from {1,··· ,n} to {1,··· ,n}. As
usual, π(x) = (x
π
−1
(1)
,··· ,x
π
−1
(n)
) and π(S) = {y ∈ R
n
|y = π(x)andx ∈ S}.
The class of all permutation operators is denoted byΠ
n
. Below, we give a list of the
axioms that will appear in this paper. We prefer to discuss another two axioms along
with the characterizations in the later sections as they look much more intuitive
along with the discussion of characterizations. F(B) refers to the point assigned to
B by solutionF .
- Pareto Optimality(PO):F(B)∈P(B).
- Symmetry(SYM): Ifπ(B) =B∀π∈ Π
n
, thenF
i
(B) =F
j
(B), ∀i,j.
- u-Symmetry(u-SYM):If π(B) = B∀π ∈ Π
n
, thenu
i
− F
i
(B) = u
j
−
F
j
(B), ∀i,j.
- Translation Invariance(TINV): For allx∈R
n
,F(B +x) =F(B)+x.
- Scale Covariance(SCOV):F(λ∗B) =λ∗F(B) for allλ∈R
n
++
.
- Independence under Common Scale Transformations(ICST):For allλ∈R
n
++
withλ
i
=λ
j
for alli6=j, and for allx∈R
n
, F(λ∗S+x) =λ∗F(S)+x.
- Independence under Affine Transformations(IAT):For allλ∈R
n
++
and for all
x∈R
n
, F(λ∗S +x) =λ∗F(S)+x.
- u-Independence of Irrelevant Alternatives(u-IIA): Suppose that S,T ∈ B
withS⊆T andu(S) =u(T). Then,F(T)∈S impliesF(S) =F(T).
7
Next, we have two widely used axioms but before introducing them, we need
some definitions. Define Δ = con{e
1
,··· ,e
n
} and Γ = con{e−e
1
,··· ,e−e
n
}.
Next, we define two other axioms:
- Individual Rationality(IR):F
i
(Δ)> 0 for alli∈N.
- Individual Fairness(IF):F
i
(Γ)< 1 for alli∈N.
Before going to the next section for our characterizations, we present two con-
crete cases of why we can’t approach the infeasible reference point approach strictly
by Nash’s approach. The first of the two theorems show that there is no solution
which will satisfy IAT and the second one will show that there is none satisfying
SCOV . For purposes of both brevity and illustrating the impossibility theorem, we
only present the proof of the second thm, which is due to CMW (2000).
Theorem 1.2.1. On classB, there exists no solution satisfying PO, SYM, u-IIA,
and IAT.
Proof. For a proof of this fact, see Roth (1979).
CMW realized that even with the weaker SCOV axiom, there is no solution we
can use to appeal to Nash’s proof.
Theorem 1.2.2 (CMW (2000)). On classB, there exists no solution satisfying PO,
SYM, u-IIA, and SCOV .
Proof. Forn = 2, consider the following problems:
A =con{(−8,0),(−8,−8),(0,−8),(−4,0),(0,−4)}
B =con{(−4,0),(−4,−12),(0,−12),(−2,0),(0,−6)}
8
By PO and SYM,F(S) = (−2,−2). Realizing thatB = λ∗A, whereλ = (
1
2
,
3
2
),
SCOV impliesF(B) = (−2,−2). Letting
C =con{(−4,0),(−1,−3),(0,−6),(−4,−6)}
, and noting that C ⊆ A, u(C) = u(A) = (0,0), and F(A) ∈ C we can apply
u-IIA so that F(C) = (−2,−2). But the following also holds: C ⊆ B, u(C) =
u(B) = (0,0) and F(B) ∈ C. Applying u-IIA again, F(C) = (−1,−3). A
contradiction.
In fact, the two proofs are essentially the same, whereas there is a redundancy
in Roth’s theorem in the sense that TINV is never used.
We close this section by noting that impossibility of SCOV requires us to work
on restricted classes on subsets-namely,sections of hyperplanes in the negative
orthant in R
n
- to characterize the solution. This implies that we have to trans-
form a set inB, so that its solution corresponds to the solution of an appropriate
hyperplane, of which there are infinitely many. In contrast, the Nash solution uses
only one such hyperplane. Both of the characterizations make heavy use of hyper-
planes and sets whose Pareto sets are sections of hyperplanes, so we will provide
the appropriate definitions in the next section.
9
1.3 Axiomatic Characterizations
1.3.1 Duality with Nash Solution and Proportional Gain vs.
Loss
As mentioned in the introduction, there is an interesting duality between the Nash
solution and EY solution. The Nash solution is the only solution that provides
proportional gains with respect to the nadir point of the bargaining set where nadir
point,n, is defined asn = (n
1
,··· ,n
n
) s.t. n
i
=min{n
i
|n∈B}. EY solution, on
the other hand, is the only solution that provides proportional losses from the utopia
point. We now introduce the proportional gain and proportional loss properties
formally.
- Proportional Gains(PGAIN):Suppose B,B
′
are sets that belong to G ,
whereG ={B∈B|P(B) =H(p,a)∩[n(B)+R
n
++
] for some p ∈R
n
++
}.
Letλ∈R
n
++
. Then, ifB
′
=λ∗B, then
[F
i
(B
′
)−n
i
(B
′
)]
[F
j
(B
′
)−n
j
(B
′
)]
=
λ
i
[F
i
(B)−n
i
(B)]
λ
j
[F
j
(B)−n
j
(B)]
for alli,j.
- Proportional Losses (PLOSS): Suppose B,B
′
are sets that belong to L
,whereL ={B∈B|P(B) =H(p,a)∩[u(B)−R
n
++
] for some p ∈R
n
++
}.
Then, ifB
′
=λ∗B, then
[u
i
(B
′
)−F
i
(B
′
)]
[u
j
(B
′
)−F
j
(B
′
)]
=
λ
j
[u
i
(B)−F
i
(B)]
λ
i
[u
j
(B)−F
j
(B)]
10
for alli,j.
The interpretation of the proportional gains statement is that the player who
gains more importance (say λ
i
> λ
j
) gains proportionally more than the other
player. For instance, if λ = (2,1), and B
′
= λ∗B, then player 1’s relative gain
would be twice the size of the relative gain of player 2. The interpretation of the
proportional loss statement is along the same lines; that is,the player who gains
more importance (say λ
i
> λ
j
) suffers less of a loss than the other player. For
instance, ifλ = (2,1), andB
′
= λ∗B, then player 1’s relative loss would be half
of the relative loss of player 2.
We are now in a position to give a dual characterization of Nash and EY solu-
tions. CMW(forthcoming) gives a characterization to both for equal weight games.
We extend their characterization to weighted games, hence completing the charac-
terization. We do not give a proof for the first theorem in this section, as it is similar
to the first part of the proof of the second theorem (characterization of the EY solu-
tion). The details of the proof is available in a working paper by CMW (1998) for
interested readers.
Theorem 1.3.1. A solutionF :B→R
n
satisfies PO, TINV , u-IIA, IR, and PGAIN
if and only if there existsw∈R
n
++
such thatF =NB
w
.
Proof. See CMW (1998).
Unfortunately, there is no direct equivalent of restating the above theorem using
PLOSS only as the Nash solution is satisfies IAT and Yu solution satisfies ICST.
Hence, we use an extended definition of PLOSS which is also meant to include
the additional requirement. We will refer to the combination of PLOSS and the
property below as simply PLOSS in the following sections. This added restriction is
11
not really desirable however, as explained ICST is weaker than IAT hence, PLOSS
has to be strengthened to make up for the more general ICST condition. This added
condition is similar to PROJ and u-CONS in spirit provided by McLean, van den
Nouweland, V oorneveld (2011).
• PLOSS in Utopia Projection: Call the cross-section of set B at any axis
(or combination of axis) C. Then a solution, F satisfies PLOSS in utopia
projection ifF(C) satisfies PLOSS, in the remaining axis where we drop the
axis mentioned in the first sentence.
It is not difficult to check that Yu solution satisfies the above property. In light
of these observations, we leave the proof of the next statement to appendix A; as it
is then much easier to understand using the statements and a crucial lemma from
the next subsection.
Theorem 1.3.2. A solutionF :B→R
n
satisfies PO, TINV , u-IIA, IF , and PLOSS
if and only if there existsw∈R
n
++
such thatF =WEY
w
.
Proof. See Appendix A.
