Available online at www.sciencedirect.com
MATHEMATICAL
AND
COMPUTER
MODELLING
SCIENCE~_____~DIRECT"
F_~EVIER
Mathematical and Computer Modelling 39 (2004) 981-989
www.elsevier.com/locate/mcm
Evolution of Populations Playing
Mixed Multiplayer Games
T. PLATKOWSKI
Institute of Applied Mathematics and Mechanics
Faculty of Mathematics, Informatics and Mechanics
University of Warsaw, Banacha 2, 02-097 Warsaw, Poland
tplatk@mimuw, edu.pl
http ://m~. mimuw, edu.pl/~tplatk
(Received and accepted November 2003)
A b s t r a c t - - W e study temporal evolution of monomorphic populations of individuals which, at each
time step participate in one of the prescribed k-player stage games, k ---- 2, 3 , . . . , n. We introduce
replicator equations for such systems, and investigate the effect of playing different stage games on
the number and stability of stationary states of the populations. When the probabilities of the stage
games depend on time, we introduce a system of replicator equations in a properly modified simplex
of the game, and discuss stationary states for such populations. (~) 2004 Elsevier Ltd. All rights
reserved.
Keywords--Mixed
N-player games, Replicator equations.
1. I N T R O D U C T I O N
In many scientific disciplines, which investigate macroscopic systems of entities (molecules, individuals, agents) with microscopic, mutual interactions, the theoretical description of such systems
is based on binary interactions. For example, in the classical game theory, which has been applied as a useful tool to describe interactions and evolution of various biological systems, the basic
interactions between entities are assumed to be pairwise [i]. In game theory there were good
reasons to limit considerations to binary contests. The multiplayer game theory is, both mathematically and conceptually, much more complicated and less understood than the theory for
two-player contests; quantitative considerations of multiplayer interactions started only recently,
cf., e.g., [2-4], and references cited therein.
An analogous reasoning can be developed when models of the mathematical kinetic theory
are proposed to describe the evolution of complex sociobiological population systems [5]. These
models are such that the microscopic state of interacting individuals includes, in addition to
mechanical variables, also a sociobiological state. The above-cited paper still uses binary interactions, although the need of multiple interactions is stated in some papers, e.g., [6], where the
kinetic theory approach is used. In the real world, the multiplayer contests, in which more than
two players are in conflict, are common. Such interactions are important in biological world (bird
The author was supported by the Polish Gov. Grant KBN No. 5P03A 025 20.
0895-7177/04/$ - see front matter (~) 2004 Elsevier Ltd. All rights reserved.
doi: 10.1016/j.mcm2003.11.002
Typeset by ~4MS-TEX
982
T. PLATKOWSKI
nesting, competition between male bees for the access to the queen) as well as as in economy
and human behavior (competition of firms on economic markets, financial markets). In the latter
case, one of the recent examples of mathematical models is the celebrated minority game, in
which an odd number o f agents compete t o b e in minority [7].
In general, the payoffs from such multiplayer contests can be different from binary ones. In the
former case, all the players take part in one, simultaneous contest. In the symmetric contests, the
payoff of each player depends on the number of other players who use specific strategies allowed in
the game, and is not directly related to two-player games, in which the same strategies would be
used. Such multiplayer contests can significantly change the time evolution and the equilibrium
properties of emerging stationary states of the population, cf. [2].
Similarly, one can corlsider strategic situations, in which the entities of the population are, at
various time steps, engaged in the contests of different order, i.e., play, at different time steps, with
a different number of opponents. In other words, they can participate at different multiplayer
games in subsequent time steps.
