Evolutionary dynamics of the delayed replicator-mutator equation: Limit cycle and cooperation
EEvolutionary Dynamics of Delayed Replicator-Mutator Equation: Limit Cycle andCooperation
Sourabh Mittal, ∗ Archan Mukhopadhyay, † and Sagar Chakraborty ‡ Department of Physics, Indian Institute of Technology Kanpur, Uttar Pradesh 208016, India
Game theory deals with strategic interactions among players and evolutionary game dynamicstracks the fate of the players’ populations under selection. In this paper, we consider the replicatorequation for two-player-two-strategy games involving cooperators and defectors. We modify theequation to include the effect of mutation and also delay that corresponds either to the delayedinformation about the population state or in realizing the effect of interaction among players. Byfocusing on the four exhaustive classes of symmetrical games—the Stag Hunt game, the Snowdriftgame, the Prisoners’ Dilemma game, and the Harmony game—we analytically and numericallyanalyze delayed replicator-mutator equation to find the explicit condition for the Hopf bifurcationbringing forth stable limit cycle. The existence of the asymptotically stable limit cycle imply thecoexistence of the cooperators and the defectors; and in the games, where defection is a stableNash strategy, a stable limit cycle does provide a mechanism for evolution of cooperation. We findthat while mutation alone can never lead to oscillatory cooperation state in two-player-two-strategygames, the delay can change the scenario. On the other hand, there are situations when delay alonecannot lead to the Hopf bifurcation in the absence of mutation in the selection dynamics.
Keywords: Evolutionary Game Theory, Replicator Dynamics, Delay Differential Equation, Hopf Bifurcation
I. INTRODUCTION
It is perplexing that cooperation [1–3] should be ubiq-uitously present in social, ecological, and biological sys-tems in spite of the fact that selfish actions by an agentfetch it relatively more benefit. While a lot of progresshas been reported in understanding this phenomenon, aunanimous consensus is still to be reached. One reasonsimply is that in the complex systems under consider-ation, comprehending every aspect of the phenomenonand trying to give one sweeping reason behind it is ex-tremely challenging if not impossible. This is where an-alyzing simple stylized strategic games like the famousPrisoner’s Dilemma game becomes useful. Such gamesallow for a drastically simplified version of the problemof the evolution of cooperation and present a transparentabstraction of the problem. Based on these games, thetheory of evolutionary games [4, 5] has been providing in-sights into the dilemma of the evolution of cooperation.In its most simple form [6], the classical game theoryassumes that two rational players—each equipped withtwo strategies (actions)—play against each other by us-ing one strategy each simultaneously to get some con-sistently quantifiable payoff (profit or loss). The infor-mation about the profit or the loss for a player corre-sponding to every strategic interaction is written downas an element in a × matrix called the payoff matrix.Here, rationality means consistency in decision-making:each decision-making player chooses the best action inaccordance with his/her set of preferences that is com-plete and transitive. One also assumes that the players ∗ [email protected] † [email protected] ‡ [email protected] have common knowledge of the rules of the game. Themost celebrated prediction of the game theory is that insuch one-shot games, theoretically, a potential outcomecorresponds to the Nash equilibrium [7] that is a pair ofstrategies (one from each player’s strategy set) such thatno player can benefit by unilateral deviating from his/herequilibrium strategy.The evolutionary game theory does away with the re-quirements of the rationality and the common knowl-edge since the biological entities it is concerned withneed not be rational. The organisms become players andthe strategies become synonymous with the phenotypesof the organisms. The elements of the payoff matricesdenote—within the Darwinian paradigm—the fitnessesthat are most conveniently interpreted as the numbers ofoffsprings. The fitness of a phenotype, thus, is defined asthe average payoff that one individual of that phenotypegets in the population of organisms of different pheno-types. In the absence of rationality, while the conceptof the Nash equilibrium becomes redundant in the evo-lutionary game theory, it turns out that the evolutionarystable strategy (ESS) must be a Nash equilibrium. ESSis a strategy that when adopted by the whole popula-tion, the host population becomes resilient against an in-finitesimal fraction of mutants playing some alternativestrategy. Since the evolution is essentially a dynamicalprocess, many dynamical equations modelling the evo-lution are in vogue [8]. One of the most investigatedevolutionary dynamical equation is the replicator equa-tion [9]. It is remarkable that its asymptotic dynamicaloutcomes are related to the underlying one-shot game’sNash equilibrium or ESS [10] so that one may predictthe dynamical outcome of the evolutionary dynamics byanalyzing the game-theoretic equilibrium concepts of thecorresponding game.Analysis of nonlinear dynamical equations modelling a r X i v : . [ n li n . AO ] F e b various aspects [11–22] of evolutionary games is an ex-citing modern interdisciplinary research area that en-compasses problems from biophysics, mathematics, eco-nomics, and sociology. In its simplest form, a strate-gic interaction in a game is supposed to lead to realiza-tion of payoff/fitness instantaneously. While most well-investigated game dynamics assume such strategic inter-actions, there are some systems where the effect of aninteraction may take some time to set in [23, 24]. Thepresence of delay affecting the course of dynamics essen-tially means that the underlying decision-making processhas a memory associated with it and is not Markovianin nature. It is well known that the effect of memory iscapable of inducing cooperation in dynamical games [25–27].There are quite a few studies on the effect of delay inselection dynamics in evolutionary game theory, specifi-cally, replicator dynamics. Inclusion of delay due to in-formation lag in the evaluation of fitness for simple two-player-two-strategy games in the continuous replicatordynamics [28] reveal that the conditions for ESS are in-dependent of delay and stability of the interior fixed pointcorresponding to mixed ESS depends on delay. Actually,two different types of delay can be envisaged [29]: onecorresponds to the information lag (delayed informationabout the population state) and the other to the delayin realizing the effect of interaction among players. Theformer one is