Extreme Correlated and Nash Equilibria in
Two-Person Games
Martin W. Cripps ∗
University of Warwick,
Coventry CV4 7AL, GB.
May 1992, First revised July 1993,
Second revision May 1995,
This version November 1995
0
* My thanks are due to Françoise Forges and to Jean-François Mertens for their encouragement,
helpful suggestions and for spotting an error in an earlier version of this paper, to Jonathan Cave who
commented on this work in its early stages, to Ehud Kalai an associate editor and referee’s who’s comments
greatly improved the exposition and finally to seminar participants at CORE.
Abstract: I show that if σ is the correlated equilibrium distribution
generated by an extreme Nash equilibrium of a bimatrix game, then σ is
an extreme point of the set of correlated equilibria of the bimatrix game.
I also describe a perturbation of a correlated equilibrium derived from an
extreme Nash equilibrium that will generate further extreme points of the set
of correlated equilibrium distributions. I use these to provide simple conditions
for the set of correlated equilibrium distributions to be different from the set
of correlated equilibrium distributions derived from Nash equilibria.
Keywords: Correlated equilibrium.
JEL Classification Numbers: .: C70.
Running Head: Extreme Correlated and Nash Equilibria
Address: Martin Cripps, Department of Economics, University of Warwick, Coventry
CV4 7AL, UK.
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1. Introduction
The set of Nash equilibrium strategy pairs for a bimatrix game is a finite union of convex polytopes, which are called maximal Nash subsets, and an extreme point of such a
polytopes is called an extreme equilibrium. I will show that each extreme equilibrium
generates a correlated equilibrium distribution that is an extreme point of the convex
polytope of all correlated equilibrium distributions for the bimatrix.
I will also provide a sufficient condition for the existence of a correlated equilibrium
distribution that cannot be written as a convex combination of extreme equilibria. Assume
the game has a quasi-strict extreme equilibrium and restrict the players to use the pure
strategies used at this equilibrium. Now choose a (mixed) strategy pair so that each player
plays a worst response to their opponent’s strategy on this sub-bimtrix ( such a strategy
pair is called an anti-Nash equilibrium here).1 I will show that provided the above two
steps are possible and yield distinct strategies, then there is a linear combination of the
anti-Nash equilibrium and a quasi-strict extreme equilibrium which generates an extreme
point of the set of correlated equilibrium distributions. Furthermore, this new extreme
point is not a convex combination of the Nash equilibria.
2. The Derived Game
Define a finite bimatrix game where the row player has the pure strategies I := {1, 2, ..., m},
the column player has the pure strategies J := {1, 2, ..., n} and the payoffs are described
by the matrices (A, B), where A := {aij }i∈Ij∈J and B := {bij }i∈Ij∈J . I will use (A, B) to
denote this game. I will also use p = (p1 , p2 , ..., pm )T ∈ ∆m and q = (q1 , q2 , ..., qn )T ∈ ∆n
to denote the players’ mixed strategy vectors.2 The set of all Nash equilibria of the game
is E(A, B) := {(p, q) ∈ ∆m × ∆n |(p − p0 )T Aq ≥ 0 p0 ∈ ∆m ; pT B(q − q 0 ) ≥ 0 q 0 ∈ ∆n }.
A correlated strategy for (A, B) induces a distribution over action pairs in I × J, the
distribution of a correlated strategy is denoted by σ = (σ11 , ..., σ1n , σ21 , ...σ2n , ...σmn ) ∈
∆nm . The set of all correlated equilibrium distributions of (A, B) is the set C(A, B) :=
1
My thanks are due to Jonathan Cave for suggesting this name to me.
The superscript T denotes the transpose of a vector and ∆k := {x ∈ <k |x ≥ 0, 1T x =
1}, where1=(1,1,... ,1)T .
2
1
{σ ∈ ∆nm |
P
j
σij (aij − akj ) ≥ 0 i, k ∈ I;
P
i
σij (b − ij − bih ) ≥ 0, j, h ∈ J}. When
two players play the strategy pair (p, q) their play induces a probability distribution over
action pairs. I define σ(p, q) to be this distribution, that is σij (p, q) := pi qj for all i ∈ I
and j ∈ J. Thus if (p, q) ∈ E(A, B) then the correlated equilibrium derived from (p, q)
is σ(p, q) and the set of correlated equilibria that use only convex combinations of Nash
equilibria is CN (A, B) := co{σ(p, q)|(p, q) ∈ E(A, B)}.3 Aumann (1974) showed that
CN (A, B) ⊆ C(A, B).
Hart Schmeidler (1989) introduce a two-person zero-sum game (G, −G) with the
payoffs derived from(A, B). The (expected) payoff function in their game (G, −G) is
P (σ, θ) where
P (σ, θ) = ρ
XXX
i∈I k∈I j∈J
σij πik (aij − akj ) + (1 − ρ)
XXX
j∈J h∈J i∈I
σij τjh (bij − bih ).