In this note, we present a theoretical setting, and study evolution equations for populations
of identical individUals (players, entities) who, at each time step, participate at one of the stage
k-player games, k = 2, 3 , . . . , n. The set of n - 1 admissible games is prescribed at the beginning
of the contest and does not change in time. Each individual plays only one such stage game at
each time step. The players can use, at each stage game, a strategy from a given, finite set S
of strategies, the same for all the players. As an example, one can have in mind the prisoner's
dilemma game with two available strategies C and D, which can be played in the groups with
various number of participants, cf., e.g., [3] for the multiplayer prisoner's dilemma. Similarly, one
can consider hunting in groups with different numbers of predators, where the individual profit
depends not only on the strategy chosen, but also on the size of the hunting group.
The choice of the game played by each individual is based only on the probability of playing the
admissible stage games in the whole population. The probabilities of the k-player stage games can
be exogenous or endogenous. In the former case, the probabilities are fixed constants, whereas
in the latter they depend on time, and are determined by the actual performance of the system.
We introduce theoretical frameworks for both cases, with a new system of replicator equations
for the stage games probabilities in the latter case. The relevant generalized state vector evolves
in an extended phase space, determined by the strategy profile and the stage games probabilities
profile vectors. We formulate the evolution equations for the frequencies of the strategies and
probabilities of the stage games in the general case, and discuss equilibrium states for the case
of the stage games of order two and three with two admissible strategies.
In both cases, we consider systems with arbitrary finite number of available strategies and
stage games of different order, and discuss in more detail the populations playing two-player and
three-player games. We find interesting situations in which two-player and three-player games,
when played separately, lead to extinction of a strategy from the population, whereas, when
played simultaneously, result, in the long run, in survival of all the strategies present initially.
In the next section, we introduce notation and define the mathematical model in the general
case of s strategies and n - 1 admissible k-player games. In Section 3, we consider the case of
s = 2 strategies. In Section 4, we study particular case of n = 3, i.e., the population playing twoand three-games games in more detail. In Section 5, we generalize the model to the case of time
dependent probabilities of the stage k-games, and study a particular case with two strategies and
two stage games (k = 2, 3). In the last section, we discuss generalizations and future work.
2. M I X E D
k-GAMES
WITH
s STRATEGIES
We consider a large population of identical individuals (players, entities), which at each time
step are randomly matched into groups consisting of k players, k = 2, 3,... ,n, and play a
prescribed k-player game. Each player, at a given time step, belongs to one group only, i.e., plays
Evolution of Populations
983
one of the admissible games. The collection of n - 1 k-player games is fixed, as well as the set S
of s strategies available to the players. Thus, S = { S l , . . . , ss}, is the same for all the players.
At each time step each player can use any of his/her s strategies, s l , . . . , ss. In other words,
we assume that the set of strategies available to the players is the same for each of the k-player
games. It follows that at each time step the population can be divided into s groups, each playing
a different Strategy si, i = 1 , . . . , s. As an example, one can consider the population playing the
prisoner's dilemma game, with n -- 3 and S -- {C, D}. The players are matched into pairs or
triples, and play the two-player (k = 2) or the three-player (k = 3) prisoner's dilemma game,
using either strategy cooperate (C) or defect (D). The payoffs in the latter game can be chosen
either as a weighted average of the binary payoffs [3], or in a more general way, conserving the
essence of the prisoner's dilemma.
We assume that each of the k-player stage games is symmetric in the following sense: the
payoffs do not depend on the positions of the players but rather on the strategies played. For
brevity, we call such games k - g a m e s in the following.
We denote c~k~the probability that an individual participates in the k-game, with ~ = 2 ak = 1,
which can be related to the frequencies of the the k-games, cf. the Appendix.
We further denote
.....
.....
(1)
- - t h e payoff of Strategy j (j = 1 , . . . , s - 1) in the stage k-game (k -- 1, 2, . . . . n), if i~ players
use the first (j = 1) strategy, ... , i j + 1 players use the j strategy (there has to be a player
using this strategy, otherwise the payoff is set to zero), . . . . , i s - 1 players use the s - 1 strategy.
