Trockel,W.

English

Research Papers in Economics

Unique Nash Implementation for a Class of
Bargaining Solutions
Walter Trockel
University of California, Los Angeles and Bielefeld University
Mai 1999
Abstract
The paper presents a method of supporting certain solutions of two-person bargain-
ing games by unique Nash equilibria of associated games in strategic form. Among
the supported solutions is the Kalai-Smorodinsky solution.
I thank Claus Jochen Haake for helpful discussions. The hospitality of the University of California,
Los Angeles and of the University of Arizona at Tucson as well as nancial support through a sabbatical
grant of the Volkswagen Foundation are gratefully acknowledged.
11 Introduction
There has been some discussion in the recent literature about the relation between the
Nash program in game theory and the theory of mechanism design [cf. Dagan and Serrano
(1998), Serrano (1997), Trockel (1999a), (1999b), and Naeve (1999)].
The Nash program aims to support axiomatic solutions of cooperative games in strategic
or extensive form. In mechanism theory one is interested in the design of game forms
implementing a prespecied cooperative solution considered as a (set of) desired social
state(s). Implementation is achieved if any population in a certain domain by playing
equilibrium strategies in the game dened by the game form and this population's prole
of utility functions establishes (an element of) the solution. That the Nash program can
be considered as a part of mechanism theory has been established by Trockel (1999a)
[see also Dagan and Serrano (1998), Serrano (1997)]. There it is shown that any support
result for a solution of cooperative NTU-games can be extended to an implementation
result.
In this paper we present a specic class of two-person games in strategic form whose unique
Nash equilibria support given bargaining solutions. In particular the Kalai-Smorodinsky
solution is among those solutions.
An interesting feature of our games under the aspect of using reasonable and sensitive
strategic models for supporting or implementation results is the fact that they enable the
players to reach any of the utility allocations of bargaining games by suitable strategic
choices.
Our games are based on the extensive form game that Trockel (1999b) has used for the
subgame perfect equilibrium implementation of the Nash solution and which has been
modied by Haake (1998) to implement the Kalai-Smorodinsky solution.
22 Background
We need a short sketch of the two-person extensive form game in Trockel (1999).
Let a two-person bargaining situation be given by some strictly convex, comprehensive
(with respect to R2
+) compact subset S of R2
+ with non-empty interior. We assume for
convenience that the bargaining game is normalized such that proj1(S) = [0;1]f0g and
proj2(S) = f0g  [0;1]. We let 1 = 2 = [0;1] be the strategy sets of the two players of
a game in strategic form.
Let B denote the set of all two-person bargaining games normalized in the way described
above. A solution is a map L : B ! R2
+ : S 7! L(S) 2 S.
Now suppose that for a given ecient solution L there exists a continuous function
zL
1 : 1 ! 1 : x1 7! zL
1 (x1) which is strictly decreasing and strictly concave such that for
all S 2 B
(MM) L1(S) = argmax
x121
min(x1;zL
1 (x1)).
Notice that by continuity of the mappings z1 and id1 we have L1(S) = z1(L1(S)).
In this case the solution L can be supported by the unique subgame perfect equilibrium
of the following two-step game in extensive form. We shall denote the ecient boundary
of S by @S.
On stage 1 player 1 chooses x1 2 [0;1]. Then on stage 2 player 2 chooses one element from
the set fX;Zg. The resulting payo vector for the strategy choices (x1;Y ) 2 [0;1]fX;Zg
is the unique payo vector (x1;x2) 2 @S or (z1(x1);z2) 2 @(S), if Y = X or Y = Z,
respectively. Note that player 1 prefers one of the points (x1;x2);(z1(x1);z2) if and
only the other player prefers the other point. Therefore backward induction leads to
(L1(S);L2(S))  L(S) as the unique equilibrium outcome for the two subgame per-
fect equilibria (L1(S);X) and (L1(S);Z). This non-uniqueness is harmless because the
uniqueness of the rst player's equilibrium strategy together with the uniqueness of the
equilibrium payo vector prevents any need for coordination, and is, therefore, for pre-
dictive purposes as good as uniqueness. The above reasoning has been used by Trockel
(1999b) and subsequently by Haake (1998) for the special cases of the Nash solution (N)
and the Kalai-Smorodinsky solution (KS). Both belong to the class of solutions allow-
ing an argmax min-characterization as described above. But only the Kalai-Smorodinsky
