Cooperators overcome migration dilemma through synchronization
Shubhadeep Sadhukhan, Rohitashwa Chattopadhyay, Sagar Chakraborty
CCooperators overcome migration dilemma through synchronization
Shubhadeep Sadhukhan, ∗ Rohitashwa Chattopadhyay, † and Sagar Chakraborty ‡ Department of Physics, Indian Institute of Technology Kanpur, Uttar Pradesh 208016, India (Dated: February 22, 2021)Synchronization, cooperation, and chaos are ubiquitous phenomena in nature. In a populationcomposed of many distinct groups of individuals playing the prisoner’s dilemma game, there existsa migration dilemma: No cooperator would migrate to a group playing the prisoner’s dilemma gamelest it should be exploited by a defector; but unless the migration takes place, there is no chanceof the entire population’s cooperator-fraction to increase. Employing a randomly rewired coupledmap lattice of chaotic replicator maps, modelling replication-selection evolutionary game dynamics,we demonstrate that the cooperators—evolving in synchrony—overcome the migration dilemma toproliferate across the population when altruism is mildly incentivized making few of the demes playthe leader game.
I. INTRODUCTION
Cooperation has strong ethical, moral, philosophical,and even theological implications for the human [1]. Itsspatiotemporal evolution in a structural arrangement offinite number of agents playing strategic games [2] is aninsightful phenomenon that exemplifies similar real-lifephenomena in social [3], economic [4, 5], physical [6], andbiological [7] systems. There are many different mecha-nisms [8, 9] of imparting cooperation. In the network ofagents, migration [10–19] could be one such mechanism:While success-driven migration [10, 13], aspiration-drivenmigration [12], expectation-driven migration [16], risk-driven migration [17], opportunistic migration [18], andmigration following a satisfying dynamic [14] lead to co-operation; random or diffusive migration of the agents isexpected to suppress cooperation by facilitating invasionby defectors [20, 21].Consider the following scenario: A population is di-vided into a finite number of groups or demes whereina very large number of individuals—cooperator anddefectors—are interacting with each other to play theprisoner’s dilemma (PD) game [22]. Furthermore, letthere be migration of individuals from one deme to theother. Effectively, what we have is a network of demes.One expects that any structured network of such demes,where the network structure represents the migrationfrom one deme to the other, should have the populationstate with only the defectors as an evolutionarily stablestate which is resilient against invasion by the coopera-tors. Now suppose that all the demes with the PD haveonly defectors left over time. With a view to establishingcooperation [23], in some of the randomly selected demesone encourages altruism by rewarding additional payoffto the cooperators who play against the defectors suchthat the PD transforms into the leader game (LG) [24]in the selected demes. The LG can be thought of as the ∗ [email protected] † [email protected] ‡ [email protected] modification of the PD such that an altruistic behaviouris rewarded, i.e., a cooperator playing against a defec-tor is given some additional payoff so that the resultantpayoff is greater than the payoff for mutual cooperation.Note that a similar idea [25] of punishing players whodefect against the cooperators transforms the PD to thestag-hunt game. With random migration in action, canthe LG induce sustained cooperation in the PD at theother demes?An interesting dilemma arises: In the population withsome demes having both the cooperators and the defec-tors playing the LG and some having exclusively defec-tors playing the PD, the cooperators would not want tomigrate to the demes playing the PD lest they shouldbe exploited. However, if they stay at the demes wherethe agents play the LG, the fraction of the cooperatorscan not increase throughout the population and theywould be surrounded by a lot of defectors present in theother demes. This means that the cooperators would al-ways be at the risk of being exploited by a free rider.To refer to this situation, we introduce the phrase, mi-gration dilemma , which is fundamentally different fromother known social dilemmas like the tragedy of com-mons (TOC) [26] and agglomeration dilemma [14]. Inview of this dilemma, it is not obvious a priori whetherthe random migration helps in increasing the cooperator-fraction of the entire population.The stylized game of the PD presents probably the sim-plest possible abstraction and visualization of the prob-lem of emergence of cooperation: The players of thePD defect to play the non-Pareto-optimal Nash equi-librium [27, 28] even though mutual cooperation wouldhave fetched more payoff. Thanks to the folk theo-rem [29, 30] of the evolutionary game theory, throughsimilar games one can also see how the game theo-retic equilibria (e.g., Nash equilibrium) and the equilibria(e.g., fixed point) of corresponding dynamical systems—especially the paradigmatic replicator equation [29, 31–35]—are related. The real world, however, is enormouslymore complex: The evolutionary dynamics have otheroutcomes like chaos [36–39] that appears in the contin-uous replicator equation with more than two strategies.Interestingly, chaotically changing payoffs [40] may en- a r X i v : . [ q - b i o . P E ] F e b Defector
Cooperator C oup l e d M a p L a tti ce Directed edge before rewiring
Broken edge with nearest neighbour
New edge with random neighbour
Initial state Intermediate state Final state (a)(b)
Leader
Game
Prisoner’s
Dilemma
FIG. 1. Schematic diagram of CML with dynamic randomrewiring. We see in top row (a) the base CML with eightdemes each having cooperators (green individuals) and/or de-fectors (red individuals). We exhibit only six representativeindividuals in each deme for illustrative purpose. Every demehas a game—say, the PD (square with lock) or the LG (orangeflag)—played by the individuals in it. The arrowheads pointtowards the destinations of respective migration. In bottomrow (b), as dynamic random rewiring is employed, some ofthe directed edges (shown by blue arrows) of the base CMLare randomly broken (as shown by red arrows with scissors)and new incoming edges (shown by green arrows) are created.The dynamic random rewiring is employed at each step of thetime evolution—initial, intermediate, or final—helping in es-tablishing enhanced cooperation throughout the CML withtime, even after starting with a very small fraction of thecooperators at only one deme. hance cooperation; and in turbulent aqueous environ-ment, chaotic flows induce migration that may facilitateevolution of colonies via cooperation [41].