We now state the equal weights version of the above theorem.
Theorem 1.3.3. A solution F : B → R
n
satisfies PO, TINV , u-IIA, IF , SYM and
PLOSS if and only ifF =EY .
Proof. See Appendix A.
1.3.2 The Nash Solution Approach
As we observed in the previous section, the main difficulty in characterizing the
EY solution in a way similar to Nash (1950) is the fact that it doesn’t satisfy IAT,
12
(0,2)
(2,0)
(1,1)
ˆ
B
xy = 1
Figure 1.1: Nash Solution
but satisfies ICST. Therefore, showing that an arbitrary solution is equal to the Nash
solution at a specific point and generalizing this result to the entire domain is impos-
sible.
By IAT, we can transform a compact and convex setB into another set,
ˆ
B, whose
disagreement point is normalized to 0, and whose solution is normalized to1 =
(1,··· ,1) inR
n
. The idea in Nash’s proof was to transform the set in such a way
that its transformed solution is equal to the Nash solution of a section of a symmetric
hyperplane in the positive quadrant. Then, one can apply PO, SYM, and IIA to
extend the solution to all the sets. Here we use a similar approach. We only deal
with the special case of Euclidean Yu solution, but the reader will easily notice that
this procedure can easily be extended to all Yu solutions. In fact, since ICST is the
13
critical property in this formulation one can define other solutions satisfying ICST
that is given by its values on hyperplane sections normalized in one coordinate. We
will make these ideas precise as we go through our discussion. We first discuss the
case where p = 2, the generalizations of this case will be given at the end of this
section.
Letp∈R
n
++
anda∈R. A hyperplane is defined asH(p,a) ={x∈R
n
|p.x =
a}. We will use the following type of sets throughout this section. LetL ={B ∈
B|P(B) = H(p,a)∩ [u(B)−R
n
++
] for some p ∈R
n
++
}. The Pareto set of any
L-set,L∈L , is obtained by taking the utopia point as the origin, and intersecting
a hyperplane with the negative orthant of the shifted space. This collection of sets
will be referred to often both in this and the following subsection.
We start by describing the solution to the weighted Yu problems on the L-sets.
This section relies on CMW(2008). Consider the weighted projection problem
min
z∈H(p,a)
P
i∈N
w
i
[u
i
(B)−z
i
]
2
has a unique solution given by
x =u(B)−
(p.u(B)−a)
P
j∈N
p
2
j
w
j
(w
−1
∗p) (1.1)
Next, we note that since p.u(B)−a ≥ 0, we have that for each i, x = u(B)−
(p.u(B)−a)
p
i
e
i
∈ [u(B)−R
n
+
], andp
i
.
u(B)−
(p.u(B)−a)
p
i
e
i
= a. This implies
thatx =u(B)−
(p.u(B)−a)
p
i
e
i
∈H(p,a)∩[u(B)−R
n
+
]. Since,x =
P
i∈N
u(B)−
(p.u(B)−a)
p
i
e
i
p
2
j
w
j
P
j∈N
p
2
j
w
j
!
is a convex combination ofx =u(B)−
(p.u(B)−a)
p
i
e
i
for each i andH(p,a)∩ [u(B)−R
n
+
] is convex, we conclude thatx ∈ H(p,a)∩
[u(B)−R
n
+
].
14
We continue our discussion with a lemma that we will use in the proof of several
theorems. Please note that, SCOV in the following lemma is a modification of
PLOSS in the new sense. That is, in the statement of Proportionality of Projection
we replace PLOSS with SCOV , but keep using SCOV notation for this new property.
Lemma 1.3.4. There exists no solution that satisfies PO, SYM, TINV , SCOV , u-IIA
such thatu(B)−F(B) / ∈R
n
++
andu(B)6=F(B). Note that, this statement extends
naturally to projections ofB, which we callP , whereP is defined as in the previous
section.
Proof. By TINV , we can normalize the utopia point to0. Then,F
i
(B) = 0 for some
i. LetM ={i∈ N|F
i
(B)6= 0}. For eachj ∈ M, pick anx
j
∈R
n
for allj ∈ M,
where the utopia point is attained. Such a point always exists as B is a compact
subset ofR
n
. Now, denote convex hull of F(B) and (x
j
)
j∈M
by D. D is contained
in B (by definition of u and convexity of B) and it is straightforward to verify that
their utopia points are the same and equal to 0. Hence, by u-IIA, F(D)=F(B). But,
by SCOV we can always transform D into a set whose pareto set is symmetric-
call this set S- by the following: P(S) = λ∗ P(D) where λ ∈ R
n
++
. (Note
that this transformation is possible by projection if needed and SCOV continues to
hold in this projection set.) By PO and SYM, F(S) > 0 while by PO and SCOV
F
j
(S) =λ
j
F
j
(D) = 0 forj∈M, a contradiction.
Remark. Note that we can use any proportionality property (such as ICST) in the
proof above as long as the 0’s remain the same. Such a property is in fact described
in the next section and will be used in the characterization proof. An illustration of
this fact is in the figure below.
15
Figure 1.2: Impossibility Result
We now start a discussion regarding our first theorem. Instead of presenting
it in a proof, we choose to illustrate the construction separately as a proof would
obscure some of the details. First, as IAT implies both TINV and SCOV , we start by
considering hyperplanes which are equal to 1 in exactly one coordinate with their
utopia points normalized to zero. Then, by PO we can start assigning values to
these hyperplanes by the following procedure: Since, these hyperplanes are L-sets,
we use equation (1) and solve it with the following to assign it a value that is equal
to the value of a u-symmetric set: (Note that herew = (1,··· ,1),u = (0,··· ,0))
x
i
=
ap
i
P
i∈N
p
2
i
Thus, the utopia point of the u-symmetric set that provides the value for our L-set
should satisfyu
i
−
ap
i
P
i∈N
p
2
i
=u
j
−
ap
j
P
i∈N
p
2
i
for alli,j∈N. A very simple example
for this kind of delegation to symmetry property is provided with the figure below.
Example 1.3.1. In this example, we consider the asymmetric L-set defined by the
convex hull of(−2,0) and (0,−1). EY solution gives(−2/5,−4/5) which is indi-
cated as pointb. Pointa is the solution of the symmetric set (con{(−1,0),(0,−1)})
16
b b −2
−
6
5
−1
2
5
(
2
5
,−
8
5
)
a
b
Figure 1.3: Delegation to Symmetry
and is equal to (−1/2,−1/2). An easy calculation shows that the supporting nor-
mal vector of the L-set is (1,2). We obtain the solution of this set as the solu-
tion of another u-symmetric set with the utopia point given by (2/5,0). Note that
u
1
−x
1
=u
2
−x
2
= 4/5 for this example.
Our final step in assigning the value to an arbitrary set is the following realiza-
tion. After fixing the values of L-sets to the values dictated by the EY solution,
we can use a nice property of the EY solution to confirm that any set can be trans-
formed so that its Pareto set coincides with a unique L-set which contain its value.
The vector of the supporting hyperplane at any solution x is equal to its negative
17
after some normalization. This can be seen as follows. For any L-set the minimiza-
tion problem implies that (u
i
−x
i
)
i∈N
is the supporting vector of hyperplane atx.
Asu
i
> x
i
, this implies that the vector is strictly positive. Now we show that any
set’s solution is contained in a unique L-set. We first show the uniqueness. Suppose
there were two points,x, y ∈ B with x 6= y such that x ∈ L
1
and y ∈ L
2
for
someL
1
, L
2
∈L . Then, as B is convex, the straight lie connecting x and y is in
B, and since x and y are optimal, the following two equations have to be satisfied:
x.(x−y)≤ 0 andy.(y−x)≤ 0. This can only happen whenx =y. A contradic-
tion. Existence follows from the fact that the functionf(x) =
P
x
2
i
is a surjection
from0−R
n
++
→ R
+
. Since we have shown that there can exist only one such
point, an easy application of u-IIA gives us the desired result since we can always
construct an L-set that contains B and both sets’ utopia points are normalized to 0.
This means that the solution of any set is supported (after a suitable normalization)
uniquely by an L-set contained in an hyperplane of the formH(−F(B),a) where a
is a multiple of (with a common scale)
P
i∈N
−(F
i
(B))
2
and where we normalized
u(B) = 0. This finishes our discussion of the first theorem together with lemma 1.
