CORE DISCUSSION PAPER
9833
ON NASH EQUIVALENCE CLASSES OF GENERIC
NORMAL FORM GAMES
Fabrizio Germano1
May 1998
Abstract
We introduce a procedure that uses basic topological characteristics of
equilibrium correspondences of standard equilibrium concepts, to define broad equivalence classes of finite generic games in normal form.
The proposed procedure is viewed as a potentially useful way of both
organizing the underlying spaces of games as well as of comparing
different equilibrium concepts with each other. The focus of the paper is mainly on equivalence classes induced by the Nash equilibrium
concept. However, equivalence classes induced by the concepts of rationalizability, iterated dominance and correlated equilibrium are also
considered.
Keywords: non-cooperative games, classification and equivalence classes,
geometry of equilibrium correspondences.
JEL Classification: C70, C72
1
CORE, Université catholique de Louvain, Belgium. E-mail: [email protected]
I would like to thank Jean-François Mertens, Joel Sobel and seminar participants at the
University of Bielefeld for valuable comments and conversations. Financial support from
CORE and the European Commission, Grant ERBFMBICT972857 is gratefully acknowledged. All errors are mine.
This text presents research results of the Belgian program on Interuniversity Poles of
Attraction initiated by the Belgian State, Prime Minister’s Office, Science Policy Programming. The scientific responsibility is assumed by the author.
1
Introduction
The representation of a strategic situation by means of a normal form game is
one of the most fundamental constructions of game theory. But, while much
work has been done devising and studying different equilibrium concepts for
normal form and extensive form games, not much has been done to systematically differentiate and classify the infinite number of normal form games that
arise. (The reader is referred to Section 5 for a discussion of related literature
and hence of some exceptions to this.)
The present paper takes some standard equilbrium concepts like rationalizability, correlated and Nash equilibrium, and studies equivalence classes of
generic normal form games that are implied by the singularities of the corresponding equilibrium correspondences. More precisely, these singularities
are defined using very basic topological characteristics of the graphs of the
equilibrium correspondences, so that they typically divide the space of games
into a finite number of connected components of regular or nonsingular games.
The equivalence classes are then defined such that all games within a component (as well as all games obtained from games in the component by applying
the usual symmetry operations of relabeling players and their strategies) are
considered to be equivalent and therefore identified with each other. This
leads to classifications of normal form games that through the geometries of
the equilibrium correspondences are naturally and essentially related to the
corresponding equilibrium concepts. The classifications obtained yield a way
of seeing which aspects of the normal form games the underlying equilibrium
concepts distinguish and which aspects they ignore. At the same time, they
provide natural organizations of the underlying spaces of games into welldefined equivalence classes, whereby, in order to keep the analysis as useful
and essential as possible, the topological characteristics inducing the equivalence classes are chosen so as to obtain as broad classes as possible. To a large
extent, this is done using transversality theory applied to the intersection of
the graphs of the equilibrium correspondences with the underlying spaces of
mixed strategies.
Not only does all this provide a better understanding of the underlying
equilibrium concepts and especially of the geometries of the corresponding
equilibrium correspondences, but it also provides a particularly transparent
way of comparing equilibrium concepts with each other. In fact, the (topological) procedure we propose to use to classify normal form games appears
to extend to many other equilibrium concepts beyond the three studied in
this paper. At the same time, the present approach to analyzing normal form
games also forces one to think about what the fundamental characteristics de1
scribing a strategic situation should be and why. Similar considerations have
received some attention recently for example in the experimental literature
(Crawford (1997), Roth and Erev (1995, 1997)) and in the literature on games
with incomplete information (Carlsson and van Damme (1993), Morris, Rob,
and Shin (1995)).
The paper is organized as follows. Section 2 introduces preliminary notation and definitions, in Section 3 the Nash equivalence classes are defined,
and Section 4 contains some general propositions for two player games as well
as classifications of small two person games using the Nash equilibrium, rationalizability, iterated dominance, and correlated equilibrium concepts. Finally,
Section 5 discusses related literature, and Section 6 concludes. The paper is to
be viewed only as a very first step towards understanding the rather complex
structures of the resulting sets of equivalence classes, with particular attention
to two person games.
2
Preliminary Notions
Let I = {1, . . . , n} denote the set of players, let Si denote player i’s strategy
space, Si the set of probability measures on Si , and let S = ×i∈I Si , S =
×i∈I Si , S−i = ×j∈I\{i} Sj , and S−i = ×j∈I\{i} Sj . Set Ki = #Si , K = Σi∈I Ki ,
κ = Πi∈I Ki , Li = {1, . . . , Ki }, and let fi : S → IR denote player i’s payoff
function and Fi : S → IR, σ 7→ Σs∈S ν(s, σ)fi (s), player i’s expected payoff
function, where ν : S × S → [0, 1], (s, σ) 7→ ν(s, σ), gives the probability of
the strategy profile s under the mixed strategy profile σ. A game in normal
form γ is defined as a pair (S, f ) or, alternatively, as a pair (S, F ) and we
say it is finite if n and each Ki , i ∈ I, are finite. In what follows, we consider
only finite normal form games and fix both the set of players I as well as the
space of strategy profiles S, so that we can identify a game γ with a point in
the Euclidean space IRκn and, accordingly, the space of games with the whole
Euclidean space IRκn . We also denote by γ i ∈ IRn the payoff array of player i.
Let γ ∈ IRκn be a finite normal form game, and let σ −i ∈ S−i be a strategy
profile of all players except i, then the set of best replies of player i to σ −i is
defined by:
BRi (γ, σ −i ) = {σ i ∈ Si : Fi (σ i , σ −i ) ≥ Fi (sik , σ −i ), ∀k ∈ Li },
and the best reply correspondence of player i is defined by:
BRi : IRκn × S−i → Si , (γ, σ−i ) 7→ BRi (γ, σ−i ).
2
We denote by βi ⊂ IRκn × S the graph of player i’s best reply correspondence.
The set of Nash equilibria of γ is defined by:
N E(γ) = {σ ∈ S : F (σ) ≥ F (sik , σ −i ), ∀k ∈ Li , ∀i ∈ I},
where a point in the set N E(γ) ⊂ S is a Nash equilibrium of the game γ,
and the Nash equilibrium correspondence is defined by:
N E : IRκn → S, γ 7→ N E(γ).
It maps to each game γ ∈ IRκn the corresponding set of Nash equilibria N E(γ).
Let ηN E ⊂ IRκn × S denote the graph of the Nash equilibrium correspondence.
Kohlberg and Mertens (1986), Theorem 1, p. 1021, show that ηN E is a manifold of dimension κn homeomorphic to the underlying space of games IRκn .1
We will need the following stronger result.
Lemma 1 For any nonempty subset of players J ⊂ I, the intersection of the
graphs of the best reply correspondences of the players in J is homeomorphic
to the product of the underlying space of games with the space of mixed strategy
profiles of the players not in J, i.e.,
∩j∈J βj ≈ IRκn × (×i∈I\J Si ).
P roof . The proof is very similar to the proof in Kohlberg and Mertens (1986),
p. 1021-1022. Let J = {1, . . . , l}, JC = {l + 1, . . . , n} and consider the
reparameterization of the payoffs of the players in J,
i
= Ais,t + ais , s ∈ Si , t ∈ S−i , i ∈ J,
γs,t
where Σt∈S−i Ais,t = 0, for all s ∈ Si , i ∈ J. Next, consider the following maps:
f : β1 ∩ · · · ∩ βl → IRκn × Sl+1 × · · · × Sn ,
i
j
Πj6=i σtjj ))i∈J
((γ i )i∈I , (σ i )i∈I ) 7→ (((Ais,t + (σsi + Σt∈S−i γs,t
s,t , (γ )j∈JC , (σj )j∈JC )
and
h : IRκn × Sl+1 × · · · × Sn → β1 ∩ · · · ∩ βl ,
j
i
j
((γ i )i∈I , (σ j )j∈JC ) 7→ ((Ais,t +gsi ((γ j )j∈J , (σ j )j∈JC ))i∈J
s,t , (γ )j∈JC , (πSi (a ))i∈J , (σ )j∈JC ),
1
We say E ⊂ IRp is a manifold of dimension n if every point in E has a neighborhood
homeomorphic to the Euclidean space IRn . See for example Greenberg and Harper (1981),
Ch. 6, p. 28.
3
where, for i ∈ J, πSi : IRKi → Si is the projection from IRKi onto the simplex
Si , and g i is defined by:
g i : IRκ × Sl+1 × · · · × Sn → IRKi ,
((γ i )i∈J , (σ j )j∈JC ) 7→ ((ais − πSi (ais ) − Σt∈S−i Ais,t Πj∈JC σtjj Πj∈J\{i} πSj (ajtj ))s ).
The maps f and h are homeomorphisms between IRκl × Sl+1 × · · · × Sn and
β1 ∩· · ·∩βl . While it is easy to see that both are continuous, it is more tedious,
although straightforward, to see that they are inverses of each other. 2
The notion of Nash equivalence that will be given in the next section relies heavily on a certain notion of regularity of a given game. The notion
of regularity is defined in terms of a transverse intersection of certain manifolds. The next definition introduces the notion of a (topologically) transverse
intersection.
Definition 1 Let E ⊂ IRp be a manifold of dimension n and let S, T ⊂ E
be submanifolds of dimensions s and t, where s + t ≥ n. We say S and
T intersect (topologically) transversally at a point x ∈ E if there is a
coordinate neighborhood U ⊂ E of x and a homeomorphism ζ : (U, U ∩ S, U ∩
≈
(S ∩ T ), U ∩ T ) → (IRn , IRs × {0}, {0} × IRs+t−n × {0}, {0} × IRt ) that maps U
to IRn , U ∩ S to IRs × {0}, U ∩ (S ∩ T ) to {0} × IRs+t−n × {0}, and U ∩ T to
{0} × IRt . We say S and T are (topologically) transverse if they intersect
transversally at each point x ∈ S ∩ T .2
Here are some standard definitions that will be useful later in the text.
Definition 2 Let (γ, σ) ∈ ηN E be a point on the graph of the Nash equilibrium correspondence, we say N E is locally continuous at (γ, σ) if there are
neighborhoods Uγ ⊂ IRκn of γ and Uσ ⊂ S of σ such that the correspondence
N E restricted to mapping from Uγ to Uσ is continuous at γ.
We say σ is an isolated Nash equilibrium of the game γ if there exists a