(1)
Here σ = (σ11 , ..., σ1n , σ21 , ...σ2n , ...σmn ) ∈ ∆mn is the mixed strategy of the row player in
(G, −G) and θT := (ρπ T , (1 − ρ)τ T ), where 0 ≤ ρ ≤ 1, π ∈ ∆mm and τ ∈ ∆nn , is the
mixed strategy of the column player. Clearly the pyaoff function P (σ, θ) is a bilinear form
and therefore can also be written in matrix form P (σ, θ) = σ T Gθ.
andcan be derived from the bimatrix game (A,B). In this derived game the set of pure
strategies for the row player (she) is IxJ, hence her mixed strategies for (G,-G) are distributions (of correlated strategies) for (A,B). I thus also use s=(s11,..,s1n;... ;sm1,...,smn)T
Dmn to describe her mixed strategies in (G,-G). A pure strategy for the column player
(he) in the derived game (G,- G) requires him to first choose a role: he can choose to be
the row player in (A,B) or he can choose to be the column player in (A,B). If he has chosen
to be the row player then he must select a pair of actions (i,k)I2 and if he has chosen to
be the column player he must select a pair (j,h)J2. His mixed strategies can therefore be
described by the triples (r,p,t)[0,1]xDmmxDnn where: r is the probability that he chooses
to be the row player, p := (p11,...,p1m;... ;pm1,...,pmm)TDmm is a probability measure
on I2, t := (t11,...,t1n;... ;tn1,...,tnn)TDnn is a probability measure on J2. I will define
his mixed strategy to be a vector q using the convention qT = (rpT,(1-r)tT). In (G,-G)
the row player plays the mixed strategy s over the rows of G and the column player uses
q over its columns.
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Where co{.} is the convex hull of the set.
2
Hart Schmeidler (1989) then define the expected payoffs in the zero-sum game (G,-G)
to be the function
P(s,r,p,t) = rSiISkISjJsijpik(aij-akj) + (1-r)SjJShJSiIsijtjh(bij-bih). (3)
This payoff function can be written as a bilinear form, P(s,r,p,t) = sTGq, where G is
the partitioned matrix G = [ G1 G2 ] given below. The submatrix G1 is played when the
column player in (G,-G) chooses the role of row player in (A,B) and the submatrix G2 is
played when the column player of (G,-G) chooses the role of column player in (A,B). The
conventions used for the players’ mixed strategies s and q give G1 and G2 their form.
G = [ G1 G2 ]
The (nmxm2) matrix G1 is block-diagonal. There are m blocks along the diagonal.
The ith diagonal block of G1 has m columns that give expected payoffs -Sjsij(aij-akj)
(k=1,2,...,m) to the column player in (G,-G). (Henceforth I will use the abbreviations
Si,Sk for SiI,SkI and Sj,Sh for SjJ,ShJ.) These payoffs are identical to the linear functions
in the definition of a correlated equilibrium distribution (2). The (nmxn2) matrix G2
does not have a block-diagonal structure because of the ordering chosen for s. In columns
kn,kn+1,...,(k+1)n-1 there are positive entries in the kth,k+mth,k+2mth,... rows, so there
is a permutation of the rows of G2 which will give it a block-diagonal structure. Once
this permutation is performed the expected payoffs in the diagonal blocks are -Sisij(bijbih) (h=1,2,...,n) which are the other linear functions in the definition of a correlated
equilibrium distribution (2). The importance of the derived game (G,-G) is that s is a
correlated equilibrium distribution of (A,B), if and only if, it is an optimal strategy for
the zero-sum game (G,-G).
Result 1 (Hart Schmeidler) : sC(A,B) iff s is an optimal strategy for the row player
in (G,-G). The value of (G,-G) is zero.
Remark 1 The strictly mixed distribution see1 is an e-correlated equilibrium (Myerson 1986 p.139) for (A,B) if sije¿e implies that the row player does not benefit by
deviating from action i if it attaches probabilities (si1e,si2e,...,sine) to the actions of the
column player (and a symmetric incentive compatibility condition applies to the column
player). Further the limit of a sequence of e-correlated equilibria s is an acceptable correlated equilibrium . But (s,q) is a perfect equilibrium of (G,-G) if there is a sequence
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of strictly mixed strategies (se,qe) such that (se,qe) (r) (s,q) as e(r)0, where the strategy
s (respectively q) is a best response to qe (respectively se) for all e (Van Damme 1987
p.27). By comparing the two definitions above it is clear that an acceptable correlated
equilibrium is a perfect equilibrium of (G,-G). The characterisation of a correlated equilibrium distribution as a Nash equilibrium of (G,-G) thus suggests another way of describing
refinements of correlated equilibria. It also provides a way of generating new refinements
of the correlated equilibrium concept by applying a refinement of Nash equilibrium to the
equilibria of (G,-G).
Remark 2 It is possible to describe the set of subjective correlated equilibrium distributions of (A,B) as the set of optimal strategies in a two-person zero-sum game. In
this new game the row player chooses two distributions: one for the row player in (A,B)
the other for the column player in (A,B). The second player in the new game verifies the
incentive compatibility conditions for both distribution. The set of a posteriori correlated
equilibria is more difficult to characterise, but this would be very useful to do because of
the relationship between a posteriori payoffs and rationalizable payoffs: Brandenburger
Dekel (1987).
Remark 3 The Hart Schmeidler (1989) game (G,-G) can be defined for general nperson games, however, it is not clear how to generalise the later sections of this paper to
the n-person case: see Chin, Parthasarathy Raghavan (1974).
