Borm, P.E.M.

Gonzalez-Alcon, C.

Hendrickx, R.L.P.

Kuijk, K. van

English

Research Papers in Economics

Top (2007) 15: 297–306
DOI 10.1007/s11750-007-0022-3
ORIGINAL PAPER
A taxonomy of best-reply multifunctions in 2×2×2
trimatrix games
Carlos González-Alcón ·Peter Borm ·
Ruud Hendrickx ·Kim van Kuijk
Received: 13 December 2006 / Accepted: 2 July 2007 / Published online: 21 September 2007
© Sociedad de Estadística e Investigación Operativa 2007
Abstract This paper provides an overview of the various shapes the best-reply mul-
tifunctions can take in 2 × 2 × 2 trimatrix games. It is shown that, unlike in 2 × 2
bimatrix games, the best replies to the opponents’ pure strategies do not completely
determine the structure of the Nash equilibrium set.
Keywords Non-cooperative games · Best-reply multifunction · Nash equilibrium
Mathematics Subject Classiﬁcation (2000) 91A06 · 91A05
1 Introduction
Vorobev (1958) introduced an easy graphical way of representing best-reply multi-
functions for 2 × 2 bimatrix games and used this method to ﬁnd all the Nash equi-
libria. These best-reply multifunction can take only a few forms. Moreover, the best-
replies to the pure strategies of one’s opponent completely determine the structure of
the set of Nash equilibria. Borm (1987) enumerated all these different shapes.
C. González-Alcón ()
Departamento de Estadística, I.O. y Computación, Universidad de La Laguna, 38071 La Laguna,
Spain
e-mail: cgalcon@ull.es
P. Borm · R. Hendrickx · K. van Kuijk
CentER and Department of Econometrics and Operations Research, Tilburg University,
P.O. Box 90153, 5000 LE Tilburg, The Netherlands
P. Borm
e-mail: P.E.M.Borm@uvt.nl
R. Hendrickx
e-mail: ruud@uvt.nl298 C. González-Alcón et al.
In 2 × 2 × 2 trimatrix games, the picture is a bit more complicated. Not only are
there simply more possibilities, but there is an added complication in that in three
dimensions, it is not sufﬁcient any more to know what happens in the extreme points.
In this short paper we provide a taxonomy of all the possible shapes of the best-reply
multifunctions and give an indication of what the set of Nash equilibria can look like.
2 Best-reply functions of 2×2 games
In this section we brieﬂy review the study of best-reply multifunctions in 2×2b i m a -
trix games in Borm (1987).
Let (N,S,P)be a 2-person game in strategic form where S = S1×S2 is the space
of strategies where each set Si of pure strategies of Player i has only two elements
denoted by 1 and 0. Functions Pi : S → R are the payoffs of Player i (i = 1,2).
Without loss of generality we develop our exposition for Player 2.
We note by aij the payoff to Player 2 when Players 1 and 2 use their pure strategies
i and j, respectively. Mixed strategies of each player are determined by a value in
the interval [0,1] giving the probability of using the ﬁrst pure strategy. So, the pure
strategies 1 and 0 can be identiﬁed with probability values of 1 and 0, respectively.
Given a mixed strategy proﬁle (p,q) ∈[ 0,1]2, the payoff function of Player 2 can
be expressed as
P2(p,q) = (p,1−p)

a11 a10
a01 a00

q
1−q

.
Denoting by U the interval [0,1], the general best-reply multifunction of Player 2
against a mixed strategy p ∈ U is given by
B2(p) =
0, if pa11 +(1−p)a01 <pa 10 +(1−p)a00,
U, if pa11 +(1−p)a01 = pa10 +(1−p)a00,
1, otherwise.
(1)
So, we can say that the shape of B2 depends on the sign of the linear function
p(a11 −a10)+(1−p)(a01 −a00). (2)
Where this function is positive, the best-reply is the pure strategy 1, whereas 0 is the
best response where it is negative; where (2) is null any strategy is a best-reply. By
linearity of expression (2) the shape of B2 is given by the signs of a11 − a10 and
a01 −a00 which indicate the best response in the extreme points of the interval [0,1],
as can be checked from (1). Since each of these amounts can be positive, negative or
zero, we have nine possible shapes for the best-reply functions as Fig. 1 shows. Note
that in almost all the cases the shape is completely determined. Only in the 0-1 and
1-0 cases a certain point of change must be calculated in which (2) is null.Best-replies in trimatrix games 299
Fig. 1 The nine different shapes of best-reply multifunction for the second player in 2×2 games, B2(p).
The best response against pure strategies of Player 1 is indicated on the bottom side of each unit square:
on the left against 0, on the right against 1
3 Best-replies in 2×2×2 games
Now, let (N,S,P)be a 3-person game in strategic form, and as above, Si ={ 0,1}
(i = 1,2,3). To simplify the discussion, we make our exposition for Player 3. We
note by
A1 =