called social delay and the latter one bio-logical delay. It has been shown that the mixed ESS isasymptotically stable for a small social delay but losesstability (unlike the case of biological delay) when thedelay increases beyond a threshold value.It is, thus, not surprising that a particular model ofdelay (that includes the social delay as one specific case)has been shown to induce limit cycle behaviour in two-player-two-strategy replicator dynamics [30]. Also, theHopf bifurcation leading to the emergence of limit cyclehas been observed in the replicator dynamics of N -personStag Hunt game [31]. It is further known [32] that thecondition, at which the stability of equilibrium pointsof replicator dynamics corresponding to two-player-two-strategy symmetric games changes, does not depend onthe distribution of delay. The consequences of delay, ingeneral, are quite nontrivial and complex. For discretedynamics, the relationship between the information lagabout the phenotypic distribution and stability of the in-terior fixed point is not monotonic [33]. The same studyalso shows that for smoothed best response dynamics inanti-coordination games, the interior fixed point is sta-ble for low probability of delay and unstable for the largeprobability of delay. In another work [34] involving a pop-ulation of finite agents with a specific memory length ofpast interactions and playing snowdrift game, one notesthat the fixed points may become unstable and give wayto limit cycles for large memory length.What is surprising is that almost all the investiga-tions (including the aforementioned works) on the effectof delay on the selection dynamics ignore the complica- tions due to the omnipresent phenomenon of mutation.While the conventional mutation can be of biological ori-gin, any shift in the strategy of an agent—assumed toplay only pure strategies—can be interpreted as muta-tion. We feel that it is of immediate pragmatic interestto study selection-mutation dynamics with delay. Themutations can be categorized into two types [35]: mul-tiplicative mutation that stands for the error in replica-tion mechanism during the birth of offspring and additivemutation that models mutation in adults. Replicator dy-namics containing the multiplicative mutation has beenstudied in the context of the problem of grammar ac-quisition [36] among other problems. The effect of theadditive mutation in the replicator dynamics for rock-paper-scissor game [35, 37] and the repeated Prisoner’sDilemma game [38] has been studied to find the Hopf bi-furcation and limit cycles therein. It should be realizedthat the existence of a limit cycle is equivalent to thepresence of cooperation in the systems as is elaboratedthroughout this paper.The combined effect of delay and mutation on theevolutionary dynamics being a relatively unaddressedproblem, we address this issue using delayed replicator-mutator dynamics for the two-player-two-strategy game.We consider both the multiplicative and the additive mu-tations and so in the two-dimensional mutation param-eter space, we find the region where a stable limit cycleemerges following the Hopf bifurcation. Our specific at-tention is on the four classes of games, viz. , the Snowdrift(SD), the Stag Hunt (SH), the Prisoner’s Dilemma (PD),and the Harmony (HG) which are traditionally studiedto understand the evolution of cooperation. Specifically,the flow of the paper is as follows: First we discuss thegames that model cooperation in Sec. II. Then, we pro-pose the models of delay in Sec. III, followed by a discus-sion on the linear stability analysis and what happens tothe evolution of cooperation in the presence of delay andmutation. We conclude our paper in Sec. IV. II. REPLICATOR-MUTATOR EQUATION
The replicator equation is one of the most widelyused models of selection dynamics in evolutionary games.For an unstructured infinite population consisting of n (pheno-)types (pure strategies), we denote the frequencyof i th type by x i ; of course, (cid:80) nj =1 x j = 1 . Let the fitnessor the expected payoff of i th type be f i . The averagefitness of the population thus is φ = (cid:80) nj =1 x j f j . Theprobability of multiplicative mutation, i.e., the probabil-ity that some of the j th type offsprings are born fromthe i th type individual is given by Q ij ( (cid:80) nj =1 Q ij = 1 ).Furthermore, we assume a constant rate µ of additive mu-tation, i.e., adults of certain type changing their strategyto another corresponding to some other type. The re-sulting replicator-mutator equation mathematically canbe expressed as, ˙ x i = n (cid:88) j =1 x j f j Q ji − φx i − µ ( nx i − . (1)Here all the terms are evaluated at the same instant oftime as there is no delay in the system and hence, we donot show time t explicitly as the argument of the vari-ables.As mentioned in Sec. I, we are interested in compre-hending the effect of delay and mutation on the evolu-tion of cooperation. The evolution of cooperation haslong been intriguing researchers [1, 2, 39, 40]. Interest-ing dilemmas result in simple one-shot two-player-two-strategy games (like the PD) when individuals defect toplay non-Pareto-optimal Nash equilibrium when mutualcooperation could have fetched them more payoff. If “co-operate” and “defect” are the only two strategies underconsideration, one may divide all the games into fourclasses [40–45], viz. , the SH, the SD, the PD, and theHG based on the fact how the symmetric Nash equilibriaare related to cooperate strategy.In other words, consider that the normal bimatrix formof one-shot symmetric two-player-two-strategy game isrepresented as follows: Player Cooperate Defect
Player Cooperate a, a b, c
Defect c, b d, d where the first element in each cell is the payoff of player and the second element is that of player . Payoffelements a , b , c , and d are real numbers. The ordinal re-lationship between the payoff elements define the afore-mentioned four classes of games:SH: c < a and a > d > b . This one-shot gamecorresponds to two symmetric Nash equilibria—cooperate and defect, and one mixed symmetricNash equilibrium. The essence of the SH (coor-dination) game is conveniently exemplified [46] asfollows: given that hunting down a stag (largestpayoff) requires cooperation between two players,a non-cooperating player can only catch a hare(smaller payoff) while the other player, being alonein the chase of the stag, returns empty-handed(least payoff).SD: c > a and b > d . This one-shot game cor-responds to one symmetric Nash equilibrium inwhich the players randomize their strategies. Theanti-coordination SD game appears in the scenariowhere two individuals are trapped in a big snow-drift that blocks a road. Either individual has thestrategy to either cooperate in clearing the block-age or to wait for the other to clear it. Of