3 Extreme correlated equilibrium distributions in two-person games
In this section the Shapley-Snow Theorem (1950) is quoted and this result in combination with Result 1 is used to describe the extreme points of the convex set C(A,B).
I also describe a result due to Kuhn (1961) and Vorobev (1958) which characterises the
extreme equilibria in E(A,B). Following this I establish the main result of the paper and
then give some simple further results.
The first step is to find necessary and sufficient conditions the distribution s to be an
extreme point of the convex polytope C(A,B) (recall that an extreme point of a convex
set is a point that cannot be written as a convex combination of other points in the set).
Result 1 is essential here because it shows that the set of correlated equilibrium distributions of (A,B) is identical to the set of optimal strategies for the row player in (G,-G). The
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Shapley-Snow Theorem (1950) provides a necessary and sufficient characterisation of the
extreme points in each player’s set of optimal strategies in a zero-sum game. Therefore,
combining these results tells us that the extreme points of C(A,B) are characterised by
the Shapley-Snow Theorem applied to the zero-sum game (G,-G). The payoffs in G are
derived from the payoffs in (A,B) so this gives a necessary and sufficient characterisation
of the set of extreme points of C(A,B) in terms of the bimatrix (A,B).
The second step is to relate the set of equilibria E(A,B) to the extreme points of
C(A,B). For a bimatrix game (A,B) the set of equilibria, E(A,B), is the union of a finite
number of closed convex sets. These are called either maximal Nash subsets or equilibrium
components and each of these sets is the Cartesian product of two convex polytopes
of Nash equilibrium strategies (Jansen 1981). Each equilibrium component is therefore
completely described by its extreme points which are called extreme equilibria . There is
a characterisation of the extreme equilibria of the bimatrix game (A,B). The result is due
to Vorobev (1958) and Kuhn (1961) and can also be found in Parthasarathy Raghavan
(1971) or Jansen (1981), it provides a necessary (but not sufficient) characterisation of
the extreme equilibria within each maximal Nash subset. To sum up, the result of Kuhn
and Vorobev can be used to characterise a property of the extreme equilibria of E(A,B)
whereas the Shapley-Snow result will provide a necessary and sufficient condition for the
extreme points of C(A,B). The equilibrium components that make up E(A,B) will now
be described in greater detail. Suppose that S is a subset of E(A,B) with the property
(p,q),(p’,q’)S implies (p,q’),(p’,q)S, such a set S is called a Nash subset . An equilibrium
component , or maximal Nash subset , is a Nash subset S with the property that there is
no Nash subset S’ that properly contains S. Each maximal Nash subset is the Cartesian
product of two convex polytopes (Jansen 1981). Thus maximal Nash subsets have a finite
number of extreme points and I will call these extreme points extreme equilibria . (Notice
that these extreme points are defined relative to the equilibrium component and are not
necessarily extreme points of the larger set E(A,B).)
Definition : The pair (p,q)E(A,B) is an extreme equilibrium if (p,q) is an extreme
point of an equilibrium component.
The main result of this paper shows that if a Nash equilibrium (p,q) is an extreme
equilibrium in E(A,B) (and thus satisfies Kuhn-Vorobev) it must generate a correlated
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equilibrium distribution s(p,q) that satisfies the necessary and sufficient conditions for an
extreme point of C(A,B). Therefore, the extreme points of CN(A,B) are extreme points
of C(A,B). If all of the extreme points of CN(A,B) are extreme points of C(A,B) it follows
that no extreme point of CN(A,B) can be written as a non-trivial convex combination of
points in C(A,B). The result is, of course, not sufficient to ensure that CN(A,B)=C(A,B),
because this is only true if all the extreme points of C(A,B) are extreme points of CN(A,B).
In general there may be extreme points of the set of correlated equilibrium distributions
that are not in CN(A,B). These properties are illustrated in the following example.
Example 1
The strategic form (A,B) has three distinct Nash equilibria: (1,0),(0,1);(0,1),(1,0);
(1/2,1/2),(1/2,1/2) and the equilibrium components are all disjoint singletons so every
Nash equilibrium of this game is also an extreme equilibrium. These equilibria induce,
respectively, the optimal strategies (0,1,0,0),(0,0,1,0),(1/4,1/4,1/4,1/4) for the row player
in (G,-G). These are also correlated equilibrium distributions in C(A,B). The typical
correlated equilibrium distribution for this game can be written as the vector (p,q,r,1-pq-r)D4. Vectors in D4 can be shown inside a tetrahedron where the vertexes represent
the distributions: p=1,q=1,r=1,p=q=r=0. For the above game I have plotted the sets
C(A,B) and CN(A,B) in such a tetrahedron. The set C(A,B) consists of the convex
hull of the shaded region and its two remaining extreme points at (1/3,1/3,1/3,0) and
(0,1/3,1/3,1/3) whereas the set CN(A,B) is the shaded region. ( Figure 1 here )
Now I quote two central results on bimatrix games that will be used repeatedly
below. Proofs of these results can be found in Parthasarathy and Raghavan (1971) or in
the original sources. The notation 1=(1,1,..,1)T is used and the dimension of this vector
should be obvious from the context.
