University of Wisconsin Centers-Marathon 518 South 7th Ave, Wausau, WI 54401 USA. T.E.S. RAaHAVAN. Department of Mathematics Statistics and Computer ...
International Journal of Game Theory (1996) 25:35-41
Game Theory
A Note on Correlated Equilibrium FE S. EVANGELISTA University of Wisconsin Centers-Marathon 518 South 7th Ave, Wausau, WI 54401 USA T.E.S. RAaHAVAN ~ Department of Mathematics Statistics and Computer Science, The University of Illinois at Chicago, 322 Science and Engineering Offices, Chicago, IL 60607-7045, USA
Abstract: The set of correlated equilibria for a bimatrix game is a closed, bounded, convex set
containing the set of Nash equilibria. We show that every extreme point of a maximal Nash set is an extreme point of the above convex set. We also give an example to show that this result is not true in the payoffspace, i.e. there are games where no Nash equilibrium payoffis an extreme point of the set of correlated equilibrium payoffs.
1
Introduction
T h e c o n c e p t of c o r r e l a t e d e q u i l i b r i u m in n o n c o o p e r a t i v e games was i n t r o d u c e d b y A u m a n n (1974). H e s h o w e d t h a t for n-person g a m e s in n o r m a l form with a finite n u m b e r of p u r e strategies, the set of c o r r e l a t e d equilibria is a c o m p a c t , convex p o l y h e d r o n which c o n t a i n s the set of N a s h equilibria (see A u m a n n (1987)). T h e set of N a s h equilibria, on the o t h e r hand, is n o t always a convex set. H o w e v e r , for a b i m a t r i x game, it is a u n i o n of a finite n u m b e r of c o n v e x p o l y t o p e s called m a x i m a l N a s h subsets ( K u h n (1961), Jansen (1981), van D a m m e (1991)). I n this p a p e r , we show t h a t every e x t r e m e p o i n t of a m a x i m a l N a s h set is an e x t r e m e p o i n t of the set of c o r r e l a t e d equilibria. W e also give an e x a m p l e to show t h a t this result is n o t true in the p a y o f f space, i.e. there are games where no N a s h e q u i l i b r i u m p a y o f f is an extreme p o i n t of the set of c o r r e l a t e d e q u i l i b r i u m payoffs. T h e necessary definitions, n o t a t i o n s a n d k n o w n t h e o r e m s a b o u t N a s h equilibr i u m a n d c o r r e l a t e d e q u i l i b r i u m are in Section 2; the discussion on equil i b r i u m follows Jansen (1981) while t h a t for c o r r e l a t e d e q u i l i b r i u m follow A u m a n n (1987) a n d M y e r s o n (1986). Section 3 c o n t a i n s the m a i n t h e o r e m a n d its proof. 1 Partially funded by NSF grant DMS-9024408 0020-7276/96/1/35-41 $2.50 9 1996 Physiea-Verlag, Heidelberg
Fe S. E v a n g e l i s t a a n d T.E.S. R a g h a v a n
36
2
Definitions and Notations
A bimatrix game (A, B) is defined by m x n matrices with real entries A = [ a J , B = [ b J , where aij (bij) is the payoffto player 1 (player 2) when player 1 chooses action i e I = {1, 2. . . . . m} and player 2 chooses a c t i o n j e J = {1, 2 . . . . . n}. Let X = {x~emlxi >__O, ~ x, = 1} i r = {yeR'lyj _> O, E Yi = 1} J
be the set of mixed strategies for players 1 and 2 respectively. A Nash equilibrium is a pair (x~ ~ such that
~_, ajax io yao >_ ij Y ' b i j x 0i 0yj> _ ij
y, aijxly j,o ij
gxeX
~bijx~ ij
Vy~Y
The set of Nash equilibria will be denoted by d~ B). A Nash set S is a subset of d~ B) in which all elements of S are exchangeable, i.e. if (xl, y 1) and (xa, y 2) are in S, then (xl, y 2) and (x2,y ~) are in S. A Nash set is called maximal if it is not properly contained in any other Nash set.