course,a free-rider (defector) has the most advantage butthere is the risk that if both keep waiting for theother to clear the blockage, then they both incur amaximum loss by being stuck forever. PD: c > a > d > b . Mutual defection is the uniqueNash equilibrium in the corresponding one-shotgame which probably is the most famous one inthe popular literature. The dilemma showcased inthe game is that although (cooperate, cooperate)strategy profile is Pareto-optimal, the (symmetric)Nash equilibrium corresponds to mutual defection.HG: c < a and b > d . Mutual cooperation is the uniqueNash equilibrium of this game.In this context, since the folk theorems [10] connectthe point attractors of replicator equation to the corre-sponding Nash equilibrium, an understanding of the se-lection dynamics is paramount. Being concerned withtwo-player-two-strategy games only, n = 2 . Let the frac-tion of cooperators be x , i.e., x = x . This implies thatthe fraction of defectors is − x , i.e., x = 1 − x . Also,the fitness of i th type is f i = (cid:80) nj =1 Π ij x j , where Π = (cid:20) a bc d (cid:21) . (2)Assuming that the fraction of accurate replication forboth the types are same, i.e., Q = Q = q ≤ , Eq. (1)can be written as, ˙ x = − x [ a − b − c + d ]+ x [ q ( a − b + c − d ) − c + 3 d − b ]+ x [ q ( b − c + 2 d ) + c − d − µ ] + d (1 − q ) + µ. (3)The mutation matrix Q , being symmetric and rowstochastic, is completely specified by the single param-eter, q . On putting q = 1 and µ = 0 , we reach thecase of no mutation, i.e., replicator dynamics. The repli-cator dynamics can have only one interior fixed point( x ∗ = x m ) along with two boundary fixed points ( x ∗ = 0 and x ∗ = 1 ). Presence of mutation (either q ∈ [0 , or µ ∈ (0 , or both) shifts the fixed points to, say, X m , X − ,and X + respectively. To have a better understanding ofthe effect of the mutation parameters on the nature ofthe fixed points for the mentioned four classes of game,it is helpful to fix some appropriate numerical values forthe payoff matrix elements as they facilitate analyticallytractable calculations. The chosen payoff matrices areshown in Fig. 1. Unless otherwise specified, we hence-forth exclusively work with these payoff matrices.In the case of the SH game, the fixed points of thedynamics are X m = 12 ,X − = 12 (cid:16) − (cid:112) q − µ − q − (cid:17) ,X + = 12 (cid:16)(cid:112) q − µ − q − (cid:17) , as shown in Fig. 2(a). Fixed points X − and X + arereal if q ≥ q = ( √ µ ) / . This means that X + and X − can exist only when ≤ µ ≤ / . It can also be CooperateDefect Cooperate DefectP1 P2 , , , , (a) SH CooperateDefect Cooperate Defect , , , , P1 P2 (b) SD
CooperateDefect Cooperate Defect , , , , P1 P2 (c) PD
CooperateDefect Cooperate Defect , , , , P1 P2 (d) HG
FIG. 1. Presented are the typical payoff matrices of (a) the Stag Hunt game, (b) the Snowdrift game, (c) the Prisoners’ dilemmagame, and (d) the Harmony game, that have been explicitly used for the calculations in this paper. P1 and P2 refer to the twoplayers playing the games. . q q . .
95 1 q +1 . . . . (a) SH X + , X m , X − . . . . . . q +1 . . . − . (c) SD X + , X m , X − . . . . . . q +2 . . − . (e) PD X + , X − . . . . . . q +1 . . − . (g) HG X + , X − . q q . .
95 1 q +0 . − . − . λ (b) . . . . . . q +3+2+10 − − − λ (d) . . . . . . q − − − − λ (f) . . . . . . q +2+10 − − − λ (h) FIG. 2. (Colour online)
Linear stability of the fixed points of the replicator-mutator equation. In the top row depicts thevariation of fixed points X m (green), X − (red) and X + (blue) with the change in mutation parameter q (for two values ofparameter µ ) when the payoff matrix corresponds to (a) the SH game, (c) the SD game, (e) the PD game, and (g) the HG.The corresponding eigenvalue, λ , of the Jacobian found in the course of linear stability analysis is plotted in the bottom row(following the colour conventions used for the top row) for (b) the SH game, (d) the SD game, (f) the PD game, and (h) theHG. The solid line stands for µ = 0 while the dashed line is for a non-zero µ , specifically, . for the SH game and . for theother three games. noted that transcritical bifurcation occurs (Fig. 2(b)) at q = q = (7 + 4 µ ) / at which X + collides with X m andtheir natures of stability get interchanged.The fixed points of the replicator-mutator dynamicsfor the SD game are X m = 12 ,X − = 12 (cid:16) − (cid:112) q − q + 4 µ + 9 − q + 3 (cid:17) ,X + = 12 (cid:16)(cid:112) q − q + 4 µ + 9 − q + 3 (cid:17) . In presence of mutation, the fixed point X + is unphysical(i.e., X + / ∈ [0 , ). The other fixed point X − is unphysicalwhenever µ (cid:54) = 0 (irrespective of the value of q ). When µ = 0 , X − = 0 for all possible values of q . The fixedpoints and their stabilities are depicted in Fig. 2(c)-(d).On solving Eq. (3) for the PD game, following fixedpoints—as illustrated in Fig. 2(e)-(f)—are obtained: X − = − (cid:112) q − qµ − q + 4 µ + 12 µ + 12 + q + 2 µ q − ,X + = (cid:112) q − qµ − q + 4 µ + 12 µ + 12 + q + 2 µ q − . The fixed point X + is unphysical when mutation of eitherkind is present. In the absence of mutation X + is phys-ical and attains the value X + = 1 . The fixed point X − always exists as an interior fixed point in the presence ofmutation. In the absence of any mutation, X − = 0 . Onemay note that mutation makes coexistence of cooperatorsand defectors possible in the PD game dynamics.Out of the four classes of games that we are consider-ing, the HG is the only one where the cooperate-strategyturns out to be a dominant strategy and hence purea Nash equilibrium. The fixed points of its replicator-mutator dynamics are given as: X − = − (cid:112) (2 − q − µ ) − µ (4 q −
5) + q + 2 µ − q − ,X + = (cid:112) (2 − q − µ ) − µ (4 q −
5) + q + 2 µ − q − . The fixed point X − is unphysical in the presence of µ -mutation. When µ = 0 , for any possible q , the fixedpoint X − = 0 . So, only X + is the fixed point of practicalimportance. We exhibit all the possible fixed points andtheir linear stabilities in Fig. 2(g)-(h).In all the four classes of games, as expected,there is no limit cycle behaviour as the phase spaceis one-dimensional for the corresponding autonomousreplicator-mutator equation. While an interior stablefixed point (like in the PD game) does imply the emer-gence of cooperation, the existence of a stable limit cy-cle provides another mechanism for the establishment ofcooperation. In general, the inclusion of delay in thereplicator-mutator equation makes the phase space effec-tively infinite-dimensional and hence there is a possibilityof limit cycles. So, does delay induce cooperation in thegames in the presence of mutation? How does the in-terplay between mutation and delay affect the dynamics? These are the main questions that we now seek to addressin the rest of this paper.
