This paper introduces the class of 2 × 2 × 2 trimatrix games with identical anonymous best-replies. For this class a complete classification on the basis of the Nash equilibrium set is provided.trimatrix games;Nash equilibrium;best-reply correspondences;symmetric games

Borm, P.E.M.

Gonzalez-Alcon, C.

Hendrickx, R.L.P.

English

Research Papers in Economics

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Nash equilibria in 2 
 
Carlos Gonz
 
 
 
 
 
 
 
 
No. 2011-138 
Nash equilibria in 2 × 2 × 2 trimatrix games
anonymous best-replies
 
 
By  
 
Carlos González-Alcón, Peter Borm, 
 
 
 
 
 
 
December 2011 
 
 
 
 
 
ISSN 0924-7815 
2 trimatrix games with identical 
replies 
, Ruud Hendrickx 
 Nash equilibria in 2 × 2 × 2 trimatrix games
with identical anonymous best-replies
Carlos Gonz´ alez-Alc´ on1,2 Peter Borm3
Ruud Hendrickx4
Abstract
This paper introduces the class of 2 × 2 × 2 trimatrix games with
identical anonymous best-replies. For this class a complete classiﬁca-
tion on the basis of the Nash equilibrium set is provided.
Key words: trimatrix games, Nash equilibrium, best-reply correspondences,
symmetric games
JEL code: C72
1 Introduction
This paper is about classifying a speciﬁc kind of strategic games on the basis
of the equilibrium set of the games under consideration. For 2 × 2 bimatrix
games, such a taxonomy was provided by Beniest (1964), Rapoport et al.
(1976) and Borm (1987) in a cardinal setting. In an ordinal framework 2×2
bimatrix games were classiﬁed by Brams (1977), Fraser and Kilgour (1986)
and Kilgour and Fraser (1988).
The aim of this paper is to classify a speciﬁc subclass of 2×2×2 trimatrix
games on the basis of their equilibrium set in a cardinal setting. As already
indicated in Gonz´ alez-Alc´ on et al. (2007), each player’s best reply correspon-
dence in a 2 × 2× 2 trimatrix games can be of 13 essentially diﬀerent types,
1Corresponding author.
2Departamento de Estad´ ıstica, I.O. y Computaci´ on (DEIOC), Universidad de La La-
guna. Campus de Anchieta. 38071 La Laguna (Tenerife), Spain. E-mail: cgalcon@ull.es.
3CentER and Department of Econometrics and Operations Research, Tilburg Univer-
sity, PO Box 90153, 5000 LE Tilburg, The Netherlands. E-mail: P.E.M.Borm@uvt.nl.
4CentER and Department of Organization and Strategy, Tilburg University, PO Box
90153, 5000 LE Tilburg, The Netherlands. E-mail: R.L.P.Hendrickx@uvt.nl.
1in principle leading to 2197 combinations. These could possibly be combined
on the basis of symmetry arguments or on the basis of the structure of the
set of Nash equilibria, but for the remaining classes possibly also a further
subdistinction could be necessary.
We restrict attention to games with identical anonymous best replies
(IABR). IABR is a symmetry property of the joint best reply correspon-
dences. In particular, every symmetric game is IABR. We provide a taxon-
omy of IABR games into 17 classes on the basis of their equilibrium set.
2 2 × 2 × 2 trimatrix games and best-replies
We denote a 2 × 2 × 2 trimatrix game by (N,{Si}i∈N,{Pi}i∈N). Here N =
{1,2,3} is the player set and for all i ∈ N we denote by 0 and 1 the two pure
strategies of player i. The set of mixed strategies Si is the set of probability
distributions on the set {0,1}. A mixed strategy can be identiﬁed with a
real number s ∈ [0,1], which is interpreted as the strategy using the pure
strategy 1 with probability s and the pure strategy 0 with probability 1−s.
In this way we set Si = [0,1] for all i ∈ N. We deﬁne S =
 
i∈N Si = [0,1]N
and for any s ∈ S, s−i = {sj}j∈N\{i} is the restriction of s to i’s opponents.
Finally, Pi : S → I R is the payoﬀ function to player i, where for strategy
proﬁle s = (s1,s2,s3) ∈ S, player i’s expected payoﬀ equals
Pi(s) =
 
T⊂N
  
j∈T
sj
 
j∈N\T
(1 − sj)Pi(1T,0N\T)
 