The case where u(B) = x can be handled as follows. It is easily seen that u(B)
is the only Pareto optimal point in B; hence by PO, F(B) = u(B) = x. This
concludes our discussion.
The discussion so far can be summarized in the following theorem.
Theorem 1.3.5. A bargaining solution is the EY solution if and only if it satisfies
the axioms PO, TINV , u-SYM, ICST, DS.
Proof. We have shown above that if a solution satisfies the four axioms, then it is
the EY solution. It is easily checked that EY solution satisfies the first three axioms.
That it satisfies DS was explained above.
18
The above discussion shows that we can extend our discussion to weighted EY
solutions (henceforth denoted as WEY); where we use an asymmetric L-set with a
utopia point of 0 to delegate all other normalized problems to. The next theorem
includes the details.
Theorem 1.3.6. A bargaining solution is the WEY solution if and only if it satisfies
the axioms PO, TINV , ICST, WDS.
Proof. We only consider the first case since the other two cases go through without
any change since the weighted solution is ICST. It is easy to verify that weighted
solutions satisfy PO, ICST. Similar to the discussion above, we can find a canonical
set (we call this procedure weighted delegation to symmetry (WDS)) as follows.
From equation (1)
u
i
−x
i
=u
j
−x
j
∀i,j∈N ⇐⇒p
i
=w
i
∀i
Hence, setting p = w (by the discussion above p > 0) we obtain an L-set. Also,
all the symmetric points are found through the canonical sets. This is the part of
the proof where we use the full force of special nature of the case p = 2. This
result is not in general true for other Yu solutions. Hence, we can characterize our
solution without resorting to any other kind of symmetry property in the weighted
case. Since this procedure determines all the canonical L-sets corresponding to the
each weighted solution, the proof parallels the arguments in the first theorem and
the details are omitted. In particular, now the solution of any set is contained in an
L-set of the formH(−w∗F(B),a) where a is a multiple of (with a common scale)
P
i∈N
−((w
i
F
i
(B))
2
).
Our discussion so far has focused entirely on the the casep = 2. One can realize
that by solving the problem all over again for 1 < p < ∞, similar conclusions
19
−1
−1 −2 −3
Figure 1.4: Yu solution on hyperplanes normalized to−1 ony coordinate
can be obtained for equal weight solutions. In fact, we can use a property of Yu
solutions to show that one can only consider L-sets normalized to −1 in only a
single coordinate. This is possible by the following property:
- Permutation Invariance: For allπ in Π
n
,F(π(B)) =π(F(B)).
The following figures clarifies the intuition behind our analysis. It shows that
the by equating an arbitrary solution to the original solution on a curve defined by
the solutions of hyperplanes normalized in one coordinate, we can use the above
procedure to show the equivalence of an arbitrary solution to the original solution
on the entire domain. By the permutation invariance property above, we only need
to consider the solutions of hyperplanes normalized in one coordinate.
20
Figure 1.5: Curves on which we set an arbitrary solution equal to the original ICST
solution
1.4 A Simple Application to Bankruptcy Problems
The previous analysis leads us to the following natural solution for bankruptcy
problems. These type of problems were first studied by Aumann and Maschler
(1985). The basic problem is the division of a resource (or a prize, estate etc.)
among parties whose claims in general sum up to more than the resource available.
Given a vector c = (c
1
,··· ,c
n
) of claims, where each c
i
≥ 0 for all i, define
C = con{(c
1
,0,··· ,0),··· ,(0,··· ,c
n
)}. u(C) = c = (c
1
,··· ,c
n
). Also, let
R > 0 denote the amount of resources available. Then, our bankruptcy solution is
equal to the following:
S =
EY
i
(C)R
P
i∈N
EY
i
(C)
21
This statement has a simple interpretation. First, we find the closest point avail-
able to the claims point on the convex combination of claims and then distributeR
in proportion to the chosen point. An easy calculation shows that for two players,
this solution is equal to division proportional to the claims. However, forn > 2 it
is not necessarily proportional to the claims.
22
1.5 Conclusion
We have given two axiomatic characterizations to the Euclidean Yu solution both of
which are closely related to Nash solution. We first established the duality between
the Nash solution and the Yu solution. The duality arises from the fact that the Nash
solution is the only solution that provides proportional gains from the nadir point of
the bargaining set, whereas the Yu solution is the only solution that provides pro-
portional losses from the utopia point of the bargaining set. Then, we provided a
characterization that is close to Nash’s idea in spirit. We first fix an arbitrary solu-
tion equal to the Yu solution on a curve (defined by the Yu solutions of hyperplanes
normalized in one coordinate), and show that the arbitrary solution is equal to the
Yu solution on the entireR
n
. We also suggest a way to generalize this idea to any
solution that satisfies ICST, in particular other Yu solutions (wherep6= 2).
We observed in the introduction that these type of bargaining problems are use-
ful in multi-decision making and bankruptcy problems. We provided a very simple
example of a solution to a bankruptcy problem. In particular, any resource division
problem with competing objectives can be resolved by using this approach. How
our solution stands among the other solutions in the bankruptcy literature and how
it can be characterized by the axioms in that literature is left for another paper.
23
1.6 Appendix A: Missing Proofs in Main Text
In this appendix, we prove the results skipped in the main text. There are a couple
of crucial lemmas that we will use.
Lemma 1.6.1. The sets Δ and Γ are included in L-sets. In particular, we can
use them as L-sets without any loss of generality with WEY solutions as the WEY
solution picks points in the L-sets containing Δ and Γ that belong to Δ and Γ,
respectively.
Proof. It is not difficult to verify that Δ and Γ are not L-sets. However, Γ is con-
tained in an L-set, namelyΓ⊂ [0−R
n
++
]∩H(e,(n−1)) whereH(e,(n−1)) =
{x ∈ R
n
|
P
n
i=1
x
i
= n− 1}. Similarly, Δ is contained in [1−R
n
++
]∩H(e,n).
From equation (1), it can be verified that
WEY
w
= 1−
1
w
i
P
j∈N
1
w
j
and
"
1−
1
w
i
P
j∈N
1
w
j
#
=WEY
i
w
(T
′
)
Note that since Δ and Γ are convex sets, Δ and Γ contain these solutions (just use
{
1
w
i
}
i∈N
for the convex combination). This finishes the proof of our lemma.
Lemma 1.6.2. There exists no solution satisfying PO,IF , TINV , PLOSS such that
u(B)−F(B) / ∈R
n
++
andu(B)6=F(B).
Proof. The proof is a slight modification of lemma 3.1. By the remark below that
lemma, we can use PLOSS instead of SCOV without loss of generality. Now, since
Γ is contained in a symmetric set S, by lemma 3.1 F
j
(S) = 1 for all j ∈ M (the
modified utopia point for the new problem). By u-IIA, F
j
(Γ) = 1 for all j ∈ M
contradicting IF. We are done.
24
Theorem 1.6.3. A solutionF :B→R
n
satisfies PO, TINV , u-IIA, IF , and PLOSS
if and only if there existsw∈R
n
++
such thatF =WEY
w
.
Proof. For allw ∈R
n
++
, the functionF : B →R
n
satisfies PO, TINV , u-IIA. To
show that EY satisfies IF, we show that
WEY
w
= 1−
1
w
i
P
j∈N
1
w
j
< 1 (1.2)
where we used equation (1), and the fact that Γ is contained in the hyperplane
H
(1,··· ,1),n− 1
. There is a problem we need to resolve here. To show that
WEY
w
satisfies PLOSS, let B,B
′
two sets in B such that B
′
= λ∗B for some
λ∈R
n
++
. Noting thatP(B
′
) = H(p
′
,a)∩[u(B
′
)−R
n
+
] wherep
′
= λ∗p. Then,
using equation (1), it can be verified thatWEY
w
satisfies PLOSS. For the next
part, assume that a solutionF satisfies the axioms stated in the theorem. For each
i, define
w
i
=
1
1−F
i
(Γ)
Now, by ifw∈R
n
++
. We will show thatF =WEY
w
. There are three cases:
Case 1. Suppose that u(B)− WEY
w
(B) ∈ R
n
++
. By TINV , we can normalize
u(B) to 0. Let x = WEY
w
(B). The set B is supported at x by the hyperplane
H(p,a) wherep =w∗(u(B)−x) =−w∗x andp∗x =a. By the assumption at
the beginning,a < 0. LetB
′
= p∗B, then the hyperplaneH(e,a) supports B’ at
the pointp∗x. Letβ be a point of equal coordinates withz >β for allz∈B
′
and
define the set T’ as follows:
T ≡{z∈R
n
++
|β≤z≤ 0 ande.z≤a}
25
Note that T’ is the hypercube inR
n
betweenβ and0. Next, we have thatP(T
′
) =
con{ae
1
,··· ,ae
n
}. We claim thatF(T
′
) =WEY
w
(T
′
).