II. THE MODEL
In order to allow for rich dynamical complexities in ourstudy and with a view to establishing the intriguing in-terplay between the random migration and the chaos, weconsider a spatiotemporal model where in each deme adiscrete replicator dynamic (replicator map) is in actionand the migration-induced coupling between the demespresents us with a coupled map lattice (CML) of thereplicator maps. The CMLs have been extensively stud-ied [42, 43] in the context of biological and computa-tional networks, fluid dynamics, ecology, chemical reac-tions, etc.
It however is the most known for the study ofspatiotemporal chaos in spatially extended systems. Itis also natural that depending on the coupling strengthbetween the lattice sites of a CML, synchronized dynam-ics [43, 44] may appear so that all the lattice sites evolvein unison.Thus, in the context of the migration dilemma, we con-struct a CML—as schematically presented in Fig. 1—where every lattice site is a deme in which the dynam- ics of the fraction of the cooperators is governed by thereplicator map corresponding to either the PD or the LG.The replicator map corresponding to the LG has chaoticdynamics implying coexistence of the defectors and thecooperators. In the presence of the random migration,fashioned by temporal rewiring [45] of the edges of theCML, we wonder if the LG induce cooperation in the PDat other demes.The one-dimensional replicator map [29, 39, 46–52], x n +1 = f ( x n ) := x n + x n [( A x n ) − ( x n ) T A x n ] , (1)such that ≤ x n ≤ for all n , for the two-player-two-strategy games is the simplest and most convenienttestbed of our idea because it models the Darwinian se-lection, its fixed points correspond to Nash equilibriaand evolutionarily stable states through the folk and re-lated theorems, and it is endowed with chaotic attrac-tors. Here, subscript ‘1’ denotes the first component ofvector A x n , n denotes the time step, where A = (cid:2) ST (cid:3) is the payoff matrix for a player in the two-player-two-strategy symmetric game. S and T are real numbers. x = ( x, − x ) is the state of the population such that x is the fraction of the cooperators and − x is the fractionof the defectors.We consider a CML that is a one-dimensional linearnetwork with N nodes/lattice sites and periodic bound-ary condition such that each lattice site or deme is con-nected to its two nearest neighbours. In order to imple-ment the random migration in the system under inves-tigation, we modify the couplings in the CML. At everytime step, any node can either allow migration from itstwo nearest neighbours or from two other demes pickeduniformly randomly. The probability of remaining cou-pled to the nearest neighbours is − p , where p is calledthe dynamic rewiring probability; ‘dynamic’ emphasizesthat the rewiring is happening at every time step. Math-ematically, the mean field equation for the CML with therandom coupling is given by, x in +1 = [(1 + S ) x in + (1 − S − T ) x in + ( S + T − x in ] × (1 − (cid:15) ) + (1 − p ) (cid:15) x i − n + x i +1 n ) + p(cid:15) x ξn + x ηn ) , (2)where, the superscript i denotes the i th lattice site and (cid:15) is the coupling strength measuring the rate of migrationto the i th node from the respective nodes. The couplingstrength (cid:15) is equivalent to a diffusion coefficient (see Ap-pendix A). Furthermore, ξ and η are the indices of thetwo randomly chosen demes and are not equal to i − , i , or i + 1 . We must restrict (cid:15) between to so that x in doesn’t become either negative or greater than one andwork with only those values [50] of S and T for which ≤ x in ≤ for all values of i and n .In our model with same A at all the nodes—all thedemes having same game (say, the LG; see later)—thereis a possibility that the dynamics at all the demes may becompletely synchronized to an interior fixed point, i.e. , x i = x j = x ∗ for all i and j . One could do a linear sta-bility analysis [53, 54] to find whether this synchronizedstate is at all stable and hence attainable. As is shownin the next paragraph, such a stable state in fact existswhen (cid:15) ≥ (cid:15) crit := [( | df /dx | − / ( | df /dx | − p )] x ∗ . Aconvenient way of quantifying the extent to which thesystem is synchronized is to define, a global order pa-rameter [55], r G := | (cid:80) Ni =1 e π √− x i | /N , that should beunity asymptotically once the system attains completesynchrony. It is easy to note that for large N and uni-formly distributed x i in the interval [0 , , r G = 0 . Hence,any partially synchronized state have a non-zero value of r G that is less than unity.To find the critical coupling strength in the presence ofdynamic random rewiring, we perform the linear stabil-ity analysis about a homogeneous fixed point by putting x in = x ∗ + δ in in Eq. (2). Expanding the resulting expres-sion up to first order to arrive at, δ in +1 = (1 − (cid:15) ) [(1 + S ) + 2(1 − S − T ) x ∗ +3( S + T − x ∗ ] δ in + (1 − p ) (cid:15) δ i +1 n + δ i − n )+ p(cid:15) δ ξn + δ ηn ) . (3)As an approximation, we consider the term consistingof random neighbours to be zero on average. This ap-proximation is valid for the interior fixed points as theperturbations about them are equally likely to be eitherside of the fixed point, and hence average out to zero.(Although not necessary, smaller values of p would fur-ther strengthen the approximation.) Consequently, be-ing interested in the average synchronization level in anensemble of different realizations of the CML, we hence-forth drop the last term— p(cid:15) ( δ ξn + δ ηn ) / —of Eq. (3) fromour calculations. Subsequently, writing the small per-turbations as a sum of its Fourier components, δ in = (cid:80) q ˜ δ qn exp (cid:0) √− qi (cid:1) , we arrive at the following expression: ˜ δ qn +1 ˜ δ qn = (1 − (cid:15) ) [(1 + S ) + 2(1 − S − T ) x ∗ +3( S + T − x ∗ ] + (cid:15) (1 − p ) cos q. (4)It is obvious that for the perturbations to die down, or inother words, for the fixed point to be stable, the modulusof the right hand side of Eq. (4) has to be less than 1.Keeping in mind that ≤ (cid:15), p ≤ , it means that if | (1 + S ) + 2(1 − S − T ) x ∗ + 3( S + T − x ∗ | ≤ , thenfor every (cid:15) between to , the fixed point is stable; butif | (1 + S ) + 2(1 − S − T ) x ∗ + 3( S + T − x ∗ | > , thento ensure that the perturbations die down, we require (1 − (cid:15) ) | (1 + S ) + 2(1 − S − T ) x ∗ +3( S + T − x ∗ | + (cid:15) (1 − p ) < . (5)A little rearrangement yields the critical couplingstrength ( (cid:15) = (cid:15) crit ), beyond which the fixed point mustbe stable, as (cid:15) crit = | dfdx ( x ∗ ) | − | dfdx ( x ∗ ) | − p , (6) where dfdx ( x ∗ ) = (1+ S )+2(1 − S − T ) x ∗ +3( S + T − x ∗ . III. MAIN RESULTS
First consider that the demes of the CML are all ofsame type, i.e. , same game is played at all the demes.For the purpose of this paper, the LG is of particularinterest to us. Its payoff matrix is characterized by theinequality—
T > S > > (compare with the PD where T > > > S ). In particular, we choose T = 8 and S = 7 as these values lead to the chaotic solutions [50]for the replicator map, given by Eq. (1), correspondingto the LG. Also, all the homogeneous fixed points, viz. , x i = x ∗ = 0 , x i = x ∗ = 0 . , and x i = x ∗ = 1 (forall i ) are unstable under replicator map dynamics whendynamic rewiring is not employed. As the strength ofmigration increases such that (cid:15) is more than (cid:15) crit (whichis . for p = 0 . ), the chaotic maps synchronize andall the demes have fifty percent cooperators in them (seeFig. 2(a)).In contrast with the above scenario, when the PD isbeing exclusively played at all the demes, we note that ithas two only homogeneous fixed points, viz. , x i = x ∗ = 0 and x i = x ∗ = 1 for all i . The former one is stable andthe later one is unstable when the dynamic rewiring isnot in action. Following a closer inspection of Eq. (2),one may intuit dynamic rewiring modelling random mi-gration to have two effects: reduced coupling strength( (cid:15) − p(cid:15) ) with nearest neighbours and multiplicative noiseof order p(cid:15) . Therefore, almost any initial condition of theCML is attracted towards the all-defect state and as thecorresponding phase trajectory approaches x i = 0 (forall i ), the noise becomes progressively weaker. Thus, thesynchronized