solution is in the class covered by our proposition.
We shall rst derive a unique Nash equilibrium support result for any solution L, satisfying
condition (MM). Then we shall apply this result to the specic function zKS
1 to support
the Kalai-Smorodinsky solution.
33 The support result
The rst attempt to achieve a Nash equilibrium support result for a solution L satisfying
condition (MM) is to mimic the players' strategic behavior in the two-stage game described
above but to let them make their choices simultaneously. The resulting game  S for a
given bargaining game would be
 S
L = ([0;1];fX;Zg;1;2)
with
1(x1;X) = x1; 2(x1;X) = x2; (x1;x2) 2 @S
1(x1;Z) = zL
1 (x1); 2(x1;Z) = z2; (zL
1 (x1);z2) 2 @S
Unfortunately, this game has no Nash equilibrium (in pure strategies). This can be seen
as follows:
case 1: (x1;X); x1  L1(S)
Player 1 can improve by choosing a larger x0
1 and getting x0
1 rather
than x1.
case 2: (x1;X); x1 > L1(S)
Player 2 can improve by choosing Z and getting z2 rather than x2.
As zL
1 (x1) < L1(S) < x1 we must have z2 > x2.
case 3: (x1;Z); x1 < L1(S)
Player 2 can improve by choosing X and getting x2 rather than z2.
As x1 < L1(S) < zL
1 (x1), we must have x2 > z2.
case 4: (x1;Z); x1  L1(S)
Player 1 can improve by choosing a smaller x0
1. He gets then zL
1 (x0
1)
rather than zL
1 (x1). As x0
1 < x1, we have zL
1 (x0
1) > zL
1 (x1).
Therefore, we consider instead the partially mixed extension   S
L of  S
L which we dene as
follows:
4  S
L := ([0;1];[0;1];  1;  2)
with
 1(x1;) := 1(x1;X) + (1   )1(x1;Z)
 2(x1;) := 2(x1;X) + (1   )2(x1;Z).
Notice that the strategy spaces of the two players, though formally identical, have very
dierent meanings, as can be seen from the payo functions. Player 1 still chooses utility
claims for himself between 0 and 1, while player 2 chooses lotteries over the set fX;Zg.
The resulting payos are therefore in terms of expected utilities.
We state now the result of the present paper.
Proposition:
Let L be a bargaining solution satisfying condition (MM). Then for any S 2 B
the game   S
L has a unique Nash equilibrium (in pure strategies) (L1(S);L(S))
whose equilibrium payo vector is L(S).
Proof:
In the rst step we show that no point (x1;) 2 [0;1]2 with x1 6= L1(S) can be an
equilibrium.
Consider such a point (x1;). The cases  2 f0;1g can be excluded because of our above
reasoning. Consider, therefore,  2]0;1[. If x1 is smaller (resp. larger) than zL
1 (x1);x2
is larger (resp. smaller) than z2. So player 2 can increase his payo by increasing (resp.
decreasing) .
Hence, the only candidates for equilibria are the points (L1(S);); 2]0;1[. All of these
result in the same payo for player 2, as L1(S) = zL
1 (L1(S)). Accordingly, each  is a
best reply of player 2 to player 1's strategy L1(S). Obviously, L1(S) is not the best reply
for player 1 to each  2]0;1[. For a very small  player 1 could improve by a smaller x1,
for an  close to 1 he could improve by an x1, close to 1. So we are looking for some  to
which L(S) is player 1's best response.
So for any  we can look at the solutions of the optimization problem argmax
x12[0;1]
f(x1),
where f(x1) := x1 + (1   )zL
1 (x1).
5As only x1 = L1(S) is consistent with the Nash equilibrium requirement for (x1;), the
question remains for which  2]0;1[ we have L1(S) 2 argmax
x12[0;1]
f(x1).
Consider the function
f : [0;1]  [0;1] ! R : (x1;) 7! f(x1):
As zL
1 is strictly concave and id[0;1] is concave the function f = f(;) is strictly concave
for any  2]0;1[. Let for  2 [0;1]
f() := max
x2[0;1]
f(x).
and
D() := argmax
x2[0;1]
f(x).
We dene the maximizer function x
1 : [0;1] ! [0;1] by fx
1()g := D() for  2 [0;1].
By the Maximum Theorem under Convexity [cf. Sundaram (1996, Theorem 9.17)] our
assumptions imply that f is a continuous strictly concave function and that D is an
upper hemicontinuous single-valued correspondence. The latter fact says that x
1 is a
continuous function.
Now observe that x
1(0) maximizes zL
1 on [0;1]. As zL
1 is strictly decreasing this implies
x
1(0) = 0 and f(0) = zL
1 (0) > 0. Similarly x
1(1) maximizes id[0;1] on [0;1]. Hence,
x
1(1) = 1 and f(1) = 1. As x
1 is continuous there exists some   2 [0;1] such that
x
1( ) = L1(S) 2 [0;1] with f( ) = L1(S). Therefore (L1(S);  ) is a Nash equilibrium of
  S
L.
To establish the uniqueness of the equilibrium we employ the strict concavity of the
function f. We have f0() = x() zL
1 (x()), for all  2 [0;1]. In particular, we have
f0(0) =  zL
1 (0) < 0;f0(1) = 1   zL
1 (0)  0 and f0( ) = x( )   zL
1 (x( ) = 0. The
concavity excludes f0(^ ) = 0 for any^  6=  , which excludes (L(S); ^ ) as an equilibrium.