III. DELAYED REPLICATOR-MUTATOREQUATION
A glance at the replicator-mutator equation for two-player-two-strategy games involving cooperators and de-fectors suggests that—whether delay corresponds tothe delayed information about the population state orin realizing the effect of interaction among players—mathematically, delay has to appear in either the state, x , or the expected fitnesses, f i ( i ∈ { , } ) . Thus, to bevery general, we speak of a doublet ( τ , τ ) , where τ and τ are respectively the characteristics delays correspond-ing to the cooperators and the defectors. How the delaysare incorporated in the dynamics is a different issue thatwe describe in what immediately follows. A. Two Types of Delay: Social and Biological
Consider the rather general case of an infinite unstruc-tured population consisting of n types of individuals. Ifthe information regarding the fitnesses in the populationis delayed by τ i (social delay) for each type, or in other words, if individuals use past information about the pop-ulation to evaluate their fitnesses, then the replicator-mutator dynamics given by Eq. (1) can be written as, ˙ x i ( t ) = n (cid:88) j =1 x j ( t ) f j ( t − τ j ) Q ji − φx i ( t ) − µ (cid:2) nx i ( t ) − (cid:3) , (4)where φ = (cid:80) nj =1 x j ( t ) f j ( t − τ j ) . If we use the payoffmatrix form given by Eq. (2), the replicator-mutator dy-namics with social delay takes the explicit form givenbelow: ˙ x = − x x τ [ a − b ] − x x τ [ d − c ] + xx τ [ q ( a − b )]+ xx τ [ q ( c − d ) − c + 2 d ] + x [ d − b ]+ x τ [ c − d + q ( d − c )] + x [ q ( b + d ) − d − µ ]+ d (1 − q ) + µ , (5)where the subscripts denote the respective delay in thecorresponding arguments, e.g. , x τ means x ( t − τ ) . Forthe case of no mutation and τ = τ (symmetric delay),this model has been introduced [29]. The case of asym-metric delay ( τ (cid:54) = τ ) case has also been studied [47] inthe absence of mutation.The second type of delay—biological delay—that in-terests us comes into action in the systems where theeffect of an interaction is not instantaneous and conse-quently there is a delay in realizing the payoff of an in-teraction. Thus, both the fitness of an individual andthe state of the population used in replicator-mutatordynamics should be calculated at the past instant whenthe interaction happened. Mathematically, the delayedreplicator-mutator dynamics should be cast as, ˙ x i ( t ) = n (cid:88) j =1 x j ( t − τ j ) f j ( t − τ j ) Q ji − φx i − µ (cid:2) nx i ( t ) − (cid:3) , (6)where φ = (cid:80) nj =1 f j ( t − τ j ) x j ( t − τ j ) . Again, for payoffmatrix given by Eq. (2), the replicator-mutator dynamicswith biological delay has the following form: ˙ x = − xx τ [ a − b ] − xx τ [ d − c ]+ xx τ [ − b ] + xx τ [ − c + 2 d ] + x τ [ q ( a − b )]+ x τ [ d − c − q ( d − c )] + x τ [ bq ]+ x τ [( c − d )(1 − q )] + x [ − d − µ ] + d (1 − q ) + µ. (7)This has also been studied [29] in the case of symmetricdelay and no mutation.Equipped with the aforementioned governing equa-tions, we want to attempt to understand the combinedeffect of mutation and delay on the evolution of cooper-ation. Specifically, in what follows, we work with sym-metric delay ( τ = τ = τ ) and two types of asymmetricdelay— (0 , τ ) and ( τ, . This choice helps us to reducethe number of delay parameters to work with only one,i.e., τ . B. Linear Stability Analysis
The next logical step in the search of stable limit cy-cle in Eq. (5) and Eq. (7) is to perform linear stabilityanalyses on the equations and look out for the Hopf bi-furcation. The eigenvalues ( λ ) dictating the stability ofthe corresponding fixed point can be obtained from thecharacteristic equation [48]: λ + αe − λτ = β, (8)where α and β are real functions of the system parame-ters, viz. , payoff matrix elements and mutation param-eters. The parameters α and β can be expressed interms of the Jacobians, J = ( d ˙ x/dx ) x = x ∗ and J τ =( d ˙ x/dx τ ) x = x ∗ . Explicitly, α and β are − J τ and J re-spectively.The infinite possible solutions [49, 50] to Eq. (8) canbe written as λ k = β + W k ( − ατ e − βτ ) τ , ∀ k ∈ Z ; (9)where W k is the k th branch of the Lambert W function.The principal branch of the Lambert W function corre-sponds to k = 0 .The use of the Lambert W function in the analysis ofstability of fixed points in delay differential equations iswidely discussed in standard literature [49–54]. Re( λ k ) —real part of λ k —is maximum for k = 0 as W has maxi-mum real part among all W k [50, 52]. Thus, one can saythat the stability of a fixed point in the presence of delayis solely determined by the eigenvalue, λ , correspond-ing to the principal branch of the Lambert W function.We expect emergence of stable limit cycle as a conse-quence of Hopf bifurcation about a fixed point when ithas purely imaginary λ and all other eigenvalues aresuch that Re( λ k ) < [55–59].Consequently, we