To see this, letT” =T
′
−ae and we haveu(T”) =ae. Furthermore,
P(T”) =con{ae
1
,··· ,ae
n
}−ae =con{(−a)(e−e
1
),··· ,(−a)(e−e
n
)} =
(ae∗Γ)
Now, let S = P(T) = (−ae)∗ Γ. It is easily checked thatL is closed under
strictly positive linear transformations; namelyλ∗L∈L for allL∈L and for all
λ∈R
n
++
. Hence, we can assume without loss of generality that Γ and S are L-sets
by lemma 1 at the beginning of the appendix. Hence, applying PLOSS, we get
[u
i
(S)−F
i
(S)][u
j
(Γ)−F
j
(Γ)] = [u
j
(S)−F
j
(S)][u
i
(Γ)−F
i
(Γ)]
This in turn implies that [a+F
i
(S)](1−F
j
(Γ)) = [a+F
j
(S)](1−F
i
(Γ)). Since
PO implies that
P
i∈N
F
i
(S) =−(n−1)a, we conclude that
F
i
(S) =−a
1−
1−F
i
(Γ)
P
j∈N
[1−F
j
(Γ)]
=−a
"
1−
1
w
i
P
j∈N
1
w
j
#
SinceS ⊆ T”,u(S) = u(T”) =−a, andF(T”)∈ P(T”) = B, we can conclude
by PO and u-IIA that
F
i
(T”) =F
i
(S) =−a
"
1−
1
w
i
P
j∈N
1
w
j
#
Applying TINV , we have the following
F
i
(T
′
) =F
i
(T”)+a =a
"
1−
1
w
i
P
j∈N
1
w
j
#
=WEY
i
w
(T
′
)
for all i.
The next step involves using the inverse transformation and PLOSS to conclude
thatF(T) =WEY
w
(B). To this end, letT =p
−1
∗T
′
and definey =F(T). Since,
26
T’ is an L-set, and P(T) can be contained in an L-set by an argument used at the
lemma at the beginning of the appendix, we can apply PLOSS again
1
p
j
[u
i
(T)−y
i
][u
j
(T
′
)−F
j
(T
′
)] =
1
p
i
[u
j
(T)−y
i
][u
i
(T
′
)−F
i
(T
′
)]
Now,u(T) =u(T
′
) = 0 andF
i
(T
′
) =a
"
1−
1
w
i
P
j∈N
1
w
j
#
yieldsp
j
y
j
w
i
=p
i
y
j
w
j
and sincep =w∗x we havep
j
y
i
x
j
=p
j
y
j
x
i
. Sincey∈F(T)∈P(T)⊆H(p,a)
andx∈H(p,a), it follows thatx
i
=y
i
for all i, henceF(T) =WEY
w
(B).
To finish this case, observe thatB ⊆ T sinceB
′
⊆ T
′
, andu(B) = U(T) = 0 and
F(T) =x∈B. An application of u-IIA one more time gives the required result.
Case 2. For this case, consideru(B)−WEY
w
(B) / ∈R
n
++
andu(B)6=WEY
w
(B).
This case is very similar to the second case in section 3.1. Using lemma 2 gives us
the desired result.
Case 3. For this case, suppose thatu(B) =WEY
w
(B). Then, the proof is the same
as in discussion of theorem 3.1.
We now state the theorem for the equal weights case.
Theorem 1.6.4. A solution F : B → R
n
satisfies PO, TINV , u-IIA, IF , SYM and
PLOSS if and only ifF =EY .
Proof. It is easily checked by above thatEY satisfies the stated properties since we
can put w = e in the above proof. Now, suppose a solution F satisfies the given
properties other than IF. By PO and SYM,
F(Γ) = (
n−1
n
)e
, hence F satisfies IF. By the previous theorem, there existsw∈R
n
++
such that
27
(
n−1
n
)e =F(Γ) =WEY
w
= 1−
1
w
i
P
j∈N
1
w
j
It follows thatw
i
=w
j
for alli,j, henceF =WEY
w
=WEY
e
=EY .
28
1.7 Appendix B: Continuity
In this appendix, we discuss the continuity of the Yu solution with respect to Haus-
dorff metric topology.
- Continuity: If a sequence of sets, B
n
, inB converges to a set B ∈ B, then
F(B
n
)→F(B).
Continuity of bargaining solutions have been investigated by Jansen, Tijs(1983),
and Salonen (1996). Their analysis is not directly related to ours, though, as the
usual IIA axiom is replaced with u-IIA in our paper. Our first result shows that all
Yu solutions with 1 < p <∞ are continuous. To show this, we introduce a lemma
which shows that ifB
n
→B in the Hausdorff topology, thenu(B
n
) = u(B) in the
usual metric. To show this result, we introduce a couple of definitions. All of these
definitions and related results can be found in, for example, Aliprantis and Border
(2005).
Definition 1.7.1. Let (X,d) be a metric space. For each pair of nonempty subsets
of A and B of X, define
h
d
(A,B) =max{sup
a∈A
d(a,B), sup
b∈B
d(b,A)}.
The extended real number h
d
(A,B) is the Hausdorff distance between A and B
relative to the metric d. By convention,h
d
(∅,∅) = 0 and h
d
(A,∅) =∞ forA6=∅.
By Aliprantis and Border (Lemma 3.73) (henceforth denoted AB), Hausdorff
distance defines a finite valued metric on the space B, the compact and convex
subsets ofR
n
. Also, by AB, lemma 3.76, there exist a ∈ A and b ∈ B such that
29
d(a,b)= h
d
(A,B), meaning that the minimal distance between the sets is actually
attained.
Definition 1.7.2. Let{E
n
} be a sequence of subsets of a topological space X. Then,
1. A point x in X belongs to topological lim sup, denoted LsE
n
, if for every
neighborhood V of x there are infinitely many n withV ∩E
n
6=∅. InR
n
, this
implies that there is a subsequencex
n
k
such thatx
n
k
→ x andx
n
k
∈ E
n
for
each n.
2. A point x in X belongs to topological lim inf, denoted LiE
n
, if for every
neighborhood V of x we have thatV ∩E
n
6=∅ for all but finitely many n. In
R
n
, this implies that there is a sequence x
n
such that x
n
→ x and x
n
∈ E
n
for each n.
3. IfLiE
n
=LsE
n
=E, then the set E is called the closed limit of the sequence
{E
n
}.
Remark. The following definition is taken from AB (2005) with a minor adaptation
to Euclidean spaces. This type of convergence is usually referred to as Kuratowski
convergence in the literature.
We need one more result before we can state and prove our lemma.
Lemma 1.7.1 (AB (2005 Theorem 3.93)). On compact subsets of a topological
space, Hausdorff convergence (regardless of the underlying metric) implies Kura-
towski convergence and vice versa.
Remark. Salinetti and Wetts (1979) prove this result in detail for normed vector
spaces, which includes the special case ofR
n
.
30
We are now ready to state our main theorem of this section.
Theorem 1.7.2. IfB
n
→ B in the Hausdorff topology, thenu(B
n
)→ u(B) in the
Euclidean metric topology.
Proof. It is sufficient to show thatu
i
(B
n
)→ u
i
(B) for alli∈ N. Pick a sequence
u
i
(B
n
). We claim that∪
∞
n=1
B
n
is compact. Denote the closedǫ neighborhood of B
byC
ǫ
(B) ={x∈R
n
|d(x,B)≤ǫ}, where d is the usual Euclidean distance. C
ǫ
(B)
is compact. Also, sinceB
n
converges toB in the Hausdorff topology,B
n
⊆C
ǫ
(B)
for alln≥N
o
for someN
o
∈N. Hence, the union of∪
No
i=1
B
n
andC
ǫ
(B) is compact
and includes∪
∞
n=1
B
n
. Thus, by passing to a subsequence if necessary,u
i
(B
n
)→x
for some ˆ x
i
∈ {x
i
|x ∈ B}. Suppose, ˆ x
i
< u
i
. Since Hausdorff convergence is
equivalent to Kuratowki convergence on compact subsets, there exists a sequence
v
n
withv
n
→u
i
. Then, there exists anm∈N such that for alln≥m,v
n
>u
i
(B
n
).