a111 a101
a011 a001

and A0 =

a110 a100
a010 a000

Player 3’s payoffs matrix from pure strategies: A1 for Player 3’s pure strategy 1, and
A0 for the pure strategy 0. The subscripts indicate the ﬁrst, second and third player’s
pure strategy, respectively.
Given a mixed strategy proﬁle (p,q,r)∈[ 0,1]3, the payoff function of Player 3
is
P3(p,q,r)= (p,1−p)

rA1 +(1−r)A0

q
1−q

= r(p,1−p)A1(q,1−q)t +(1−r)(p,1−p)A0(q,1−q)t. (3)300 C. González-Alcón et al.
Expression (3) gives us an easy way to obtain the best-reply multifunction for
Player 3, given the Player 1 and 2’s strategies p,q ∈[ 0,1]:
B3(p,q) =
0, if (p,1−p)(A1 −A0)(q,1−q)t < 0,
U, if (p,1−p)(A1 −A0)(q,1−q)t = 0,
1, otherwise.
(4)
The unit square [0,1]2 in which the pair (p,q) moves can be divided into two
regions in which the best reply is 1 and 0, respectively. The border between these
regions is given by the points (p,q) satisfying
(p,1−p)(A1 −A0)

q
1−q

= 0. (5)
On these points the best reply is any point in the interval [0,1].
Equation (p,1−p)(A1 −A0)(q,1−q)t = r gives a hyperbolic paraboloid in the
space (p,q,r). It is one of the three doubly ruled quadrics: through every point on
the surface there are two distinct lines that lie on the surface. In our case, for each
point on the surface, the projections on the pq plane of these two lines are parallel to
the p and q axes, respectively (see Fig. 2). As a degenerate case, the surface can be a
plane (the other ruled surface together with the hyperboloid of one sheet).
In order to study the best-reply multifunctions we have to determine the sign that
the left side of (5) takes for each point (p,q) in the unit square. For points with pos-
itive sign, the best reply is the pure strategy 1. If the sign is negative, the best-reply
of Player 3 is the second pure strategy, 0. The expression is zero on the indifference
points: both pure strategies are equally good responses and indeed, each mixed strat-
egy (r,1−r)is a best-reply strategy.
The level curves of a hyperbolic paraboloid are hyperbolas. Furthermore, in this
case its axes are parallel to the coordinate axes (and the axes are indeed level curves
for some level)—see bottom part of the two boxes in Fig. 2. In the degenerate case in
Fig. 2 Player 3’s payoff function is a section of an hyperbolic paraboloid (left) and or a plane (right). On
the bottom, some level curves are shownBest-replies in trimatrix games 301
Fig. 3 Different possibilities of sign (best replies) regions on the unit square. Plus and minus signs can
be interchanged to obtain other four possibilities. Of course, a trivial case with no change of sign could be
added
which the surface is a plane, the level curves are straight lines. Figure 3 shows differ-
ent possibilities of regions. The different situations of the unit square with respect to
the zero-level curves determine the different forms the best-reply multifunction can
adopt. However, the trimatrix case is more complicated than the previous 2-person
games.
As a starting point, we could have a look at the signs of the four elements of the
matrix A1 − A0, as they give the best response to the four possible combinations of
pure strategies for Player 1 and 2: (0,0), (1,0), (0,1), and (1,1).F r o mt h et h r e e
possible signs of these differences (positive, negative, or unsigned) we have 34 = 81
cases to consider. Each corner of the strategy square can be labeled with 0, 1, or U
by the best reply against it. Since many of these cases can be obtained from others by
relabeling the strategies of the players, we group them into thirteen classes in Table 1.
Although we classify the best-reply multifunctions by the best response to the four
combinations of pure strategies (the four corners of the unit square), the exact shape
of that function can vary. Figure 4 shows an example of three different shapes be-
longing to the same best-reply pattern against combinations of pure strategies. These
three (sometimes four, as in class 6) possibilities are indicated in Table 1 by gray
lines.
4 Nash equilibria
An interesting aspect of studying the best-reply multifunctions is to obtain the Nash
equilibria. We have a Nash equilibrium at each point at which the best replies of all
the players coincide. In trimatrix games a point (p,q,r)∈[ 0,1]3 constitutes a Nash
equilibrium if the best-reply multifunctions satisfy
p ∈ B1(q,r), q ∈ B2(p,r), and r ∈ B3(p,q).
Given the three best-reply multifunctions of the players of a game, we are able to
obtain the Nash equilibrium set. For example, suppose that the three best-reply mul-
tifunctions belong to the ﬁrst case of class 7, which we represent by 0U0U. Figure 5
shows consecutively the best replies of player 1, 2, and 3 and the intersection with the