64 Application to the Kalai-Smorodinsky solution
x
0
1
x
00
1
x
0
1
x2
x
00
1
x1
z
KS
1 (x
00
1) z
KS
1 (x
0
1)
(1;1)
utopia point
0
Let S be again a normalized bargaining game in the class B.
In the case that the solution L is the Kalai-Smorodinsky solution KS we dene the
function zKS
1 : [0;1] ! [0;1] as follows:
Any point x1 2 [0;1] denes a unique point (x1;x1) on the diagonal of R2
+. Projecting this
point along the two axes to the ecient boundary generates two dierent points (x1;x2)
and (zKS
1 (x1);x1). They are identical only if (x1;x1) = KS(S). The function zKS
1 dened
in this way satises the assumptions of our proposition. In particular, we get:
z
KS
1 (0) = 1;z
KS
1 (KS(S)) = KS1(S);z
KS
1 (1) = 0:
Accordingly, the proposition asserts the support of the Kalai-Smorodinsky solution by the
unique Nash equilibrium of the game   S
KS.
75 Concluding Remarks
The present paper presents a unique Nash support result for a class of solutions of two-
person bargaining games containing the Kalai-Smorodinsky solution. Application of the
imbedding principle in Trockel (1999a) immediately transforms this contribution to the
Nash program into a result on unique Nash implementation in the sense of mechanism
theory. The class of solutions satisfying the assumptions of our proposition is charac-
terized by the fact that each of the solutions can be implemented in subgame perfect
equilibrium in some divide-and-choose-mechanism. This yields an interesting alternative
to implementations in the literature based on auction mechanisms [cf. for instance Moulin
(1984)].
8References
[1] Dagan, N. and R. Serrano (1997):\Invariance and Randomness in the Nash Pro-
gram for Coalitional Games", Brown University, mimeo
[2] Haake, C.-J. (1998): \Implementation of the Kalai-Smorodinsky Bargaining Solu-
tion in Dominant Strategies", IMW Working Paper No. 301, Bielefeld University
[3] Moulin, H. (1984): \Implementing the Kalai-Smorodinsky Bargaining Solution",
Journal of Economic Theory, 33, 32{45
[4] Naeve, J. (1999): \Nash Implementation of the Nash Bargaining Solution using
Intuitive Message Spaces", Economics Letters, forthcoming
[5] Serrano, R. (1997): \A Comment on the Nash Program and the Theory of Imple-
mentation", Economics Letters, 55, 203-208
[6] Sundaram, K.R. (1996): \A First Course in Optimization Theory", Cambridge
University Press
[7] Trockel, W. (1996): \A Walrasian Approach to Bargaining Games" Economics
Letters, 51, 295{301
[8] Trockel, W. (1999a): \Integrating the Nash Program into Mechanism Design",
UCLA Working Paper No. 787
[9] Trockel, W. (1999b): \On the Nash Program for the Nash Bargaining Solution",
UCLA Working Paper No. 788
9

Unique Nash implementation for a class of bargaining solutions

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Unique Nash implementation for a class of bargaining solutions

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