A contradiction. This concludes our proof.
Remark. In the proof above, we only used the existence of maximal points. Bound-
edness from above in all coordinates and closedness is enough to guarantee this
result. Hence, it is straightforward to modify the above proof closed, convex, and
comprehensive sets that converge in Hausdorff topology. Also, note that the results
easily extends to nadir points by a simple modification of the proof.
Continuity of the Yu solutions can be established by using Theorem 6. Since
u(.) is Hausdorff continuous, any continuous function of x and u(.) is also con-
tinuous. Hausdorff convergent sequences define a nonempty, compact, and convex
valued correspondence. By the above theorem, this correspondence continuous by
definition and hence the minimizers of a continuous function on this set is also
continuous by Berge’s maximum theorem.
31
One question that arises from this analysis is whether continuity implies or is
implied by any of the simple axioms mentioned at the beginning of the second
section. Our answer is no in general. An open question at this point is whether
PLOSS implies continuity or is implied by continuity. However, we show in the
following two examples that PO, u-IIA, and IAT is neither sufficient nor necessary
for continuity.
Example 1.7.1. First, we show that continuity doesn’t imply either of PO and u-
IIA. To see this consider the solutionF :D
n
→R
n
, which picks the nadir point of
the set whereD
n
is the subset ofB such that a memberD
n
∈D
n
, includes its nadir
point. This solution is obviously not PO, and it is not difficult to show that it is not
u-IIA either. However, it is easy to see that the nadir point solution is IAT, which
leads us to the second example. For n=2, consider a variation of the Yu solution
given by
F(B) =argmin
x∈B
(u
1
−x
1
x
2
)
This solution is continuous by the arguments given above (as it is continuous in
u(.) for each coordinate), and the function is strictly concave (asD
2
(F(x)) is neg-
ative definite) on the domain u
1
= 0 ofB, hence the solution defines a continu-
ous function on this domain by Berge’s theorem. However, the following simple
argument shows that it is not SCOV , hence not IAT. Forcon{(−
3
2
,−
1
2
),(0,−2)}},
the solution picks (−1,−1). This is the straight line between (−2,0) and (0,−2)
with the northwest corner cut. For λ = (4,4), this set is transformed into
con{(−6,−2),(0,−8)}. Since the only interior solution (−4,−4) is not in the
feasible set one of the corner solutions, (−6,−2) is picked. Now, it follows that
SCOV is not satisfied.
32
Figure 1.6: Level Sets and Impossibility Result
Example 1.7.2. Our next example shows that PO, u-IIA, IAT doesn’t imply conti-
nuity. Consider the correspondence given by
F(B) =argmin
x∈B
max{|u
i
−x
i
|}
i∈N
The level sets of this correspondence is illustrated in figure 3.a. This solution cor-
responds to the Yu solution withp =∞. Consider the 3 cases we discussed in our
characterization proofs. In the first case, the solution above picks a single point. The
second case is impossible, by theorem 2.1. This fact is illustrated in figure 3.b. For
the third case, the Yu
p=∞
solution is actually a correspondence; but we can always
pick the utopia point, u, as it is available and minimizes the function above. This
solution is obviously PO, u-IIA, and IAT. To see that it is discontinuous consider the
sequence of sets defined byB
n
=con{(0,−1),(−1,1−
1
n
)} forn∈N inR
2
. It is
easy to see that this sequence Hausdorff converges toB =con{(0,−1),(−1,−1)}.
However,F(B
n
)→ (−1,−1) whileF(B) = (0,−1).
33
Chapter 2
Optimism in Bargaining Games
without Common Priors: Further
Results on Immediate Agreement
2.1 Introduction
The non-cooperative approach to the bargaining problem proved to be one of the
more fruitful approaches to bargaining problem. The cooperative approach, while
simple and elegant, doesn’t take into account the ways in which a typical point of
agreement is reached. The origins of this approach date back to Stahl (1972)and
Rubinstein (1982). In those models, it is shown that under a strict alternating offers
procedure, the players will reach an agreement in the first period. However, as
Yildiz (2003) demonstrated, when we relax the strict bargaining procedure to allow
for beliefs about recognition at any time period, there is not necessarily immediate
agreement. As the players are now allowed to hold any belief, when the players are
highly optimistic, they will not agree immediately. In fact, if each player is highly
optimistic about their chances of getting selected, they will wait in the hopes of
getting selected in future periods obtaining a higher rent.
As there is no more a strict procedure determining who moves when, the only
source of bargaining power comes from the differences in belief as opposed to the
34
first mover advantage in the classical model of bargaining. Yildiz (2003) argues that
these differences in beliefs can either arise from a lack of common knowledge or a
lack of common priors. Since common knowledge is assumed in his model, the only
source of the differences in beliefs will be a lack of common priors. In his model,
the selected player will extract a positive rent and at any time period the present
value of the rents extracted will constitute the players’ continuation values in that
game. In this paper we relax the transferable utility assumption and demonstrate
that these rents are higher than the ones in transferable utility case. This means that
the players will find it more attractive to wait in the hopes that they will get selected
in future periods, thereby acquiring more rent in the future.
We will intuitively give a relation between optimism and immediate agreement
to illustrate some of the points we want to make in this paper. In this model, we have
optimism if one player’s assessment of probability of getting selected is higher than
the other player’s assessment. This amounts to the fact that the level of optimism is
greater than zero:
y
t
=p
1
t
+p
2
t
−1≥ 0
Note that in this setting when we talk about optimism, we are talking about
optimism for both players. Hence, when the level of optimism is very high, each
player will think that the other player will also be very optimistic and as a result
she will lower her expectations about higher rents in future periods. This is the
fundamental insight for our discussion in the next paragraph.
Yildiz(2003) shows that with perpetual optimism, players will reach an agree-
ment immediately in a sufficiently long game. When the level of optimism stays
high enough, each player will think that the rents he will be able to extract in future
periods will be lower, hence agrees immediately. Without transferable utility these
35
rents are higher as we will demonstrate in the following sections. Therefore, play-
ers will agree immediately only if the level of optimism is so high that none of them
thinks they can extract a huge rent in future periods. To put it another way, when
the players are risk averse they are more willing to wait unless they are really opti-
mistic. We will discuss an example in section 3 where there is perpetual optimism
but where it is also common knowledge that players will not reach an agreement
in the first period. This example demonstrates that it is impossible to extend the
immediate agreement under perpetual optimism result to bargaining without trans-
ferable utility. Despite this, as we will discuss in the next sections, there is a strictly
positive level of optimism for which immediate agreement occurs with risk averse
players.
Another reason why agreement may be delayed is periods of optimism followed
by pessimism. If players are very optimistic for a certain time period in the future,
but very pessimistic for the next period then the agreement might be delayed. In
short games, players will wait until the moment where optimism is really high and
after which the optimism is much lower. This is observed in bargaining experiments
(Roth et al. 1988) where players reach an agreement just before the deadline.
1
Our
second result illustrates this point for risk averse players. The bound on the drop in
optimism in one period to the next required for immediate is agreement is tighter
in the risk averse case. That is, sudden drops in the level of optimism will make
it more likely that players will delay the agreement. Intuitively, higher levels of
optimism must be sustained throughout for the immediate agreement to hold.
Obviously, both lines of reasoning discussed above rely on the assumption of
no learning. If a player observes that his opponent has made more offers in the past,
1
The level of optimism is 0 at the deadline.
36
he might reduce his level of optimism, hence our immediate result won’t hold since
perpetual optimism will not be sustained. Therefore, both of the arguments above
rely on the implicit requirement that players’ beliefs are firm.
Before introducing the model and the preliminary results we provide a review
of the literature. The first to papers to investigate the bargaining game of alternating
offers were Stahl (1972) and Rubinstein (1982). In their models, there is immedi-
ate agreement resulting and the bargaining power arises from the order of players.
In other precursors of the model Rubinstein and Fishburn (1982) characterize the
time preference representation that is critical to the standard bargaining framework.
Shaked and Sutton (1984) provides a simple argument for calculating the subgame
perfect equilibria of a bargaining game, but they consider a bargaining framework
between employers and workers.