previous player. The last one is the whole Nash equilibrium set: four isolated corner
points.302 C. González-Alcón et al.
Table 1 Classiﬁcation of best-reply multifunctions of 2×2×2 games. The 4-tuples represent best reply
to the (0,0), (0,1), (1,1),a n d(1,0) pure strategies combinations of the other players. Indications of how
the indifference curves could go are given in the square in the same way as in Fig. 4. The 81 possible
shapes of the best-reply curves are grouped into 13 classes on the basis of symmetry (pure strategies labels
could be interchanged)
Class Best-reply pattern Class Best-reply pattern
1 0000 1111 2
000U
U000
0U00
00U0
111U
U111
1U11
11U1
3
0001
1000
0100
0010
1110
0111
1011
1101
4
00UU
U00U
UU00
0UU0
11UU
U11U
UU11
1UU1
5
00U1
100U
U100
0U10
001U
U001
1U00
01U0
11U0
011U
U011
1U01
110U
U110
0U11
10U1
6
0011
1001
1100
0110
7
0U0U
U0U0
1U1U
U1U1
8
0U01
10U0
010U
U010
1U10
01U1
101U
U101
9
0UUU
U0UU
UU0U
UUU0
1UUU
U1UU
UU1U
UUU1
10
0UU1
10UU
U10U
UU10
1UU0
01UU
U01U
UU01
Figure 6 shows an example of how the Nash equilibria can adopt different combi-
nations of isolated points, lines, and surfaces.
As the interaction among the best-reply functions gives the Nash equilibrium set
of the game, one could think that the best-reply functions can be used to classify
games by their equilibria. This argument works perfectly for 2 × 2 games, in whichBest-replies in trimatrix games 303
Table 1 (Continued)
Class Best-reply pattern Class Best-reply pattern
11
0U1U
U0U1
1U0U
U1U0
12 0101 1010
13 UUUU
Fig. 4 Three different shapes of best-reply function with the same best responses against the pure strate-
gies. It is the case 1110, from class 3
Fig. 5 Left to right: best-reply multifunctions of Players 1, 2, and 3 showing a 0U0U pattern. The dark
zones in the second graph are the intersection of Player 1 and 2’s best-replies; in the third one the Nash
equilibrium set is given304 C. González-Alcón et al.
Fig. 6 Left to right: best reply multifunctions of Players 1 (0UUU), 2 (0101), and 3 (0UU1). Now, in the
third graph it can be seen that the Nash equilibrium set can be a composition of points, straight segments,
and surfaces
Fig. 7 The ten different shapes of Nash equilibria in 2×2 games produced by the 81 combinations of the
nine best-reply multifunctions of Fig. 1. Each drawing represents the whole group of its symmetries
Fig. 8 Examples of shapes of the Nash equilibrium sets from the interaction of different best replies from
class 7
the shape of the interaction between the same class of best-reply functions of the two
players presents the same structure. For instance, when the players have both a best-
reply multifunction of type 1-0, the equilibrium set has always three elements (twoBest-replies in trimatrix games 305
Fig. 9 Different Nash equilibrium sets (dark region in the right-hand graphs) from the same best-reply
functions combination: a 010U pattern for all players
pure strategy equilibria (1,0) and (0,1), plus a mixed equilibrium) no matter where
the multifunctions change from 1 to 0. From nine classes of best-reply function, the
81 interactions can be classiﬁed into ten shapes shown in Fig. 7.306 C. González-Alcón et al.
However, in the 2 × 2 × 2 case, although the classes group best-reply multifunc-
tions showing similar shapes, the interaction among functions in the same class can
lead to different equilibria sets. Figure 8 shows examples of shapes of the Nash equi-
librium sets from the interaction of different best replies patterns all belonging to
class 7.
Moreover, we can obtain different equilibria set conﬁgurations from the interac-
tion of best-reply functions with the same pattern, as Fig. 9 shows. The three players
have a best-reply multifunction with pattern 010U and indifference curves with sim-
ilar curvatures, but for slightly different parameters the Nash equilibrium set varies.
Acknowledgements Part of this research was conducted while the ﬁrst author was visiting Tilburg Uni-
versity thanks to the support of Dirección General de Universidades e Investigación del Gobierno de Ca-
narias. Financial support from the Ministerio de Ciencia y Tecnología, under project TIC2000-1750-C06-
02 is gratefully acknowledged. Ruud Hendrickx acknowledges ﬁnancial support from the Netherlands
Organisation for Scientiﬁc Research (NWO). The authors thank two anonymous referees for their helpful
comments.
References
Borm P (1987) A classiﬁcation of 2×2 bimatrix games. Cah Cent Études Rech Opér 29:69–84
Vorobev N (1958) Equilibrium points in bimatrix games. Theory Probab Appl 3:279–309

A taxonomy of best-reply multifunctions in 2x2x2 trimatix games.

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