The payoff of Strategy s is denoted by P ikl ..... i j ..... i~-l,s" Note that with this notation for the
payoffs, the number is of players which use the last Strategy s is given by the formula is =
k - (1 - 6is ) - il - i2 . . . . .
i s - l , where 5is denotes the Kronecker's delta.
Denote x j - - t h e frequency of Strategy j (j = 1, 2 , . . . , s) in the whole population, with
=1.
(2)
j=l
The state of the population is determined by the profile vector x -- (xl, x 2 , . . . , xs), which belongs
to the s-dimensional unit simplex.
We also denote W ~ - - t h e average payoff of Strategy sj, j = 1, 2 , . . . , s in the stage k-game.
The expression for W~ is obtained from the polynomial formula. Let us consider a player using
his/her jth strategy. The remaining k - 1 players are distributed among all available s strategies
in arbitrary ways. With the notation
Nk_ 1
=
~1..... ~ - 1
illi2!...is-l!(k-
(k - 1)!
l - il .....
(a)
is-l)!'
one obtains, assuming statistical independence of the players, cf. [1].
k--1 k - 1
Wk = E
Q=0
k--1
E...
i2=0
E
hrk-1
,,~k
xil~i~..,
"~'il,...,is_lFil,...,ij~-lp..,is-l~j
1 2
Xsis-Sis "
(4)
is-l=0
The relevant expressions for W~ for the k-games with s = 2 strategies are discussed in the next
section.
Let W j be the mean payoff of Strategy j, averaged over all types of the games played in the
population
Wj - - ' E akW~,
k~2
j ---- 1 , 2 , . . . ,s.
(5)
984
T. PLATKOWSKI
Finally, let us denote W - - t h e mean payoff in the whole population for the given game profile
x = (Xl,...,~s)
8
w= xjw
(8)
j=l
We assume, cf. [1], that the rate of change of the frequency of each strategy depends on the
difference between its mean payoff and the mean payoff in the whole population. The system of
replicator equations for the frequencies xj reads
d
~ x j = xj (Wj - W ) ,
j = 1,...,s.
(7)
It is a system of s nonlinear ODE with polynomial rhs. Thus, with the Lipschitz property of
the rhs the theorem of existence and uniqueness of the solutions is straightforward. The strategy
profile vector follows a curve in the unit simplex of the strategy space. Singular points of the
system determine equilibrium states of the population. Their analytical determination is in
general a difficult task. In this paper, we consider the case of s = 2 strategies. In Section 3, we
specify the evolution equations for. s = 2 and arbitrary n. In the next two sections, we carry out
analytical considerations and discuss numerical examples for n = 3, i.e., the population playing
two- and three-games.
3. k - G A M E S
WITH
TWO
STRATEGIES
Consider the k-game with s = 2 strategies. In this case, the mean payoff of j = 1, 2 strategies
in the k-game reads
k--1
Wk ----E Nk-Z-k
"
i pk-i,i Xk--Z-i~l
i
~ - zi) ~,
(S)
i=0
k-1
W2k
E
,,rk-1
Iv
i p kk _ l _ i , 2 x ik - l - i , . ,(i - xl)',"
(9)
i=0
with p0k,1 = pk,2 ----0.
The above expressions are particular cases of the general expression for W~ given in the previous
section. Thus, for the k-game with two strategies,, the considered model is determined by 2k
payoffs (k for each strategy): P~,r is the payoff for Strategy 1 if exactly r (r e {1,2,...,k})
players play Strategy 1, prk,2 is the payoff for Strategy 2 if exactly r (r e {0, 1 , . . . , k - 1}) players
play Strategy 1 (note different sets of indices, which corresponds to the fact that if all the players
play Strategy sl then the payoff for the Strategy s2 is zero).
After some algebraic manipulations, changing in an adequate way indices in the sums, and
using the relation N~ = N~_k, the equation for evolution of Strategy Xl can be rewritten as
d
~-~Xl = X l
n
n
k--1
(1 - ml) E C~k(Wt - W~) = m1(1- mz)E c~kE a~m~(1-ml) k-l-i,
k=2
k=2
(10)
i=0
where
ak
i : . Y ~k-i
_ i _ 1 (pk+l;1
- -
k
Pi,2)
i
=
0, 1 , . . . , k -
1.