Our model is a sense more related to bargaining in stochastic environments that
originated with the work of Merlo and Wilson (1995). Merlo and Wilson (1995) and
(1998) analyze a multilateral bargaining game where the surplus and recognition
are stochastic but where agents agree over the distribution of the stochastic process.
They find that the unique stationary equilibrium may involve delayed agreement but
such delays are necessarily Pareto-efficient as the unanimity rule implements the
optimal stopping-time. Delay in bargaining with complete information is obtained
in models of Sakovics (1993), Motty and Perry (1987) obtains delays in bargaining
with complete information, however in their models, delay occurs through strategic
play rather than an uncertainty. Feinberg and Skrzypacz (2005) consider a model
with delayed bargaining where the buyer has private information about the value of
the object and the seller has private information about his beliefs about the buyers
valuation. They show that delay is caused by this one sided uncertainty.
37
Yildiz (2001) and (2003) are the two papers that we improve on and provide
a benchmark for our analysis. His model is a significant departure from the usual
alternating offers with a strictly determined procedure. As we have seen, the pro-
cess determining who is going to make the offers are determined by nature and
players have subjective preferences about the probability that they are going to get
selected. There are several other papers which use these papers as benchmarks in
their model. Ali (2005) considers the bargaining problem without a common prior
for playersn≥ 3 and finds that most of the results fromn = 2 does not carry over
to n player case. Specifically, he studies a case where agents disagree about their
bargaining power. He shows that if agents are extremely optimistic, there may be
costly delays in an arbitrarily long finite games but if the optimism is moderate, then
all sufficiently long games end in immediate agreement. There are two other issues
that he deals with that we will not talk about in this paper. The first is the behavior
of the equilibrium in the infinitely long games. He shows that in case of extreme
optimism, the game is highly unstable in the finite horizon. Another issue discussed
in the paper is the voting rule in bargaining. His paper shows that majority rule may
be a better option that unanimity rule when agents are optimistic.
Yildiz (2011) reviews the literature regarding optimism in bargaining and is a
good reference for the combined results of papers by Yildiz (2003) and Yildiz and
Simsek (2007). Yildiz and Simsek (2007) mainly discuss bargaining with stochastic
deadlines. Their first main finding is that the deadline effect naturally occurs in
equilibrium of bargaining models with optimistic players. This is mainly because
at the deadline date, there is a wide variety of individually rational outcomes and
high bargaining power of the agent who gets to make the final offer allows him
to wait until the last period where he is so optimistic about his chances of being
38
selected that he is not willing to agree immediately. The crucial point driving all
these type of results is that fact that the players will not obtain anything after the
deadline, hence excessive optimism near the deadline will make the agents settle in
periods closer to the deadline.
The rest of the paper is organized as follows. In section 2, we outline the model
in detail and provide several results about bargaining without common priors with
TU assumption for purposes of discussion and comparison. In section 3, we provide
our results discussed in the introduction mathematically. We conclude in section 4.
We discuss our construction of the equilibrium results, in particular the discussion
of the agreement and disagreement regimes in the appendix.
39
2.2 Model and Preliminary Results
In this section, we construct our general model. There are two players,N ={1,2}
and we will occasionally index them by players i and j. We writeR
n
for the n-
dimensional Euclidean space and N for the set of nonnegative integers. In this
paper, we only consider finite games, hence our time grid is T = {t ∈ N|t < t}
for somet <∞ with the interpretation thatt is where the game ends. The players
bargain over a compact and convex set whose Pareto frontier inR
2
+
is given by the
following. Letx∈ [0,1] andf(x) = (1−x
a
)
1
a
for somea> 1. Note that we have
chosen our function so thatf(0) = 1 andf(1) = 0, and the bargaining setU whose
Pareto frontier is defined by such a function is symmetric and convex. Symmetry
is useful in writing the expression for rents as will be explained shortly. Note also
that the Pareto frontier is strictly decreasing, hence the bargaining set satisfies the
free disposal assumption.
As in Yildiz (2003), we analyze the gameG
t
[δ,ρ]. At eacht∈T , nature recog-
nizes a playeri ∈ N; i offers an alternativeu = (u
1
,u
2
) ∈ U. If the other player
accepts the offer, then the game ends with a payoff vector δ
t
u = (δ
t
u
1
,δ
t
u
2
for
someδ∈ [0,1]; otherwise, the game proceeds to datet+1 for0≤t≤t−2. Att−1,
the game ends and both players get 0. We writeρ ={ρ
t
}
t∈T
for the recognition pro-
cess which is observed by both players. We letρ
t
= (ρ
0
(ω),ρ
1
(ω),··· ,ρ
t−1
(ω))∈
N
t
for a list or recognized players at timet and P
i
(.|ρ
s
) for playeri’s conditional
belief at any historyρ
s
∈ N
s
. We assume perfect recall, and everything described
so far is common knowledge.
In this paper we assume that no learning takes place. This implies that ρ is
independently distributed under both P
i
and P
j
. Writing p
i
t
(ρ
s
) = P
i
(.|ρ
s
), this
40
implies that we can drop the argument ofp
i
t
(ρ
s
) and denote the probability of player
i getting selected simply byp
i
t
.
If p
1
t
+ p
2
t
> 1, then each player’s assessment of his probability of getting
selected is higher than what the other player assesses. We say that players are
optimistic (pessimistic) if p
1
t
+ p
2
t
≥ (≤)1. The level of optimism at time t is
denoted byy
t
=p
1
t
+p
2
t
−1.
Now, we describe and discuss the subgame perfect equilibria of an arbitrary
game; the results in this section will be used later for comparison purposes. For any
gameG
t
[δ,ρ], writeV
i
t
(ρ
s
) for the equilibrium continuation value of playeri at any
t with historyρ
s
. Then,V ={V
i
t
}
i,t
is defined by the following recursive equation
V
i
t
(ρ
t
) =p
i
t
m
i
(δV
t+1
(ρ
t
,i))+p
j
t
δV
i
t+1
(ρ
t
,j)) (2.1)
and the boundary conditionV
t
≡ 0. Here,m
i
:R
n
→R
+
is defined by
m
i
(v) =max[{v
i
∪{u
i
|(u
1
,u
2
)∈U,u
j
≥v
j
}]
at eachv∈R
2
fori6= j. Note that herem
i
(v) is the maximum utility that playeri
will enjoy if he has to give at leastv
j
to playerj. Hence, player i obtains the consent
of player j, and obtainsm
i
(v), otherwise in case of disagreement, the payoff vector
v will be realized. Hence, equation 1 is an expression of the expected continuation
value of the playeri at timet depending on the recognition at timet. The boundary
conditionV
t
≡ 0 states that players automatically get 0 att. The following propo-
sition is adapted from Yildiz (2001) for finite games and characterizes the subgame
perfect equilibrium strategies in our game.
41
Proposition 2.2.1. In a gameG
t
[δ,ρ], at any subgame perfect equilibrium and at
any datet withρ
t
, the following are true: the vector of continuation values at the
beginning oft is V
t
(ρ
t
); ifδV
t+1
(ρ
t
,i) is in the interior of U, andi is recognized,
then they reach an agreement that givesδV
j
t+1
(ρ
t
,i) toj andm
i
(δV
t+1
(ρ
t
,i)) toi;
and ifδV
t+1
(ρ
t
,i) / ∈U andi is recognized, then they do not reach an agreement at
t.
In the equilibrium, the rents are defined by what a player gets when he gets
selected and there is agreement minus what he gets when the other player gets
selected. Hence player i’s rent is given by
R
i
t
(ρ
t
) =m
i
(δV
t+1
(ρ
t
,i))−δV
i
t+1
(ρ
t
,i)) (2.2)
It is also true that for our game, the continuation value of a playeri at any given
date t is given by the present value of the rents from t onwards. For our general
functional form introduced at the beginning of this section, we use the fact that
m
1
(v) =f(v
1
) andm
2
(v) =f(v
2
) which can be easily verified. This has the effect
of reducing the expression for rents to
R
i
t
=f(δV
i
t+1
)−δV
i
t+1
Note also that the rents are strictly higher than the rents in the transferable utility
case, which is equal to1−δV
i
t+1
−δV
i
t+1
.
We will introduce two more results on bargaining games without common priors
that we extend on in this paper. The first of these results is concerned with perpetual
42
optimism and immediate agreement. It states that with transferable utility and no
learning perpetual optimism leads to immediate agreement.