state of the CML is the state where none ofthe demes have even a single cooperator. If, however, insome demes the PDs transform to the LGs, we have theinteresting situation where the CML has now mixed typesof demes. With random migration in play, can demeswith the LG induce cooperation in demes with the PDand hence help in overcoming the migration dilemma?In the CML with the two types of demes ( i.e. , A , andhence S and T depend on i [56, 57]), let the fraction of thedemes with the LG be denoted by φ . It is quite evidentthat for any φ , such that < φ < , there should be com-petition between the demes playing the LG and the onesplaying the PD in order to sustain cooperation. However,since the dynamics is chaotic, there always must be somecooperator at all times at all the demes even if to beginwith the cooperators were present only in the demes withthe LG. The dynamics would go towards a chaotic attrac-tor that is being constantly perturbed due to the small ef-fective noise generated by the dynamic random coupling.With the increase in the migration rate or the couplingstrength, it may be noted that the cooperator-fractionincreases in the demes playing the PD, and beyond thethreshold, (cid:15) crit , it increases synchronously so much so . . . . . . (cid:15) . . . . . . x n (a) φ = 1 . . . . . . . (cid:15) (b) φ = 0 . . . . . . . (cid:15) . . . . . . x n (c) φ = 0 . . . . . . . (cid:15) (d) φ = 0 . . . . . . . . . . . . x P ( x ) (e) (cid:15) = 0 . (cid:15) = 0 . (cid:15) = 0 . (cid:15) = 0 . (cid:15) = 0 . . . . (cid:15) . . . x n φ = 0 . FIG. 2. Bifurcation diagrams: The demes with the LG inducecooperation in the demes with the PD. We note in subplot (a)that if all the demes are playing the LG, after a critical valueof coupling strength, (cid:15) = (cid:15) crit (vertical dashed line) all the 100chaotic trajectories (orange dots) synchronize onto the fixedpoint x ∗ = 0 . of the CML. As we introduce the PD in someof the demes with no (or some) cooperators, then the corre-sponding trajectories (blue dots) are pulled onto the synchro-nized state x ∗ ≈ . for all demes beyond (cid:15) crit as exhibited insubplots (b)-(d) for the LG game fraction, φ = 0 . , . , and . respectively. In subplot (e), having φ = 0 . , we note howthe normalized the probability density function ( P ( x ) ) of thecooperator-fraction ( x ) for two randomly selected demes—one with the LG (solid lines) and the other the PD (dashedlines)—peak together near x = 0 . as (cid:15) increases. that all the nodes of the CML have almost of co-operators. This is shown in Fig. 2; especially, Fig. 2(e)where one sees that the probability density functions ofcooperator-fractions for two arbitrary demes—one play-ing the PD and the other the LG—almost merge as thecoupling strength increases. What is remarkable is thefact that even with low values of φ (say, φ = 0 . ; seeFig. 2(d)) and with no initial cooperators in the demesplaying the PD, random migration leads to strong emer-gence of cooperation in all the demes. Without any lossof generality, for the sake of concreteness, we have chosen S = − . and T = 1 . for the PD.In short, beyond the critical migration strength, oncethe dynamics of demes are all almost synchronized and . . . . . . φ . . . . h ¯ x i p = 0 . p = 0 . p = 0 . FIG. 3. Numerically validated analytical estimation of coop-eration in the CML. The cooperation level—characterized by (cid:104) ¯ x (cid:105) , the average cooperator fraction at each deme averagedover realization—in statistically steady state of the CML isplotted against the leader game fraction, φ . Black solid lineis the analytical estimation given by Eq. (7), whereas themarkers denote the numerically calculated values; the cyantriangles, the magenta squares, and the green circles respec-tively indicate the dynamic rewiring probabilities, p = 0 . , . , and . . the average cooperation is about fifty percent, the co-operation is sustained at all times. The final value of (cid:104) ¯ x (cid:105) ≈ . is an enormous increase when compared withthe initial value of (cid:104) ¯ x (cid:105) ∼ − used in some of the runsof our simulations (see Appendix B).Here the