focus on the solution for λ . We firstnote that λ is real if W is real. This implies − ατ e − βτ ≥− /e or α ≤ e βτ /eτ . Therefore, if α ≤ [ e βτ /eτ ] ∀ τ > —or in other words, α ≤ min τ [ e βτ /eτ ] —then λ is alwaysreal ∀ τ > . If, however, α > min τ [ e βτ /eτ ] , then λ is complex for some τ > . It is easy to note thatcondition α > min τ [ e βτ /eτ ] is equivalent to condition α > max { , β } .Now, consider that α > max { , β } and assume that ∃ τ = τ H > at which λ = iω , a purely imaginarynumber. By putting this in Eq. (8), and separating thereal and the imaginary parts, we get, ω = α − β , (10) τ H = 1 (cid:112) α − β cos − (cid:18) βα (cid:19) . (11)For τ H to be real, ω must be real; in this case it meansthat in addition to α > β (trivially satisfied because ofthe condition: α > max { , β } ), α + β > must alsohold. These considerations give us the recipe to find the stable limit cycle solutions in the two-player-two-strategyreplicator-mutator equation with social and biological de-lays, either symmetric or asymmetric: All one has to do isto find the system parameters such that the correspond-ing α and β obey both the inequalities— α > max { , β } and α + β > . C. Limit Cycle and Cooperation
Since the existence of a stable limit cycle in the dy-namics means coexistence of cooperators and defectors,the level of cooperation in the game is closely associatedwith limit cycles. This is especially important in thosecases where defect strategy corresponds to an attractingfixed point of the dynamics in the absence of mutationand delay that when present induce cooperation in thegames through limit cycles. Having already fixed thepayoff matrix elements (see Fig. 1), it is the set of mu-tation parameters, q and µ , which decide when stablelimit cycles are possible in accordance with the recipeoutlined in the immediately preceding subsection. Afterinvestigating all the four classes of games for both thesymmetric and the asymmetric delays of both the socialand the biological types, below we report only the caseswhen a stable limit cycle could be firmly established an-alytically and numerically. Interested readers may referAppendix A to find the summary of what happens whenthere is no mutation but delay is in play.To begin with, we consider the SH game. We find thatall the three fixed points ( X m , X − , and X + ) undergothe Hopf bifurcation leading to the emergence of a stablelimit cycle for social asymmetric delay where delay is onlyin the defector’s fitness (refer Fig. 3). It is clear from thefigure that delay alone, in the absence of any mutation,cannot lead to any limit cycle. A stable limit cycle isalso possible in the case of symmetric biological delaywhen the fixed point X m undergoes Hopf bifurcation asillustrated in Fig. 4).Recall that when mutation is present, X m is the onlyphysical fixed point of the replicator-mutator dynamicsfor the SD game. This fixed point changes stability andgives way to the Hopf bifurcation when social delay—whether symmetric or asymmetric (delay only in defec-tor’s fitness)—is in play. The illustrative limit cycles andthe corresponding region of mutation parameter spaceare presented in Fig. 5. As an aside, we point out thatunlike the SH game, in the absence of any mutation, i.e., q = 1 and µ = 0 , we can find stable limit cycle [30] inthe SD game.The game of the PD is different from both the SH andthe SD games in the sense that it allows the emergenceof a stable limit cycle when symmetric biological delayis incorporated in the replicator-mutator equation (seeFig. 6). The Harmony game is similar to the PD game inthe sense that it also requires biological delay for exhibit-ing limit cycle behaviour, however, unlike the PD game,the biological delay has to be asymmetric such that the . . . . q . . . . µ (a) . . . . x . . − . − . ˙ x (b) . . . . . x . . . − . − . ˙ x (c) . . . . . . x − . − . − . . . . ˙ x (d) FIG. 3. (Colour online)
Stable limit cycle emerges following the Hopf bifurcation in the SH game for social asymmetric delay (0 , τ ) : Colour-filled areas in subplot (a) marks the total region in the mutation parameter space where stable limit cycle emergesat some threshold value of delay. The fixed points that undergo the Hopf bifurcation in the green, the yellow, the cyan, and theblue colours are X m , X m and X − , X − and X + , and X + respectively. Subplots (b), (c), and (d) showcase the phase diagramcorresponding to some point belonging to the green, the cyan, and the blue region respectively in q - µ space. In particular, wefix q , µ , and τ as . , , and . ; . , . , and . ; and . , . , and respectively in subplots (b), (c), and (d). Thefilled circle, unfilled circle, and cross in the phase plots respectively represent stable focus, unstable focus, and saddle. The red,the blue, and the green curves are representative phase trajectories approaching attractors. .
00 0 .
05 0 .
10 0 .