Proposition 2.2.2. With transferable utility, no learning, and perpetual optimism,
we have an agreement regime at eacht≤t−L(δ)−2.
Here, L(δ) is the length of the disagreement regime. We will characterize the
disagreement regime for our game in the appendix. Yildiz (2001) shows that it is
not possible to extend this result to risk averse players with an example we will
discuss at the beginning of next section. However, we have the following result, on
which we expand.
Proposition 2.2.3. Given anyδ∈ (0,1) and any nonnegative integert
∗
∈N, there
exists somey
∗
∈ (−1,1) such that, for every game witht≥t
∗
+L(δ)+3 and with
y
t
≥y
∗
at eacht∈T , we have an agreement regime at everyt≤t
∗
.
This theorem extends the above theorem in a weaker form. As we will see we
will show the existence of a strictly positive y
∗
such that the our version of this
result holds.
Finally, before we finish this section, we provide the final result we need.
Proposition 2.2.4. Assume transferable utility, no learning and thaty
t
−y
t+1
≤
1−δ
δ
at each t ∈ T . Then, we have an agreement regime at each t ∈ T with t ≤
t−L(δ)−2.
This proposition establishes that, we do need the optimism to stay high, all we
need is that it doesn’t drop too fast. Usually, there is a disagreement regime at the
end of the game, because when the game ends y
t
is identically−1, and thus we
may have a substantial drop in optimism at the end of the game. This proposition
43
implies that if the transition to−1 is smooth enough, there will be no disagreement
regime at the end of the game either.
44
2.3 Results
We saw in the previous section that perpetual optimism and immediate agreement
cannot be generalized to the case of risk averse players. In fact, as the follow-
ing example shows, it is also impossible to obtain a y
∗
with y
∗
≤ 0. We will
demonstrate the existence of a strictly positivey
∗
such that the a new version of the
perpetual agreement result will hold.
Example 2.3.1. Assume no learning and take the bargaining set with the Pareto
frontier given by f(x) =
√
1−x
2
for x ∈ [0,1]. Also, let t > 3, δ ∈ (0.907,1),
and p
1
= (1,1,1/2,0,··· ,0) and p
2
= (0,1,1/2,1,··· ,1) which implies that
y = (0,1,0,··· ,0). By the fact that continuation values are the present values of
the rent at each time t ∈ T , we have that V
3
= (0,1). Hence, at t=2, if recog-
nized, players 1 and 2 will offer (
√
1−δ
2
,δ) and (0,1) respectively which will be
accepted. Hence,
V
2
= (1/2)(
√
1−δ
2
,δ)+(1/2)(0,1) =
√
1−δ
2
2
,
1+δ
2
!
At the beginning of date1, player1 is certain that she will be recognized and hence
will offer(
p
1−(δV
2
2
)
2
,δV
2
2
) which will be accepted. Thus, his continuation value
will be, V
1
1
=
r
1−δ
2
1+δ
2
2
. Similarly, for the second player we have V
1
1
=
r
1−δ
2
1−δ
2
4
. There is a disagreement regime att = 0 if(V
1
1
)
2
+(V
2
1
)
2
> 1/δ
2
.
Using the values given above, we have disagreement if 2−
δ
2
2
(1 +δ)−
1
δ
2
> 0.
Lettingf(δ) equal to the right side of the expression, we have f(1) = 0, f
′
(1) =
−1/2< 0 which implies thatf(δ) is sufficiently close to1.
45
This example shows that takingy∗≤ 0 will not suffice for an immediate agree-
ment result. Hence, we have the following strengthening of Proposition 2.3 for risk
averse players.
Theorem 2.3.1. For any game G
t
[δ,ρ] and for any a > 1, there exists a y
∗
> 0
such that whenever y ≥ y
∗
, we have an agreement regime for each t ∈ T with
t≤t−L(δ)−2.
Proof. See appendix
As we have noted before, this theorem strengthens the immediate agreement
under perpetual optimism result that was given by Yildiz (2003). The example
above shows that when the players are risk averse, they will not reach an agreement
immediately even under perpetual optimism. Our results state that to reach such an
agreement, each player must be very optimistic throughout the game so that their
expectations about rents in further periods are further scaled downwards. This is the
key insight to the immediate agreement result under transferable utility assumption.
We show that when the agents are risk averse, with a given level of optimism it
makes more sense to wait to extract a higher rent rather than agreeing immediately.
Our second result provides another counterpart to another explanation for imme-
diate agreement. With this explanation, it may be that the players are not always
entirely optimistic, but they don’t lose their optimism abruptly. This result implies
that as long as the optimism stays within a bound, we will have an immediate agree-
ment result throughout the game, regardless of whether the optimism is positive or
negative. The theorem below provides such a bound for games with risk averse
players.
46
Theorem 2.3.2. For any gameG
t
[δ,ρ] and for anya> 1, the drop in optimism from
one period to the next required for immediate agreement is bounded by
(1−δ)/δ
a
Forδ > 1/2, this expression is less than(1−δ)/δ, which is the bound on the drop
from Yildiz (2001).
Proof. See appendix.
Intuitively, given a change in optimism from one period to the next, if the players
are risk averse since their expectations about rents in the next period will be affected
more than in the case when they are risk neutral (transferable utility). Hence, one
of the players will have higher expectations about rents in future periods so he
will decide to wait instead of agreeing in this period. In a sense, as the players
become more and more risk averse, the bounds on optimism from one period to
next required for immediate agreement becomes tighter. In fact, we conjecture but
do not prove that in the limit
2
, the players become so risk averse that immediate
agreement can only be attained unless the optimism stays throughout. The fact that
optimism is zero at the end of the game gives us an insight about this. This implies
loosely that infinitely risk averse players can only agree at identically a 0 level of
optimism. We do not make this insight precise at the moment.
2
Namely,a→∞
47
2.4 Conclusion
In this paper, we have extended the model of Yildiz (2003) to allow for risk averse
players. We do that by analyzing the equilibrium of the bargaining game on a
strictly convex and symmetric set. Our analysis can be extended to arbitrary convex
sets by obvious modifications of our construction. We have established two results
that were discussed in his paper but were left as an open question.
First, we discussed the case of perpetual optimism. With transferable utility
(without risk aversion) and no learning as long as the level of optimism stays posi-
tive indefinitely, players will reach an agreement immediately. If both players think
that the their probability of getting selected is high, in very long games they will
lower their expectations about extracting high rents. Thus, they will reach an agree-
ment immediately. When the players are risk averse, as we have shown the rents
are higher than the transferable utility case, meaning that at the same levels of opti-
mism, the players will now be more optimistic about getting higher rents in the
future, so sustaining immediate agreement will be harder. In fact, as the example
at the beginning of section 3 demonstrated, risk averse players will not reach an
agreement immediately with perpetual optimism. To sustain immediate agreement,
both players have to be so optimistic that they must lower their expectations more
about higher rents in the future. This is our first result regarding the immediate
agreement.
Yildiz (2003) also demonstrates another way immediate agreement can be
attained. Instead of requiring perpetual optimism, he shows that as long as the
drop in optimism from one period to the next stays sufficiently low, there will be
immediate agreement. We show that if the players are risk averse, then the bound
48
in drop has to be narrower than the one in transferable utility case. Intuitively, if the
players more risk averse and there is a sudden drop in optimism from one period to
the next, they will have more incentive to settle before the last period where there
is a sudden drop in optimism.
Our results are also related to the explanations of delay in agreement that occur
in various models of bargaining such as strategic uncertainty. In those models,
delay is caused by various forms of uncertainty and it is not observed as soon as
uncertainty vanishes. In our model, too, delay is less likely when the agents are risk
neutral.
49
2.5 Appendix: Missing Proofs in Main Text
In this appendix, we characterize the disagreement and agreement regimes in our
equilibria and show the results that were stated without a proof in section 3. For any
finite given finite horizon game, G
t
[δ,ρ], we construct our equilibrium as follows.
We writeV
i
t+1
for the continuation value of playeri at beginning of datet. For our
specific form of the bargaining set introduced in section 2, S
t
= (V
1
t
)
a
+ (V
2
t
)
a
for a > 1 gives us the perceived size of the pie
3
If they have not reached an
agreement before t each player will get 0; hence, we set V
1
t
= V
2
t
= S
t
= 0.