neighborhood Uσ ⊂ S of σ such that (γ, σ) = ({γ} × Uσ ) ∩ ηN E .
2
See for example Greenberg and Harper (1981), Ch. 31, p. 290-291. Notice that if E is
with boundary, then the coordinate neighborhoods U may also have a boundary, in which
case, if neither U ∩ S nor U ∩ T are contained in the boundary of U , then the homeorphism
≈
takes the form ζ : (U, U ∩ S, U ∩ (S ∩ T ), U ∩ T ) → (IH n , IH s × {0}, {0} × IH s+t−n × {0}, {0} ×
t
n
n
IH ), where IH ⊂ IR denotes the halfspace {x ∈ IRn |xn ≥ 0}. If, say, U ∩ S is contained
in the boundary of U , then IH s × {0} and {0} × IH s+t−n × {0} are replaced by IRs × {0}
and {0} × IRs+t−n × {0}.
4
We say σ is an essential Nash equilibrium of the game γ if for any arbitrarily small neighborhood Uσ of σ there exists a neighborhood Uγ of γ such
that ({γ 0 } × Uσ ) ∩ ηN E is nonempty, for all γ 0 ∈ Uγ .
Finally, we say σ is a locally unique Nash equilibrium of the game γ, if
there exists a neighborhood U ⊂ ηN E of (γ, σ) such that the projection map
π : IRκn × S → IRκn , (γ, σ) 7→ γ, restricted to U is a homeomorphism onto
π(U ).
The following relation follows immediately from either Kohlberg and Mertens
(1986), Theorem 1, p. 1021, or from Lemma 1 above, for J = I since ηN E =
∩i∈I βi .
Lemma 2 Let (γ, σ) ∈ ηN E be a point on the graph of the Nash equilibrium
correspondence, then N E is locally continuous at (γ, σ) if and only if σ is an
isolated and essential Nash equilibrium of γ.
In Section 4 we will consider further relations between the concepts defined in
this and the following section. In the next section, we introduce the notion of
Nash equivalence for generic games.
3
Nash Equivalence Classes
Generally speaking, the procedure we use to classify normal form games consists of two steps. In the first step, we use the equilibrium correspondence
of the underlying equilibrium concept to decompose the space of games into
typically finitely many connected components. This is done via basic topological characteristics of the equilibrium correspondences such as regularity in
the case of Nash equivalence, or the dimension of the set of correlated equilibria in the case of correlated equivalence. Since, by our very definition of the
equivalence classes, all games within a given component are considered to be
equivalent, this first step allows to reduce the infinite number of games to a
typically finite number.
In the second step, the connected components obtained in the first step are
further identified by identifying games that are obtained one from the other
by reordering either the strategies of the players or by reordering the players
themselves. This does not make use of any equilibrium concept but, rather,
of fundamental assumptions on the behavior of the players by which they all
act in the same way, selecting strategies that maximize payoffs regardless of
the labels attached to them and regardless of the labels attached to themselves as players in the game. This second step generally substantially reduces
the number of different connected components to be distinguished. After all
5
identifications are made, i.e., after the connected components of the first step
are identified according to the identifications of the second step, we associate
to each of the different sets of connected components an equivalence class of
games. Two games that belong to the same equivalence class are then said to
be equivalent. The classification, finally, is determined by the resulting set of
classes of equivalent games. Now we formally describe the procedure for the
case of Nash equilibrium. Before defining the notion of Nash equivalence, we
first introduce the notions of a regular Nash equilibrium and of a Nash regular
game.
Definition 3 Let γ ∈ IRκn be a finite game in normal form, we say a Nash
equilibrium σ ∈ N E(γ) is (topologically) regular if ηN E intersects {γ} × S
transversally at (γ, σ) ∈ ηN E . Furthermore, a game in normal form is said
to be Nash regular if all its Nash equilibria are (topologically) regular. It is
said to be Nash singular if it is not Nash regular.
The notion of (topologically) regular Nash equilibrium is related although,
generally speaking, weaker than the corresponding notion defined for example
in Harsanyi (1973), p. 241, and van Damme (1991), p. 38. More precisely, by
requiring certain full rank conditions on certain Jacobian matrices, Harsanyi
and van Damme’s definition requires that the intersection of ηN E with {γ 0 }×S
be transverse for all γ 0 in an arbitarily small neighborhood of γ, thus implying
local uniqueness of the given regular equilibrium. In contrast, our definition
requires the intersection of ηN E with the vertical space {γ 0 }×S to be transverse
only at γ 0 = γ, thereby allowing isolated equilibria to be regular even if the
correspondence N E is locally not a function.
Harsanyi (1973), Theorem 3, p. 249, also shows that the space of games
with only regular Nash equilibria in his sense is dense in the space of games,
and Blume and Zame (1994), Theorem 3, p. 792 and p. 785, strengthen this
result and show that the class of games with discontinuous Nash equilibrium
correspondence is contained in a closed, lower dimensional, semi-algebraic subset of the space of games, which divides the space of games into a finite number
of connected components. (See also Schanuel, Simon, and Zame (1991), p. 21.)
We will see in the next section how the Nash singular two person games coincide with games with discontinuous Nash equilibrium correspondence, which
shows that the number of components of Nash regular games will also be
finite. More generally, given the semialgebraic nature of all equilibrium correspondences considered in this paper, we can use the Generic Local Triviality
Theorem (see for example Blume and Zame (1994), p. 785, or Schanuel et al.
(1991), p. 20) to show that the number of maximal connected components of
6
equivalent games - in any of the senses to be defined below - will always be
finite.
The second part of the definition of Nash equivalence can be formalized
by means of certain maps, which we call symmetry operations, following Nash
(1951), p. 288-289, that identify games through relabeling of the players’
strategies and/or of the players themselves.3 More precisely, let Pa be the set
of all permutations pa : I → I satisfying Kpa (i) = Ki , i ∈ I, let Ps be the set
of all permutations ps = (pis : Si → Si )i∈I , and let:
p (i)
Ψa = {ψa : IRκn → IRκn |ψa (γ) = (γk a−1
pa (1)
,..,k
p−1
a (n)
)i∈I , pa ∈ Pa , γ ∈ IRκn },
Ψs = {ψs : IRκn → IRκn |ψs (γ) = (γpi 1s (k1 ),..,pns (kn ) )i∈I , ps ∈ Ps , γ ∈ IRκn }.
We say a map ψ : IRκn → IRκn is a symmetry operation within the class of
n-person K1 × · · · × Kn games, if ψ = ψa ψs for some ψa ∈ Ψa , ψs ∈ Ψs , and
denote by Ψ the set of all such symmetry operations. Notice that this space
depends on the number of players and the cardinality of the strategy spaces,
i.e., on K1 , . . . , Kn , which, to save notation, we often leave out. The maps
pa ∈ Pa , ps ∈ Ps correspond respectively to relabeling of the players and of the
players’ strategies. The maps ψa ∈ Ψa , ψs ∈ Ψs are the maps induced on the
space of games by corresponding maps pa and ps .
Lemma 3 The space Ψ with the composition operation forms a subgroup of
the group of linear maps A : IRκn → IRκn with determinant + or −1.
P roof . Clearly, the space of linear maps A : IRκn → IRκn with determinant
+ or −1 forms a group with the usual composition of matrices. We need to
show, that the elements of Ψ are indeed matrices with determinant + or −1,
and that they form a subgroup of the space of all such linear maps.
(i) Since the maps pa , ps are permutations, the induced maps ψa , ψs are
also permutations of the entries of the given game γ. Hence they are linear
maps representable by (κn) × (κn) matrices that moreover have, for every row
and column, exactly one entry equal to 1 and all other entries equal to 0.
This readily implies that they are invertible and have determinant + or −1.
Clearly, all these properties carry over when composing ψa , ψs .
(ii) To see that Ψ contains the unit element, notice that taking pa , ps
to be the identities, one obtains ψa , ψs , and hence also ψ = ψa ψs to be the
identity matrix. To see that ϕψ ∈ Ψ whenever ϕ, ψ ∈ Ψ, notice that, for every
3
For a thorough treatment of such symmetries within a more general class of games, the
reader is referred to Mertens (1987) and the notions of ordinality defined there.
7
ψa ∈ Ψa , ψs ∈ Ψs , there exists ψ̃s ∈ Ψs such that ψs ψa = ψa ψ̃s . (Take ψ̃s to
p−1 (i)
be the map induced by p̃s = (ps a
)i∈I .) Then we have:
ϕψ = (ϕa ϕs )(ψa ψs ) = ϕa (ϕs ψa )ψa = ϕa (ψa ϕ̃s )ψa
= (ϕa ψa )(ϕ̃s ψa ) = ψ̃a ψ̃s ∈ Ψ,
for some ψ̃a ∈ Ψa , ψ̃s ∈ Ψs . Finally, to see that each element ψ ∈ Ψ has an
inverse element ψ −1 ∈ Ψ, notice that, if ψ = ψa ψs , then ψ −1 = ψs−1 ψa−1 is
such an inverse that is contained in Ψ since ψs−1 ψa−1 = ψa−1 ψ̃s and ψa−1 ∈ Ψa ,
ψ̃s ∈ Ψs . 2
Next we introduce the notion of Nash equivalence.
Definition 4 Let IRκn be the space of K1 ×· · ·×Kn games in normal form with
n players, we say the games γ0 , γ1 ∈ IRκn are Nash equivalent, which we
denote by γ0 ∼N E γ1 , if there exists a symmetry operation ψ ∈ Ψ(K1 , . . . , Kn ),
and a continuous path υ : [0, 1] → IRκn , with υ(0) = γ0 and υ(1) = ψ(γ1 ),
such that, for all γ ∈ υ([0, 1]), γ is Nash regular.
Notice that since we require all games along the path υ to be Nash regular,
including the games at the endpoints, γ0 , γ1 , we in fact define Nash equivalence
only for Nash regular games. The following shows that this indeed defines an
equivalence relation.
Lemma 4 The relation ∼N E defines an equivalence relation on the space of
Nash regular games in IRκn .
P roof . We need to show that ∼N E is reflexive, symmetric, and transitive.
(i) ∼N E reflexive: take the identity map in Ψ and the constant path,
υ : [0, 1] → IRκn , t 7→ γ, and it follows that γ ∼N E γ for any Nash regular
game γ ∈ IRκn .
(ii) ∼N E symmetric. Let γ0 , γ1 ∈ IRκn and suppose γ0 ∼ γ1 , then there
exist ψ ∈ Ψ, υ : [0, 1] → IRκn , with υ(0) = γ0 and υ(1) = ψ(γ1 ), such that, for
all γ ∈ υ([0, 1]), γ is Nash regular. Let ψ̃ = ψ −1 ∈ Ψ be the inverse of ψ, and
consider the path υ̃ : [0, 1] → IRκn , t 7→ ψ̃(υ(1 − t)), then υ̃(0) = ψ̃(υ(1)) =
ψ −1 (ψ(γ1 ) = γ1 and υ̃(1) = ψ̃(υ(0)) = ψ̃(γ0 ). Moreover, since all games in