15 0 . q . . . . . µ (a) . . . . x − . − . . . . ˙ x (b) FIG. 4. (Colour online)
Stable limit cycle emerges followingthe Hopf bifurcation in the SH game for biological symmetricdelay ( τ, τ ) : Grey area in subplot (a) marks the region in themutation parameter space where stable limit cycle emerges atsome threshold value of delay. The fixed point that undergoesthe Hopf bifurcation is X m . Subplot (b) showcases an illus-trative phase diagram corresponding to q = 0 . , µ = 0 . ,and τ = 10 picked from the grey region. The unfilled cir-cle represents unstable focus, X m ; and the red and the bluecurves are representative phase trajectories approaching thelimit cycle from inside and outside respectively. delay is only in the cooperator’s fitness (see Fig. 7). X − and X + undergo the Hopf bifurcation in the PD gameand the HG respectively. IV. DISCUSSION AND CONCLUSION
Before we conclude, let us first succinctly point out thesalient features of aforementioned scenarios of the Hopfbifurcation: Firstly, it is interesting to note that whilethe stable limit cycles appear at relatively low values ofadditive mutation, it is not always so with the multiplica-tive mutation; for a limit cycle to appear in the case ofsocial delay, the multiplicative mutation has to be rel- . . . . . q . . . . µ fixed point: X m (a) . . . . q . . . . . µ fixed point: X m (c) . . . . x . . . − . − . ˙ x (b) . . . . . . x − . − . − . . . . ˙ x (d) FIG. 5. (Colour online)
Stable limit cycle emerges follow-ing the Hopf bifurcation in the SD game for social delay:Grey areas in subplot (a) and (c) mark the regions in themutation parameter space where stable limit cycle emergesat some threshold value of asymmetric delay (0 , τ ) and sym-metric delay ( τ, τ ) respectively. The fixed point that under-goes the Hopf bifurcation is X m . Subplot (b) and (d) respec-tively showcases illustrative phase diagrams corresponding to q = 0 . , µ = 0 . , and ( τ , τ ) = (0 , τ ) = (0 , ; and q = 1 , µ = 0 . and ( τ , τ ) = ( τ, τ ) = (30 , picked from the greyregions. The unfilled circle represent unstable focus, X m ; andthe red and the blue curves are representative phase trajec-tories approaching the limit cycles from inside and outsiderespectively. .
00 0 .
05 0 .
10 0 .
15 0 . q . . . . . µ (a) . . . . . . x − . − . . . . ˙ x (b) FIG. 6. (Colour online)
A stable limit cycle emerges followingthe Hopf bifurcation in the PD game for biological symmetricdelay ( τ, τ ) : Grey area in subplot (a) marks the region in themutation parameter space where stable limit cycle emerges atsome threshold value of delay. The fixed point that undergothe Hopf bifurcation is X − . Subplot (b) showcases an illus-trative phase diagram corresponding to q = 0 . , µ = 0 , and τ = 20 picked from the grey region. The unfilled circle repre-sent unstable focus, X − ; and the red and the blue curves arerepresentative phase trajectories approaching the limit cyclefrom inside and outside respectively. .
00 0 .
02 0 .
04 0 .
06 0 . q . . . . . µ (a) . . . . . . x . . − . − . ˙ x (b) FIG. 7. (Colour online)
Stable limit cycle emerges followingthe Hopf bifurcation in the HG for biological asymmetric de-lay ( τ, : Grey area in subplot (a) marks the region in themutation parameter space where a stable limit cycle emergesat some threshold value of delay. The fixed point that un-dergoes the Hopf bifurcation is X + . Subplot (b) showcasesan illustrative phase diagram corresponding to q = 0 . , µ = 0 . , and τ = 20 picked from the grey region. The filledcircle represents unstable focus, X + ; and the red and the bluecurves are representative phase trajectories approaching thelimit cycle from inside and outside respectively. atively weak (high q -values). Whenever a limit cycle isborn in the presence of biological delay, the multiplica-tive mutation is needed to be quite high (low q -values).Secondly, out of all the cases studied, the HG is the onlycase where the delay has to be solely in the fitness ofthe cooperators for limit cycles to exist and it should berecalled that in the HG, cooperate-strategy is the soledominant Nash strategy (hence, ESS) unlike the otherthree classes of games. Thirdly, in the presence of opti-mal delay, the SH game, the PD game, or the HG canpossess stable limit cycles only if there is a finite non-zeromutation in the system. Fourthly, it is worth noting that the SH is the only one game where a stable limit cycleemerges for both social delay and biological delay, andthe SD is the only game where biological delay doesn’tlead to a stable limit cycle.We remind the readers that to arrive at the re-sults listed above, we have modified the replicator-mutator dynamics corresponding to two-player-two-strategy games—in particular, the SH, the SD, the PD,and the HG—to include the social and the biological de-lays. To the best of our knowledge, the insightful in-terplay between the mutation (both additive and mul-tiplicative) and the delays has never been investigatedeither analytically or numerically as undertaken in thispaper. It is obvious that the search for a limit cycle insuch systems is linked with the bigger question of theevolution of cooperation through simple but instructivegames. Also, seen from another perspective, the delayand the mutation leads to coexistence of (pheno-)typesin a population.Nevertheless, an important question may and shouldbe raised: Two-player-two-strategy symmetric games canbe classified into twelve ordinal classes [60] that can becollected into four game-types—the SH, the SD, the PD,and the HG—based on how cooperate strategy fares or,equivalently, what the Nash equilibria are. While thePD game has only one type of ordinal game, the SH, theSD, and the HG has respectively three, three, and fiveordinally inequivalent payoff matrices and correspondinggames. Moreover within each ordinally distinct game,there can be games that are connected by some cardi-nal transformation. In simpler terms, while this paperdeals with two free parameters, q and µ , actually eventhe payoff matrix elements could be treated as param-eters. So, how would all the results qualitatively andquantitatively change if even the payoff matrix elementswere varied? One immediately notes that a study span-ning the full six-dimensional