This paper provides an overview of the various shapes the best- reply multifunctions 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.non-cooperative games;best-reply multifunction;Nash equilibrium

Borm, P.E.M.

Gonzalez-Alcon, C.

Hendrickx, R.L.P.

Kuijk, K. van

English

Research Papers in Economics

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
No. 2007–11 
 
A TAXONOMY OF BEST-REPLY MULTIFUNCTIONS IN 
2 X 2 X 2 TRIMATRIX GAMES 
 
By Carlos González-Alcón, Peter Borm, Ruud Hendrickx,  
Kim van Kuijk 
 
 
January 2007 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
ISSN 0924-7815 A taxonomy of best-reply multifunctions in
2 × 2 × 2 trimatrix games∗
Carlos Gonz´ alez-Alc´ on1 Peter Borm2
Ruud Hendrickx2 Kim van Kuijk2
Abstract
This paper provides an overview of the various shapes the best-
reply multifunctions 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.
Key words: non-cooperative games, best-reply multifunction, Nash
equilibrium
JEL code: C71
1 Introduction
Vorobev (1958) introduced an easy graphical way of representing best-reply
multifunctions for 2 × 2 bimatrix games and used this method to ﬁnd all
the Nash equilibria. These best-reply multifunction can take only a few
forms. Moreover, the best-replies to the pure strategies of one’s opponent
∗Part of this research was conducted while the ﬁrst author was visiting Tilburg Univer-
sity thanks to the support of Direcci´ on General de Universidades e Investigaci´ on del Gob-
ierno de Canarias. Financial support from the Ministerio de Ciencia y Tecnolog´ ıa, under
project TIC2000-1750-C06-02 is gratefully acknowledged. Ruud Hendrickx acknowledges
ﬁnancial support from the Netherlands Organisation for Scientiﬁc Research (NWO).
1Corresponding author. Departamento de Estad´ ıstica, I.O. y Computaci´ on Universidad
de La Laguna, 38071 La Laguna, Spain. E-mail: cgalcon@ull.es.
2CentER 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,
ruud@uvt.nl.
1Best-replies in trimatrix games /January 25, 2007 2
completely determine the structure of the set of Nash equilibria. Borm (1987)
enumerated all these diﬀerent shapes.
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 × 2 bimatrix 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 [0,1] interval giving the probability of using
the ﬁrst pure strategy. So, the pure strategies 1 and 0 can be identiﬁed with
values for p 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 [0,1] interval, 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 < pa10 + (1 − p)a00
U, if pa11 + (1 − p)a01 = pa10 + (1 − p)a00
1, in other case
. (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)Best-replies in trimatrix games /January 25, 2007 3
0 0 0 U 0 1
U 0 U U U 1
1 0 1 U 1 1
Figure 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 in the botton side of each unit square: on the left against
0, on the right against 1.
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, ie the best response in the extreme points
of the [0,1] interval. Since each of these amounts can be positive, negative
or zero, we have nine possible shapes for the best-reply functions as Figure 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 /January 25, 2007 4
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
 