υ([0, 1]) are Nash regular so must be all games in υ̃([0, 1]) since applying a
symmetry operation to a game certainly does not affect whether the game is
Nash regular or not.
(iii) ∼N E transitive. let γi ∈ IRκn , i = 0, 1, 2 and suppose γ0 ∼N E γ1
and γ1 ∼N E γ2 , then there exist ψi ∈ Ψ, υi : [0, 1] → IRκn , with υi (0) = γi ,
8
υi (1) = ψi (γi+1 ), i = 0, 1, such that, γ is Nash regular, for all γ ∈ υi ([0, 1]),
i = 0, 1. Let ψ = ψ0 ψ1 ∈ Ψ, and consider the path:
(
υ : [0, 1] → IR , t 7→
κn
υ0 (2t),
0 ≤ t ≤ 1/2
.
ψ0 (υ1 (2t − 1), 1/2 ≤ t ≤ 1
Then we have, υ(0) = υ0 (0) = γ0 , υ(1/2) = υ0 (1) = ψ0 (γ1 ) = ψ0 (υ1 (0)) and
υ(1) = ψ0 (υ1 (1)) = ψ0 ψ1 (γ2 ) = ψ(γ2 ), where again all games along the path
are Nash regular. This implies γ0 ∼N E γ2 and completes the proof. 2
The following, stronger notion of regularity is useful when computing Nash
equivalence classes.
Definition 5 Let βi ⊂ IRκn × S be the graph of player i’s best reply correspondence, i ∈ I, let γ ∈ IRκn be a finite game in normal form, we say the
game γ is regular if, for all J ⊂ I, JC ⊂ I \ J, and Ti ⊂ Si for i ∈ JC ,
(∩j∈J βj ) intersects {γ} × (∩i∈JC ,ti ∈Ti {σ ∈ S|σtii = 0}) transversally. We say
it is singular if it is not regular.
Roughly, a game γ is regular if the graphs of the best reply correspondences of
all players as well as all their intersections are transverse or in general position
with respect to all the possible faces and intersections of faces of the space of
mixed strategies S. This is a particularly strong notion of regularity, which,
in Section 4, will be seen to have a particularly nice characterization for two
player games.
The following lemma gives an idea of some of the implications of requiring
tranversal intersections of graphs of best reply correspondences with the corresponding spaces of mixed strategies. First, recall that a strategy sik ∈ Si of
player i is said to be redundant if there exists a convex combination of the
other strategies of player i that leads to precisely the same payoffs for player
i at all possible strategy profiles s−i ∈ S−i of the other players. A strategy
sik ∈ Si of player i is said to be weakly dominated if there exists a convex
combination of the other strategies of player i that leads to payoffs for player
i at least as large as all payoffs of strategy sik at all possible strategy profiles
s−i ∈ S−i of the other players and strictly larger for at least one strategy
profile s−i ∈ S−i of the other players, and it is said to be strictly dominated if there exists a convex combination of the other strategies of player i
that leads to payoffs for player i that are strictly larger for all strategy profiles s−i ∈ S−i of the other players. The following is a characterization of
transversal intersections of βi with {γ} × S.
9
Lemma 5 Let γ ∈ IRκn be a finite game in normal form. Then, for i ∈ I, βi
is transverse to {γ} × S if and only if, at γ, player i has neither redundant
nor weakly dominated strategies that are not strictly dominated.
P roof . (⇒): Suppose sik ∈ Si is redundant or weakly but not strictly dominated, then there exists some mixed strategy profile σ −i of the other players
that makes player i indifferent between strategy k and some other mixed strategy σ i ∈ Si that is either payoff equivalent to or weakly dominates sik . This
implies that sik is a best reply to σ −i . Consider the perturbation of γ that
decreases the payoffs of player i at all entries of strategy k against the strategy
profiles in the support of σ −i by some arbitrarily small amount. Then it is
clear that, at any such arbitrarily close game, sik is not a best response to
neither σ −i nor to any mixed strategy profile close to σ −i . But this implies
that βi cannot be transverse to {γ} × S at (γ, (sik , σ −i )).
(⇐): Suppose that, at γ, player i has neither redundant nor weakly dominated strategies that are not strictly dominated. Then for each strategy of
player i there exists at least one mixed strategy profile σ −i ∈ S−i of the other
agents such that player i’s payoff of playing sik against σ −i is strictly greater
than that of any other strategy in Si . In view of Lemma 1 and since the set
of best replies to any mixed strategy profile σ −i ∈ S−i of the other agents is
always a face of the simplex {γ} × Si × {σ −i }, it suffices to show that, for each
point (γ, σ) ∈ βi ∩ ({γ} × S), there exists, for each γ 0 arbitrarily close to γ, a
point (γ 0 , σ 0 ) ∈ βi ∩({γ 0 }×S) arbitrarily close to (γ, σ). In particular, this will
show that at any γ 0 arbitrarily close to γ there are strategy profiles of the other
0
players σ −i that are arbitrarily close to σ −i such that player i’s set of best
0
replies to σ −i at γ 0 is a simplex of the same dimension as the one against σ −i
at γ. Pick (γ, σ) ∈ βi ∩ ({γ} × S), and let γ 0 be any game arbitrarily close to
γ. Since i has neither redundant nor weakly dominated strategies that are not
strictly dominated, there exists, for each sik in the support of σ i at least one
mixed strategy τ −i (k) ∈ S−i such that F (sik , τ −i (k)) > F (sil , τ −i (k)), l 6= k.
Moreover, each of the vectors τ −i (k) can be chosen from an open subset of S−i
therefore providing sufficiently many directions in which to adjust (if neces0
sary) the other agents’ mixed strategy from σ −i to σ −i so as to make player i
indifferent between all strategies in the support of σ i against the new strategy
0
profile σ −i at the perturbed game γ 0 . This then shows that the set of pure
0
best responses against σ −i at γ 0 coincides with the set of pure best responses
to σ −i at γ. In particular, there exists, for each γ 0 arbitrarily close to γ, a
point (γ 0 , σ 0 ) ∈ βi ∩ ({γ 0 } × S) arbitrarily close to (γ, σ), which concludes the
proof of the lemma. 2
10
Corollary 1 Let γ ∈ IRκn be a finite game in normal form, then, if γ is
regular, it is also Nash regular, and there are neither redundant nor weakly
dominated strategies that are not strictly dominated.
P roof . This follows immediately from the definitions and Lemma 5 above,
after noticing that ηN E = ∩i∈I βi . 2
This implies that the set of singular games divides the space of games into at
least as many components as does the set of Nash singular games. Further
characterizations of the notions of regularity and Nash regularity will be given
in the next section.
4
Two Person Games
In this section we consider two person normal form games. To save on notation,
we write m = K1 , n = K2 , and κ = mn. Occasionally, we will also describe
two person games by means of two m × n matrices A and B that denote the
payoffs respectively of players 1 and 2. Before considering the general case,
we first consider some simple examples.
4.1
2 × 2 Games
The space of 2 × 2 two person games which we identify with IR8 is divided by
the singular 2 × 2 games into 16 connected components of regular games. By
Corollary 1 the set of Nash singular games is contained in the set of singular
games, so that all games within any given one of the 16 connected components
are Nash equivalent. Applying the symmetry operations of relabeling the
players’ strategies further identifies some of the 16 components and leads to 5
classes of Nash regular games, where all games within a given class are Nash
equivalent. The 5 classes can be unambiguously represented by the following
games:
Ã
γ1 =
(1, 0) (0, 1)
(0, 1) (1, 0)
Ã
γ20
=
!
Ã
, γ2 =
(1, 1) (1, 0)
(0, 1) (0, 0)
(1, 1) (1, 0)
(0, 0) (0, 1)
!
Ã
, γ200
=
!
Ã
, γ3 =
(1, 1) (0, 0)
(0, 0) (1, 1)
(1, 1) (0, 0)
(0, 1) (1, 0)
!
,
!
.
Applying the symmetry operation of relabeling the players allows to further
identify γ20 and γ200 . Moreover, it is easy to see that the games γ2 , γ20 , γ200 can all
be connected by paths of Nash regular games, so that they are in fact all Nash
11
equivalent, i.e., [γ2 ] = [γ20 ] = [γ200 ], where [γ] denotes the Nash equivalence
class containing the game γ. This shows that the space of Nash regular 2 × 2
games can be decomposed into the three Nash equivalence classes [γ1 ], [γ2 ],
and [γ3 ]. As one may expect, these are precisely the Nash regular games with
only one mixed strategy Nash equilibrium, [γ1 ], with only one pure strategy
equilibrium, [γ2 ], and with one mixed and two pure strategy equilibria, [γ3 ].
Examples of games in the different classes are the matching pennies games
for [γ1 ], the prisonners’ dilemma games for [γ2 ], and games like battle of the
sexes, pure coordination games, and chicken for [γ3 ].
4.2
2 × 3 Games
Similarly, it can be shown that the space of 2 × 3 two person games can be
decomposed into five Nash equivalence classes of Nash regular games of which
three are essentially represented by γ1 , γ2 , and γ3 , (after adding a strictly
dominated strategy for player 2), and that the other two classes, which are
not present among the 2 × 2 games, have representatives:
Ã
γ4 =
(1, 3) (0, 2) (1, 0)
(0, 0) (1, 2) (0, 3)
!
Ã
, γ5 =
(1, 3) (1, 2) (0, 0)
(0, 0) (0, 2) (1, 3)
!
.
These are representatives of Nash regular games with one pure strategy and
two mixed strategy equilibria, [γ4 ], and with one mixed strategy and two pure
strategy equilibria, [γ5 ], where all pure strategy profiles are in the support of
at least one of the Nash equilibria.
Notice that within the 2×2 and the 2×3 Nash regular games, knowing the
equilibrium distributions of the Nash equilibria is sufficient to place the game
unambiguously within its Nash equivalence class. This already fails with the
3 × 3 games.
4.3
3 × 3 Games
It can also be shown, and this is quite tedious, that the space of 3 × 3 two
person games, which we identify with IR18 , can be decomposed into 32 Nash
equivalence classes of Nash regular games of which five are essentially represented by γ1 , . . . , γ5 , (again, after adding a strictly dominated strategy for
player 1), and that the other 27 can be found in the appendix.4
4
To give an idea of the difficulties that arise in performing the complete classification
already for the 3 × 3 games, notice that while in the 2 × 2 case the singular games divided
the space of games into 16 connected components of regular games, in the 3 × 3 case the
corresponding number of connected components is well over 100,000.
12
As mentioned above, with 3×3 games it can happen that two Nash regular
games that have exactly the same Nash equilibria need not be Nash equivalent.
This is illustrated by the following pair of games.