(11)
Note that the coefficients in the second sum in the above equation depend, for given k, on k
coefficients which are the differences between the relevant payoffs rather than explicitly on the 2k
payoffs coefficients pik,j. This reduces the dimension of the corresponding parameter phase space
of the problem.
Evolution of P o p u l a t i o n s
985
4. P O P U L A T I O N
PLAYING
TWOAND
THREE-GAMES
WITH
TWO
STRATEGIES
In this section, we study a particular case: n = 3, s = 2, i.e., the population playing two-players
and three-players games, with two available strategies, the same for both games.
1. For k = 2: we consider the general payoff matrix for two-players symmetric games, with
the payoff matrix aij, which satisfies, for simplicity, the symmetry condition a12 = a21.
We obtain p2,1 = a n = A1, P0,2
2 --= a22 = A2, p,2,1 ~--Pl,2
2 -~ a12 = A 3 .
2. For k = 3, i.e., for three-players games, with the notation p3a,1 = ea, p2a,, = elh, p31,1 ~ e2h,
3
a
P2,2
= eld, Pl,2
= e2d, p03,2 = e4, the contribution of the corresponding payoffs to the
replicator depends on three parameters only: a0a = e2h--e4, ela = 2(elh--e2d), aa2 = e3--eld.
As an example, one can consider a three-players version of the hawk-dove game, with two
strategies: hawk (H), and dove (D), and with the following payoffs from the three-players payoffs:
ea-
the payoff for H from the contest in which all three players use the Strategy H,
elhthe payoff for H from the 2 H and 1 D contest;
e2ht h e H payoff from the 1 H and 2 D contest;
e l d - - the D payoff from 2 H and 1 D contest;
e2d - - the D payoff from the 1 H and 2 D contest,
e4 - - the payoff of D from the 3 D contest.
Note that in the case of the so-called super-symmetric games [2] e l h = e l d and e2h = e2d , which
reduces the number of the free parameters in the model. However, the final equations depend on
the differences of the payoffs rather than on the actual payoffs, and thus the distinction between
the super-symmetric and symmetric games becomes irrelevant for our purposes.
With this notation, in the case of two- and three-games with two strategies, the replicator
equation reads
d
~-~Xl = Xl ( X l ' - - 1) ( E x 2 + F X l + ( 7 ) ,
(12)
where
E
=
F=a3
c
-3
-
a0 -
_=
3A,
(2a 3 - a 3) - ( 1 --aa)-~l ~- oe3B - ( 1 - a a ) - ~ l ,
= -aaa
a +
(1 - aa)22 = a3C + (1 - ~a)A2,
A1 = A1 + Au - 2Aa,
A2 = A2 - A3.
The number of stationary points in the physically relevant simplex (here the interval [0, 1])
depends on the coefficients defining E, F, G, i.e., on three coefficients which are differences of
the relevant three-players game payoffs, two coefficients A1, A2, depending on the payoffs of the
two-players game, and the frequency c~3 o f the three-players game.
Thus, apart from xl = 0, xl = 1 equilibria, there can be one or two additional internal
equilibrium points. For brevity, we introduce notation SEP for stable internal equilibrium points,
and UEP for unstable ones.
REMARK 1. For ol 3 = 0 (i.e., pure two-player game) E = 0, therefore, as expected, there exists
at most one internal equilibrium point, given by xo = A 2 / A 1 , cf., e.g., [1]. For ~3 = 1 (pure
three-player games) the equilibrium points are discussed, e.g., in [2,8].
In generic cases, if there are two internal equilibrium points, one of them is stable (SEP),
another one unstable (UEP). The sufficient condition for the existence of two internal equilibrium
points in the unit simplex in this case is given by three relations
A = F 2 - 4 E G > 0,
E # 0,
0<
- F TE v ~
< 2.