Result 2 (Shapley-Snow) : In a zero-sum game (A,-A) with the value v0 : (i)T hesetE(A, −A)isconvexan
singularsubmatrixA0 suchthat : therowsof A0 containthesupportof p, thecolumnsof A0 containthesupportof
v1, (p0 )T A0 = v1T, wherep0 , q0 arevectorsobtainedf romp, qwithappropriatedeletionsof zeros.
Result 3 (Kuhn-Vorobev) : If (p,q) is an extreme equilibrium of (A,B), then there
exists square submatrices A’,B’ of A and B such that: (i) The columns of A’ are the
support of q and the rows of B’ are the support of p. (ii) A’q’=a1, (q’)TB’=b1T where
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p’,q’ are vectors obtained from p,q with appropriate deletions of zeros. (iii) The matrices
H,K below are non-singular
The first proposition applies the Shapley-Snow result to (G,-G) to characterise the
extreme points of the set of correlated equilibrium distributions C(A,B). The problem that
this presents is that the value of the derived game (G,-G) is zero and so Shapley-Snow
cannot be directly applied. To avoid this problem I add a constant d to all the payoffs in
G; let U be the matrix with unity in every entry then G+dU is the payoff matrix G with d
added to each element. This does not alter the set of equilibria, E(G,-G) = E(G+dU,-GdU), because the inequalities defining an equilibrium are linear in the payoffs. The value
of (G,-G) is zero, so the value of (G+dU,-G-dU) will be d.
Proposition 1: s is an extreme point of C(A,B) if and only if s is an optimal strategy
for the row player in (G,-G) and there exists a non-singular submatrix R of G+dU with
rows that contain the support of s such that (s’)TR=d1T, where s’ is s with appropriate
zero elements deleted.
Proof: If s is an extreme point of C(A,B) then it is an extreme point of the set of
optimal strategies for the row player in (G+dU,-G-dU) (by Result 1). The existence of R
now follows from Result 2. Let s be an optimal strategy for the row player in (G,-G) and
(s’)TR=d1T where R is a non-singular submatrix of G+dU. (i) If R contains a column
of d’s then there exists an optimal strategy q for the column player (play the column of
d’s) such that Rq’=d1. By Result 2 (s,q) is an extreme point of E(G+dU,-G-dU) and
E(G,-G). This set is a Cartesian product so s is an extreme point of the set of optimal
strategies for the row player in (G,-G) and by Result 1 s is an extreme point of C(A,B). (ii)
If R does not contain a column of d’s and (s’)TR=d1T then R has at least two columns.
Call the last two columns of R the vectors r1 and r2: R = [ Y r1 r2 ]. Let S be the matrix
R with the last column replaced by d1 and let T be the matrix R with the second from
last column replaced by d1. Both S and T are also submatrices of G+dU since d1 is the
first column of G+dU. One of the matrices S and T is non-singular. Suppose not, then
there exists weights (a1,a2,b1,b2) such that
Ya1 + r1a2 + d1 = Yb1 + r2b2 + d1 = 0.
(The weight on the vector d1 can be normalised to unity because if it were zero the
7
columns of R would be linearly dependent.) Subtract the middle from the left above to
show that the columns of R are linearly dependent: a contradiction. Thus at least one of
S or T is non- singular call the non-singular one R’. This gives a non-singular submatrix
R’ of G+dU and an optimal strategy q (play the column of d’s) for the column player
such that: (s’)TR’=d1T; R’q’=d1. Now repeat the above argument to show that s is an
extreme point of C(A,B). Q.E.D.
Example 1 Continued
In Figure 1 five extreme points are given for C(A,B) these are: (0,1,0,0),(0,0,1,0),
(1/4,1/4,1/4,1/4), (1/3,1/3,1/3,0),(0,1/3,1/3,1/3). Proposition 1 can be easily verified for
the first two of these strategies. A submatrix of G+dU can also be found for the fourth,
because the first, second, fourth and sixth columns of G+dU are linearly independent.
The submatrices for the fourth and fifth strategies use columns one, two, six and columns
one, four, seven respectively.
Proposition 2: Let (p,q)E(A,B) be an extreme equilibrium, then s(p,q) is an extreme
point of C(A,B).
Proof: Let (p,q)E(A,B) be an extreme equilibrium, without loss of generality assume pi¿0 iff i=1,2,...,cm and qj¿0 iff j=1,2,...,dn. Thus the distribution s(p,q) plays rows
1,2,...,d,n+1,n+2,...,n+d, ...,cn+1,cn+2,...,cn+d of the matrix G+dU with positive probability, cd rows in total. Suppose that there exists weights : m1,...,md,mn+1,...,mn+d,...,mcn+d
so that a linear combination of these cd rows sums to zero. As the first column of G+dU
is d1 we have Stmt=0 (where the summation ranges over all the weights mt). Consider
the rows 1,2,...,d of G+dU. These are the only rows in the linear combination (md) that
have non-d entries in the first d columns. Inspection of G1 shows that the first d entries
of the first d rows of G+dU are the vectors: (d+a11)1T-(a.1)T, (d+a12)1T-(a.2)T, ...,
(d+a1d)1T-(a.d)T, where a.j is a vector containing the first d entries of the jth column of
A. If the cd rows of G+dU with weights mt sum to zero, then the first d columns satisfy
m1[(d+a11)1-(a.1)] + m2[(d+a12)1-(a.2)] + ...+md[ (d+a1d)1-(a.d) ] + St¿dmtd1
= 0,
or equivalently m1a.1 + m2a.2 + ...+mda.d - ( Stda1tmt )1 = 0.