Taking this as final values, can determine the values of V and S recursively. First,
we discuss the disagreement case, which is characterized byS
t+1
> (1/δ)
a
. They
will then agree on a utility pair u = (u
1
,u
2
) only if u
i
≥ δV
1
t+1
, which requires
that (u
1
)
a
+(u
2
)
a
≥ (δV
1
t+1
)
a
+(δV
2
t+1
)
a
= (δ)
a
S
t
> 1 showing that suchu is not
feasible. Hence, the continuation value of eachi at the beginning of datet will be:
(V
i
t
)
a
= (δV
i
t+1
)
a
, and summing up this expression fori andj, we obtain
S
t
= (δ)
a
S
t+1
(2.3)
As in Yildiz (2001), the beliefs about recognition att is immaterial to the problem.
The nice point about the expression above is that it allows us to find a bound on
the disagreement regimes for eacha > 1 just as in Yildiz (2003). Hence, in a dis-
agreement regime, the players belief about which player is going to be recognized
at t is immaterial, and the perceived size of the pie will shrink as we go back in
3
From here on when we talk about a gameG
t
[δ,ρ], we will be talking about a game with a given
value ofa > 1. Thus, we do not specify the value ofa unless necessary .
50
time. Letting
ˆ
t be the first time period that they reach an agreement after a period
of disagreement regimes, the perceived size of the pie at any time periodt will be
S
t
=δ
a(
ˆ
t−t)
S
ˆ
t
. Hence, the length of the disagreement regime will be
L(S
ˆ
t
,δ) =
&
alogS
ˆ
t
log(1/δ)
'
−1 (2.4)
where the operator⌈.⌉ finds the smaller integer that is greater than or equal to the
argument. Here, one more observation is worth noting. As a increases the length
of the disagreement regimes also increases. Thus, in general it is harder to reach
an immediate agreement than the transferable utility case. Also, sinceS
ˆ
t
can be at
most 2, the length of the disagreement regime is also uniformly bounded. This is
denoted by
L(δ) =
&
alog2
log(1/δ)
'
−1 (2.5)
. Simple calculations reveal that the length of the delay can be arbitrarily large for
arbitrarily largea. In fact, one can easily show the following expression to hold:
(1/2)
1/a
≤δ
L(δ)
≤ (1/2δ)
1/a
In above, asa→∞, the length of the delays will be arbitrarily close to the size of
the pie.
Now consider the caseS
t+1
≤ (1/δ)
a
. Any playerj accepts an offer if and only
if it gives her at least δV
j
t+1
. If the recognized player i offers δV
j
t+1
to j and he is
left withf(δV
j
t+1
)−δV
j
t+1
, which is going to be at least as high as his continuation
51
value which is given byδV
i
t+1
. Here, f(.) is defined as at the beginning of section
2. To see why the above statement holds, consider the following inequalities:
δV
i
t+1
+δV
j
t+1
≤ (V
i
t+1
)
a
+(δV
j
t+1
)
a
=S
t+1
≤ (1/δ)
a
≤f(δV
j
t+1
)
, where the first inequality follows from Lemma 1 below, and the last inequality
follows from the definition of f. Given this, the continuation value of player i at
timet is equal to
V
i
t
=p
i
t
(f(δV
j
t+1
)+(1−p
i
t
)δV
i
t+1
(2.6)
=p
i
t
(f(δV
j
t+1
)−V
i
t+1
)+δV
i
t+1
(2.7)
In the expression above, f(δV
j
t+1
)−V
i
t+1
is rent for player i, what he expects to
get in the event he gets recognized minus what he gets when the other player gets
recognized. Unfortunately, deriving a relation betweenS
t
andS
t+1
is algebraically
intractable. However, we can use derive a relation that provides bounds on the
relation between those two. We first state a lemma that will be useful in establishing
the bounds of the size of the pie from one period to the next.
Lemma 2.5.1. For any x,y ∈ [0,1] and any a > 1, the following relation holds:
x+y≤x
a
+y
a
≤ (x+y)(1−(1/2)
a
).
Proof. The first inequality is immediate and the second one follows from the fact
that the graphf(.) as defined in Section 2 lies below the linex+y = (1−(1/2)
a
).
Next, we have the bounds on the size of the pie att andt+1.
52
Lemma 2.5.2. For any gameG
t
[δ,ρ] and for anya> 1,
Σ(V
t
)≤ Σ(V
t+1
)+(3+y
t
)min
f(δV
i
t+1
),f(δV
j
t+1
.
Proof. By Lemma 1, the following holds:
Σ(V
t
)≤
X
i∈N
p
i
t
(f(δV
j
t+1
)−V
i
t+1
)+δV
i
t+1
(1−(1/2)
a
)
≤
(1+y
t
)max
f(δV
i
t+1
),f(δV
j
t+1
)
+(1−δ)(V
i
t+1
+V
j
t+1
)
(1−(1/2)
a
)
≤
(1+y
t
)max
f(δV
i
t+1
),f(δV
j
t+1
)
+2(1−δ)
(1−(1/2)
a
)
Also,
Σ(V
t
)≥
X
i∈N
p
i
t
(f(δV
j
t+1
)−V
i
t+1
)+δV
i
t+1
≥
X
i∈N
p
i
t
(f(δV
j
t+1
)+(1−p
i
t
)δV
i
t+1
≥ (1+y
t
)min
f(δV
i
t+1
),f(δV
j
t+1
We also know that
Σ(V
t+1
) = (V
1
t+1
)
a
+(V
2
t+1
)
a
= 1−f(V
1
t+1
)+1−f(V
2
t+1
) = 2−(f(V
1
t+1
)+f(V
2
t+1
)).
Hence,2[1−max(f(δV
1
t+1
),f(δV
2
t+1
))]≤ 2−f(δV
1
t+1
)−f(δV
2
t+1
)≤ 2−f(V
1
t+1
)−
f(V
2
t+1
) = Σ(V
t+1
), where the first inequality follows from the fact that f(.) is a
strictly decreasing function. Combining the two equations above, we get
Σ(V
t
)≤ Σ(V
t+1
)+(3+y
t
)min
f(δV
i
t+1
),f(δV
j
t+1
(2.8)
. The result is established.
53
We now state and prove our theorems in the results section.
Theorem 2.5.3. For any game G
t
[δ,ρ] and for any a > 1, there exists a y
∗
> 0
such that whenever y ≥ y
∗
, we have an agreement regime for each t ∈ T with
t≤t−L(δ)−2.
Proof. The first task is to realize that levels of y
∗
≤ 0 will not lead to immediate
agreement by Example 3.1. It is straightforward to extend this example wherea = 2
to any strictly positive arbitrarya. Then, the proof follows by realizing thaty ≥ 1
which means that y
∗
= 1 identically yields immediate agreement. This fact is
proved in the appendix of Yildiz (2011).
Theorem 2.5.4. For any gameG
t
[δ,ρ] and for anya> 1, the drop in optimism from
one period to the next required for immediate agreement is bounded by
(1−δ)/δ
a
Forδ > 1/2, this expression is less than(1−δ)/δ, which is the bound on the drop
from Yildiz (2001).
Proof. We first write the expression for the bounds of the pie from lemma 2.
Σ(V
t
)≤ Σ(V
t+1
)+(3+y
t
)min
f(δV
i
t+1
),f(δV
j
t+1
)
Σ(V
t+1
)≤ Σ(V
t+2
)+(3+y
t+1
)min
f(δV
i
t+2
),f(δV
j
t+2
)
Then, proceeding as in Yildiz (2003), it is not difficult to show that the theorem
holds.
(1−δ)/δ
a
< (1−δ)/δ forδ > 1/2 providing the required result.
An interesting observation from this result is that when a → ∞ and δ > 1/2,
the agreement is sure to be reached at constant levels of optimism; that isy =y
∗
∈
[−1,1]. However, as with all the results in this section this is only a sufficient con-
dition. Whether constant levels of optimism is required for immediate agreement
as the risk aversion level approaches infinity is left unanswered by our analysis.
54
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57
Abstract (if available)
Abstract
This dissertation is centered around bargaining games. I study both the cooperative and the non-cooperative approaches to the bargaining games. The two chapters of my dissertation deal with each of these approaches. ❧ Coming back to the bargaining problem
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Ertas, Yusuf
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Essays on bargaining games
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Economics
Publication Date
11/28/2012
Defense Date
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