Figures
Abstract
A direct sum form is proposed for constructing a composite game from two games, prisoner's dilemma and snowdrift game. This kind of direct sum form game is called a multiple roles game. The replicator dynamics of the multiple roles game with will-mixed populations is explored. The dynamical behaviors on square lattice are investigated by numerical simulation. It is found that the dynamical behaviors of population on square lattice depend on the mixing proportion of the two simple games. Mixing SD activities to pure PD population inhibits the proportion of cooperators in PD, and mixing PD activities to pure SD population stimulates the proportion of cooperators in SD. Besides spatial reciprocity, our results show that there are roles reciprocities between different types of individuals.
Citation: Ma C, Cao W, Liu W, Gui R, Jia Y (2013) Direct Sum Matrix Game with Prisoner's Dilemma and Snowdrift Game. PLoS ONE 8(12): e81855. https://doi.org/10.1371/journal.pone.0081855
Editor: Matjaž Perc, University of Maribor, Slovenia
Received: April 2, 2013; Accepted: October 17, 2013; Published: December 20, 2013
Copyright: © 2013 Ma et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: This work was supported by the National Natural Science Foundation of China under 11175068 (YJ). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
Introduction
As mathematical framework, evolutionary game theory has been used to model evolutions in social, economical and biological systems widely [1]–[3]. One of the fundamental problems in this theory is the evolution and maintenance of cooperation among selfish individuals [4]–[8]. Two prominent mathematical metaphors have attracted most attention in theoretical and experimental studies of cooperation, the prisoner's dilemma and the snowdrift games [9]–[11]. In both games, each player decides whether to cooperate or defect. In the prisoner's dilemma(PD), cooperation of the players results to the highest payoff which is equally shared among the two players, yet individual defectors will do better if the opponent decides to cooperate. Since the selfish players want to maximize their own income and they both decide to defect. None of them gets a profit and instead of equally sharing the payoff received by mutual cooperation, they end up almost empty-handed. The snowdrift(SD) game has different payoffs matrix compare with the PD. It is an interesting alternative for the study of cooperation, and individuals in such game can gain access to benefits for the pair at one individual cost. Cooperators have to bear the costs whereas defectors are not. Both games represent social dilemmas [12], in which defectors are prone to exploit cooperators, and have an evolutionary advantage over cooperators in populations. But cooperation is widespread in the real world and required for many levels of biological organization ranging from genes to groups of animals. Cooperation is also the decisive organizing principle of human societies. Therefore, the underlying mechanisms of cooperation are much needed and have been investigated extensively in different contexts, such as kin selection [13], direct reciprocity [4], [9], [14]–[16], indirect reciprocity [17]–[19], group selection [20]–[22] and network reciprocity [23], [24], which are summarized as five rules [25]. The network reciprocity could also be regarded as a generalization of spatial reciprocity.
To study the dynamical behaviors of a spatial game, the following points should be considered: the spatial structure on which the game runs, the interactive ways of individuals, the updating and mutation rules. The most common spatial structure is the square lattice [23], in which the cooperators can form clusters to protect themselves against the exploitation by defectors. Small-world networks, scale-free graphs and evolving networks may also serve as the spatial structures of a game [26]. By adopting coevolutionary rules, coevolutionary games can rearrange network structures [27]. For instance, Chen et al. [28] have proposed a coevolutionary rule in spatial public goods games. The interactive ways are determined by payoff functions, which are known as payoff matrices in matrix games, and the scope of opponents, such as Von Neumann and Moore neighborhood in square lattice, pairwise interaction, group interaction, etc. Strategy updating and mutation rules may also affect evolutionary dynamics. Some examples include the birth-death and imitation rule [29], the proportional imitation rule [30], the reinforcement learning adoption rule [31], or the Fermi rule [32]. Wang et al. [33] have recently considered an adaptive strategy-adoption rule in which the focal player evaluates its strategy by comparing the average payoff of each strategy in the neighborhood. They have shown that the survivability of cooperators has a significant increment in contrast with that of pairwise strategy updating. Based on win-stay-lose-shift rule, Liu et al. [34] introduced a win-stay-lose-learn strategy updating rule in spatial prisoner's dilemma game, where the focal player attempts to update her strategy only when her payoff is less than her aspiration.
Most previous works about PD and SD game have been referred to the consideration of the individual interaction being one fold, which means that all interactions adopt either PD or SD payoff matrix [32]–[35]. That is all individuals participate in the same kind of game. However, the situation that individuals involved in multiple activities is happened frequently in economic and social activities or biological behaviors. In the context of game theory, individuals participate in multiple games and act different roles in these games. Such games can be called multiple roles game(MRG).
In the present work, a simple MRG model with PD and SG payoff matrix is constructed. In the modeling approach section, we describe the model for multiple roles games with prisoner's dilemma and snowdrift game. In the results section, the replicator dynamics of the multiple roles game with will-mixed populations is explored, and the dynamical behaviors on square lattice are calculated in a numerical way over the parameter space. In the last section, conclusions are presented and the potential clues to other general cases are predicated.
Modeling Approach
To elaborate the idea of MRG, let's focus on two-strategy games. The two strategies can be denoted as cooperation and defection, abbreviated as and
, respectively. The payoff matrix has the form
(1)Here, a player with
strategy obtains profit
from another
player,
from a
player. Similarly, a
player obtains
from a
player,
from another
player. The Prisoner's dilemma(PD) game starts on the condition as following:
(2)The relationship
implies that mutual cooperation is superior to mutual defection, while the relationships
and
imply that defection is the dominant strategy regardless of opponent's strategy. The SD game has a different condition as shown below:
(3)In contrast to PD, the best strategy now depends on the opponent's strategy. The relationship
implies that cooperation is better if the opponent defects, while
implies that defection is better if the opponent cooperates.
The general payoff matrix form(1) has too many parameters to be analyzed. To prevent this kind of situation, the payoff matrix form with two parameters is given for two-strategy game as following [36], [37],(4)with
and
. If
, (4) indicates PD payoff matrix, and if
, it indicates SD payoff matrix. If a MRG model with PD and SD is constructed with the payoff matrix (4), it would have four parameters which will lead to complexity of analysis. Therefore, it is necessary to construct single parameter payoff matrices of PD and SD. One can rescale the payoff matrix (4) to single parameter forms [10], [11], [35], [38] as shown in Eq.(5)
(5)for the PD game, and
(6)for the SD game. Both
and
are constrained to the interval
. Obviously, (5) and (6) satisfy the corresponding inequalities (2) and (3) respectively.
Now consider a population where each individual has two identities: prisoner for PD game and driver for SD game. Each individual participates in two games simultaneously. When two individuals interplay, the process is divided into two steps. In the first step, they have to determine what kind of game they will play, and in the second step, they play the selected game according to their own strategy. Let us assume that in a pairwise interplay they choose PD game with probability , SD game with probability
. In this way, a simply MRG model is constructed(Fig. 1). The left panel of Fig. 1 shows that there are two channels of interaction between the individual A and B, the PD channel with probability
and the SD channel with probability
. The right panel of Fig. 1 shows the details of the payoff matrix of MRG. Here we denote strategies with capital letters
and
in PD game, and lowercase letters
and
in SD game. Formally, an extended payoff matrix
can be obtained by direct sum for the MRG,
(7)
There are two interaction channels between individual A and B. The channel PD has a probability to be selected and the channel SD has a probability
to be selected.
Since it is impossible that an individual who opts for PD game interacts with the individual who has made a choice of SD game with or
strategy, we express these cases with two sub-matrices 0 in (7).
To construct a spatial model of MRG, let's place each individual on one cell in a two-dimensional square lattice with periodic boundary conditions being used. There are no empty cells on lattice. An individual interacts with its Moore neighborhoods, and its total payoff is the sum of payoffs obtained from its all neighbors. The population is asynchronously updated by the rule known as the replicator rule or proportional imitation rule [30], [36], [39]. Randomly selecting a focal individual
, let
denotes its total payoff,
its category, or its strategy set. Individual
updates its strategy set by comparing its total payoff
to the total payoff
of a randomly selected neighbor
. The focal individual
adopts
's strategy set
with a probability
proportional to the payoff difference, provided that
. So the probability
is given by
(8)where
denotes a normalization constant to ensure
. Here
(9)where
represents the number of
's Moore neighbors. After updating its strategy set, the payoff of the focal individual
with that of its all neighbors are reset to zero. Note that the current MRG model does not take into account mutations in updating strategy sets, and individuals propagate every strategy of its each role to the next generation perfectly.
Results
Replicator dynamics of the multiple roles game
According to the strategies individuals adopt, they can be divided into four categories, which can be recorded as ,
,
,
. With this convention,
means that the individual adopts strategy of cooperation in PD games, and defection in SD games, and the other three categories are similar to
. In the replicator dynamics,
,
,
,
represent the fractions of
,
,
,
respectively. Without arising confusions, sometimes
,
,
,
also represent the fractions of the corresponding types of the individuals. The payoff matrix for the four strategies is:
(10)
The deterministic evolutionary game dynamics are given by the replicator equation [40], [41] . Here
, and
. Let
, it is not difficult to get eight equilibrium points:
,
,
,
,
,
,
,
. Apparently, only the last two are related to the parameter
. Consideration of the constraint
, the seventh and eighth are only valid as they meet criterion
and
, respectively. It shows that the mixed probability
of PD and SD can influence the number of the equilibrium points.
Effect of the mixed probability 
on population structure
Simulation runs on a lattice with four categories individuals initial randomly uniform distributing over it equiprobably. To ensure a correct convergence,
time steps are employed [36], where one time step means each individual updates its strategies one time averagely. Preliminary simulation results shows that, for
from
to
step
, the two-dimensional
parameter plane can be divided into four areas roughly as shown in Fig. 2. As expected, individuals tend to cooperate for smaller
or
values in both games, to defect for larger
or
values. As a result,
individuals locate mainly in left bottom region on
plane,
in left up region,
in right bottom region, and
in right up region. For more details, we set the parameter
at the interval
with step
,
at
with step
, and for
from
to
step
. For each
, we introduce average fractions of
,
,
,
over the region
and
in the
phase plane. These global indexes fall in the range
. Fig. 3 shows the simulation results of the influence of
on the average fractions of the four categories of individuals. The left endpoints of the curves correspond to the case
, which indicate the population structure with pure SD game. The right endpoints of the curves correspond to the case
, which indicate the population structure with pure PD game. Comparing with these reference endpoints the average fractions of
and
form two convex curves, and the average fractions of
and
form two concave curves. It is quite obvious that from the global perspective, mixing SD and PD games will stimulate
and
, and inhibit
and
individuals.
Simulation runs with the parameters as step
,
step
.
For each , the simulations run on
square lattice with
in discrete
parameter plane. The initial average fractions of the four types are all
.
For comparison with the pure PD and SD games, let represents the average fraction of players with
strategy in PD games, which equals to
, and
represents the average fraction of players with
strategy in SD games, which equals to
. The average fractions of
and
are given in Fig. 4 as functions of
. The
curve increases with the increasing mixed probability
, which indicates cooperators of SD increase as
increases. It means that mixing PD game in SD population can stimulate cooperate fraction in SD. The average fraction of
forms a concave curve and stills bottoming out in the area
to
. From the view of PD, mixing SD game in PD population tends to inhibit the cooperate
strategy in PD game. The extent of inhibition reaches a maximum at
to
. The population fractions distributing in the phase plane can reveal more details of the evolutionary dynamics. Fig. 5 shows the fraction of
distributing in
parameter plane. As a frame of reference, Fig. 5A(
) represents the case of pure SD population, in which the
individuals are mainly distributed in the low value zone of
, or the bottom of
plane, and have no association with the parameter
. Fig. 5B,C,D(
respectively) show that with the increase of
, the
region expands slowly to the zone of the larger
value. This is in consistency with the
curve in Fig. 4. Fig. 6 shows the the fraction of
distributing in
plane. As a frame of reference, Fig. 6A(
) represents the case of pure PD population, in which the
individuals are mainly distributed in the low value zone of
, or the left of
plane, and have no association with the parameter
. Fig. 6B,C(
respectively) show that with the decrease of
, the
region compresses to the zone of the smaller
value inhomogeneously. But one should note that for
(Fig. 6 D), the
region has an anomalous expansion comparing with that of
(Fig. 6 C). This kind of anomalous expansion is consistent with the curve
in Fig. 4 which decreases first and then increases with
from
to
. To explore the origin of the
anomalous expansion at lower
values, it should be noted that
. The fractions of
and
in
parameter plane are shown in Fig. 7 A–F for several different
values. From Fig. 7 A,D we know that the anomalous expansion of
for
is originated from that of
.
For each , the simulations run on
square lattice with
in discrete
parameter plane. The initial average fractions of the four types are all
.
The initial average fractions of ,
,
,
are all
and the individuals are randomly distributed on lattices evenly in the beginning.
The initial fractions of ,
,
,
are all
and the individuals are randomly distributed on lattices evenly in the beginning.
For the first column p = 0.1; the second column p = 0.4;the third column p = 0.7.
It is well known that if there is only one kind of game in population, such as PD or SD game, there exists effect of spatial reciprocity in lattice space [23], [24]. But if an individual in lattice space is drawn into multiple games, its different strategies in each game may lead up to the situation: what one loses on the swings, she gets back on the roundabouts. In the current situation, it becomes that what one loses as a prisoner in PD, she gets back as a driver in SD, or vice verse. This phenomenon can be called as roles reciprocity. This effect superimposes on that of spatial reciprocity to influence the fractions of individuals on payoff matrix parameter space plane.
In the PD and SD MRG case, from comparing the fractions of and
(Fig. 7 D–F, G–I) in
parameter plane, it is found that the regions of
and
extend to the side of each other. This phenomenon is a typical demonstration of roles reciprocity. Roles reciprocity strengthens the survival ability of
and
players. Although individuals with strategies
or
have no maximal fractions in four categories(Fig. 3), they have more tolerance in unfavorable parameters(Fig. 4
curve) in appropriate parameter intervals of
,
and
.
To illustrate the effect of roles reciprocity, we cancel the interaction between the and
individuals for simulation with
. Comparing with roles reciprocity(the case of existing interaction between the
and
), the average fractions of
and
descend in the absence of interaction between them(Fig. 8). In
parameter plane, the effect of roles reciprocity become ever more evident(Fig. 9). Without the roles reciprocity, the cross domain of
and
vanished completely(Fig. 9 B). It can be say that
and
individuals establish a ecological chain in a small parameter region with about
and
at
. The fact that the region is too small reflects the frangibility of that toy ecosystem.
Without the interaction between and
, their average fractions descend obviously.
A) The fraction of with the interaction between
and
. The cross domain of
and
obviously exists. B) The fraction of
without the interaction between
and
. The cross domain of
and
vanishes completely.
Discussion
In the real world, it is common that an agent acts as multiple identities. It can be say that the agents participate in multiple games and have multiple roles, and it can be called multiple roles game, or MRG. To imitate these cases, a evolutionary game model is introduced, in which each agent has two identities, one for PD game and another for SD game. A parameter is introduced to indicate the probability to select PD game for each pair interaction. It shows that the mixed probability
of PD and SD can influence the number of the equilibrium points in the deterministic evolutionary game dynamics. In the spatial MRG, the agents are placed on patches in a two-dimensional lattice with periodic boundary conditions being used. The number simulation shows that mixing SD and PD games will stimulate
and
, and inhibit
and
individuals from the global perspective. Comparison with the pure PD or SD games shows that SD cooperators ratio(
) increases as
increase, and PD cooperators ratio(
) is inhibited for
.
The population fraction distributions in the phase plane reveals that besides spatial reciprocity, there exist roles reciprocity, which means for some kind of agents their multiple identities will have help them to obtain benefits in population. In the MRG model with PD and SD games, because of the roles reciprocity,
and
individuals form a ecological chain in appropriate parameter region. These toy ecosystems are fragile on account of too small of the appropriate parameter areas.
To configure our multiple roles game model, we introduced direct sum form to construct the payoff matrix from simple game's payoff matrix. This method gives a way to construct complex MRG model from simple sub games. In game theory experiments, participators often influence by various factors. Or in biology, a group may have more than one function. If one need to construct a game theory frame in these cases, the MRG modeling method could be an option to consider.
Acknowledgments
We thank Lijian Yang, Jun Ma and Jun Tang for inspiring discussions and helpful comments on earlier versions of the manuscript.
Author Contributions
Conceived and designed the experiments: CM WC WL RG YJ. Performed the experiments: CM WC WL RG YJ. Analyzed the data: CM WC WL RG YJ. Contributed reagents/materials/analysis tools: CM WC WL RG YJ. Wrote the paper: CM.
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