angularbrackets denoting average over many realizations of ran-dom migration and overbar denoting the average overdemes. Thus, synchronization overcomes the migrationdilemma.Since the level of cooperation can at most be fiftypercent, the final synchronized state achieved in thesystem—apart from rendering much needed emergenceof the cooperators—establishes the co-existence of strate-gies and hence, promotes biodiversity. In such states ofthe population modelled by the CML with the mixedtypes of demes, we can estimate the average cooperator-fraction. Since we are interested in a statistically steadyhomogenous state, suppose that the average state of eachdeme is by (cid:104) ¯ x (cid:105) . Furthermore, let f LG and f PD denote thereplicator maps respectively corresponding to the LG andthe PD. Therefore, we expect that the effective replicatormap for any deme should be a weighted average of thesemaps, and hence we expect (cid:104) ¯ x (cid:105) = φf LG ( (cid:104) ¯ x (cid:105) ) + (1 − φ ) f PD ( (cid:104) ¯ x (cid:105) ) . (7)On solving this equation, we get a non-trivial solution for (cid:104) ¯ x (cid:105) that is plotted in Fig. 3 as a function of the fraction, φ , of demes playing the LG. We note that it remarkablymatches with the numerical simulations’ results done atthree different dynamic rewiring probabilities— . , . ,and . . For the numerical simulations, we chose (cid:15) =0 . so that for all the three aforementioned values of p ,synchronization is effected. It is interesting to note thatthe results are statistically independent of the exact valueof p . IV. SYNCHRONIZATION SUPPRESSESMIGRATION DILEMMA: A ROBUSTMECHANISM
The way cooperators overcome migration dilemmathrough synchronization is actually a very robust phe-nomenon. With a view to justifying this claim, we showin this section that how the phenomenon is independentof the other payoff matrices (that lead to chaotic dynam-ics) and also how the phenomenon shows up even if weuse an evolutionary dynamic other than the replicatormap.
A. Other payoff matrices
Two-player–two-strategy symmetric games can be clas-sified into twelve distinct games [58] in terms of ordi-nality as shown in the Fig. 4. We are mainly inter-ested in the anti-coordination games (that consists of theleader game, the snowdrift/chicken game, and the battle − S − T PD SD LG BSSH
H1H2
D2C1 C2 H3 D1 T = S T = 1 S = S = T = 0 FIG. 4. Two-player–two-strategy symmetric gamescatagorised into twelve games in S - T space. The greyshaded part of the picture denotes the anti-coordinationgames. Here PD: Prisoners Dilemma, SD: Snowdrift, LG:Leader game, BS: Battle of sex, SH: Stag Hunt, H1: HarmonyI, H2: Harmony II, D2: Deadlock 2, C1: Coordination I, C2:Coordination II, D1: Deadlock I. The interior of the bluedashed curve denotes the allowed range of parameters thatlead to physical solutions for the replicator map [50]. of sexes) as they are capable of giving rise to chaotic dy-namics [39, 50] when used in the replicator map. Werecall that we have used the payoff matrix in the form A = (cid:2) ST (cid:3) . We used S = − . and T = 1 . for the pris-oner’s dilemma (PD) game. We rewarded the altruisticbehaviour in a fraction of the lattice sites such that theyare effectively involved in a leader game. To check therobustness of our model against the nature of the game,we have simulated our model with the two other anti-coordination games—the chicken game and the battle ofsexes.We have used T = 8 and S = 7 for the leader game forwhich the Lyapunov exponent, λ = n (cid:80) n − i =0 log | f (cid:48) ( x ) | ,is approximately 0.49 when used in replicator map. Thepositive Lyapunov exponent indicates chaos. For the bat-tle of sexes game, we have used T = 6 . and S = 6 . ;the corresponding λ ≈ . . In order to get chaos inthe chicken game, we have taken a linear transformationof the payoff matrix— A → γ A , such that it remains achicken game but gives rises to chaos. We have used T = 1 . , S = 0 . , and γ = 25 for the chicken game [50]which then shows chaos as verified by a positive value ofthe Lyapunov exponent ( λ ≈ . ).Now we have chaos for all these three games. In theFig. 