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This will be true if the weights m1,m2,...,md are all zero. Suppose instead that at least
one of the weights mt is non-zero. Recall that (p,q) is an extreme equilibrium and (a.1,a.2,
...,a.d) are the columns of the matrix A’ described in Result 3. Thus the above expression
gives a linear combination of the columns that make up the first d rows of the matrix H
in Result 3. As H is non-singular there is no non-trivial combination of the columns
of H that sum to zero so this implies Stdmt0 .Def inel = (l1, ..., ld)T satisf yinglt =
mt/Stdmtthenf romaboveA0 l = a1wherea = Stda1tlt.T hepair(p, q)isanextremeequilibriumsobyResult3A
a1, whereq0 isqwithzeroelementsdeleted.T hevector((q 0 −l)T, a−a)theref oregivesalinearcombinationof the
lorqj = mj/Stdmt.T husif oneof theweightsm1, m2, ..., mdisnon−zerotheymustallhavethesamesignandbe
dU thatareinthesupportof s(p, q).T hisshowsthatf ori = 1, 2, ..., ctheweights(min+1, min+
2, ..., min+d)areeitherallzeroorallhavethesamesign.Nowlinkthesecblocksof drowstogether.T hereisaperm
diagonalf orm.Af terthispermutationperf ormtheanalysisaboveonthef irstblockof crowsof G2+
dU.If bi.isthevectorthatconsistsof thef irstcelementsof theithrowof Btheweights(m1, md+
1, ..., m(c − 1)d + 1)mustsatisf y
m1b1.+md+1b2.+...+m(c-1)d+1bc. + Sicm(i-1)d+1bi1 = 0.
The matrix K, of Result 3, is non-singular, so again it can be shown that either the
vector (m1,md+1,...,m(c-1)d+1) is zero or all of its elements have the same sign. Thus all
the mt’s in all of the c blocks of d rows either have the same sign or are zero. If one of the
mt’s is non-zero and all of the mt have the same sign then this contradicts Stmt=0. Thus
mt=0 for all t and the rows of G+dU in the support of s are linearly independent. This
not only shows that the rows in the support of s are linearly independent but it is also
true that only columns of G giving a payoff of d to the column player in (G+dU,-G-dU)
were used in the calculation. Thus, because (p,q) is a Nash equilibrium, there exists a
non- singular cd-dimensional submatrix R of G+dU such that (s’)TR=d1T (where s’ is s
with appropriate deletions). By Proposition 1 s is an extreme point of C(A,B). Q.E.D.
This establishes the main result; the extreme points of the set of simple correlated
equilibrium distributions, CN(A,B), are also extreme points of the set C(A,B). Some
consequences of the relationship between the extreme points of C(A,B) and the extreme
equilibria are the two propositions below. In Proposition 3 I give a lower bound on
the number of constraints (2) that are binding at an extreme point of C(A,B). And
Proposition 5 is a consequence of Result 1; it is a simple result on the uniqueness of
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correlated equilibrium distributions which follows from a well known result on unique
Nash equilibria. (A correlated equilibrium distribution s is said to be of full support
if sij¿0 for all iI,jJ.) The result gives a way of checking whether a game has a unique
correlated and/or Nash equilibrium. To extend this result one could investigate when the
existence of a unique Nash equilibrium of full support ensures the existence of a unique
correlated equilibrium distribution. This is obviously true for zero-sum games (see Forges
(1990)), but is not clear for the non-zero-sum case.
Proposition 3: If sDnm is an extreme point of C(A,B), then at least (i,j) – sij¿0
-1+m+n of the constraints (2) are binding at s.
Proof: If s is an extreme point of C(A,B), then there exists a non-singular submatrix
R of G+dU such that (s’)TR=d1T, where R has dimension greater than (i,j) – sij¿0 .
Each column of G+dU that has a payoff equal to d represents one of the constraints (2)
that binds at s. As one of the columns of R can be d1 and there are m+n constraints
in (2) that are trivially satisfied there will be at least (i,j) – sij¿0 -1+m+n binding
constraints. Q.E.D.
Proposition 4: If all of the correlated equilibrium distributions of the bimatrix game
(A,B) are of full support and if rank(G)=nm-1, then the game (A,B) has a unique correlated equilibrium distribution.
Proof: Suppose that all of the optimal strategies for the row player in the game (G,G) are of full support. By Parthasarathy Raghavan (1971) (Theorem 3.1.5 p.61) if all the
optimal strategies for the row player in an rxt zero-sum game with rank(r-1) and a zero
value are of full support, then there is a unique optimal strategy for the row player. Thus
there is a unique optimal strategy for the row player in (G,-G) and a unique correlated
equilibrium distribution. (Since CN(A,B) is contained in the set C(A,B) the corollary
below follows.) Q.E.D.
