Norde, Henk

Potters, Jos

Reijnierse, Hans

Vermeulen, Dries

English

Research Papers in Economics

GAMES AND ECONOMIC BEHAVIOR 12, 219–225 (1996)
ARTICLE NO. 0014
Equilibrium Selection and Consistency
HENK NORDE,J OS POTTERS,H ANS REIJNIERSE, AND DRIES VERMEULEN
Department of Mathematics, University of Nijmegen,
The Netherlands
Received October 24, 1994
In this paper we show that, for two important classes of strategic games, namely
the class of mixed extensions of ﬁnite games and the class of games with compact
and convex strategy spaces and continuous-concave payoff functions, equilibrium
selection is incompatible with One Person Rationality, Consistency and (restricted)
Non-Emptiness. Ó 1996 Academic Press, Inc.
1. INTRODUCTION
For two important classes of strategic games, the existence of Nash
equilibria has been shown. The ﬁrst class is the class of mixed extensions
of ﬁnite games (Nash, 1951). The second class is the family of strategic
games with compact convex strategy spaces and continuous payoff func-
tions with the property that the payoff function of each player is concave
in his strategy (Glicksberg, 1952). In fact, in many cases the set of
Nash equilibria is too large and contains—from a strategic point of
view—unreasonable equilibria (e.g. equilibria using weakly dominated
strategies). This observation has been the starting point of the theory
of equilibrium selection (cf. Harsanyi and Selten, 1982) as well as the
theory of reﬁnements (cf. Van Damme, 1987) and stable sets (cf. Kohlberg
and Mertens, 1986). These theories have in common that they try to
ﬁnd for each game in a certain class of strategic games one equilibrium
(equilibrium selection) or a nonempty subset of Nash equilibria (reﬁne-
ments or stable sets) as a collection of more advisable equilibria. For
the two classes mentioned before we will prove that these attempts are
bound to lead to inconsistent solutions.
So we will investigate solution rules w deﬁned on the class Gf of all mixed
extensions of ﬁnite games or deﬁned on the class Gc of games wherein the
219
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Copyright Ó 1996 by Academic Press, Inc.
All rights of reproduction in any form reserved.220 NORDE ET AL
strategy spaces are convex and compact subsets of ﬁnite-dimensional vector
spaces and the payoff functions are continuous-concave functions. We as-
sume that the solution rule w assigns to each game G [ Gf (or Gc)anon-
empty set of strategy proﬁles (NEM) and to each one-person game a subset
of the set of strategies where the payoff function takes its maximum (UM).
We shall prove that w assigns to each game G [ Gf (or Gc) the Nash
equilibrium set NE(G), if w satisﬁes, moreover, the consistency property
introduced by Peleg and Tijs (1992).
First we recall the deﬁnitions of reduced game and a consistent solu-
tion concept.
Let G: 5 hNG, hAi, Uiji[NGj be a strategic game in a class G of strategic
games, let x be a strategy proﬁle in G and let S be a proper subcoalition
of the player set NG of G (i.e., S ? NG, B).
The reduced game GS,x of the game G with respect to S and x is a strategic
game with player set S. Each player i [ S has the same strategy space Ai
as in G and the payoff functions hU9 iji[S are deﬁned by U9 i(yS): 5
Ui(yS, xN\S) for every strategy proﬁle yS. In the reduced game GS,x the
players outside S stick upon their strategies in x and only the players in S
are free to reconsider their choice.
A solution rule w deﬁned on G is called consistent (CONS) if for every
game G [ G, every proper coalition S , NG and every element x [ w(G),
(i) GS,x [ G, (ii) xS [ w(GS,x).
So, under a consistent solution rule the players of a reduced game do
not need to reconsider their strategy choice: they can play the same strategy
as in the original game.
The main theorem of the paper states that there is only one solution
rule w deﬁned on Gf (or Gc), satisfying non-emptiness (NEM), utility max-
imizing for 1-person games (UM), and consistency (CONS): the Nash equi-
librium solution NE.
THEOREM 1. If G5G for Gc and w is a solution rule deﬁned on G with
the properties (NEM) (i.e., w(G) ? B for all G [ G), (UM) (i.e., w(G) ,
arg max U if G is a 1-person game in G and U is the payoff function) and
(CONS), then w(G) 5 NE(G) for all games G [ G.
Note that the theorem gives not only a characterization of the Nash
