It is shown that in any affine space of payoff matrices the equilibrium payoffs of bimatrix games are generically finite.Bimatrix Games, Generic Finiteness

Andreu Mas-Colell

English

Research Papers in Economics

 
 
 
 
 
 
GENERIC FINITENESS OF EQUILIBRIUM PAYOFFS FOR BIMATRIX GAMES 
“In Honour of Martin Shubik” 
Andreu Mas-Colell 
Universitat Pompeu Fabra, and  
Barcelona GSE  
 
 
ABSTRACT  
 
It is shown that in any affine space of payoff matrices the equilibrium  
payoffs of bimatrix games are generically finite. 
 
 
 
KEY WORDS  Bimatrix Games, Generic Finiteness 
 
 
 
 
 
 
  
PROLOGUE (2003)  
 
The paper that follows is based on notes written in December of 
1990. I’m indebted to A. McLennan and W. Zame for helpful comments of 
those. See also Govindan and McLennan [2001]. The paper was written up 
and presented at the Cowles Foundation in February 1994. The text 
below is the text of 1994. The topic belongs to a general area that 
has always been of interest to Martin Shubik and, thus, it is for me a 
pleasure and an honour to offer this contribution to him.  
I. Introduction 
It is well known that if the entries of the normal form of a 
bimatrix game (in fact, of games with any number of players) can be 
perturbed independently then generically there is a finite number of 
equilibria (see Van Damme [1983] and Fudenberg-Tirole [1991]).  
Kreps-Wilson [1982] criticized this result as not being very 
helpful when the normal form is derived from an extensive form since 
then many strategies lead to the same final node. In fact, even if 
the payoffs of the final nodes can be perturbed independently the 
finiteness of the number of equilibria is not a generic property, yet 
Kreps-Wilson showed that the finiteness of the number of equilibrium 
payoffs is a generic property in this case. In many applications this 
suffices.  
However, the Kreps-Wilson criticism to the normal form result 
can be reiterated: it is very strong to assume that final nodes 
utilities can be perturbed independently. Typically, many final nodes 
will correspond to the same final position of a game and the most 
 natural assumption may be that the final utilities depend on the 
position and not on the particular history of play (i.e. the node). 
  In this note it is showed that for bimatrix games it is still 
true that the number of equilibrium payoffs is generically finite, in 
whatever way the payoffs of final nodes are tied together by (linear) 
constraints. 
  The interest of this result is limited (or enhanced?) by the 
fact that it does not extend to more than two players.  A. McLennan 
[1990] has a clever example to that effect. 
 
II.  The Result 
  
 Proposition:  Let U  be an arbitrary affine space of matrices.  
Then for a.e. payoff matrices  U ∈ 2 1,U U  for two players the number of 
equilibrium payoffs of the corresponding bimatrix game is finite. 
 
 Remark: It is indispensable for this result that U  be the same 
for the two players (interpretation:  if two final nodes correspond to 
two different positions then one can perturb the payoffs of both 
players).  This is clear: take two payoffs matrices  2 1,U U  with a 
continuum of payoff equilibrium.  Then  {} 1 1 U = U  and  {} 2 2 U = U  are 
affine spaces violating the desired property. 
 
III.  Proof of the Proposition 
 
  Without loss of generality it is enough to show that almost 
every pair of payoff matrices has at most a unique completely mixed 
equilibrium payoff.  In general we should choose  2 1,U U  such that all 
of its submatrices belong to the generic set for the projection of U  
on the corresponding coordinates.   First two preliminary Lemmas. 
 
 Lemma  1:  Let A  be a convex set of matrices.  Then the function 
A → rankA is constant a.e. on A . 
 
  Proof:  This follows from the analyticity of the 
determinant function.  Let k  be the maximal rank over the matrices in 
A .  Suppose that we fix a  A ∈ A  with maximal rank.  Say that its 
first k  columns are linearly independent.  By Fubini’s theorem it is 
enough to show that for all  A ∈ B  and a.e.  [] 1 , 0 ∈ α  we have that 
() () k B A rank = − + α α 1 .  Let  α C  be the matrix formed by the first k  
columns of  () B A α α − + 1 .  Define  () ( ) α α α C C f
T det = .  Note that  () 0 1 ≠ f  
and that  f  is a polynomial on α .  Hence for  () 0 ≠ α f  for a.e. α  and 
so  () () k B A rank = − + α α 1  for a.e. α . 
 
 Lemma  2:  Let A  be a convex set of matrices.  Then either  
  (1)  () () 1 ,..., 1 = = e Av  has a solution for a.e.  A ∈ A , or 
  (2) e Av =  does not have a solution for a.e.  A ∈ A . 
 
  Proof:  Let k  be the rank that prevails a.e. over A      
(Lemma 1).  Consider the (isometric to A ) convex set of matrices 
[] {} A A ∈ = ′ A e A : , .  Then either the a.e. rank of A ′ is k  and then (1) 
holds, or it is  1 + k  and then (2) holds. 
 
We now prove the Proposition. 
 Consider any pair of payoff matrices  2 1,U U  with the property 
that, first, they belong to the generic set of Lemma 2 and, second, 
T T U U 2 1 ,  belong to the generic set of Lemma 2 for  { } U U ∈ = A A
T T : . 
  If possibility (2) of Lemma 2 holds for  1 U and 
T U 2  then either 
there is no completely mixed solution of the unique equilibrium payoff 
is (0,0). 
  Suppose that possibility (1) of Lemma 2 holds for  1 U (hence for 
2 U ).  Then there is  2 v  such that  e v U = 2 2 .  Take now any solution 
() α , 1 p  to  1 , 1 1 2 = = •e p e p U
T α .  Then: 
    () e v p U v p e
T T
• • • = = = 2 1 2 2 1 1 α  
and so the only (completely mixed)  equilibrium payoff for player 2 is 
    () e v •
=
2
1
α . 
 Since  then  0 ≠ α  this also implies that possibility (1) of Lemma 
2 holds for 
T U 2  (hence for 
T U1 ).  We now repeat the argument.  Choose 
1 v  such that  e v U
T = 1 1 .  Consider any solution () β , 2 p  to 
1 , 2 2 1 = = •e p e p U β .  Then: 
    () e v p U v p e • • • = = = 1 2 1 1 2 1 β  
and so the only (completely mixed) equilibrium payoff for player 1 is 
    () e v •
=
1
1
β . 
  Finally, the argument is completely symmetric (in fact just a 
matter of labelling) if possibility (1) of Lemma 2 holds for 
T U 2 . 
 
References 
 
Fudenberg, D. and J. Tirole [1991] Game Theory Cambridge:  MIT Press. Kreps, D and R. Wilson [1982] Sequential equilibrium, Econometrica, 
50: 863-94. 
 
Govindan, S. and A. McLennan [2001]. On the Generic Finiteness of 
Equilibrium Outcome Distributions in Game Forms,Econometrica, 69,
2: 455-472.
 
McLennan, A. [1990] Private Communication. 
van Damme, E. [1983]  Refinements of the Nash equilibrium concept, 
Springer-Verlag. 

Generic finiteness of equilibrium payoffs for bimatrix games

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