,
where (1T,0N\T) denotes the strategy proﬁle where the players in T play
their pure strategy 1 and the remaining players play 0. For player i ∈ N, the
best-reply correspondence Bi : S ։ Si is deﬁned by
Bi(s) = {t ∈ Si |Pi(s−i,t) ≥ Pi(s−i,t
′) for all t
′ ∈ Si}.
Because Pi is multilinear, for any s ∈ S, Bi(s) can take only three possible
outcomes: {0}, {1} and [0,1]. Note moreover that Bi(s) does not depend on
si.
A strategy proﬁle consisting of best replies against itself is a Nash equi-
librium:
NE(N,S,P) = {s ∈ S |∀i ∈ N : si ∈ Bi(s)}.
So, a Nash equilibrium is a strategy proﬁle in which a unilateral deviation of
one player cannot be proﬁtable to this player.
In this paper we focus on games with all the players having “identical”
best-reply correspondences. As a ﬁrst step we focus on a player’s opponents.
2We say that in a game the best-reply correspondence of a player i ∈ N
exhibits opponent anonymous best-replies (OABR) if for every strategy proﬁle
s ∈ S and bijection π : N → N such that π(i) = i,
Bi(s) = Bi(π(s)),
where π(s) = (sπ(j))j∈N. That is, the best response of player i depends on
the strategies used by the other players as a whole, but not on which player
uses what strategy.
Example 1 Consider the 2 × 2 × 2 trimatrix game in which the payoﬀs
corresponding to the pure strategy proﬁles are represented below:
player 2 player 2
1 0 1 0
player 1 1 1,1,1 1,0,1 1 1,0,0 0,0,0
0 0,1,0 0,0,0 0 0,0,1 1,1,1
player 3: 1 player 3: 0
The best-replies of the players are, for any s ∈ S, given by
B1(s) =

   
   
{0} if s2 + s3 − s2s3 < 1
2,
[0,1] if s2 + s3 − s2s3 = 1
2,
{1} if s2 + s3 − s2s3 >
1
2;
B2(s) =

   
   
{0} if s1 + s3 − s1s3 < 1
2,
[0,1] if s1 + s3 − s1s3 = 1
2,
{1} if s1 + s3 − s1s3 > 1
2;
B3(s) =

   
   
{0} if s1 < 1
2,
[0,1] if s1 =
1
2,
{1} if s1 > 1
2.
The best-reply correspondence B1 is OABR: the roles of players 2 and 3
can be interchanged. Similarly, B2 is OABR, but B3 is not.
In this paper we consider games with identical anonymous best-replies
(IABR), that is, for each strategy proﬁle s ∈ S and each bijection π : N → N,
Bi(π(s)) = Bπ(i)(s),
3for all i ∈ N. Note that whereas OABR is a property of a single best-reply
correspondence, IABR relates to all best-reply correspondences in a game
simultaneously. Obviously, if the best-reply correspondences in a game are
IABR, then each of them separately is OABR.
Example 2 Consider the 2 × 2 × 2 trimatrix game, which only diﬀers from
Example 1 in P3, represented by
player 2 player 2
1 0 1 0
player 1 1 1,1,1 1,0,1 1 1,0,0 0,0,0
0 0,1,1 0,0,0 0 0,0,0 1,1,1
player 3: 1 player 3: 0
For all s ∈ S, B1(s) and B2(s) are as given in Example 1 and
B3(s) =

 
 