Lemma: Every maximal convex subset of g(A, B) is a Nash set. Proof: See Jansen (1981), Theorem 2.6. Lemma: Any maximal Nash set is a convex and closed subset of X x Y Proof." See Heuer and Millham (1976), Theorem 3.3. An element (x, y) of g(A, B) is called an extreme equilibrium if it is an extreme point of some maximal Nash set. A correlated strategy for the game (A, B) is a probability distribution on the set I x J. It is denoted by an mn-tuple p = (Pij)ieI,jeJ, Plj >- O, 7~.ijPlj = t. Such a strategy can be implemented in a game as follows: a referee chooses a pair (i,j) using the distribution p, then suggests secretly to player 1 (player 2) that he use action i (action j). Each player then decides whether to follow the referee's suggestion or choose some other action. The correlated strategy p is a correlated equilibrium if no player Can gain by deviating from the referee's suggestion, assuming that the other player obeys it. The set of correlated equilibria will be denoted by (g(A, B). Note that Cg(A,B) is a convex set in R"".
A No te on Correlated Equilibrium
37
Theorem: For a bimatrix game (A, B), p = (pij)eCg(A, B) if and only if ~" (aij--aki)pij>_ O,
Vi, k~I
(1)
Vj, leJ
(2)
~. Pij = 1
(3)
j=l
~" (bij
-
bil)pij > O,
-
i=1
Pij >- O,
ij
Proof See Aumann (1987), Proposition 2.3. One can show that if (x,y)~8(A, B), then p = (Pii)= (x~yj)ecg(A, B). Thus, we can think of g(A, B) as a subset of cg (A, B).
3
Result
Under the mapping n:X x Y--* R"" given by n(x, y) = p, where pij = xiy~, we can view g(A, B) as a subset of ~(A, B), and study the structural relation between the two sets. Since 8(A, B) is the union of maximal Nash sets, we focus our attention on what happens to extreme points of maximal Nash subsets of these sets under the mapping n. We now state our main result.
Theorem: Every extreme equilibrium is an extreme point of the set of correlated equilibria.
Proof Since (x , y ) is a Nash equilibrium, one can reindex the rows and columns of A and B so that x* > 0 , i = 1,2 . . . . . s; y* > 0, j = l , 2 . . . . . r
E aijY~ = max ~. akjY*, J
i = 1,2 . . . . . t
J
2 bijx* = max 2 bux*, i
l
i
j = 1, 2 ..... u
with s _< t and r < u. Let p * - - x ' y * and suppose p * = (p*) is not an extreme point of Cg(A,B). Then there exist pl, p2 ~Cg(A' B) such that
P~
i
=
i
i
2
-2Pu "}- "~Plj
Viii, j~J
Note that p~ -- pii2= 0 whenever i > s o r j > r.
38
F e S. E v a n g e l i s t a a n d T.E.S. R a g h a v a n
F o r i, k = 1, 2 .... , t,
0 = ~. (aij - aki)p* = 1/2 Z ( a i j - - akj)P~j + 1/2 • (a u -- aki)p 2 J
J
J
F r o m (1) and (2), Z j ( a u - - a k j ) p ~ j > 0 and Z j ( a u - - a k j ) p 2 > 0 , Vi, k e I . 1 1 2 2 Y'.jauPij = ~4akjpU, ~ j a u p u = Y ' T a k j P i j , Vi, k = 1, 2 . . . . . t, and so
E 2 aijP~ i
=
j
Thus,
L akjP.~ j
~', ~, aUp2 = 2 akjP.5 i
J
J
Vk = 1, 2 . . . . . t where p j = ~-,iPlj" t l = 1, 2. a t is the payoff to player 1 when the correlated Let a 1 = Y,i Z j aUPij, strategy p~ is used. Then (PI1, P.2,z... , PI,, at), l = 1, 2 are solutions to the system of equations
i aijzj=a, • Zj:1
i=
1, 2 , . . . ,
t
j=l
j=l
zj>_O
j = 1,2 .... ,r.
Since (x*, y*) is an extreme equilibrium, z i = y * , j = 1, 2 . . . . . r, a = x ' r A y * is the unique solution of the above system. (See L e m m a 3.3 and T h e o r e m 3.8 in Jensen (1981), or L e m m a 7.6.3 in P a r t h a s a r a t h y and R a g h a v a n (1971). Hence, p.j1 __ p.j2 = y , , j = 1, 2 , . . . , r. In a similar way, one can show that pi1. = p2 = x*, i = 1, 2 . . . . . s where p~. = ~2jPuWe consider two cases:
Case 1: r < s Let/5* -- (Pu),* !Sl = ( p i1j ) ,1i = . . . . . s ; j = l .... , r; / = 1, 2. N o t e that/5",/51,/52 are all solutions of the system of equation s
• (a u - akj)P q = O, i = 1. . . . . S; k = 1.... , t, i # k
(4)
j=l
~, Pij
= x * , i = 1, 2 . . . . , s
j=l
We claim that the coefficient matrix of this system has rank rs.
(5)
A N o t e on C o r r e l a t e d E q u i l i b r i u m
39
The rank is _ s
Consider the system of equations
• (bij-bl)pi~=O,
j= l,...,r;l=
l ..... u , j # l
(6)
i=t
Pij=Y*, i=1
J = l , 2 ..... r
(7)
40
Fe S. Evangelista and T.E.S. Raghavan The (u + 1) x (s + 1) matrix
2=
'bli
...
bsi
b12
.--
bsz - 1
-1\
bis
..-
bss
-1
bi.
---
bs.
-1
1
.-.
1
0
)
has rank s + 1. The argument in Case t can then be used to show that the system of equations (6) and (7) has a unique solution9 Therefore,/5* =/51 =/~2, and so p* is an extreme point of Cg(A,B). As a consequence of the above theorem, one can always find a Nash equilibrium among the extreme points of the set of correlated equilibria9 Being the image of a convex polyhedron under a linear map, the set of correlated equilibrium payoffs
is also a convex set in R 2. However, as the following game shows, the set of extreme points of the set of correlated equilibrium payoffs need not contain a Nash equilibrium payoff9 The example is adapted from Nau and Cardle (1988) and is due to R. Aumann.
Example: Let A=
B=
0 30 60 \40
60 0 30 40
30 60 0 40
40) 40 40 41
6o)
0 3O 6O 0 30 0 30 60 40 40 40
40 40 40 41
The unique Nash equilibrium is the pair of pure strategie s giving payoff (41, 41). This payoff is a convex combination of the correlated equilibrium payoffs (45, 45), (50, 40), (40, 40~-) corresponding to the correlated equilibrium strategies q i = 1/6(0, 1, 1,0, 1,0, 1,0, 1, 1,0,0,0,0,0,0),
q2=1/63(0,32,1,0,4,0,2,0,8,16,0,0,0,0,0,0) q3 = 1/693 (0, 11, 22, O, 4, O, 2, O, 8, 16, O, O, O, 210, 420, O)
A Note on Correlated Equilibrium
41
respectively, where the ordering ofthe vectors q is (q~l,
I l l q12, q13, q i 4 , q21' q22' l q 2t 3 ' ' ' " q i 4 ) "
Acknowledgement: We would like to thank an anonymous referee for very helpful comments on the paper.
References
Aumann RJ (1974) Subjectivity and correlation in randomized strategies. J Math Economics 1:67-95 Aumann RJ (1987) Correlated equilibrium as an expression of Bayesian rationality. Econometrica 55: 1-18 Heuer G, Millham C (1976) On nash subsets and mobility chains in bimatrix games. Naval Res Logist Quart 23:311-319 Jansen MJM (1981) Equilibria and optimal threat strategies in two-person games. PhD Thesis, Katholieke Universiteit Nijmegen. The Netherlands Kuhn H (1961) An algorithm for equilibrium points in bimatrix games. National Academy of Sciences Proceedings 47:1657-1662 Myerson R (1986) Acceptable and predominant correlated equilibria. IJGT 15:133-154 Nau R, McCardle K (1988) Coherent behavior in noncooperative games. Working paper 8701, Fuqua School of Business Duke University Parthasarathy T, Raghavan TES (1971) Some topics in two-person games. American Elsevier Publishing Co New York Van Damme EEC (1991) Stability and perfection of Nash equilibria. Springer Verlag, New York
Received November 1991 Revised version Sept. 1992 Second revised version Sept 1993 Final version June 1994