equilibrium solution on Gf and Gc; it also frustrates beforehand every at-
tempt to ﬁnd a solution rule for Gf or Gc (unequal to NE) that extends the
idea of utility maximization of 1-person games, assigns to each game a
nonempty subset of strategy proﬁles, and satisﬁes consistency. All (proper)
reﬁnements of the Nash equilibrium solution are therefore violating consis-
tency or nonemptiness. As a consequence of the theorem, perfectness,EQUILIBRIUM SELECTION AND CONSISTENCY 221
properness, and persistency must give inconsistent solution rules, as there
are existence theorems for these types of Nash equilibria (cf. Selten, 1975;
Myerson, 1978; and Kalai and Samet, 1984). Quasi-strictness (cf. Harsanyi,
1973), strongness (cf. Aumann, 1959) and coalitional proofness (Bernheim
et al., 1987) are consistent solution rules satisfying (UM) (this is easy to
check for quasi-strictness; see Peleg and Tijs (1992) for the last two solution
rules) and must therefore violate nonemptiness on Gf. Of course these
results are not new but what is new is that they all follow immediately from
one theorem.
In the following proposition we prove that every solution rule deﬁned
on any class of strategic games that satisﬁes (UM) and (CONS) is a subsolu-
tion of the Nash equilibrium solution.
PROPOSITION 2. If w is a solution rule deﬁned on a class G of strategic
games and w satisﬁes (UM) and (CONS), then w(G) , NE(G) for all games
G [ G.
Proof (cf. Peleg and Tijs, 1992). Proof is by induction to uNGu, the
number of players in G [ G.I fGis a 1-person game in G, the Nash
equilibrium set consists of the points where the payoff function U attains
its maximum value (if anywhere). Then (UM) gives w(G) , NE(G).
Suppose that the inclusion w(G) , NE(G) was proved for games
G [ G with less than n players. Let G be a game in G with n players and
x [ w(G). Then, by (CONS) for w, GS,x [ G and xS [ w(GS,x) for all proper
coalitions S , NG. From the induction hypothesis we infer that
xS [ NE(GS,x). By (COCONS) for NE (Peleg and Tijs, 1992), we ﬁnd
x [ NE(G).
Q.E.D.
The following proposition studies the conditions that (NEM), (UM), and
(CONS) characterize the Nash equilibrium solution.
PROPOSITION 3 (cf. Peleg et al., 1993). If w is a solution concept deﬁned
on a class G and w has the properties (NEM), (UM) and (CONS), if for
every game G [ G and every Nash equilibrium x ˆ [ NE(G) a game H [ G
can be constructed such that the following conditions are satisﬁed:
(i) NH . NG,
(ii) yNG 5 x ˆ for every y [ NE(H),
(iii) HNG,y 5 G for every y [ NE(H),
then w 5 NE.
Proof. From Proposition 2 we know that w(G) , NE(G) for all games
G [ G. Suppose that G [ G and x ˆ [ NE(G). Take a game H [ G satisfying
the properties mentioned in the proposition. Take y [ w(H)( wsatisﬁes222 NORDE ET AL
(NEM)). Then x ˆ 5 yNG [ w(HNG,y) by (CONS) and HNG,y 5 G. Therefore,
x ˆ [ w(G). Q.E.D.
Proposition 3 can be extended to classes of games wherein the Nash
equilibrium set is sometimes empty. Then there is no solution rule w satis-
fying (NEM), (UM), and (CONS) (cf. Proposition 2). But, if we replace
(NEM) by restricted nonemptiness (r-NEM) saying w(G) ? B if NE(G) ?
B, we can prove the same result.
InthefollowingsectionweprovethatintheclassesGfandGctheconstruc-
tion (G, x ˆ) R H is possible.
2. THE CONSTRUCTION OF H
In this section we construct for each game G [ Gf (of Gc) and each Nash
equilibrium x ˆ [ NE(G) a game H [ Gf or Gc respectively, satisfying the
conditions (i), (ii), and (iii) of Proposition 3.
(A) Let G be the mixed extension of a ﬁnite game and let x ˆ be a
Nash equilibrium of G. The game H will have 3n players where n 5 uNGu.
The player set NH consists of hi, i1, i2 u i [ NGj. The players i, i1, and i2
have the same strategy set Di as player i has in the game G.
The payoff functions of the players i, i1, and i2 are denoted by
U* i , V* i , and W* i and the strategies are called xi, yi, and zi, respectively.
The deﬁnition of the payoff functions is
U* i (x, y, z): 5 Ui(xi u x2i 1 x ˆ2i 2 y2i) 2 kxi 2 x ˆi, yi 2 x ˆil,
where x2i 1 x ˆ2i 2 y2i is a shorthand notation for [(xj 1 x ˆj 2 yj)j?i] and
k.,. ldenotes the Euclidean inner product. Notice that xj 1 x ˆj 2 yj is not
necessarily an element of Dj but it is an element of the afﬁne hull of Dj. The
multilinearity of Ui, the payoff functions in G, allows a unique multilinear