{0} if s1 + s2 − s1s2 < 1
2,
[0,1] if s1 + s2 − s1s2 = 1
2,
{1} if s1 + s2 − s1s2 > 1
2.
Hence, this is a game with IABR.
IABR is a property of best-reply correspondences and does not necessarily
imply symmetry in the corresponding payoﬀs1. This is illustrated in the
following example.
Example 3 Consider the 2 × 2 × 2 trimatrix game represented by
player 2 player 2
1 0 1 0
player 1 1 5,4,4 3,1,2 1 2,2,1 1,2,2
0 2,2,1 3,2,0 0 2,1,1 4,4,3
player 3: 1 player 3: 0
This game is not symmetric in terms of payoﬀs, but the best-reply corre-
spondences exhibit IABR.
1So IABR does not imply Pi(π(s)) = Pπ(i)(π(s)) for all i ∈ N. Obviously, every
symmetric game is a game with IABR. In fact, it is not diﬃcult to show that also every
weighted potential game with a symmetric potential function is a game with IABR.
43 Nash equilibria of IABR 2 × 2 × 2 games
In this section we provide a full description of the set of Nash equilibria in
2×2×2 trimatrix games satifying IABR. To do this, we classify all such games
on the basis of the shape of their (identical) best-reply correspondences. Our
classiﬁcation diﬀers from the one in Gonz´ alez-Alc´ on et al. (2007) because, on
the one hand they do not impose IABR, and on the other hand symmetries
in terms of only a single best-reply correspondence may break down when
considering the interaction between all best replies.
To illustrate our classiﬁcation, consider player 3. His best reply corre-
spondence may look as follows:
Note again that against any strategy proﬁle of players 1 and 2, player 3’s
best response is either {0}, {1}, or [0,1]. Moreover, as a result of IABR, it
is symmetric around the diagonal (0,0) − (1,1) ⊂ S1 × S2. To get a more
compact overview, we consider the projection of the picture above on the
opponents’ joint strategy space S1 × S2, where in the four extreme points
player 3’s best reply is indicated (with U = [0,1]):
U
0 1
0
(0,0)
(0,1)
(1,0)
(1,1)
Note that as a result of IABR, the best reply against (1,0) and (0,1)
must be the same.
If we relabel the pure strategies of all players simultaneously by switching
0 and 1, the best-reply correspondences still exhibit IABR. Basically the only
thing that changes is that all best-reply correspondences (and hence, the set
of Nash equilibria) are inverted trough point (1
2, 1
2, 1
2). Applying this to the
best-reply correspondence depicted above yields:
50
1 U
1
(0,0)
(0,1)
(1,0)
(1,1)
Table 1 shows our classiﬁcation of IABR 2 × 2 × 2 games. Column 2
depicts the best replies projected onto the opponents’ joint strategy space.
Columns 3, 4 and 5 depicts the whole strategy space S with in it the graphs
of B1, B2 and B3, respectively. Their respective intersection is depicted in
black, so the black points in the ﬁnal column are all Nash equilibria. An
isolated Nash equilibrium is depicted as a square when it is in pure strategies
and as a circle if it is mixed. Equilibria “hidden” by Player 3’s best-reply
correspondence are hollow circles or squares, or thick dashed lines.
In various cases an indication of the diﬀerent shapes that the indiﬀerence
curve (the locus of U) can have is given. This shape only aﬀects the equilib-
rium set in the exact location of some of the mixed-strategy equilibria, except
for classes 14, 15 and 16 where because of its further-reaching implications
we indeed distinguish between three separate classes.
Table 1: Classiﬁcation of IABR 2 × 2 × 2 games
class best replies player 1’s
best replies
player 2’s
best replies and
interaction with 1’s
player 3’s
best replies and
equilibria
1
1
1 1
1
0
0 0
0
2
1
1 U
1
U
0 0
0
6Table 1: Continuation
class best replies player 1’s
best replies
player 2’s
best replies and
interaction with 1’s
player 3’s
best replies and
equilibria
3
U
1 1
1
0
0 U
0
4
0
1 1
1
0
0 1
0
5
1
1 0
1
1
0 0
0
6
U
1 U
1
U
0 U
0
7
1
U 1
U
0
U 0
U
8
0
1 U
1
U
0 1
0
7Table 1: Continuation
class best replies player 1’s
best replies
player 2’s
best replies and
interaction with 1’s
player 3’s
best replies and
equilibria
9
1
0 U
0
U
1 0
1
10
U
U 1
U
0
U U
U
11
U
U 0
U
1
U U
U
12 0
U 1
U
13 1
U 0
U
14
1
0 1
0
0
1 0
1
8Table 1: Continuation
class best replies player 1’s
best replies
player 2’s
best replies and
interaction with 1’s
player 3’s
best replies and
equilibria
15
1
0 1
0
0
1 0
1
16
1
0 1
0
0
1 0
1
17 U
U U
U
The equilibrium sets present several structures: some with only pure
strategy equilibria (classes 1, 3 and 7), other with pure and mixed but al-
ways isolated equilibria (4, 5, 9, 12, 14, 16). In some cases there appear a
continuum of equilibria in lines (2, 6, 8, 10, 13, 15), in surfaces (11), apart
of the degenerate case of class 17. Moreover, note that all the classes are
diﬀerent in the composition of their Nash equilibria set except classes 4 and
12 (two pure, one mixed isolated equilibria) and 6 and 10 (a pure isolated
plus the three edges with common vertex the opposite point).
References
Beniest, W. (1964). Une Classiﬁcation des Points d’Equilibre du Jeu Matriciel
2×2. Cahiers du Centre d’Etudes de Recherche Op´ erationnelle, 6, 44–55.
Borm, P. (1987). A classiﬁcation of 2×2 bimatrix games. Cahiers du Centre
d’´ Etudes de Recherche Op´ erationnelle, 29, 69–84.
Brams, S. (1977). Deception in 2 × 2 Games. Journal of Peace Science, 2,
171–203.
9Fraser, N. and D. Kilgour (1986). Non-strict ordinal 2 × 2 games: A com-
prehensive computer-assisted analysis of the 726 possibilities. Theory and
Decision, 20, 99–121.
Gonz´ alez-Alc´ on, C., P. Borm, R. Hendrickx, and K. van Kuijk (2007). A tax-
onomy of best-reply multifunctions in 2×2×2 trimatrix games. TOP, 15,
297–306.
Kilgour, D. and N. Fraser (1988). A taxonomy of all ordinal 2 × 2 games.
Theory and Decision, 24, 99–117.
Rapoport, A., M. Guyer, and D. Gordon (1976). The 2×2 Game. Ann Arbor:
The University Michigan Press.
10

A classiﬁcation of 2×2 bimatrix games.

A taxonomy of all ordinal

A taxonomy of best-reply multifunctions in 2×2×2 trimatrix games.

games: A comprehensive computer-assisted analysis of the 726 possibilities. Theory and Decision,

The 2×2 Game. Ann Arbor:

Une Classiﬁcation des Points d’Equilibre du Jeu Matriciel 2×2. Cahiers du Centre d’Etudes de Recherche Op´

Nash Equilibria in 2 × 2 × 2 Trimatrix Games with Identical Anonymous Best-Replies

Nash Equilibria in 2 × 2 × 2 Trimatrix Games with Identical Anonymous Best-Replies

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics