An N-person Battle of Sexes Game
Jijun Zhao1, Miklos N. Szilagyi2 and Ferenc Szidarovszky1
Abstract
The n-person Battle of Sexes game is first introduced. The game’s properties are discussed and payoff
functions are modeled under the main assumption that players’ payoffs are based on whether they
likes their choices or not and also on how many other players have the same choices. Linear payoff
functions are assumed and the existence of the Nash equilibrium is examined. The dependence of the
equilibrium on model parameters is also analyzed.
Keywords: battle of sexes, n-person game, payoff functions, Nash-equilibria
1. Introduction
As an interesting example of noncooperative games with multiple Nash equilibria, the
Battle of Sexes (BOS) game has been studied by many researchers. They considered it as
a 2-person one-shot game in both theoretical studies and laboratory experiments
([1]—[5]). There has been a large number of theoretical discussions based on Forward
induction [3]. Experiments have been conducted for analyzing the effects of
1
Systems and Industrial Engineering Department, The University of Arizona, Tucson, Arizona 85721-0020, U.S.A.
2
Electrical and Computer Engineering Department, The University of Arizona, Tucson, Arizona, 85721-0104, U.S.A.
1
communication [2][4], personality and perception [1], and prior experience [5]. However,
no study has been conducted about the multi-person and dynamic extensions, and all
recent studies are limited to the use of simple payoff matrices. In this paper, we will
introduce and study BOS games with arbitrary number n of players. The ‘n-person Battle
of Sexes’ game is a straightforward extension of the 2-person case, however this
generalization is not trivial. As we will see later, we have to be very careful about
preserving the essential properties of the game in the general case. In Section 2 we will
therefore discuss the fundamental properties of 2-person BOS games and then they will
be extended to the n-person case.
A large number of studies on n-person dilemma games (Prisoner’s Dilemma and Chicken
game) has been published in the literature recently including both payoff function
modeling and computer simulation ([6]—[10]). The payoff functions were always
dependent on the number of cooperators in the game. A BOS game is different than
dilemma games since in the dilemma games we are interested in when and why players
cooperate, and how to manipulate the players to make them cooperate, while cooperation
itself in BOS games is not important. In a 2-person BOS game, when both players
cooperate, they will receive the lowest possible payoffs. Therefore we are interested in
finding out when and how the players could have the same choice; and the fact that how
the players have the same choices should be reflected in the payoffs. As we will see in
Sections 3 and 4, the payoffs of n-person BOS games are therefore modeled as functions
2
of the number of players with the same choices. Special linear payoff functions are
introduced and analyzed in Section 5. Final conclusions are given in Section 6.
2. 2-person BOS games
A 2-person BOS game is usually introduced by a story as follows. A man and his wife
want to spend an evening together, so they try to choose an entertainment of common
interest. The husband (Player 1) prefers a football game, while the wife (Player 2) prefers
a ballet performance. However, both of them prefer going out together to going to either
entertainment alone. The 2-person BOS game is a coordination game with conflicting
preferences. There are two choices for the players, and the preferred choices of the two
players are conflicting with each other. The player’s decision is one of the choices.
Depending on how the two players’ decisions are combined, the players obtain payoffs
according to the payoff matrix shown in Table 1. The columns of the matrix correspond
to Player 2’s decisions, and the rows correspond to Player 1’s decision. The first entry in
the parenthesis of each cell represents the payoff of Player 1, and the second entry
represents the corresponding payoff of Player 2.
Table 1
Payoff Matrix of 2-person BOS games
Player 2
Football
Ballet
(Cooperate)
(Defect)
(Cooperate)
(R, R)
(S, T)
Football
(T, S)
(P, P)
Ballet
Player 1
3
(Defect)
Usually in 2-by-2 games (2 people, 2 choices), the players’ decisions are interpreted as
‘defection’ or ‘cooperation’. For example, in a Prisoner’s Dilemma (PD), ‘cooperation’ is
not to confess the crime (so cooperation with the partner) and ‘defection’ is to confess; in
a Chicken game, ‘defection’ is not to avoid crashing the car, and ‘cooperation’ is to avoid
it. When utilizing the terms ‘defection’ and ‘cooperation’ to discuss general BOS games
in which the two choices are not specified, ‘defection’ is the choice that the player likes
and ‘cooperation’ is the choice that the player dislikes. In the above described BOS game,
football is the entertainment that Player 1 prefers; however this choice shows Player 1’s
not cooperating behavior, the player’s ‘defection.’ ‘Cooperation’ represents the decision
that Player 1 chooses Player 2’s preference, the ballet performance, and not his own
preferred type of entertainment. Player 2 has a similar situation. In 2-by-2 symmetric
games, the payoff matrix can have four possible values T, S, R, P. These notations
originate from 2-person PD games in which defection is a ‘temptation’ when the other
cooperates, cooperation is a ‘sucker’s’ choice when the other defects, ‘reward’ is given to
both cooperators, and ‘punishment’ for both defectors. Hence, T represents ‘temptation’
payoff, S represents ‘sucker’ payoff, R represents ‘reward’, and P represents
‘punishment’. The four values have to satisfy relation T>R>P>S to represent PD games.
When the relative values of T, S, R, P change, the matrix can represent different games. A
BOS game is defined by relation T>S>P>R. Inequality (T and S) > (P and R) requires that
4
a player must get higher payoff when both players are together than in the case when they
are separated. Relations T>S and P>R guarantee that a player will get higher payoff if the
choice is what he likes than in the case when the choice is what he does not like.
There is a fundamental difference between the ‘cooperation’ and ‘defection’ in BOS and
those in PD or Chicken games. In PD or Chicken games, the meaning of ‘defection’ is the
same for both players—to confess or to crash. The meaning of ‘cooperation’ is also the
same for both players. In a BOS game, ‘defection’ is the choice what the player likes,
however since players have different preferences, ‘defection’ of Player 1 is to choose
football, and ‘defection’ of Player 2 is to choose ballet. The conflict between the
preferences in the BOS game gives ‘defection’ and ‘cooperation’ a deeper meaning than
in other games. The 2-player PD and Chicken games have four decision combinations:
(defect, cooperate), (cooperate, defect), (cooperate, cooperate) and (defect, defect). In a
2-person BOS game we have theoretically 16 decision combinations since the players’
preferences are also involved in the payoff matrix (see Table 2). It can be divided into
four parts. The upper right and lower left corners represent situations when the players
have the same preferences. We have to exclude these cases since the lack of conflict in
the players’ interests would violate the basic condition of BOS games. Hence these two
parts of the matrix are left blank. The other eight combinations form two submatrices,
one in the upper left corner and one in the lower right corner. Only one of the submatrices
has been used in earlier literature, since they are symmetric and by interchanging the two
5
players one submatrix becomes the other one. In our story, the preferences of the players
are predetermined: Player 1 likes football and Player 2 likes ballet. Hence only the upper
left submatrix is used (as in Table 1), and the players’ preferences are default and not
shown. The 2-person BOS game is then reduced to a 2-by-2 game with the same payoff
notation and matrix structure than the one being used in dilemma games. However, as it
will be discussed later, in the case of n players (n>2), the players’ preferences cannot be
ignored.
Table 2
Payoff Matrix with preference of 2-person BOS games
Player 2
like
football
ballet
ballet
(R, R)
(S, T)
Football
football
(T, S)
(P, P)
like
like Football
ballet
football
football
(R, R)
(S, T)
ballet
(T, S)
(P, P)
like
Player 1
Ballet
Ballet
The preferences of the players are capitals: Football and Ballet.
The decisions are lowercase: football and ballet.
For the sake of convenience, we will keep using ‘football’ and ‘ballet’ as the two choices
in the examples and analysis of n-person BOS games.
Based on the payoff matrix shown in Table 1, the main properties of a 2-person BOS
game are summarized as follows:
1) A player has two choices (i.e. football, ballet), each player likes one of them and
dislikes the other;
6
2) The preferences of the 2 players are opposite;
3) A player will receive higher payoff when both players have the same choice than in
the case of different choices.
4) A player will receive higher payoff if the common choice is what he likes than if the
choice is what he dislikes.
Properties 3) and 4) require that the payoff values satisfy T>S>P>R.
3. n-person BOS games
An n-person BOS game must have very similar properties as a 2-person BOS with the
only difference that the number of players is larger. There are however some difficulties
in extending the major properties of 2-person BOS games directly to n-person cases
without certain modifications. Before defining and discussing n-person BOS games,
some notations have to be introduced. The total number of players is n, the number of
players who like football (dislike ballet) is F, so there are n-F players who like ballet
(dislike football). Denote the number of cooperating players by y, then the number of
defecting players is n-y. The number of players who choose football is x, so the number
of players who choose ballet is n-x.
Property 1) of a 2-person BOS game clearly must apply to the n-person case:
1)
Each player has two choices (i.e. football, ballet), each player likes one of them and
dislikes the other;
7
Next, we will discuss and extend properties 2) to 4) and in the process we will introduce
some additional assumptions about the configuration of the game.
In general, there are two types of players in terms of their preferences: players of the first
type prefer football, and players of the second type prefer ballet. In an n- player 2-choice
situation, some players must have the same preference; therefore we cannot say that the
preferences are conflicting for all pairs of players. Similarly to the 2-person case, we will
exclude the situation when all players have the same preference. Otherwise there would
be no conflict; all players could have the same preferred choice. Under this assumption,
there are two extreme cases about the players’ preferences. The first case occurs when
only one player’s preference is opposite to that of all the others. We have the second case
when the preference of one half of the players is the same, and it is opposite to the
preference of the other half of the players. There are many additional cases in between.
The number of players of either type is between 1 and n-1. Property 2) is now
reformulated as follows:
2) For each player we can find at least one other player with different preference.
Properties 1) and 2) are assumptions about the players’ preferences. Properties 3) and 4)
will give conditions about the players’ decisions and their payoffs. Before continuing our
discussion, we need to make an important additional assumption: every player’s decision
8
has the same influence on the payoffs of the others. This assumption can be interpreted as
a symmetry of the game, when all players have identical role. Hence, the players can be
divided into two groups according to their choices, one (who choose football) has group
size of x and the other (who choose ballet) has group size of n-x. Clearly, 1  x  n  1 .
The symmetry of the game implies that only the numbers of players in the two groups
determine the payoffs. A player is considered to be happier, therefore must receive higher
payoff, when he has the same choice with more players. The payoffs of the players do not
depend on which other players are in the same groups with them, since only the size of
the group matters. Therefore the payoffs of the players are determined by the group sizes
x and n-x. In each group, there are defectors and cooperators. When all players have the
same choice, then, similarly to the 2-player case, defectors should receive the highest
payoff. The lowest payoff should be given to the player who is in the worst situation. If a
player whose decision is opposite to all other players is called a loner, then a loner
cooperator should have the lowest payoff in the game since he is alone and his choice is
against his own preference. Hence lowest payoff occurs when x=1 or n-x=1 and the
player is a cooperator. The worst situation of a loner cooperator is different than that of
the cooperators in the other group. They should receive higher payoff than the lowest
payoff of the game if there are more than one cooperators in that group.
The player’s payoff therefore depends on whether the player’s decision is his preferred
choice or not, and also on the number of other players with the same choice. The payoff
9
of a player who is with all the others but chooses against his preference must not exceed
the highest possible payoff of the player who chooses what he likes.
The above conditions can be summarized as follows:
3) With any given choice, when the number of players in a group increases, the payoff
of any player of the group will also increase;
4) With any given number of players in a group, a player will receive higher payoff
when his choice is what he likes compared to players whose choice is opposite to
their preference.
In addition to these conditions, we also require the following:
5) Any player, who dislikes his choice and this choice is the same as that of all other
players, has higher payoff compared to the case when he likes his choice and being
alone.
4. Payoff functions of n-person BOS games
In n-person games, when the number of players becomes large, the number of choice (or
decision) combinations grows exponentially. There are four choice combinations for
2-person 2-choice PD or Chicken games and when there are n players, the number of
combinations grows up to 2 n . In an n-person BOS game, let i denote the number of
players who like football, clearly 1  i  n 1 as a consequence of property 2). There are
i+1 possible situations (or decision combinations) reflected by the number of cooperators,
10
which can be 0, 1, 2, …, i. In the other group there are n-i players with n-i+1 possible
situations reflected by the number of cooperators in that group. Hence in n-person BOS
games there are altogether
 n  6  n2  1
1 3
2
 i  1 n  i  1   n  6n  n  6 

6
6
i 1
n 1
(1)
possible combinations.
Payoff functions will be used instead of payoff matrices in modeling n-person BOS
games. In n-person Prisoners’ Dilemma and n-person Chicken games, the payoff
functions are modeled as the functions of the number of cooperators. They are based on a
cooperation function and a defection function. If y is the number of cooperators, then a
cooperating player receives C(y) and a defector receives D(y). When the order of the
values T, S, R, P changes, different n-person games are obtained. Figure 1 shows the case
of a 2-person game.
T
S
P
D(y)
C(y)
R
y
0
Figure 1
Cooperation and defection functions
11
However for n-person BOS games, the situation is more complicated. By using only the
number of cooperators as the only variable in modeling n-person BOS games, there
would be only n+1 different game situations to be analyzed. This is much less than the
total number of n-person BOS game situations given in equation (1). The use of functions
C and D alone might also lead to confusion in many situations. Take a simple case when
n=3, and denote the three players by A1, A2, and A3. The 12 unrepeated combinations
(from equation (1)) are listed in Table 3. In columns 2, 4, 6, the capital letter of each
element represents the player’s preference; the lower case letter represents the player’s
decision. Columns 3, 5, and 7 reflect the players’ states: defection (denoted by d) or
cooperation (denoted by c). If we use only the C and D functions (see Figure 2(a)), cases
of rows 7, 8, 10 and 12 repeat those of cases 1, 2, 4 and 6, however they are different
situations. Consider rows 1 and 7 as an example. Row 1 has the combination (Bf, Bf, Fb),
and row 7 has the combination (Bf, Fb, Fb). The number of cooperators in rows 1 and 7
is the same; in both cases the state is (c, c, c), hence the payoffs for the three players are
the same, (R, R, R) for both cases. Another alternative is the following. Consider an
arbitrary player, and let z denote the number of players with the same choice (including
himself). If this player’s choice is what he likes, then his payoff is L(z), otherwise his
payoff is DL(z), where L and DL are given functions. If we use these L and DL functions,
then the payoffs for the players in case 1 are (DL(2), DL(2), DL(1)), the payoffs for
players in case 7 are (DL(1), DL(2), DL(2)). The payoffs of players A1 and A3 are
12
interchanged. In situation 3, A2 is a defector and still receives the highest possible payoff
T of the game with using the C and D functions. The highest payoff T should be given to
the defector only when he is with all other players. Hence the C and D functions alone
cannot reflect this situation right. However, by using the L and DL functions, A2’s payoff
in situation 3 is L(2), which is less than the highest payoff L(3) of the game. Hence the
use of the L and DL functions can describe this situation better. Payoffs from the C and D
functions in situation 5 could violate property 4), since D(1) is always less than S.
L(3)
T
S
L(z)
DL(3)
D(y)
DL(z)
C(y)
P
R
0
1
(a)
2
y
3
z
0
1
C and D functions
(b)
2
3
L and DL functions
Figure 2 Payoff functions for 3-person BOS games
Table 3 Comparison of payoffs based on the C and D functions and those obtained from the L and
DL functions
A1
A2
state
A3
state
Number of
Payoffs by C, D
Payoffs by L and DL
state
cooperators
A1
A2
A3
A1
A2
A3
1
Bf
c
Bf
c
Fb
c
3
R
R
R
DL(2)
DL(2)
DL(1)
2
Bf
c
Bf
c
Ff
d
2
C(2)
C(2)
T
DL(3)
DL(3)
L(3)
3
Bf
c
Bb
d
Fb
c
2
C(2)
T
C(2)
DL(1)
L(2)
DL(2)
4
Bf
c
Bb
d
Ff
d
1
S
D(1)
D(1)
DL(2)
L(1)
L(2)
5
Bb
d
Bb
d
Fb
c
1
D(1)
D(1)
S
L(3)
L(3)
DL(3)
6
Bb
d
Bb
d
Ff
d
0
P
P
P
L(2)
L(2)
L(1)
7
Bf
c
Fb
c
Fb
c
3
R
R
R
DL(1)
DL(2)
DL(2)
13
8
Bf
c
Fb
c
Ff
d
2
C(2)
C(2)
T
DL(2)
DL(1)
L(2)
9
Bb
d
Fb
c
Fb
c
2
T
C(2)
C(2)
L(3)
DL(3)
DL(3)
10
Bf
c
Ff
d
Ff
d
1
S
D(1)
D(1)
DL(3)
L(3)
L(3)
11
Bb
d
Fb
c
Ff
d
1
D(1)
S
D(1)
L(2)
DL(2)
L(1)
12
Bb
d
Ff
d
Ff
d
0
P
P
P
L(1)
L(2)
L(2)
Therefore, it is more reasonable to model the payoffs based on the number of players who
have the same choice rather than based on the number of cooperators in the game.
We have already divided the players into two groups. All players in group 1 choose
football; all players in the other group choose ballet. In each group, there are two types of
players, one type likes his choice, and the other type dislikes it. Hence there are four
kinds of players: like football and choose football, like ballet and choose football, like
football and choose ballet, like ballet and choose ballet. We will use Ff, Bf, Fb, Bb
accordingly to represent these four kinds of players. The first capital letters show the
players’ preferences, the second lower case letters represent the players’ decisions. F or f
indicates football; B or b represents ballet. For the players who choose football we define
two functions. If a such player likes football, then his payoff is L(x), otherwise DL(x),
where x is the number of players who select football. Both functions L(x) and DL(x)
should be monotonically increasing in x. The graph of these functions are shown in
Figure 3. Similar analysis applies to the group who chooses ballet. Hence we have four
payoff functions. For the sake of simplicity, assume that the two groups have identical
payoff functions. Now we have four possible payoffs: L(x), DL(x), L(n-x) and DL(n-x) for
14
the four kinds of players Ff, Bf, Bb and Fb, respectively. Similar payoff functions can be
used to model 2-person BOS games as well, however the resulting payoff values can be
represented as payoff matrices.
Properties 1)–5) of BOS games can be mathematically expressed as follows. As before,
let x denote the number of players who select football, then 1  x  n 1 , functions L and
DL are increasing, furthermore
L(x) > DL(x)
for all x,
(2)
and
DL(n) > L(1).
(3)
We could further generalize the game by introducing different L and DL functions for
players who like football and for those who like ballet. However in our equilibrium
analysis only the symmetric case will be considered.
L(n)
L(x)
L(1)
DL(n)
DL(x)
DL(1)
x
0
1
Figure 3
n
Graphs of functions L and DL
15
5. Linear Payoff Functions
Linear payoff functions can be generally modeled as
L(x)=ax+b
DL(x)=cx+d
x 1
(4)
L’(x)=a’(n-x)+b’
DL’(x)=c’(n-x)+d’,
where x is the number of players who choose football. L and DL are the payoffs of
players who select football and L’ and DL’ are the payoffs of those who select ballet. For
the sake of simplicity, we only consider the symmetric situation when a=a’, b=b’, c=c’
and d=d’. According to the games properties given in relations (2) and (3), the following
conditions have to be satisfied:
a, c >0
d<b
cn + d < an + b
cn + d > a + b.
(5)
As before, let F be the number of players who like football, then 1  F  n 1 , and n-F is
the number of players who prefer ballet. Assume that the state of the system is in an
equilibrium with x players selecting football. Consider first a player with football
preference. His choice is football, if and only if it is not his interest to change his choice
from football to ballet, which can be written as
16
L  x  D L x1 ,
(6)
where we assume that the players’ motivation is to obtain the highest possible payoff, and
if there is no payoff difference between the choices, then the player selects his preference.
This relation can be rewritten as
x
c  n 1  d b
.
ac
(7)
A player with football preference selects ballet if it is not his interest to change his choice
from ballet to football:
D L  x  L x1 ,
(8)
c n d a b
.
ac
(9)
that is
x
Consider next a player with ballet preference. His best choice is football, if
D L x  L x1 ,
(10)
which can be rewritten as
x
a  n 1  b d
;
ac
(11)
and his choice is ballet, if
L  x  D L x1 ,
(12)
a n b c d
.
ac
(13)
which can be rewritten as
x
For the sake of simplicity, introduce the notation
17
x1* 
c n d a
ac
b
x2* 
and
a  n  1  b  d
.
ac
(14)
Clearly,
x1*  x2*  n ,
and relations (5) imply that x1*  0 , furthermore,
1
 n  a  c   2a  2  b  d  
ac 
1
b  d  2a

 0,
 d  b   2a  2  b  d   
ac
ac
x2*  x1* 
so x1*  x2* .
Selects
football
Selects
ballet
Prefers
football
0
Prefers
ballet
x1*
Selects
ballet
x2*
n
x
Selects
football
Figure 4. Selection of the players as function of x
We therefore have the following cases.
Case 1.
If x  x1* , then all players’ best choice is ballet, so x=0 with all players
choosing ballet is an equilibrium.
Case 2.
If x  x2* , then all players choose football, so x=n with all players
selecting football is also an equilibrium.
Case 3.
If x1*  x  x2* , then all players select their preference, so if x1*  F  x2* ,
then x = F is also an equilibrium with all players choosing what they like.
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The differences of the two sides of inequalities (6), (8), (10) and (12) can be interpreted
as follows.
L(x)-DL’(x-1): incentive of those who like football to choose football;
DL’(x)-L(x+1): incentive of those who like football to choose ballet;
DL(x)-L’(x-1): incentive of those who like ballet to choose football;
L’(x)-DL(x+1): incentive of those who like ballet to choose ballet.
In addition,
L(x)-DL(x): payoff difference of those who like football and those who like ballet in the
football-choosing group;
L’(x)-DL’(x): payoff difference of those who like ballet and those who like football in the
ballet-choosing group;
L(x)-L’(x): payoff difference of defectors who like football and defectors who like ballet;
DL(x)-DL’(x): payoff difference of cooperators who dislike football and cooperators who
dislike ballet.
As we have seen earlier, the critical value x1* has the following property. If x  x1* , then
L(x+1)-DL’(x)>0, players who like football have the incentive to choose what they like.
Their incentive values increase when x increases. If x  x2* , then L’(x-1)-DL(x)>0
showing that players who like ballet have the incentive to choose it. Their incentive
values increase when x decreases. We can call the effect of a player’s choice on his own
19
payoff the internality, then L(x+1)-DL’(x) or L’(x-1)-DL(x) is the player’s internality. The
effect of a player’s choice on other players’ payoffs is called the externality.
6. Conclusions
In this paper, n-person BOS games were defined and their properties were analyzed. The
players of n-person BOS games were divided into two groups, in each group the players
have the same choice. The players in the two groups were then further divided into two
subgroups based on their defection or cooperation. Four payoff functions were therefore
defined based on the two possible choices and the two possible preferences. These payoff
functions were modeled as functions of the number of players with the same choice. We
have illustrated by a three-person case that the payoffs of BOS games cannot be
accurately described by using only two payoffs, one for cooperating and one for defecting
players, as it is usual in analyzing prisoners’ dilemma game. For the sake of simplicity,
we examined the case of linear payoff functions with symmetric players. A complete
equilibrium analysis was performed, in which we have shown that there are always two
equilibria, and a third equilibrium exists if and only if the number of football preferring
players is between x1* and x2* .
Acknowledgement. The authors are grateful to the U.S. Air Force Office of Scientific
Research (MURI Grant N00014-03-1-0510) for supporting this research.
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