(5, 3) (0, 0) (5, 2)
(5, 3) (0, 0) (5, 2)




γ6 =  (4, 3) (2, 2) (2, 0)  , γ7 =  (0, 3) (5, 2) (0, 0)  .
(0, 0) (5, 2) (0, 3)
(4, 0) (2, 2) (2, 3)
Both games have the same Nash equilibria, namely the pure strategy equilibrium, where both players play strategy 1, the mixed strategy equilibrium
where players 1 and 2 mix between respectively strategies 2 and 3, and 1 and
2, and, finally, the completely mixed equilibrium. To understand why the two
games are not Nash equivalent, we point out that the indices of the two mixed
strategy equilibria are reversed in the two games, where, following Shapley
(1974), p. 184, we define the index of a regular Nash equilibrium σ ∈ S of a
two person game γ = (A, B), as the number (+ or −1) given by:
ind(σ, γ) = (−1)k+1 sgn(det(A0 )det(B 0 )),
where det and sgn denote respectively the determinant and the sign, and A0 , B 0
are the k × k submatrices of A, B that are obtained from A, B by deleting all
rows and columns that do not correspond to strategies that are in the support
of σ, and possibly also by adding a positive constant to all payoffs so as to make
all entries in A0 , B 0 strictly positive. It is easily verified that the completely
mixed strategy equilibrium, in γ6 for example, has index +1, while it has index
−1 in γ7 , and that the other mixed strategy equilibrium has index −1 in γ6
and +1 in γ7 .
4.4
Some General Properties
In this section, we consider general finite two person games and derive, through
some lemmas and propositions, a necessary condition for two games to be Nash
equivalent. The following is a characterization of regular Nash equilibria in
two person games.
Theorem 1 Let γ ∈ IR2κ be a two person game in normal form and let σ ∈
N E(γ) be a Nash equilibrium of γ. Then the following are equivalent:
(a) σ is locally unique,
(b) σ is regular,
(c) N E is locally continuous at (γ, σ).
13
P roof . We show (a) ⇒ (b) ⇒ (c) ⇒ (a).
(a) ⇒ (b): Suppose σ is locally unique. Then the projection map π :
IR2κ × S → IR2κ , (γ, σ) 7→ γ, restricted to U is a homeomorphism onto π(U ).
This readily implies that ηN E is transverse to {γ} × S at (γ, σ) since the
product of the projection maps from a coordinate neighborhood of (γ, σ) to
IR2κ and to S essentially yield the desired homeomorphism.
(b) ⇒ (c): Let Uγ ⊂ IR2κ and Uσ ⊂ S be as in Definition 2. If σ is regular,
then, in particular, N E restricted to map from Uγ to Uσ will be single-valued
at γ, and the homeomorphism ζ from the definition of regularity yields the
rest.
(c) ⇒ (a): This follows directly from van Damme (1991), Theorem 3.4.4, p.
55, and Lemma 2 above. (Notice that, in van Damme’s terminology, strongly
stable corresponds to what we call locally unique.) 2
This characterization should not be surprising in view of the above mentioned theorem of van Damme. In particular, it shows that, when there are just
two players, the Nash equilibrium correspondence is sufficiently well-behaved
that the three notions, local uniqueness, (topological) regularity, and local
continuity, that are different in general, in fact coincide for N E. While the
implications (a) ⇒ (b) ⇒ (c) continue to hold for more than two players, (for
the same reasons given in the proofs above), this need not be the case for the
reverse directions.
An immediate consequence of Theorem 1 is the following.
Corollary 2 Let γ ∈ IR2κ be a two person game in normal form, then if γ is
Nash regular, all its Nash equilibria are locally unique.
This implies that the classification of Nash equivalence classes for generic
two person games does not change if one requires games along the paths υ
of Definition 4 to have all equilibria locally unique in addition to being Nash
regular. The following theorem gives a characterization of regular games,
which form a subset of the Nash regular games. The characterization is useful
for the computation of Nash equivalence classes.
Theorem 2 Let γ ∈ IR2κ be a two person game in normal form. Then the
following are equivalent:
(a) γ is regular,
(b) βi is transverse to {γ} × (∩t∈Tj {σ ∈ S|σtj = 0}), for all Tj ⊂ Sj , j 6= i,
and for i = 1, 2.
14
(c) any mixed strategy x ∈ S1 of player 1 has at most |supp(x)| pure best
responses, and the same holds for any mixed strategy of player 2.
P roof . We show (a) ⇒ (b) ⇒ (c) ⇒ (a).
(a) ⇒ (b): This follows directly from the definitions.
(b) ⇒ (c): We show ¬(c) ⇒ ¬(b). We make use of the following well-known
lemma. (See for example von Stengel (1996), Theorem 2.4, p. 9.)
Lemma 6 Let γ = (A, B) ∈ IR2mn be a two person m × n game in normal
form, and let y ∈ S2 , then x ∈ IRm is a best reply to y if and only if there
exist u ∈ IR such that,5
1Tm x = 1
(1)
1m u − Ay ≥ 0
(2)
x (1m u − Ay) = 0
(3)
x ≥ 0.
(4)
T
Suppose (c) does not hold, i.e., suppose that player 1 has more best replies to
y than there are strategies in the support of y. We show that β1 cannot be
transverse to {γ}×(∩t∈T2 {σ ∈ S|σt2 = 0}), for some T2 ⊂ S2 . Suppose without
loss that exactly the first s inequalities of (2) hold with equality and suppose
that the first r strategies of player 2 are exactly the strategies in the support
of y. The set of best replies to y forms a simplex of dimension s. Moreover,
if player 1 has more pure best responses to y than there are strategies in the
support of y, then s > r. Let T2 = {s2r+1 , . . . , s2n }, then we claim that if s > r,
β1 cannot be transverse to {γ} × (∩t∈T2 {σ ∈ S|σt2 = 0}). To see this, notice
that, by Lemma 1, the dimension of β1 is 2mn + n − 1. Also, the dimensions of
IR2mn × S and {γ} × (∩t∈T2 {σ ∈ S|σt2 = 0}) are respectively 2mn + m + n − 2
and m + r − 1(= (m + n − 2) − (n − r − 1)). Now, recall that the set of vectors
y 0 ∈ IRn that satisfy (2.1)-(2.s) for some value u0 ∈ IR form a linear subspace
of IRn , which for generic A has dimension n − s. Requiring 1Tn y = 1 leads to
a further linear subspace, which for generic A has dimension n − s − 1, and,
finally, requiring y ≥ 0 leads to a subset of S2 which again for generic A has
dimension n − s − 1 and moreover has exactly s nonzero entries. Therefore,
for generic A, the dimension of β1 ∩({γ} × (∩t∈T2 {σ ∈ S|σt2 = 0})) is equal to
r(= (2mn + n − 1) + (m + r − 1) − (2mn + m + n − 2)). But, if at the game
γ, player 1 has s pure best responses to y ∈ ∩t∈T2 {σ ∈ S|σt2 = 0}), then the
5
A superscript
T
denotes the transpose, and 1r ∈ IRr stands for the r-vector of ones.
15
latter number will be s. Since s > r, this shows that β1 cannot be transverse
to {γ} × (∩t∈T2 {σ ∈ S|σt2 = 0}), and hence (b) cannot hold.
(c) ⇒ (a): Suppose (c) holds. Pick (γ, σ) ∈ β1 ∩ β2 , then (c) implies
that σ ∈ S is uniquely determined by linearly independent equations, (see von
Stengel (1996), Theorem 2.7, p. 20), which readily implies β1 ∩β2 is transverse
to {γ}×S. Next we show that β1 is transverse to {γ}×(∩t∈T2 {σ ∈ S|σt2 = 0})
for all T2 ⊂ S2 . Fix T2 ⊂ S2 and pick (γ, σ) ∈ β1 ∩ ({γ} × (∩t∈T2 {σ ∈ S|σt2 =
0})). As in the proof of Lemma 5, it suffices to show that, for all γ 0 arbitrarily
close to γ there exists (γ 0 , σ 0 ) ∈ β1 ∩ ({γ} × (∩t∈T2 {σ ∈ S|σt2 = 0})) arbitrarily
close to (γ, σ). Again, this will show that for all γ 0 close to γ there exists
0
σ 2 ∈ ∩t∈T2 {σ ∈ S|σt2 = 0} arbitrarily close to σ 2 such that player 1’s set of
best replies is a simplex of the same dimension as the one against σ 2 at γ. By
(c), player 1 has at most |supp(σ 2 )| pure best responses against σ 2 at γ, which
0
in view of Lemma 6 readily implies that there exists σ 2 ∈ ∩t∈T2 {σ ∈ S|σt2 = 0}
0
such that player 1 has the same best responses against σ 2 at γ 0 as against
σ 2 at γ. This in turn implies that for all γ 0 arbitrarily close to γ there exists
(γ 0 , σ 0 ) ∈ β1 ∩ ({γ} × (∩t∈T2 {σ ∈ S|σt2 = 0})) arbitrarily close to (γ, σ). The
same argument applies also for player 2 and β2 and therefore concludes the
proof of the theorem. 2
The interpretation of (a) ⇔ (b) means in particular that, in order to verify
regularity of a two person game, it suffices to verify that the payoff matrices
A, B yield best-replies that are transverse to {γ} × (∩t∈Tj {σ ∈ S|σtj = 0}),
for all Tj ⊂ Sj , j 6= i, individually for i = 1, 2. In other words, checking
transversality of β1 , β2 with respect to the intersections of faces of S automatically guarantees that the intersection β1 ∩ β2 will be transverse to {γ} × S.
This property no longer holds with more than two players. There, one cannot verify regularity by considering the payoff matrices of the different players
independently of each other. The equivalence with condition (c) strengthens
this result and gives a particularly operational characterization of regularity
for the two player case.6
Our next theorem provides a necessary condition for two games to be Nash
equivalent. Its proof makes use of the following lemma.
Lemma 7 Let γ ∈ IR2κ be a two person game in normal form, and (γ, σ) ∈
ηN E . If N E is locally continuous at (γ, σ), then there exists a neighborhood
U ⊂ ηN E of (γ, σ) such that for any two points (γ0 , σ0 ), (γ1 , σ1 ) ∈ U , we have,
supp(σ0 ) = supp(σ1 ).
6
For more on this notion of regularity with two players, the reader is referred to von
Stengel (1996), Section 2.6, p. 19-25.
16
P roof . From van Damme (1991), Theorem 3.4.4, p. 55, and Lemma 2 above,
it follows that, if N E is locally continuous at (γ, σ), then every player has
exactly as many best replies to σ i as there are strategies in the support of σ i ,
i = 1, 2. (In van Damme’s terminology, an equilibrium with such a property
is called quasi strict.) But, if such a property holds at γ, it must also hold
in a sufficiently small neighborhood of γ, and since σ is isolated, it must also
hold in a neighborhood of (γ, σ). 2
Theorem 3 Let γ0 , γ1 ∈ IR2κ be two Nash regular two person games in normal form. If γ0 is Nash equivalent to γ1 , then there exist a symmetry operation
ψ ∈ Ψ(m, n) and a support and index preserving bijection from the set of Nash
equilibria of γ0 to the set of Nash equilibria of ψ(γ1 ).
P roof . If γ0 and γ1 are Nash regular and γ0 ∼N E γ1 , then γ0 , γ1 , and ψ 0 (γ1 ),
for any ψ 0 ∈ Ψ(m, n), must have the same (finite) number of Nash equilibria.
Hence it is easy to see that there exists a bijection between N E(γ0 ) and
N E(ψ 0 (γ1 )), for any ψ 0 ∈ Ψ(m, n). To see that there must exist a bijection that
preserves the supports of the elements of N E(γ0 ), N E(ψ(γ1 )), for some ψ ∈
Ψ(m, n), notice that otherwise it would not be possible to join the games γ0
and ψ(γ1 ) by means of a path of Nash regular games. This follows immediately
from Lemma 7, since it would not be possible to join them by means of a path
of games along which N E is locally continuous at all Nash equilibria, and by
Theorem 1, which implies that local continuity at all Nash equilibria of a given
game is equivalent to Nash regularity of the game. Similarly, if, in addition,
one could not require the bijection to preserve the indices of all Nash equilibria,
it would also not be possible to join the games γ0 and ψ(γ1 ) by means of a
path of Nash regular games, since all indices are invariant (individually) along
such a path. This completes the proof of the theorem. 2
The converse of the theorem is false, as the following pair of 3 × 4 games
show:


(4, 3)
(3, 4) (−3, −2) (−3, −3)


(4, 3)
(1, 2)
(−1, −8)  ,
γ8 =  (3, 4)
(−6, −2) (0, 0)
(5, 1)
(5, 3)


(3, 4)
(4, 3)
(−3, −2) (−3, −3)


(3, 4)
(1, 2)
(−1, −8)  .
γ9 =  (4, 3)
(−6, −2) (−1, −1)
(5, 1)
(5, 3)
Both games have the same Nash equilibria, namely the pure strategy equilibrium where the players play respectively strategies 3 and 4, and the two mixed
17
strategy equilibria where both players mix between the strategies 1 and 2, and
2 and 3. Not only do the equilibria of the two games coincide, but the indices
of the equilibria across the games coincide as well. Nonetheless, it is possible
to show that the two games are not Nash equivalent. To see why, observe that,
in some sense, the orientation of the mixed strategy equilibrium where both
players mix over strategies 1 and 2 is reversed across games, while all other
entries are essentially unchanged. As a consequence it is no longer possible
to join the two games by means of a path of Nash regular games. Applying
symmetry operations to the games does not make any difference.
This counterexample raises the question of whether and why the games
γ8 and γ9 should be considered as strategically different, (especially from the
viewpoint of Nash equilibrium). For example one can conceive of equivalence
classes where the necessary condition of Theorem 3 is also sufficient for two
games to be equivalent. We leave this as well as the question of what is a
sufficient condition for Nash equivalence (as in Definition 4) for future research.
The theorem above also does not extend to more than two players. This
is illustrated by the following example, which is a slight variant of the three
person 2 × 2 × 2 game given in Kojima, Okada, and Shindoh (1985), p. 662.
ÃÃ
γ10 =
(1, 1, 1) (2, 1, 0)
(2, 2, 1) (0, 3, 1)
! Ã
(1, 1, 1) (5, 0, 2)
(3, 2, 0) (3, 3, 0)
!!
.
The game has a unique Nash equilibrium σ = ((.5, .5), (.5, .5), (0, 1)), which can
be shown to be locally unique and hence also regular and locally continuous.
However, it is not the case that all equilibria close by have the same support
as σ. In particular, there are games arbitrarily close to γ10 with a unique
completely mixed equilibrium arbitrarily close to σ. Therefore, since such close
by games will be Nash equivalent, this would violate the necessary condition
of Theorem 3 as well as Lemma 7 showing that these do not extend to more
than two players.
Next, we consider the relationship of the Nash equivalence classes with
equivalence classes induced by other equilibrium concepts for simple classes of
games.
4.5
Other Equilibrium Concepts
The analysis carried out in the previous sections for the Nash equilibrium
concept can be extended to other equilibrium concepts. In this section, we
define equivalence classes for the concepts of rationalizability, iterated strict
dominance, and correlated equilibrium. The reader is referred to, for exam18
ple, Fudenberg and Tirole (1991), Ch. 2, p. 45-59, for definitions of these
equilibrium concepts.
4.5.1
Rationalizability and Iterated Strict Dominance
The cases of rationalizability and iterated strict dominance are straightforward. A natural definition of equivalence with respect to these concepts is to
say that two n-person games γ0 , γ1 ∈ IRκn are equivalent, if, for each player,
the number of either the rationalizable strategies or of the strategies that remain after all strictly dominated strategies have been eliminated, coincides
across the games γ0 and γ1 . As one expects, equivalence in this sense amounts
to simply counting, for each player, the number of either rationalizable strategies or of strategies that survive iterated strict dominance. By considering
the games at which these numbers change, one obtains a subset of the singular games that also divides the space of games into finitely many connected
components.
In the case of the Nash regular 2 × 2 games, the equivalence classes with
respect to iterated strict dominance are the two classes of games consisting
of the games with respectively one and two undominated strategies for each
player. In particular, unlike the case of Nash equivalence, equivalence classes
with respect to rationalizability or iterated strict dominance do not distinguish
between matching pennies and coordination games. (Notice that the latter two
concepts coincide for two player games, see for example Fudenberg and Tirole
(1991), Theorem 2.2, p. 51.)
However, one should not deduce from the 2 × 2 games nor from the 2 × 3
games that two games that are Nash equivalent will also be equivalent with
respect to rationalizability or iterated strict dominance. This is illustrated by
the following pair of 3 × 3 games.