5, we can see how the order parameter (cid:104) r G (cid:105) and theaverage cooperation (cid:104) ¯ x (cid:105) vary with the fraction φ corre-sponding to the anti-coordination game and the couplingstrength (cid:15) . In the Fig. 5(a)–5(c), it is shown how thesystem gets synchronized as the coupling strength is in-creased. The lower panel, Fig. 5(d)–5(f), the averagecooperation is shown for three different games. We havecalculated the critical coupling strengths for the synchro-nization calculated in the case of the CMLs with exclu-sively the anti-coordination games under investigation.The critical coupling strengths are . , . and . for the chicken game, the battle of sexes, and the leadergame respectively. The interior fixed point correspond-ing to these there games are . , . and . respec-tively. So one can see that beyond the critical couplingstrength, the average cooperation levels are fixed at theinterior fixed point of the chaotic game when the fraction φ is not very low. B. Another evolutionary dynamics
The mechanism of synchronization overcoming migra-tion dilemma is not restricted to the replicator dynamics.It is far more general. To establish this claim it suits usto present our studies with the imitative-logit (or i-logit)map [59, 60]. For two strategy games, the discrete-timei-logit dynamic is given by the one-dimensional map, x n +1 = g ( x n ) := x n x n + (1 − x n ) e β ( π D − π C ) , (8)where β , π C , π D and n are the rationality factor, ex-pected payoff for player using first strategy (cooperating), . . . . . φ (a) Chicken Game (b)
Battle of Sexes (c)
Leader Game . . . (cid:15) . . . . . φ (d) 0 . . . (cid:15) (e) 0 . . . (cid:15) (f) 0 . . . . h ¯ x i . . . . . . h r G i FIG. 5. In the upper panel (a)–(c), the order parameter r G is shown as a function of the coupling strength (cid:15) and the gamefraction φ for the chicken game, the battle of sexes and the leader game respectively. We have shown the corresponding averagecooperation over all the demes in the lower panel (d)–(f). Black dashed lines indicate the critical coupling strength when φ = 1 . p = 0 . in these simulations. . . . . . . x n . . . . . . x n + Replicatori-logit, β = 1i-logit, β = 13i-logit, β = 30Best Response FIG. 6. Discrete time i-logit map’s dependence on themyopic rationality parameter, β . S = 1 . and T =1 . so that it corresponds to a leader game. At low β , resemblance with a version of replicator map, x n +1 = x n ( π C + k ) / [ x n π C + (1 − x n ) π D + k ] (where k is some back-ground fitness in the absence of any competition); and athigher β , the resemblance with the best response dynamics, x n +1 = H ( π C − π D ) (where H is the Heaviside step function),are quite obvious. expected payoff for the player using second strategy (de-fecting), and time step respectively. The expected payoffsfor two strategies π C and π D are given by, x n + (1 − x n ) S and T x n respectively at the time step n .Our main reason for choosing this model is the factthat i-logit map has a parameter, the myopic rationalityfactor β , that when varied can mimic various dynamicsranging from a version of the replicator map ( β → ) tothe best-response dynamics ( β → ∞ ). This rationalityfactor models the ability of a player to choose strategy insuch a way that maximizes her payoff. Higher β implieslesser chance of mistake in choosing the wrong strategy.Thus, it is not surprising that high β corresponds to thebest response dynamics (myopically rational), while forlow β i-logit somewhat follows the replicator dynamics(low rationality) as shown in Fig. 6.Now we use this dynamics at the each site of the CML.We have mean field equation for the i th site is given by, x n +1 = (1 − (cid:15) ) g ( x in )+ (cid:15) (1 − p )2 ( x i +1 n + x i − n )+ (cid:15)p x ξn + x ηn ) . (9)where, p and (cid:15) are the rewiring probability and the cou-pling strength respectively. ξ and η are the randomlychosen node index except i − , i , and i + 1 . Using thisi-logit model we get similar outcome as the replicator dy-namics. We use the same parameters for the prisoner’s .
00 0 .
25 0 .
50 0 .
75 1 . (cid:15) . . . . . . φ (a) .
00 0 .
25 0 .
50 0 .