(13)
986
T. PLATKOWSKI
Below we discuss several examples. First, we discuss a case for which both two-game and
three-game do not possess interior equilibrium points, however, an arbitrarily small frequency of
three-game in the population playing a two-game results in emergence of an SEP. In this case,
both strategies are present in the stationary state of the population.
Denote V = (A1, A2,A, B, C, a3) the vector of parameters determining the model. The first
two parameters correspond to the relevant combinations of payoffs in the two-games, the next
three to those in the three-games, whereas a3 defines probability of the three-game. Note that
each such vector describes a whole class of two- and three-games, with relevant combination of
payoffs.
EXAMPLE 1. Let us consider the class of games defined by the parameters vector V = (-1, 0, 1,
1.5, - 1 , a3). One can easily show that for the pure two-game (a3 = 0) the only equilibrium points
are boundaries x = 0, x = 1. For as > 0, one obtains the following.
/
LEMMA 1. For V = ( - 1 , 0 , 1 , 1 . 5 , - 1 , a 3 ) , and any as 6 (0,1] there ex/st exactly one SEP
e (0, 1).
Thus, for as > 0 the boundary x = 0 equilibrium point "bifurcates" into SEP in (0, 1). The
proof follows by calculating the roots of the relevant second-order polynomial on the rhs of the
replicator equation and verifying the sufficient conditions for the existence of the relevant roots.
We omit details.
In Example 2, below we discuss an interesting situation in which both two- and three-games,
when considered individually, do not have equilibrium points inside the unit simplex, however the
mixed two- and three-game with well-specified proportions of both games possesses an internal
SEP. Thus, if the proportion of two stage games is adequate, the temporal evolution of the
system, as governed by the relevant replicator equations, does not eliminate any strategy from
the population in the long run; for the initial frequencies of both strategies in the relevant basin
of attraction both strategies are present in the stationary state of the populationl opposite to the
long run limits for both games considered separately.
EXAMPLE 2. For the parameters vector V = (0, 1, - 6 , 4, - 1 , as), both pure two-game and threegame have boundary equilibrium points x = 0, x -- 1 only. Then, from the analytical expressions
for the roots of the corresponding polynomial one obtains the following.
LEMMA
2.
1. For ~3 E [0, 0.25] and ~3 e (0.75, 1], there are no internal equilibrium points.
2. For a3 E (0.25, 0.5], there exists one, internal UEP.
3. For a3 E (0.5, 0.75) there exists two internal equilibrium points: SEP and UEP.
Thus, the adequate proportion of three-games can have a stabilizing effect on the population-by creating an internal stable equilibrium point, not existing in the case of pure two- and threegames. In particular, the adequate proportion of the three-game results in surviving of both
strategies in the long run; both strategies have nonzero frequencies in the stationary state of the
population for properly chosen initial data.
Similar phenomena can occur in the "opposite" case of populations playing mostly three-games,
with small probabilities of two-games. Below we discuss a numerical example.
EXAMPLE 3. We take the parameter vector: V = ( ' 1 , 1, 1 , - P - 1, P, a3) with
(A) P = - 2 .
Then the pure three-games (as = 0) have boundary equilibrium points only, whereas small
enough proportions of two-games result in emergence of an SEP. More in particular, one can
prove the following.
LEMMA 3 A .
1. There exists a3 6 (0.33, 0.34) such that for any 1 > a3 > oz3 there is one SEP in (0, 1).
2. For any as 6 [0, ~3) there are no internal equilibrium points.
Evolution of Populations
987
Thus, small enough p e r t u r b a t i o n s of the three-game w i t h o u t internal equilibria by the twog a m e w i t h o u t internal equilibria can result in a m i x e d g a m e with an SEP.
(B) P = 0.5.
Here the pure three-game possesses an internal UEP.