γ11



(5, 3) (5, −1) (0, 0)
(5, 3) (5, 2) (0, 0)




=  (2, 2) (2, −1) (4, 3)  , γ12 =  (2, 2) (2, 0) (4, 3)  .
(0, 0) (0, −1) (5, 3)
(0, 0) (0, 2) (5, 3)
Both games have three Nash equilibria, two pure where both players play
strategies 1 and 3, and a mixed strategy equilibrium where they mix respectively between strategies 1 and 2 and 1 and 3. Notice that in neither of the
games is player 2’s strategy 2 used, and, in γ11 it is actually strictly dominated.
While moving from game γ11 to γ12 does not affect the Nash equilibrium correspondence, the set of rationalizable strategies goes from containing only
strategies 1 and 3 for player 2 at γ11 to containing strategies 1, 2, and 3 at
γ12 .
19
The following captures a basic link between the Nash and the rationalizable
or the iterated strict dominance equivalence classes.
Lemma 8 Let γ ∈ IR2κ be a two person game in normal form. Then, if γ
is regular, the set of rationalizable strategies or of strategies surviving iterated
strict dominance is locally constant. In fact, if γ ∈ IRκn is a regular n-person
game then the set of strategies surviving iterated strict dominance is locally
constant.
P roof . Consider the set strategies surviving iterated strict domonance at
γ and denote this set by U = U1 × U2 . Suppose moreover that this set changes
at γ. Then there exists at least one player, say 1, who has a strategy that is
either redundant or weakly but not strictly dominated within U . But then,
by Lemma 5 this implies that β1 restricted to IR2κ × U is not transverse to
{γ} × U, where U is the set of probability measures on U . Hence β1 is not
transverse to {γ} × S1 × ({0} × U2 ), which is equal to {γ} × (∩t∈S2 \U2 {σ ∈
S|σt2 = 0}). By definition of regularity this implies that γ cannot be regular.
For rationalizability, use the fact mentioned above that, for two players, the
set of rationalizable strategies coincides with the set of strategies surviving
iterated strict dominance. Finally, the case of iterated strict dominance where
γ is an n-person game is analogous to the two person case. 2
4.5.2
Correlated Equilibria
The case of correlated equilibria is somewhat less straightforward than the
cases of rationalizability and iterated strict dominance. If we view a correlated
equilibrium of a given n-person game γ ∈ IRκn as a probability measure over
the set of pure strategy profiles S, which unlike with the Nash equilibria, we
do not restrict to being a product measure, then we can view the correlated
equilibrium correspondence ηCE as a subset of IRκn × ∆(S), where ∆(S) is
the set of all probability measures on S. Moreover, since the set of correlated
equilibria CE(γ) ⊂ ∆(S) of any given game γ ∈ IRκn is a compact, convex
polyhedron described by a finite number of linear inequalities, it has a welldefined dimension. We use this to define a notion of equivalence with respect
to correlated equilibrium.
Definition 6 Let IRκn be the space of K1 × · · · × Kn games in normal form
with n players, we say the games γ0 , γ1 ∈ IRκn are correlated equivalent,
which we denote by γ0 ∼CE γ1 , if there exists a symmetry operation ψ ∈
Ψ(K1 , . . . , Kn ), and a continuous path υ : [0, 1] → IRκn , with υ(0) = γ0 and
υ(1) = ψ(γ1 ), such that, for all γ ∈ υ([0, 1]), the dimension of the set of
correlated equilibria CE(γ) is constant.
20
Notice that unlike the definition of Nash equivalence, the notion of correlated
equivalence is defined over all games, i.e., not just over the Nash regular ones.
Nonetheless, we will continue to focus only on Nash regular games. An alternative definition would have been to require the set of correlated equilibria of
all games along the path υ to have the same number of vertices rather than
the same dimension. However, we believe that this would lead to too many
different equivalence classes to be distinguished and we do not pursue the idea
further. (For example, the sets of correlated equilibria of the 3 × 3 games γ6
and γ7 have respectively 170 and 107 vertices, while the dimension of the sets
is 8 in both cases.)
With the above definition of correlated equivalence, it can be shown that
the space of Nash regular 2 × 2 games is decomposed into exactly the same
equivalence classes using corelated equivalence as using Nash equivalence.
That is, one obtains again the three representative games γ1 , γ2 , and γ3 , that
are representative of the same classes of games, i.e., [γi ]CE = [γi ], for i = 1, 2, 3,
where [γ]CE denotes the correlated equivalence class that contains the game
γ. To see that γ1 and γ2 represent different correlated equivalence classes although the sets of correlated equilibria are singletons in both cases, we point
out that there is no way to join the two games without passing through games
whose set of Nash equilibria and hence also of correlated equilibria has dimension at least 1.
With the 2 × 3 games, it can be shown that the two classifications no
longer coincide. In fact, the games γ4 and γ5 that are distinguished by Nash
equivalence are no longer distinguished by correlated equivalence as can be
seen by considering the straight paths between the games γ4 and γ40 , between
γ40 and γ400 , and between γ400 and γ5 , where:
Ã
γ40
=
(1, 3) (0, 2) (0, 0)
(0, 0) (1, 2) (0, 3)
!
Ã
, γ400
=
(1, 3) (0, 2) (0, 0)
(0, 0) (0, 2) (1, 3)
!
.
It is easily verified that the set of correlated equilibria remains full dimensional along the entire path. This shows that two games that are correlated
equivalent need not be Nash equivalent. Further examples of this can be found
among the 3 × 3 games.
Among the 3 × 3 games one also finds examples of games that are Nash
equivalent but not correlated equivalent. This can be seen from the games γ11
and γ12 of the previous section. Although both games are Nash equivalent,
they are not correlated equivalent, as the dimension of the set of correlated
equilibria changes from 5 to 8.
However, it is possible to show that the analogue to Corollary 1 and
Lemma 8 holds also for correlated equivalence. This provides a basic com21
mon property of all equivalence concepts considered in this paper at least for
the two player case.
Theorem 4 Let γ ∈ IR2κ be a two person game in normal form. If γ is
regular, then the dimension of the set of correlated equilibria CE(γ) is locally
constant.
P roof . We make use of the following well-known lemma. (See for example
Fudenberg and Tirole (1991), Ch. 2, p. 57.)
Lemma 9 Let γ = (A, B) ∈ IR2mn be a two person game in normal form.
Then p ∈ IRmn is a correlated equilibrium of γ if and only if it is a solution to
the linear inequalities:
Cp ≥ 0, 1mn p = 1, and p ≥ 0,
(5)
where C is the (m(m − 1) + n(n − 1)) × mn matrix defined by:

a11 − a21

..

.


 a11 − am1












 b11 − b12

..


.

 b11 − b1n






0

..


.
0
···
···

a1n − a2n
..
.
0
a1n − amn
..
.
am1 − a11
..
.
0
am1 − am−1,1
bm1 − bm2
..
.
···
0
..
.
···
..
.
···
0
bm1 − bmn
b1n − b11
..
.
0
..
.
· · · b1n − b1,n−1
···
···
0
···
amn − a1n
..
.
· · · amn − am−1,n
···
0
..
.
···
..
.
···
···
0
bmn − bm1
..
.
