parameter space would bedaunting. Nevertheless, as an indication of what to ex-pect, we present a detailed analysis of the three ordi-nally inequivalent SD games and the cardinally relatedSD games in Appendix B and Appendix C respectively.It is worth pointing out that there are many possi-ble future research directions that can be pursued fol-lowing this paper: We have limited our analysis only totwo-player-two-strategy games because our intention hasbeen to bring the intricacy of the delayed dynamics tothe fore and for this purpose, these games are the sim-plest yet conveniently non-trivial. Extension of our workto the games with more strategies— e.g. , the rock-paper-scissors game [61] and the problem of grammar acquisi-tion [36]—would definitely be exciting. It is known thatthe replicator equation with additive mutation leads tocoexistence in the game [35] but how the game behavesunder multiplicative mutations and delay remains unex-plored. Following the route of investigation delineatedherein one should be able to attack similar problemsin the discrete replicator equations [60], in the repeatedgames, and also in other types of selection-dynamics [8].Furthermore, one could start with a model of selection-replication-mutation dynamics in a finite population andstudy the effect of the delay therein with a view to con-trasting the resulting stochastic dynamics with the cor- responding deterministic dynamics obtained in the limitof infinite populations. Last but not the least, the si-multaneous effects of the delay and the mutations in astructured population may spring a few surprises. [1] R. Axelrod and W. D. Hamilton, Science , 1390(1981).[2] R. Axelrod, The Evolution of Cooperation , Basic books(Basic Books, 1984).[3] A. Bourke,
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In this appendix, we briefly present when stable limitcycles can emerge in two-play-two-person symmetricgames under the replicator equation with social and bi-ological delay, i.e., Eq. (5) and Eq. (7) but with q = 1 and µ = 0 . It is known that there are actually twelveordinally distinct games [60, 62, 63] whose payoff ma-trices, owing to positive affine transformations, can becompactly represented by a two-parameter matrix: Π = (cid:20) ST (cid:21) ; T, S ∈ R . (A1)Since we are interested only in physical limit cycles, i.e.,the limit cycles which do not go beyond the interval [0 , ,we should consider only those games that have a physi-cal interior fixed point ( x m ) that may undergo the Hopfbifurcation. This happens only for the SH class ( T < and S < ) and the SD class ( T > and S > ), eachof which consists of three ordinally inequivalent games.Subsequently, following the method outlined in the maintext of this paper, we find when the games within theseclasses undergo the Hopf bifurcation and a stable limitcycle emerges about x m when delay is above some thresh-old value. Fig. 8 exhibits the conclusions for all the kindsof delays employed in this paper. We remark that ourresults regarding the emergence of stable limit cycles forsocial symmetric delay (refer Fig. 8c) matches with thatfound in literature [30]; other results are unreported else-where. Also, note that there is no limit cycle behaviourpossible for the SH game in the absence of mutation. S - axis T - a x i s (a) T =1 Social delay: (0 , τ ) S - axis T - a x i s (b) T =1 Social delay: ( τ, S - axis T - a x i s (c) T =1 Social delay: ( τ, τ ) S - axis T - a x i s (d) T =1 Biological delay: (0 , τ ) FIG. 8. (Colour online)
Emergence of limit cycle in the ab-sence of mutation: The grey region schematically marks theregion in S - T space where the interior physical fixed point, x m , can undergo the Hopf bifurcation for the cases of (a)social asymmetric delay (0 , τ ) , (b) social asymmetric delay ( τ, , (c) social symmetric delay ( τ, τ ) , and (d) biologicalasymmetric delay (0 , τ ) . Other cases are not shown as thereis no possibility of a limit cycle in those cases. The red regionemphasizes that no interior physical fixed point and hence, aphysical limit cycle around it, is impossible. [ ] . SD - I [ ] SD - II [ ] SD - III S = S = S = T T = 1 FIG. 9. Three ordinally inequivalent SD games: The lines S = 0 , T = 1 , S = 1 , and S = T divide the SD class of gamesin three distinct games, SD-I, SD-II, and SD-III. We show arepresentative payoff matrix for each of the games. Appendix B: Ordinal Inequivalent SD Games
Since based on what the Nash equilibria are, two-player-two-strategy symmetric games can be classifiedinto twelve ordinal classes, we compare the delayed repli-cator dynamics of the inequivalent ordinal games. Tothis end, the class of the SD games that has three ordi-nally distinct games as shown in Fig. 9. The representa-1 . . . . . . q . . . . (a) SD-II X + , X m , X − . . . . . . q . . . (c) SD-III X + , X m , X − . . . . . . q − − − λ (b) . . . . . . q − − λ (d) FIG. 10. (Colour online)
Linear stability of the fixed points ofthe replicator-mutator equation corresponding to SD-II andSD-III. The top row depicts the variation of fixed points X m (green), X − (red) and X + (blue) with the change in mutationparameter q (for two values of parameter µ ) when the payoffmatrix corresponds to (a) SD-II game and (c) SD-III game.The corresponding eigenvalue, λ , of the Jacobian found in thecourse of linear stability analysis is plotted in the bottom row(following the colour conventions used for the top row) for (b)SD-II game and (d) SD-III game. The solid line stands for µ = 0 while the dashed line is for a non-zero µ , specifically, . for SD-II game and . for SD-III game. tive payoff matrix that we have studied earlier within theSD class (refer Fig. 1) belongs to SD-I ordinal structure.To make our study complete, we study the dynamics ofSD-II and SD-III ordinal class of games in the delayedreplicator-mutator models.In Fig. 10 we showcase the variation of fixed points andtheir corresponding eigenvalues (on doing linear stabilityanalysis) for the non-delayed case as a function of mul-tiplicative mutation ( q ) for two given additive mutation( µ ) values. We note that qualitatively the behaviours ofSD-II and SD-III are same as that of SD-I: In the pres-ence of mutation, the fixed point X + is unphysical (avalue outside [0 , ). The other fixed point X − is un-physical whenever µ (cid:54) = 0 (irrespective of the value of q ).When µ = 0 , X − = 0 for all possible values of q . X m is the only fixed point that is always existent irrespec-tive of how much mutation is in action. Furthermore, asdone for SD-I game earlier, we find the region in muta-tion parameter space where a stable limit cycle emergesfollowing Hopf bifurcation (refer Fig. 11) for SD-II andSD-III ordinal games. We also illustrate the limit cyclesusing the phase plots corresponding to one point fromthe corresponding mutation parameter space in Fig. 11. . . . q . . . . µ (a) . . . . q . . . µ (c) . . . . . x +0 . . − . − . ˙ x (b) . . . . x +0 . . − . − . ˙ x (d) FIG. 11. (Colour online)