Corollary: If all the correlated equilibrium distributions of the game (A,B) are of full
support and if rank(G)=nm-1, then (A,B) has a unique Nash equilibrium.
In this section I have established that the extreme equilibria in the set E(A,B) generate correlated equilibrium distributions that are extreme points of the set C(A,B) and
derived some easy consequences of this. Now I investigate when the sets CN(A,B) and
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C(A,B) are equivalent. That is, when are correlated equilibrium distributions only convex
combinations of Nash equilibria? As a consequence I present some results on what causes
the differences between the set of correlated equilibrium distributions and CN(A,B).
4 Correlated Equilibrium Distributions and CN(A,B)
The previous section shows that the set of correlated equilibria derived from public correlation, CN(A,B), is a convex polytope with extreme points that are also extreme
points of C(A,B). In this section I give one explanation for why CN(A,B)C (A, B)andthusthatingeneralsome
The perturbations of CN(A,B) described in this section take as a starting point a
particular quasi-strict extreme equilibrium (p,q)E(A,B). The correlated equilibrium distribution s(p,q)CN(A,B) is perturbed by taking a linear combination of s(p,q) and one
other point in Dmn. The precise form of the linear combination used is (1-m)-1s(p,q)ms(u,v). Thus the perturbation is determined by a pair of strategies (u,v)DmxDn and a
small weight 0¡m¡1. This linear combination reduces the probability s(p,q) attaches to the
action profiles played by s(u,v) whilst increasing the probability attached to the action
profiles not played by s(u,v). Since the pair (u,v) are particularly ”bad” strategies for the
two players in (A,B) the effect of this perturbation is, therefore, to place less probability
on a bad outcome.
The direction of the perturbation of s(p,q) is determined by a particular pair of
strategies (u,v)DmxDn. I will now describe how this pair is determined. Let ( , ) be the
game (A,B) where the rows and columns not used by (p,q) are deleted, so the bimatrix ( ,
) contains only the action profiles (i,j) that are played with positive probability by s(p,q).
Then define ( , )E(- ,- ), thus ( , ) is a Nash equilibrium for this subgame with the signs
of the payoffs switched. I call the pair ( , ) an anti-Nash equilibrium of ( , ), because ( , )
is a pair of strategies for ( , ) where each player uses a worst response to their opponent’s
action.
( , )E(- ,- ) T (u’)T for all u, T T v’ for all v’ .
We can, therefore, interpret the pair ( , ) as a pair of ”bad” actions in the subgame (
, ). Now the direction of the perturbation s(u,v) is given by the pair (u,v)DmxDn which
is ( , ) with appropriate zeros added.
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The effect of the linear combination (1-m)-1s(p,q)-ms(u,v) is to reduce the weight
put on worst responses when the players use the Nash equilibrium (p,q). It is difficult to
suggest a simple story behind the correlation devices that generate the distribution (1m)-1s(p,q)- ms(u,v) because the linear combination used is not a convex. The correlated
equilibrium distribution could perhaps be generated by a two stage device: At the first
stage a public signal is announced and this is used to coordinate the players’ behaviour
on one extreme Nash equilibrium. At the second stage the players receive private signals
that implement the distribution (1-m)-1s(p,q)-ms(u,v). Two examples below show how
the perturbations work in practice.
Example 1 Continued There is one completely mixed Nash equilibrium (p,q)=((1/2,1/2),(1/2,1/2))
for this game which is also a quasi-strict and extreme equilibrium in E(A,B). As (p,q) is
completely mixed (- ,- ) = (-A,-B). The game (-A,-B) has two pure strategy Nash equilibria (u,v)=((1,0),(1,0)) and (u’,v’)=((0,1),(0,1)) and these are also extreme equilibria in
E(- ,- ). One can describe (u,v) and (u’,v’) as anti-Nash equilibria of (A,B) since both
player is using a worst response to its opponent. If we take m=1/4 the linear combinations (1-m)-1 s(p,q) - ms(u,v) and (1-m)-1s(p,q) - ms(u’,v’) will give the correlated
equilibrium distributions (1/3,1/3,1/3,0) and (0,1/3,1/3,1/3) which are the two extreme
points of C(A,B) that are not extreme points of CN(A,B) (see Figure 1). In this example
using the anti-Nash equilibria to perturb (p,q)=((1/2,1/2),(1/2,1/2)) is sufficient to find
all the extreme points of C(A,B) that are not extreme points of CN(A,B).
Example 2 A game (A,B) in strategic form is given below together with one correlated
equilibrium distribution s.
The symmetric, quasi-strict Nash equilibrium p=(1/3,1/3,1/3), q=(1/3,1/3,1/3) generates a correlated equilibrium distribution s(p,q) which is an extreme point of C(A,B)
(by Proposition 2). The pair of strategies u=(0,0,1), v=(0,0,1) are an extreme equilibrium
of (-A,- B) and hence they are also an anti-Nash equilibrium of (A,B). The linear combination (9/8)(s(p,q)-(1/9)s(u,v)) gives the correlated equilibrium distribution s above.
By Proposition 5 this correlated equilibrium distribution is an extreme point of C(A,B).