extension of Ui to afﬁne hulls.
Furthermore,1
V* i (x, y, z): 5 kyi, xi 2 zil, W* i (x, y, z): 5 kzi, yi 2 xil.
This ﬁnishes the deﬁnition of H. It is immediately clear that H [ Gf.
1 The referee pointed us to the following interpretation of the game H. Player i1 obtains a
positive payoff for as far as he matches the strategy of player i and loses payoff for as far as
he matches the strategy of player i2. Player i2 loses if he matches player i and wins payoff if
he matches the strategy of player i1.EQUILIBRIUM SELECTION AND CONSISTENCY 223
We prove that all equilibria (x, y, z)o fHsatisfy x 5 y 5 x ˆ. Then we
ﬁnd (x, y, z)NG 5 x 5 x ˆ and, if we substitute y 5 x ˆ, U* i (x, y, z)y5x ˆ 5
Ui(xi u x2i) 5 Ui(x) for every player i [ NG.
Suppose (x, y, z) is a Nash equilibrium of H and xi ? yi for some index
i [ NG.
If zij . 0 (player i2 puts positive weight on his jth strategy), then yij .
xij. For, as (x, y, z) is an equilibrium and yik . xik for at least one pure
strategy k, player i2 uses only strategies k with yik 2 xik . 0. Then yij . 0
and therefore xij $ zij by the same argument but now for player i1. Then
O
j:zij.0
yij . O
j:zij.0
xij $ O
j:zij.0
zij 5 1.
This is impossible: x 5 y.
Suppose that xi 5 yi ? x ˆi for some i [ NG. If player i deviates from xi
to x ˆi, the value of his payoff goes from Ui(xi u x ˆ2i) 2 ixi 2 x ˆii2 to Ui(x ˆ).
This is an improvement as Ui(x ˆ) $ Ui(xi u x ˆ2i)( x ˆis an equilibrium) and
ixi 2 x ˆii2 . 0i fx i?x ˆ i. Therefore, x 5 y 5 x ˆ. Q.E.D.
(B) Let G be a strategic game with compact convex strategy spaces
hAG
i ji[NG, subsets of a ﬁnite-dimensional vector space and payoff functions
hUiji[NG satisfying:
(i) the functions Ui: AG 5 Pi[NG AG
i R R are continuous
(ii) the functions Ui(d u x2i): xi [ AG
i R Ui(xi u x2i) is concave for
every complementary strategy proﬁle x2i [ Pj?i AG
j .
Let x ˆ be an equilibrium of G. To construct H we take NH: 5
hi, i1 u i [ NGj and AH
i 5 AH
i1: 5 AG
i . So, H has 2n players. Before we can
introduce the payoff functions we need the following observations:
(1) The strategy space AG
i may be assumed to have the same dimen-
sion as the surrounding ﬁnite dimensional vector space Ei.
(2) If xi, x ˆi, and yi are elements of AG
i , then xi 1 x ˆi 2 yi is perhaps
not an element of AG
i but it is an element of Ei.
(3) There exists a continuous retraction of Ei to AG
i , i.e., there is a
continuous map fi: Ei R AG
i with fi(x) 5 x for x [ AG
i .
Let (x, y, z) be a strategy proﬁle in H. Using the shorthand notation
f2i(x2i 1 x ˆ2i 1 y2i) for (fj(xj 1 x ˆj 2 yj)j?i), we deﬁne
U* i (x, y,): 5 Ui(xi u f2i(x2i 1 x ˆ2i 2 y2i)) 2 ixi 2 x ˆiiiy i2x ˆ ii .
Note that U* i is a continuous function and concave in xi. For the other224 NORDE ET AL
players i1 the payoff functions are V* i (x, y, z): 52 iy i2x ii . These functions
are again continuous in (x, y) and concave in yi.
Suppose that (x, y) is an equilibrium of H. Then x 5 y (as follows from
the maximality of the payoff V* i ). If xi ? x ˆi for some i [ NG, the deviation
from xi to x ˆi gives player i a change in payoff Ui(x ˆ) 2 Ui(xi u x ˆ2i) and
ixi 2 x ˆii2. The ﬁrst term is nonnegative, the second term is positive. So, if
x ? x ˆ, we are not in an equilibrium. Therefore, the game H has one
equilibrium, namely (x 5 x ˆ, y 5 x ˆ). The properties (i), (ii), and (iii) of
Proposition 3 can now be checked immediately. Q.E.D.
If G9 c is the class of strategic games with compact convex strategy spaces
and continuous payoff functions (so we delete the concavity condition),
the same construction (B) can be used to prove
THEOREM 1.bis. If w is a solution rule deﬁned on G9 c satisfying (r-NEM),
(UM), and (CONS), then w 5 NE.
ACKNOWLEDGMENTS
The seminal idea of how to attack the problem discussed in this paper we owe to Bezalel
Peleg. During the visit of one of the authors (JP) to the Hebrew University in Jerusalem he
discussed the problem with Peleg and we tried to solve it for different classes of games (see
the references Peleg et al., 1994, and Peleg and Sudho ¨lter, 1994). We thank the Center of
Rationality for their hospitality and Peleg for his stimulating cooperation.
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Equilibrium selection and consistency.

Equilibrium selection and consistency.

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