75 1 . (cid:15) . . . . . . φ (b) . . . . . . h r G i . . . . . h ¯ x i FIG. 7. (a) Order parameter and (b) average cooperation are shown as a function of the coupling strength (cid:15) and the chaoticgame fraction φ . Here we have used the parameters S = 1 . and T = 1 . for the leader game, the rationality factor β = 13 ,and dynamic rewiring probability p = 0 . . dilemma as used in the replicator map. We find a set ofparameters for the leader game which put the dynamicsof the i-logit map in the chaotic regime: Consequently, weset T = 1 . , S = 1 . , and β = 13 so that it shows chaosin the absence of any coupling; the Lyapunov exponentis found to be . (approximately).We find the order parameter and the average coopera-tion with the varying coupling strength (cid:15) and the fractionof the leader game φ and exhibit them in Fig 7(a) andFig 7(b) respectively. Here again we can see that beyonda critical value of coupling strength we have synchroniza-tion in cooperator fractions at every deme. (The criticalcoupling strength for φ = 1 which is (cid:15) crit ≈ . .) As φ increases towards unity the synchronised value is tend-ing to the interior fixed point of the leader game whichis ≈ . in this case. V. MIGRATION DILEMMA AND ITSSUPPRESSION BY SYNCHRONIZATION: ACLOSER LOOK
The system we have studied in this work may remindone of the interdemic models specified by a migrationpattern (island model [61], stepping stone model [62], ora mixture of the two) and a competition model (differ-ential proliferation [63], differential extinction [64], etc.)in the set of demes. However, in the conventional stud-ies, a deme has finite number of individuals and hencegenetic drift [65] plays an important role. By choosingto work with the deterministic dynamical equation, weare effectively working with the set of demes each havingpractically infinite number of individuals; hence, the ge-netic drift is out of the consideration. Thus, we are ableto exclusively focus on the intriguing interplay betweenthe chaotic dynamics at the demes and the migrationbetween the demes.Moreover, while the cooperators and the defectors can have different rates of migration [11], we have kept therates same in this paper. This simplifying assumptionnot only reduces the number of parameters in the prob-lem, but also has the advantage that it neutralizes theeffect of a possible mechanism of bringing forth cooper-ation in which the cooperators outrun the defectors [66].Moreover, we have implemented symmetric migration,i.e., same migration rates between different allowed pairsof demes; the asymmetric migration is mostly known toalter the stability and the resilience of the populationstate [19]. While models with the migration of defectorsexploiting a population of cooperators have been found tosuppress cooperation in the overall population [20, 21], amodel with success-driven migration of individuals haveshown to enhance cooperation in the overall popula-tion [10]. In such models, either migration of only defec-tors occurs (in the former), or individuals’ migration isvoluntary (in the latter). Our model employs random mi-gration of individuals—both cooperators and defectors—between any two arbitrarily chosen demes, which leadsto synchronization between all the demes in the network,thereby enhancing overall cooperation.
A. Importance of random migration
The incorporation of the random migration is very cru-cial for the results obtained in our paper. In order toexplicitly show it, we present Fig. 8 where we have simu-lated the evolution of the CML with demes playing eitherthe LG or the PD but with the nearest neighbour migra-tion only (i.e., no random migration); mathematically,the evolution is governed by Eq. (2) with p put to zero.Fig. 8 shows the cooperator fraction as a function ofthe coupling strength (cid:15) in all the demes of our network ata particular time step for given fractions ( φ ) of the demeswith the LG. We see that for both the low ( φ = 0 . ) and . . . . . . (cid:15) . . . . . . x n (a) φ = 0 . . . . . . . (cid:15) (b) φ = 0 . FIG. 8. Bifurcation diagrams: Absence of synchronized coop-eration in the CML with 100 lattice sites. Whether the frac-tion of LG is high (a) φ = 0 . or low (b) φ = 0 . , the chaoticevolution of the cooperator fraction at the demes with the LG(orange dots) is not settling down into a stable synchroniziedstate with demes where the PD (blue dots) is being played.The rewiring probability is fixed to p = 0 . . the high ( φ = 0 . ) values of the fractions, a majorityof demes have fraction of cooperators way less than 0.5for the entire range of the coupling strength. In contrastwhen random migration is present (refer Fig. 2) the co-operation fraction of all the demes converges towards 0.5as (cid:15) is increased. This happens because, with randommigration between the demes, the fixed point x ∗ = 0 . ,becomes stable while the other two fixed points—0 and1—remain unstable. Thus, the random migration drivesup the cooperator fraction from a low initial value to ap-proximately 0.5 in the deme playing the LG, which inturn pulls up the cooperator fraction in the demes play-ing the PD to approximately 0.5. Interestingly, this syn-chronization to a fixed point is very robust, which doesnot get affected by all the defectors coming in from theneighbouring demes playing the PD. The migration ofcooperators from the deme with the LG drives up theentire system’s overall cooperation fraction to a higherlevel, starting from a low initial value; the system needsto be beyond the critical coupling strength that leads tothe synchronization induced cooperator fraction’s suste-nance in all the demes to a value of around 0.5 for alltime steps. This synchronization mechanism is absentwhen the migration is not random (recall Fig. 8).At first glance, one may be tempted to conclude thatin a CML with both the LG and the PD, cooperationshould trivially be established based on the following ar-gument: Given that both cooperation and defection canbe found long-term in the LG, but only defection in thePD, the defectors migrating