The corollary to the proposition shows that sC(A,B), thus for the game above we can be
certain that some correlated equilibria cannot be achieved by public correlation.
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Proposition 5: Suppose that (p,q)E(A,B) is an extreme equilibrium and that it is
a quasi- strict equilibrium. Let ( , ) be the submatrix of (A,B) defined by the support
of s(p,q). If ( , )E(- ,- ) is an extreme equilibrium satisfying (u,v)( p, q), then(i)s :=
(1−m)−1(s(p, q)−ms(u, v))C(A, B)f or0mmf orsomem > 0, (ii)s := (1−m)−1(s(p, q)−
ms(u, v))isanextremepointof C(A, B), (where(u, v)is(, )withappropriatezerosadded).
Proof: Let (p,q),( , ),(u,v) satisfy the conditions in the proposition. Without loss of
generality assume: pi¿0 for i=1,2,...,c; ui¿0 for i=c’+1,c’+2,...,c; qj¿0 for j=1,2,...,d; vj¿0
for j=d’+1,d’+2,...,d. For m¡1 define s := (1-m)-1(s(p,q) - ms(u,v)), then
Sjsij(aij-akj) = Sj(1-m)-1(piqj - muivj)(aij-akj) = (1-m)-1piSjqj(aij-akj) - m(1-m)1uiSjvj(aij-akj).
(p,q)E(A,B) is a quasi-strict equilibrium so if pi¿0 and pk=0 then Sjqjaij¿Sjqjakj and
if pi¿0 and pk¿0 then Sjqjaij=Sjqjakj. Thus the first term above is either strictly positive
or zero. The second term above is zero if ui=0. If ui¿0 row i is a worst response to v for
the game ( , ), thus SjvjaijSjvjakj when pk¿0, but when pk=0 the sign of Sjvjaij-Sjvjakj is
indeterminate. Thus we have the following cases: (i) if pi¿0 and pk¿0 both piSjqj(aij-akj)
and -uiSjvj(aij-akj) are non- negative. (ii) If pi¿0 and pk=0 then piSjqj(aij-akj)¿0 and
-uiSjvj(aij-akj) is indeterminate. (iii) If pi=pk=0 then both sums are zero. Thus it is
possible to find m1¿0 such that Sjsij(aij-akj) 0 for all i,kI and all 0mm1. An identical
argument allows one to find m2¿0 such that Sisij(bij- bih)0 for all j,hJ and all 0mm2.
Finally define m3 so that min(i,j)Supp(s(p,q))(piqj-m3uivj)=0, where supp(s):=(i,j)–
sij¿0. Now let m:=minm1,m2,m3. Then sDmn for all 0mm and sC(A,B) because it
satisfies (2). Let (- , ) denote the zero-sum game derived from (- ,- ). ( , )E(- ,- ) is an
extreme equilibrium so by Propositions 1 and 2 there exists Y, a non-singular submatrix of
+dU, such that [s( , )’]TY = d1T (where s( , )’ is s( , ) with the appropriate zero elements
deleted). By construction is a submatrix of G, thus Y is also a non-singular submatrix of
G+dU such that [s(u,v)’]TY = d1T (again where s(u,v)’ is s(u,v) with the appropriate
zero elements deleted). There are cd-(c-c’)(d-d’) rows of G+dU that are played by s(p,q)
but that are not played by s(u,v). Let (p,q) be s(p,q) restricted to these rows of G+dU.
The construction used in Proposition 2 applied to these rows alone gives X, a non-singular
submatrix of G+dU, such that (p,q)TX = e1T where e=d1T . The columns of G+dU
that are included in X have d’s in the rows played by s(u,v)’. It follows that the matrix
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R, given in partitioned form below, is a submatrix of G+dU
R := ,
where Z is an unspecified collection of elements of G+dU. Since Y is a submatrix
of +d it follows that (s(p,q)’)TR=d1T, (where s(p,q)’ is s(p,q) with all zero elements
deleted). Consider a vector l such that lTR=0. Partition the vector lT=(l1T,l2T) so that
l1 is multiplies the matrix X and l2 multiplies the matrix Y. Then lTR=0 iff
(i) l1TX + l2T(dU) = 0 and (ii) l1TZ + l2TY = 0.
First notice that (i) and (ii) are solved with l0 if f l10 andl20 sinceXandY arenon −
singular.AlsosinceXisnon−singularand(p, q)T X = e1T if li0 wemusthavel1 = h(p, q)wherewithoutlossof
d1T implies((p, q))T X+((p, q))T (dU ) = d1T and((p, q))T Z+((p, q))T Y = d1T.Nowsubtractthesef rom(i)a
(p, q)
( (p,q)-l2)T(dU) = 0 ( (p,q)-l2)TY=d1T
From above Y is non-singular and [s( , )’]TY = d1T, therefore from the second
equation above (p,q)-l2=s( , )’. If this is then substituted into the first equation above we
get [s( , )’]T(dU) = d1T=0 a contradiction. This has established that R is non-singular
and therefore that s is an extreme point of C(A,B). Q.E.D.