LEMMA 3B. There exists a bifurcation p o i n t 53 C (0.965, 0.966) such t h a t for any 03 E (53, 1)
there are two internal equilibrium points. For c~3 = 53, the equilibrium p o i n t s merge, and for
~3 < 53, there are no internal equiIibria.
A n analogous situation occurs for the same class o f three-games (i.e., P = - 2 or P = 0.5) and
A1 = - 2 , fit2 = - 1 , i.e., w h e n the two-game has an internal unstable equilibrium point.
5. k-GAMES W I T H VARIABLE F R E Q U E N C I E S
In the theory presented above, probabilities of the stage games were constants, exogenously
fixed at the beginning of the game. In this section, we generalize the model, and allow the
probabilities of various stage games to depend on time. Moreover, we shall assume that the
time dependence is regulated by the evolutionary forces of the system itself. The mathematical
description of this assumption will be provided by adequate systems of replicator equations.
We consider the population with s strategies, with time-dependent probabilities of the stage
k-games
n
(14)
k=2
and assume that the rate of change of the probability of the k-game is proportional to the difference between the mean success of the k-game and the average success of the whole population.
In other words, the games with higher mean payoffs have evolutionary advantage, as measured
by the average fitness of the particular game, and are played more often in the next round.
Let W k be the mean payoff of the k-game, averaged over all the available strategies
8
xjW?,
k=
(15)
j=l
The corresponding evolution equation for ak reads
d
a~
= ok ( w k - w ) ,
k = 2, 3 , . . . , , .
(16)
We limit our considerations to the case of s = 2 strategies, and n = 3, i.e., k = t w o - and threegames, leaving more general cases for future work. Denoting the frequency of the first strategy
by x = Xl, and inserting the relevant expressions for Wjk into the above equation we obtain, after
some algebraic manipulations
d
~-~a2 = a2 (1 - a2) ( I x 3 + J x 2 + K x + L ) ,
(17)
where the coefficients I, J, K do not depend on ~2,
I = cl - dl,
J = c 2 - d2 + d l ,
Cl -~ 2elh -- e2h -- e3,
dl =- 2e2d -- e4 -- eld,
and we put A3 = 0 for simplicity.
K = ca - d3 + d2,
c2 • 2e2h -- 2elh -]- A1,
d2 = 2e4 -- 2e2d -- A2,
L = d3,
53 : --e2h,
da = A2 - e4,
988
T. PLATKOWSKI
The replicator equation for the time evolution of xl = x reads (cf. Section 4)
~X
~.
x(x - - 1)(Ex2+Fx+G)
(18)
where
E = E (a2) = (1 - a2) (a~ - ao~ - a~),
F = F (a2) = (1 - a2) (2a30 - a 3) - a2 (A1 + A2),
G = G (a2) = - (1 - a2) a~ + a2A2.
In this way, we have obtained a system of two coupled ODE for x(t), a2(t) with polynomial rhs.
The phase space is a unit square here, with the vertices being the equilibrium points. There
might also be equilibrium points on the edges, corresponding to a2 -- 0 or a2 -- 1, as well as
equilibrium points inside the unit simplex.
The system has an interesting structure, which allows to determine the internal equilibrium
points analytically. Let (Xo, ao) be an internal equilibrium point. Then Xo is a root of the thirdorder polynomial in the rhs of equation for a2, and of the second-order polynomial in the rhs
of the equation for x (note that coefficients I, J, K, L do not depend on a2). Inserting xo into
the second-order polynomial we obtain a linear algebraic equation for a2. We look for solutions
in the interval (0, 1). In such a way, all the equilibria can be determined. For example, for the
payoff vector: P -~ (A1, A2, eli, e.2j, elg, e2g, e3, e4) = (--5, --1, 3, --5, --2, --5, 3, 1), there exist two
internal equilibria: (x0, a2) -- (0.175, 0.911), and (0.415, 0.739). We leave more general discussion,
and stability analysis of such systems for future work.