.















bmn − bm,n−1
Let E be the matrix consisting of all rows of (C 1mn Imn )T , that hold with
equality for all vectors p ∈ IRmn satisfying (5) at a given game γ.7 Then,
as a direct application of a well-known result from polyhedral theory, (see for
7
We say a row of C or Imn holds with equality or strict inequality at γ, if the corresponding
inequality of (5) holds with equality or with strict inequality.
22
example Nemhauser and Wolsey (1988), Proposition 2.4, p. 87), it follows that
the dimension of the set of correlated equilibria of γ is equal to mn−rank(E).
Therefore, it suffices to show that the rank of E is locally constant at γ. To
avoid confusion, we will fix E to consist of all rows holding with equality at
the given, regular game γ, so that as we change γ, we keep the set of rows
of E fixed while allowing the entries to vary. This means that the only way
the dimension of the correlated equilibria can change as γ changes is (i) that
the rank of E changes or (ii) that there exists a row of (C 1mn Imn )T not
belonging to E at γ that holds with equality for all correlated equilibria of a
close by game γ 0 and moreover is linearly independent from the rows of E at γ 0 .
We show that neither case can arise if γ is regular. First, notice that if a row
of (C 1mn Imn )T does not belong to E, then it must be linearly independent
from the rows of E at γ. And if it is linearly independent from E, and the
span of the rows of E does not increase with small perturbations of γ, it will
continue to be linearly independent from the rows of E, and moreover, there
will be correlated equilibria of the perturbed games at wich such a row will
continue to hold with strict inequality. Therefore, it remains to show that the
span of the rows of E cannot be increased by slight perturbations of γ when
γ is regular.
Since the number of linearly independent rows of E cannot decrease locally,
it suffices to show that it cannot increase either. To show this, we make use
of the following lemma.
Lemma 10 Let γ = (A, B) ∈ IR2mn be regular two person game in normal
form and C as above, let D be any submatrix obtained from (C 1mn )T by
deleting all rows corresponding to strictly dominated strategies and possibly
more,8 then D has full row rank, i.e., rank(D) = min {mn, number of rows
of D}.
Before proving this lemma, we conclude the proof of the theorem. Notice
that any strictly dominated strategy has probability zero at any correlated
equilibrium, which implies that any row of C corresponding to a strictly dominated strategy either holds trivially with equality for all correlated equiliria
(i.e., because all corresponding entries of the correlated equilibria are zero), or
it holds with strict inequality for some correlated equilibrium. (This can be
seen by inspection of the matrix C above.) This implies that all the rows of
E that correspond to strictly dominated strategies are ones where all corresponding entries of all correlated equilibria are zero, which in turn implies that
8
We say a row of C corresponds to a strategy sik ∈ Si if it contains aks , for some s =
1, . . . , n, when i = 1 or if it contains btk for some t = 1, . . . , m, when i = 2.
23
the corresponding rows of Imn must also be contained in E. In other words
the rows of E that correspond to strictly dominated strategies are spanned by
corresponding rows of Imn . Hence the span of these rows together with the
corresponding rows of Imn cannot be increased by varying γ locally. It remains
to see whether it is possible to increase the span of the remaining rows, i.e.,
whether it is possible to increase the rank of the matrix D obtained from E
after deleting all rows corresponding to strictly dominated strategies as well
as the corresponding rows of Imn . By the lemma above, such a matrix has
full row rank. This implies that it is not possible, by perturbing its entries to
increase its rank. As a consequence, perturbing γ will not increase the rank
of such a submatrix D. But then, we have shown that perturbing γ will not
affect the rank of E, and hence the dimension of the set of correlated equilibria
must remain locally constant. 2
We come back to the proof of Lemma 10
P roof of Lemma 10. It suffices to consider the statement of the lemma
for the case where D is the full matrix C, assuming that the original game has
no strictly dominated strategies. For, if a game is regular, then, the reduced
game obtained after iteratedly eliminating all strictly dominated strategies will
continue to be regular. Moreover, it will have neither redundant nor weakly
dominated strategies. But then, a close inspection of the matrix C shows that
it must have full row rank since the columns of any set of at least mn rows of
C are linearly independent. To see this it may be useful to rewrite the matrix
C as the difference of two matrices C1 and C2 in the obvious way, and recall
that regularity of a game with no strictly dominated strategies implies that
the submatrices of A and B T appearing in C2 will have full row rank (see von
Stengel (1996), Theorem 2.7, p. 20). This implies that any set of at most
mn rows of C are linearly independent, and the statement of the lemma then
follows. 2
5
Related Literature
It was already mentioned in the introduction that the literature concerning
equivalence classes or classifications of normal form games is astonishingly
small. Besides the classic distinctions such as between zero-sum and non-zerosum or constant-sum and non-constant-sum games, or between coordination,
common interest games and purely competitive games, or simply between
normal form and extensive form games, there has been very little work that
has, as its object, the classification of games into typically finitely many classes
24
using standard (non-cooperative) equilibrium concepts.
Some exceptions are the works of Rapoport, Guyer, and Gordon (1976)
who classify 2 × 2 two person games into 78 strategically different classes
of games and suggest that these can be further identified into 24 different
classes. However, it is not clear how the distinctions made relate to standard
non-cooperative equilibrium concepts and, as Bárány et al. (1992), p. 268,
point out, it is not clear how their procedure extends to larger games.
Bárány, Lee, and Shubik (1992) classify ordinal two person m × n games
according to the outcome sets generated by the payoff matrices, although they
also mention the possibility of using properties of the best reply or other noncooperative equilibrium correspondences to derive alternative classifications.
In particular, they obtain asymptotic bounds for the number of different classes
of games (≈ (mn)2mn ) and for the number of different classes of outcome sets
(≈ (mn)mn ). Their classification distinguishes 78 strategically different two
person 2×2 games and already over 65 billion strategically different two person
3 × 3 games. Also here, it is also not immediate to us how and to which noncooperative equilibrium concepts the shapes of the outcome sets relate to.
In the mathematics literature, Conway (1976) addresses the question of
classifying games within the particular class of so-called nim-type games.
These are two person zero sum games where agents make sequential moves,
one after the other, choosing from a given set of strategies that decreases as
agents make moves. Conway shows that the set of equivalence classes of these
nim-type games can be identified with the space of nonstandard numbers. Although the class of nim-type games does not seem to be of particular relevance
to the social sciences, its classification shows that equivalence classes of games
may indeed have a very particular structure.9
However, besides several papers (such as for example Carlsson and van
Damme (1993) and Morris, Rob, and Shin (1995)) that define classes of games
in order to show that certain properties hold over all games within the class,
no other literature was found that addresses the question of systematically
classifying normal form games according to the more standard equilibrium
concepts used in (non-cooperative) game theory such as correlated or Nash
equilibrium.
9
For further literature on nim-type and related games see also Berlekamp et al. (1982),
Conway (1991), and Guy (1991).
25
6
Conclusion
The present paper has introduced a procedure for identifying games into broad
equivalence classes that are defined using very basic topological characteristics
of equilibrium correspondences. This was seen to lead to an organization of
the spaces of games into a typically finite number of well-defined equivalence
classes, which also constitute a classification of the underlying space of games,
according to the given equilibrium concept. We find the proposed procedure
as providing a useful way of thinking about srategic situations and of comparing equilibrium concepts. However, many questions are left unanswered, (see
especially Sections 4.4 and 4.5), and we view the present paper as a first step
in a possibly large field.
A question we think is particularly important is that of finding sufficient
conditions that can be imposed directly on a pair of games to decide whether
they are equivalent with respect to one or the other of the equivalence concepts
introduced. In fact, the presented procedure of defining equivalence classes can
be viewed as being in some sense the reverse of the more direct procedure of
defining equivalence classes based on properties only of the pairs of games
to be compared, i.e., without invoking properties of further games along a
path. One would then require all games within an equivalence class to be
invariant with respect to the given properties. We view such an approach
as complementary to the approach taken in this paper, where the mentioned
sufficient conditions for deciding equivalence between two games play the role
of the invariance properties relative to the underlying equivalence concept.
Some further questions we think are important are to obtain a more operational characterization of Nash regularity for more than two players, to
understand the structure of the set of all equivalence classes for Nash and
correlated equilibria for a given space of games, and to see how these change
when strategies and/or players are added. It would also be interesting to obtain a clearer understanding of the relationship between the Nash equivalence
classes and the correlated and rationalizability equivalence classes.
Another question that has not been addressed at all in the present paper,
is whether one can use the classifications to obtain insights about singular
games in order to say something about robustness and refinements of equilibria. What we have in mind is to use the equivalence classes of games around a
given component of Nash equilibria to say something about the component and
about some of the individual equilibria contained in it. A further possibility
would be to define equivalence classes for Nash singular games, using different refinements of the Nash equilibrium concept and to compare the resulting
equivalence classes with each other. In fact, the equivalence classes defined
26
with respect to rationalizability, iterated dominance, and correlated equilibria
were defined for arbitrary normal form games and hence also for singular and
Nash singular games and may turn out to be useful in this context.
References
[1] Bárány, I., J. Lee, and M. Shubik (1992) “Classification of Two-Person
Ordinal Bimatrix Games,” International Journal of Game Theory, 21:
267-290.
[2] Berlekamp, E.R., J.H. Conway, and R.K. Guy (1982) Winning Ways for
your Mathematical Plays, Academic Press, London.
[3] Blume, L.E. and W.R. Zame (1994) “The Algebraic Geometry of Perfect
and Sequential Equilibrium,” Econometrica, 62: 783-794.
[4] Carlsson, H. and E. van Damme (1993) “Global Games and Equilibrium
Selection,” Econometrica, 61: 989-1018.
[5] Conway, J.H. (1976) On Numbers and Games, Academic Press, London.
[6] Conway, J.H. (1991) “Numbers and Games,” Proceedings of Symposia in
Applied Mathematics, 43: 23-34.
[7] Crawford, V.P. (1997) “Theory and Experiment in the Analysis of Strategic
Interaction,” in D.M. Kreps and K.F. Wallis (eds.), Advances in Economics
and Econometrics: Theory and Applications, I, Seventh World Congress
of the Econometric Society, Cambridge University Press, Cambridge.
[8] Fudenberg, D. and J. Tirole (1991) Game Theory, MIT Press, Cambrgidge,
MA.
[9] Greenberg, M.J. and J.R. Harper (1981) Algebraic Topology: A First
Course, Addison-Wesley, New York.
[10] Guy, R.K. (1991) “What is a Game?,” Proceedings of Symposia in Applied
Mathematics, 43: 1-22.
[11] Harsanyi, J. (1973) “Oddness of the Number of Equilibrium Points: A
New Proof,” International Journal of Game Theory, 2: 235-250.
[12] Kohlberg, E. and J.F. Mertens (1986) “On the Strategic Stability of Equilibria,” Econometrica, 54: 1003-37.
27
[13] Kojima, M., A. Okada, and S. Shindoh (1985) “Strongly Stable Equilibrium points of N -Person Noncooperative Games,” Matematics of Operations Research, 10: 650-663.
[14] Mertens, J.-F. (1987) “Ordinality in Non-Cooperative Games,” CORE
Discussion Paper No. 8782.
[15] Morris, S., R. Rafael, and H.S. Shin (1995) “p-Dominance and Belief
Potential,” Econometrica, 63: 145-157.
[16] Nash, J. (1951) “Non-Cooperative Games,” Annals of Mathematics, 54:
286-295.
[17] Nemhauser, G.L. and L.A. Wolsey (1988) Integer and Combinatorial Optimization, Wiley and Sons, New York.
[18] Rapoport, A., M.J. Guyer, and D.G. Gordon (1976) The 2 × 2 Game,
University of Michigan Press, Ann Arbor.
[19] Roth, A.E. and I. Erev (1995) “Learning in Extensive-Form Games: Experimental Data and Simple Dynamic Models in the Intermediate Term,”
Games and Economic Behavior, 8: 164-212.
[20] Roth, A.E. and I. Erev (1997) “Modeling how people play games: Reinforcement Learning in Experimental Games with Unique Mixed Strategy
Equilibria,” mimeo, University of Pittsburgh.
[21] Schanuel, S.H., L.K. Simon, and W.R. Zame (1991) “The Algebraic Geometry of Games and the Tracing Procedure,” in R. Selten (ed.), Game
Equilibrium Models, II: Methods, Morals, and Markets, Springer Verlag,
Berlin.
[22] Shapley, L.S. (1974) “A Note on the Lemke-Howson Algorithm,” Mathematical Programming Study, 1: 175-189.
[23] Van Damme, E. (1991) Stability and Perfection of Nash Equilibria, 2nd
ed., Springer Verlag, Berlin.
[24] Von Stengel, B. (1996) “Computing Equilibria for Two-Person Games,”
ETHZ, Institut für theoretische Informatik, Technical Report # 253, forthcoming in R. Aumann and S. Hart (eds.), Handbook of Game Theory, III,
North-Holland, Amsterdam.
28
7
Appendix
The following is a classification of 3 × 3 Nash regular games. With the help of
a computer program we obtained 32 distinct Nash equivalence classes, each of
which is unambiguously represented by a representative game ρi , i = 1, . . . , 32.
Games with one Nash equilibrium.
Zero pure strategy equilibria, zero completely mixed:10