Corollary: Let (p,q)E(A,B) be a quasi-strict extreme equilibrium. Let ( , ) be (A,B)
restricted to the support of s(p,q). If there exists an extreme equilibrium ( , ) E(- ,) satisfying (u,v)( p, q)(where(u, v)is(, )withappropriatezerosadded), thens = (1 − m) −
1s(p, q) − ms(u, v)CN(A, B)andCN(A, B)C (A, B).
Proof: Suppose that the corollary is false and that sCN(A,B). By the proposition s
is an extreme point of C(A,B) so it is also an extreme point of CN(A,B). However, s is
an extreme point of CN(A,B) only if s=s( , ) for some ( , )E(A,B) and (1-m)-1(s(p,q)
- ms(u,v)) = s( , ) only if supp(p)=supp(u) and supp(q)=supp(v). If supp(u)=supp(p)
and supp(v)=supp(q), where supp(p):= i – pi¿0 then (p’,q’),( , )E(- ,- ) (where p’
(respectively q’) is p (respectively q) with zero elements deleted). But ( , ) is an extreme
equilibrium and, by Result 3, the matrix below is non- singular.
This gives a contradiction because T =a1T and (p’)T =b1T and up impliesthat(T −
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(p0 )T, b−a)isanon−zerolinearcombinationof thecolumnsof thatmakesthemsumtozero.Q.E.D.
The perturbation used in the Proposition gives a correlated equilibrium distribution
outside the set CN(A,B), only if the anti-Nash equilibria E(- ,- ) are different from the
original extreme equilibrium quasi-strict equilibrium (p,q). For the above perturbation
to be feasible we therefore require (- ,- ) to have at least two equilibria. Thus Proposition
5 can be used to deduce that if CN(A,B)=C(A,B) then the quasi-strict extreme Nash
equilibria of (A,B) are square.
Proposition 6: If CN(A,B)=C(A,B) and (p,q)E(A,B) is a quasi-strict extreme equilibrium, then i – pi¿0 = j – qj¿0.
Proof: If (p,q) is a quasi-strict extreme equilibrium and CN(A,B)=C(A,B), then there
is a unique anti-Nash equilibrium ( , )(- ,- ) where (respectively )is the vector obtained by
deleting the zero elements of p (respectively q). From Jansen (p. 541 1981) there exists
a game (- ,- ) with (u,v) as its unique equilibrium point, if and only if, i – ui¿0 = j –
vj¿0. It follows that i – pi¿0 = j – qj¿0.
Remark 4 Forges (1985 p.143) presents an example of a game (A,B) where an action
profile (i,j) is played with positive probability at a correlated equilibrium distribution but
the game (A,B) has no Nash equilibrium that plays (i,j) with positive probability. The
perturbations suggested in this section will therefore not find all the correlated equilibrium
distributions of a game. This is because the class of perturbations here always has the
property that the players play action profiles that are used with positive probability at
some Nash equilibrium.
References
Aumann, R.J.(1974). ”Subjectivity and Correlation in Randomized Strategies,”
Journal of Mathematical Economics 1, 67-96. Aumann, R.J. (1987). ”Correlated Equilibrium as an Expression of Bayesian Rationality,” Econometrica 55 , 1-18. Brandenburger,
A , and Dekel, E. (1987). ”Rationalizability and Correlated Equilibria,” Econometrica
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of Equilibria in n- Person Cooperative Games,” International Journal of Game Theory 3,
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15
”Correlated Equilibrium in Two-Person Zero-Sum Games,” Econometrica 58, 515. Hart,
S. and Schmeidler, D. (1989). ”Existence of Correlated Equilibria,” Mathematics of Operations Research 14, 18-25. Jansen, M.J.M. (1981). ”Regularity and Stability of Equilibrium Points of Bimatrix Games,” Mathematics of Operations Research 6 , 530-550. Kuhn,
H.W.(1961). ”An Algorithm for Equilibrium Points in Bi-Matrix Games,” Proceedings of
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Figure 1 : The sets C(A,B) and CN(A,B).
FOOTNOTES
1 Dm := xRm – x0, 1Tx=1 where 1T := (1,1,1,... ,1). 2 My thanks are due
to Jonathan Cave for suggesting this name to me. 3 . denotes the number of elements
in a set. 4 By Van Damme (1987 p.23) at a quasi-strict equilibrium both players play
all of their best replies, that is, if (p,q)E(A,B) is quasi-strict then it satisfies pi¿0 iff
(ei)TAq=pTAq and qj¿0 iff pTB(ej)=pTBq (where ei denotes a vector with unity in the
ith entry and zeros elsewhere).
Dm := xRm – x0, 1Tx=1 where 1T := (1,1,1,... ,1). My thanks are due to
Jonathan Cave for suggesting this name to me. . denotes the number of elements in a set.
By Van Damme (1987 p.23) at a quasi-strict equilibrium both players play all of their
best replies, that is, if (p,q)E(A,B) is quasi-strict then it satisfies pi¿0 iff (ei)TAq=pTAq
and qj¿0 iff pTB(ej)=pTBq (where ei denotes a vector with unity in the ith entry and
zeros elsewhere).
16
References
Fudenberg, D. (1993): “Explaining Cooperation and Commitment in Repeated Games”,
forthcoming in J.-J. Laffont (ed.), Advances in Economic Theory, Sixth World
Congress, Cambridge: Cambridge University Press.
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