6. C O N C L U S I O N S
We presented a mathematical model and analytical and numerical results for monomorphic
populations of entities which, at each time step, play one of the prescribed stage games, with
exogenously or endogenously determined probabilities of the particular games. In this note,
we focused on general strategic settings rather than on particular applications to evolutionary
systems.
We found interesting classes of stage games, which, when played separately, do not have internal
stable equilibrium points, therefore, for large time scales, lead to extinction of one of the strategies,
whereas when played together in well defined "proportions", lead the population to the states in
which all the strategies are present in the long run.
In a similar way, one can consider other classes of games, e.g., perturbations of two-games by
four-games, etc.
For games with more than two strategies, the phase space is at least two dimensional. We
discussed here only the systems of players with two admissible strategies, leaving a general analysis
for future research. We also note that in the case of time dependent probabilities of the stage
games one could consider evolution equations for the frequencies f~k of the stage k-games rather
than on c~k, cf. the Appendix. Still another possibility would be to assume t h a t the probabilities of
the stage k-games are proportional to their mean payoffs. The relevant equations, and analytical
treatment of the corresponding equilibria is left for future research.
Similar phenomena as discussed in the paper occur, e.g., in mechanical engineering, when two
unstable systems, when merged together, become a stable one. One could also think of the
discussed results (in particular, when two stage games without stable internal equilibrium points,
when played sequentially with the adequate frequencies, result in an extended game, in which
stable internal equilibria exist) in connection with the recently discovered Parrondo paradox of
the game theory, where two losing games, when play one after another, become winning [9].
Evolution of Populations
APPENDIX
989
A
Let us denote Nk--the total number of individuals playing the k-game, N - - t h e total number
of individuals in the population. Thus, for each k ~ {2, 3 , . . . , n} there are Nk/k stage k-games
n N k. We introduce
played in the whole population, and N = ~k=2
Nk/k
(19)
I=2
--the frequency of the k-game, k = 2, 3 , . . . , n, with }-]2=2 flk = 1.
There is a relation between the frequencies ~k and the parameters ak = Nk/N, denoting the
percentage of the total population, playing the k-game (probability of the k-game)
~k--
n
c~k/k
k = 2,...,n
(20)
/=2
(note that ak can be viewed as the probability, that a randomly chosen player participates at the
stage k-game). This system of n - 1 equations can be solved for ak.
REFERENCES
1. J.W. Weibull, Evolutionary Game Theory, MIT Press, (1995).
2. M. Broom, C. Cannings and G.T. Vickers, Multi-player matrix games, Bull. Math. Biol. 59 (5), 931-952,
(1997).
3. M. Matsumura and T. Ikegami, Evolution of strategies in the three-person iterated prisoner's dilemma game,
J. Theor. Biol. 195, 53-67, (1998).
4. T. Platkowski, Stationary self-organized states in many-particle systems with ternary interactions, Applied
Mathematics and Computations 146, 701-712, (2003).
5. L. Arlotti, N. Bellomo and F. DeAngelis, Generalized kinetic (Boltzmann) models, mathematical structures
and applications, Math. Mod. Meth. Appl. Sci. 12, 567-591, (2002).
6. B. Carbonaro and N. Serra, Towards mathematical models in psychology: A stochastic description of human
feelings, Math. Mod. Meth. Appl. Sei. 12, 1465-1500, (2002).
7. D. Challet and Y.C. Zhang, Emergence of cooperation and organization in an evolutionary game, Physiea A
246, 407, (1997); http ://~rww.unifr, ch/econophysics.
8. J. Bukowski and J. Miekisz, Evolutionary and asymptotic stability in multi-player games with two strategies,
Inst. Appl. Math. Mech. R W 02-15, (Preprint).
9. G. Harmer and D. Abbot, Losing strategies can win by Parrondo's paradox, Nature 402, 864, (1999).