the Player 3’s payoﬀs matrix from pure strategies: A1 for the Player 3’s ﬁrst
pure strategy (1) and A0 for the second 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)
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, in other case
. (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 replies are any point in the [0,1] interval.
Equation (p,1− p)(A1 − A0)(q,1− q)t = 0 gives a hyperbolic paraboloid
in the space (p,q,r). It is one of the three ruled surfaces: for each point
on the surface there are two lines on the surface passing it. In our case,
these two lines are parallel to the pq axis (see Figure 2). As a degenerate
case, the surface can be a plane (the other ruled surface together with the
hyperboloid).Best-replies in trimatrix games /January 25, 2007 5
0
0.25
0.5
0.75
1
0
0.25
0.5
0.75
1
-5
0
5
0
0.25
0.5
0.75
1
0
0.25
0.5
0.75
1
0
0.25
0.5
0.75
1
0
0.25
0.5
0.75
1
-5
0
5
0
0.25
0.5
0.75
1
0
0.25
0.5
0.75
1
player 1
player 2
player 1
player 2
Figure 2: Player 3’s payoﬀ function is a section of an hyperbolic paraboloid
(left) and or a plane (right). In the bottom, some level curves are shown.
In order to study the best-reply multifunctions we have to determine the
sign that the left side of Equation (5) takes for each point (p,q) in the unit
square. For points with positive 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 response and indeed, each mixed strategy (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
Figure 2. In the degenerate case in which the surface is a plane, the level
curves are straight lines. Figure 3 shows diﬀerent 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). From the three possible signs of these diﬀerences (positive, negative
or unsigned) we have 34 = 81 cases to consider. Each corner of the strategyBest-replies in trimatrix games /January 25, 2007 6
Figure 3: Diﬀerent possibilities of sign (best replies) regions on the unit
square. Plus and minus signs can be interchanged to obtain other four pos-
sibilities. Of course, a trivial case with no change of sign could be added.
square can be labelled with 0, 1 or U by the best reply against it. Since
many of these cases can be obtained from others by relabelling the strategies
of the players, we group them in the thirteen classes of 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 belonging to the same best-reply pattern against
combinations of pure strategies. These three (sometimes four, as in case 4)
possibilities are indicated in Table 1 by grey lines.
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) and (1,0) pure strategies combinations
of the other players. In the square how could go the
indiﬀerence curves is indicated.
class best-reply pattern class best-reply pattern
1
0
0 0
0
0000 1111 2
0
0 0
1
0001
1000
0100
0010
1110
0111
1011
1101Best-replies in trimatrix games /January 25, 2007 7
Table 1: Continuation
class best-reply pattern class best-reply pattern
3
0
0 0
U
000U
U000
0U00
00U0
111U
U111
1U11
11U1
4
0
0 1
1
0011
1001
1100
0110
5
0
0 1
U
001U
U001
1U00
01U0
00U1
100U
U100
0U10
110U
U110
0U11
10U1
11U0
011U
U011
1U01
6
0
0 U
U
00UU
U00U
UU00
0UU0
11UU
U11U
UU11
1UU1
7
0
1 0
1
0101 1010 8
0
1 0
U
010U
U010
0U01
10U0
101U
U101
1U10
01U1
9
0
U 0
U
0U0U
U0U0
1U1U
U1U1
10
0
U 1
U
0U1U
U0U1
1U0U
U1U0
11
0
U U
1
01UU
U01U
UU01
1UU0
10UU
U10U
UU10
0UU1
12
0
U U
U
0UUU
U0UU
UU0U
UUU0
1UUU
U1UU
UU1U
UUU1Best-replies in trimatrix games /January 25, 2007 8
Table 1: Continuation
class best-reply pattern class best-reply pattern
13
U
U U
U
UUUU
4 Nash equilibria
An interesting aspect of studying the best-reply multifunctions is to obtain
the Nash equilibria. We have a Nash equilibrium in each point in 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 equilibria set. For example, suppose that the
three best-reply multifunctions belong to the ﬁrst case of class 9, 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 equilibria set: four isolated corner points.
Figure 6 shows an example of how the Nash equilibria can adopt diﬀerent
combinations of isolated points, lines and surfaces.
As the interaction among the best-reply functions gives the Nash equilib-
ria 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 which 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 equi-
libria set has always three elements (two pure strategy equilibria (1,0) and
(0,1), plus a mixed equilibrium) no matter where the multifunctions changeBest-replies in trimatrix games /January 25, 2007 9
1
1 1
0
(0,0)
(0,1)
(1,0)
(1,1)
1
1 1
0
(0,0)
(0,1)
(1,0)
(1,1)
1
1 1
0
(0,0)
(0,1)
(1,0)
(1,1)
Figure 4: Three diﬀerent shapes of best-reply function with the same best
responses against the pure strategies. It is the case 1110, from class 2.
Figure 5: Left to right: best-reply multifunctions of Players 1, 2 and 3 show-
ing a 0U0U pattern. The dark zones in the second graph is the intersection
of Player 1 and 2’s best-replies; in the third one is the Nash equilibria set.Best-replies in trimatrix games /January 25, 2007 10
Figure 6: Left to right: best reply multifunctions of Players 1 (0UUU), 2
(0101) and 3 (0UU1). Now, in the third graph can be seen that the Nash
equilibria set can be a composition of points, straight segments and surfaces.
1 2 3 4 5
6
s
7
s
s
s
8
s
s
9
s
10
s
Figure 7: The ten diﬀerent shapes of Nash equilibria in 2×2 games produced
by the 81 combinations of the nine best-reply multifuncions of Figure 1. Each
draw represent to the whole group of its symmetries.Best-replies in trimatrix games /January 25, 2007 11


0U0U
0U0U
0U0U


￿
￿
￿
￿
￿
￿
￿
￿
s
s
s
s


0U0U
0U0U
U1U1


￿
￿
￿
￿
￿
￿
￿
￿
s


0U0U
0U0U
1U1U


￿
￿
￿
￿
￿
￿
￿
￿


0U0U
U0U0
1U1U


￿
￿
￿
￿
￿
￿
￿
￿
s


0U0U
0U0U
U0U0


￿
￿
￿
￿
￿
￿
￿
￿
s


0U0U
U0U0
U1U1


￿
￿
￿
￿
￿
￿
￿
￿
Figure 8: Examples of shapes of the Nash equilibria sets from the interaction
of diﬀerent best replies from class 9.
from 1 to 0. From nine classes of best-reply function, the 81 interactions can
be classiﬁed in the ten shapes shown by Figure 7.
However, in the 2 × 2 × 2 case, although the classes group best-reply
multifunctions showing similar shapes, the interaction among function in the
same class can lead to diﬀerent equilibria sets. Figure 8 shows examples of
shapes of the Nash equilibria sets from the interaction of diﬀerent best replies
patterns all belonging to class 9.
Moreover, we can obtain diﬀerent equilibria set conﬁgurations from the
interaction of best-reply functions with the same pattern, as Figure 9 shows.
The three players have a best-reply multifunction with pattern 010U and in-
diference curves with similar curvatures, but for slightly diﬀerent parameters
the Nash equilibria set varies.
References
Borm, P. (1987). A classiﬁcation of 2×2 bimatrix games. Cahiers du Centre
d’´ Etudes de Recherche Op´ erationelle, 29, 69–84.
Vorobev, N. (1958). Equilibrium points in bimatrix games. Theory Probab.
Appl., 3, 279–309.Best-replies in trimatrix games /January 25, 2007 12
Figure 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.

A Taxonomy of Best-Reply Multifunctions in 2x2x2 Trimatrix Games

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