(2, 1) (1, 2) (1, 0)


(1,
2) (2, 1) (2, 0)  .

(0, 2) (0, 1) (0, 0)
Zero pure strategy equilibria, one completely mixed:


(5, 1) (5, 0) (0, 3)


 (2, 0) (2, 3) (2, 0)  .
(0, 3) (0, 1) (5, 0)
One pure strategy equilibrium, zero completely mixed:


(2, 2) (2, 1) (2, 0)


 (1, 2) (1, 1) (1, 0)  .
(0, 2) (0, 1) (0, 0)
Games with three Nash equilibria.
Zero pure strategy equilibria, zero completely mixed:

 

(5, 0) (0, 3) (5, 2)
(5, 0) (0, 3) (5, 2)

 

 (2, 0) (4, 2) (2, 3)  ,  (2, 0) (2, 2) (4, 3)  .
(0, 3) (5, 2) (0, 0)
(0, 3) (5, 2) (0, 0)
Zero pure strategy equilibria, one completely mixed:


(0, 3) (5, 0) (0, 1)


 (5, 0) (0, 3) (5, 1)  .
(2, 1) (2, 0) (4, 3)
10
All remaining equilibria are mixed strategy equilibria with two pure strategies in each
player’s support.
29
One pure strategy equilibrium, zero completely mixed:

 


 


 

(2, 3) (2, 0) (1, 2)
(5, 3) (0, 0) (5, 2)

 

 (1, 0) (1, 3) (2, 2)  ,  (2, 3) (4, 2) (2, 0)  ,
(0, 3) (0, 2) (0, 0)
(0, 0) (5, 2) (0, 3)
(5, 3) (0, 0) (5, 2)
(3, 5) (1, 2) (1, 0)

 

 (2, 0) (4, 2) (2, 3)  ,  (0, 0) (3, 4) (0, 5)  ,
(0, 3) (5, 2) (0, 0)
(1, 5) (0, 2) (3, 0)
(5, 5) (0, 0) (5, 4)
(5, 3) (0, 2) (0, 0)

 

 (2, 0) (4, 5) (2, 2)  ,  (4, 0) (2, 2) (2, 3)  .
(0, 5) (5, 0) (0, 2)
(0, 3) (5, 0) (5, 2)
One pure strategy equilibrium, one completely mixed:

 


 

(5, 3) (0, 0) (5, 2)
(5, 3) (0, 0) (5, 2)

 

 (4, 3) (2, 2) (2, 0)  ,  (0, 3) (5, 2) (0, 0)  ,
(0, 0) (5, 2) (0, 3)
(4, 0) (2, 2) (2, 3)
(5, 3) (0, 0) (5, 1)
(5, 3) (0, 0) (5, 2)

 

 (4, 1) (2, 3) (2, 0)  ,  (2, 0) (2, 2) (4, 3)  ,
(0, 1) (5, 0) (0, 3)
(0, 3) (5, 2) (0, 0)


(3, 3) (3, 1) (0, 0)


(2,
1) (0, 3) (3, 0)  ,

(0, 1) (2, 0) (2, 3)
Two pure strategy equilibria, zero completely mixed:

 


 

(2, 2) (1, 1) (2, 0)
(2, 3) (2, 2) (1, 0)

 

 (1, 1) (2, 2) (1, 0)  ,  (1, 0) (1, 2) (2, 3)  ,
(0, 2) (0, 1) (0, 0)
(0, 3) (0, 2) (0, 0)
(5, 5) (5, 2) (0, 0)
(5, 3) (5, 2) (0, 0)

 

 (2, 5) (2, 2) (4, 0)  ,  (2, 0) (2, 2) (4, 3)  .
(0, 0) (0, 4) (5, 5)
(0, 2) (0, 0) (5, 3)
Two pure strategy equilibria, one completely mixed:


(5, 3) (5, 1) (0, 0)


(4,
1) (2, 3) (2, 0)  .

(0, 1) (0, 0) (5, 3)
30
Games with five Nash equilibria.
One pure strategy equilibrium, zero completely mixed:


(3, 5) (0, 2) (5, 0)


(2,
0) (4, 4) (2, 5)  .

(0, 5) (5, 2) (0, 0)
Two pure strategy equilibria, zero completely mixed:

 

(5, 3) (5, 0) (0, 2)
(5, 3) (5, 0) (0, 2)

 

(2,
0)
(2,
3)
(4,
2)
(2,
0) (4, 3) (2, 2)  .
,

 
(0, 2) (0, 0) (5, 3)
(0, 0) (0, 2) (5, 3)
Two pure strategy equilibria, one completely mixed:

 


 

(5, 5) (5, 4) (0, 0)
(5, 3) (5, 2) (0, 0)

 

 (4, 5) (2, 2) (2, 0)  ,  (4, 2) (2, 0) (2, 3)  ,
(0, 0) (0, 2) (5, 5)
(0, 0) (0, 2) (5, 3)
(5, 3) (5, 1) (0, 0)
(5, 5) (5, 2) (0, 0)

 

 (2, 1) (4, 3) (2, 0)  ,  (2, 0) (4, 4) (2, 5)  .
(0, 0) (0, 1) (5, 3)
(0, 0) (0, 2) (5, 5)
Three pure strategy equilibria, zero completely mixed:

 

(3, 3) (0, 2) (0, 0)
(3, 3) (2, 2) (2, 0)

 

(1,
0)
(3,
3)
(1,
2)
(2,
2) (3, 3) (0, 0)  .
,

 
(0, 0) (1, 2) (3, 3)
(0, 0) (0, 2) (3, 3)
Games with seven Nash equilibria.
Three pure strategy equilibria, one completely mixed:


(3, 3) (1, 1) (1, 0)


 (1, 1) (3, 3) (0, 0)  .
(0, 1) (0, 0) (3, 3)
31