Stable limit cycle emerges followingthe Hopf bifurcation in SD-II game for social delay: Greyareas in subplot (a) and (c) mark the regions in the mu-tation parameter space where stable limit cycle emerges atsome threshold value of asymmetric delay (0 , τ ) and symmet-ric delay ( τ, τ ) respectively. The fixed point that undergoesthe Hopf bifurcation is X m . Subplot (b) and (d) respec-tively showcases illustrative phase diagrams corresponding to q = 0 . , µ = 0 . , and ( τ , τ ) = (0 , τ ) = (0 , ; and q = 0 . , µ = 0 . and ( τ , τ ) = ( τ, τ ) = (5 , picked from the grey re-gions. The unfilled circle represent unstable focus, X m ; andthe blue and the red curves are representative phase trajec-tories approaching the limit cycles from inside and outsiderespectively. . . . q . . . . . . µ (a) . . . . . x +0 . . − . − . ˙ x (b) FIG. 12. (Colour online)
Stable limit cycle emerges followingthe Hopf bifurcation in SD-III game for social symmetric de-lay ( τ, τ ) : Grey area in subplot (a) marks the region in themutation parameter space where stable limit cycle emerges atsome threshold value of delay. The fixed point that undergoesthe Hopf bifurcation is X m . Subplot (b) showcases an illus-trative phase diagram corresponding to q = 0 . , µ = 0 . , and τ = 12 picked from the grey region. The unfilled circle repre-sents unstable focus, X m ; and the red and the blue curves arerepresentative phase trajectories approaching the limit cyclefrom inside and outside respectively. − . − . . . δ γ FIG. 13. Emergence of limit cycle depends on scaling andshifting of the payoff matrix: We fix q = 0 . and µ = 0 . ,and grey the region in δ - γ space (see Eq. (C2)) where X m un-dergoes the Hopf bifurcation for SD-I game with social asym-metric delay, (0 , τ ) . It is interesting to observe that stable limit cycle emergesin SD-II for both social asymmetric and symmetric de-lay (Fig. 11a and Fig. 11c respectively), whereas SD-IIIshows stable limit cycle only for social symmetric delay(Fig. 12a).
Appendix C: Cardinal Equivalent SD Games
We know that positive affine transformations, γ (cid:20) a bc d (cid:21) + δ (cid:20) (cid:21) = (cid:20) a (cid:48) b (cid:48) c (cid:48) d (cid:48) (cid:21) ; γ ∈ R + , δ ∈ R , (C1)leads to cardinally equivalent games within a distinctordinal structure of the payoff matrix. The same ordi-nal structure implies the same rational outcome for agiven one-shot game. However, it doesn’t guarantee thatthe solutions of the delayed replicator-mutator equationbased on the payoff matrix is independent of the parame-ters, γ and δ , of the affine transformation. Hence, it is im-portant to understand what significant change happens inthe dynamics when one deals with cardinally equivalentgames. For the sake of convenience and continuity, we yet again consider the SD class of games to find the regionin γ - δ space where a stable limit cycle emerges followingthe Hopf bifurcation for a given mutation strength.Our line of argument to find stable limit cycle is thesame as discussed in the main text. First, we considerSD-I game and use the payoff matrix given in Fig. 1).The matrix is scaled and shifted through the followingpositive affine transformation: Π = γ (cid:20) (cid:21) + δ (cid:20) (cid:21) ; γ ∈ R + , δ ∈ R . (C2)The fixed points of the replicator-mutator dynamics forthis cardinally transformed game when q = 0 . and µ =0 . are X m = 12 ,X − = 110 γ (cid:16) γ − (cid:112) γ + 10 δγ + 10 γ (cid:17) ,X + = 110 γ (cid:16) γ + (cid:112) γ + 10 δγ + 10 γ (cid:17) . The values of the mutation parameters are so chosenbecause they do give rise to stable limit cycles for so-cial delay before any transformation is effected. On do-ing linear stability analyses about the fixed points andfinding the characteristic equations having same formas in Eq. (8), e.g, about X m , we get α = 6 γ/ and β = ( − / γ + 2 δ + 2) . Now, we find the region in γ - δ space where the Hopf bifurcation leads to the emergenceof stable limit cycle (see Fig. 13) by imposing the con-ditions α = max { , β } and α + β > . We do similarcalculation using the payoff matrices for SD-II and SD-III given in Fig. 9. We show the corresponding results inFig. 14 where only those cases of delay are shown wherethere is any possibility of stable limit cycle. In conclusion,there is no denying the fact that detailed dynamics de-pends on the exact values of the payoff matrix elementsalthough there is a qualitative similarity in the resultsthat the limit cycles, if at all, are still seen only for thesocial delay case in the SD game irrespective of whetherit is SD-I, SD-II or SD-III. However, it must be kept inmind that we have not explored what happens for all pos-sible payoff matrices that is a daunting task but it doesnot appear to us to be leading to any fundamentally newinsight.3 − − δ . . . . . . . γ (a) − − δ . . . . . . . γ (b) − − δ . . . . . . . γ (c) − − δ . . . . . . . γ (d) − − δ . . . . . . . γ (e) FIG. 14. The emergence of a limit cycle depends on scaling and shifting of the payoff matrix: We fix q = 0 . and µ = 0 . , andgrey the region in δ - γ space where X m undergoes the Hopf bifurcation for the SD games with payoff matrices given in Fig. 9.In particular, subplots (a)-(e) respectively correspond to the case of SD-I game with social asymmetric delay, (0 , τ ) ; SD-I gamewith social symmetric delay, ( τ, τ ) ; SD-II game with social asymmetric delay, (0 , τ ) ; (d) SD-II game with social symmetricdelay, ( τ, τ ) ; and SD-III game with social symmetric delay, ( τ, τ ))