from a deme playing the PDinto a deme playing the LG would not threaten cooper-ation in the latter; on the other hand, given that coop-eration can be found transiently in the PD, a continualarrival of cooperators from the LG to allow them to befound long term on the PD sites. The immediately pre-ceding paragraph hints that it is not so straightforwardbecause the aforementioned argument does not take intoaccount the facts that how crucial the random nature ofmigration and the synchronization beyond a critical cou- pling strength are in the establishment and sustenanceof cooperation in the CML; any arbitrary kind of mi-gration and unsynchronized dynamics cannot result insustenance of non-negligible cooperator fraction acrossall the demes at all times. While the continuous arrivalof cooperators from the demes playing the LG demes in-deed allows them to be found long term on the PD sites,we must appreciate that this continuous generation ofcooperators at the LG demes and their arrival of coop-erators in the demes playing the PD—thereby leading tothe sustenance of cooperation—is achieved via synchro-nization that is established only when random migrationis in action. B. Migration dilemma
In the population with some demes having both thecooperators and the defectors playing the LG and somehaving exclusively defectors playing the PD, the cooper-ators would not want to migrate to the demes playingthe PD lest they should be exploited. However, if theystay at the demes where the agents play the LG, thefraction of the cooperators can not increase throughoutthe population and they would be surrounded by a lotof defectors present in the other demes. This means thatthe cooperators would always be at the risk of being ex-ploited by a free rider. The term “migration dilemma"has been introduced by us to refer to this situation. Aswe discuss below, this is a very distinct kind of dilemmathat has not been investigated earlier. First we recall afew well-known dilemmas.The tragedy of commons (TOC) [26] and its two-playerversion, arguably, the prisoner’s dilemma [22], constitutethe most well known dilemma. The dilemma arises whenthe players choose to either cooperate (to various de-grees) or defect with a plan to exploit a finite publicgood. The players get the maximum combined payoffif they cooperatively and restrainedly utilize the good.But if a player exploits (i.e., she defects while otherscooperate) the good without restrain, she receives themaximum individual payoff. Therefore, all players woulddefect to lead to destruction of the public good. Anotherdilemma of interest is the agglomeration dilemma [14]. Inthe model that leads to the agglomeration dilemma [14],the individuals in a network play the public goods game(PGG) [11]. Some lattice points in the network are va-cant for individuals to migrate to if they are not satis-fied with the payoff received at their current locations.Now the dilemma the individuals are facing is whetherthey should agglomerate into large groups or not. Thisis because for a single PGG, with every individual’s con-tribution, the benefit obtained decreases with increasedgroup size. The benefit of an individual in large groupsbecome more dependent on others’ contributions, whichis a risky proposition. Despite this risk, the potentialfor higher benefits also increases in a large group as anindividual can get involved in more PGGs with its neigh-bours, thereby constituting a dilemma. The migrationdilemma is different from these dilemmas.The TOC anyway does not involve any migration tohave any direct resemblance with the migration dilemma.However, the migration dilemma in intertwined withthe TOC: In the CML if the migration dilemma is notaverted, the demes playing the PD game will be exploitedoff its resources and so a partial TOC will be effectedin the demes. Random migration, assisted by synchro-nized dynamics, develops groups of cooperators compet-ing with the defectors at every deme, so that the TOCmay be averted.Although it involves migration of individuals in a net-work, the agglomeration dilemma is fundamentally dif-ferent from the migration dilemma. Unlike the case withthe latter, there is no usage of chaos or synchronizationin the formulation or the resolution of the agglomera-tion dilemma. Migration is a voluntary decision takenby an individual in the setup constituting the agglomer-ation dilemma, and all individuals do not need to migrateto form large groups. Whereas, migration of individualsfrom all the demes is a built-in feature of the setup show-casing the migration dilemma. Another essential featurein our model is that the migration between the demes israndom (not based on any decision-process). In the caseof agglomeration dilemma, the individuals can only mi-grate to specific empty sites in a lattice that fall withina certain predefined range from the individual’s currentlocation.Probably, the most appealing feature of the migrationdilemma is that it brings together the ideas of synchro-nization, cooperation, and chaos in the context of evolu-tionary game dynamics.
VI. CONCLUSIONS
Summarizing, we have provided a macroscopic descrip-tion of the emergence of cooperation owing to the syn-chronization of chaos in a population split into a net-work of demes having random migration among them.Specifically, we find that, by inducing synchronization,the random migration can actually increase cooperationin such a structured population. Interestingly, our modelis seen bringing forth a stable biodiversity in the formof a heterogenous population (mixed cooperator-defectorstate) from an initially homogeneous population (all de-fector state) where some defectors in a few select demesare incentivized to cooperate. To the best of our knowl-edge this is the first demonstration of such constructiveoutcomes of chaos and synchronization in the theory ofevolutionary games. This general mechanism is quite ro-bust against the change in the payoff matrix and thegame dynamics as long as chaos is observed and dis-tinct from the other migration induced cooperation mod-els [10, 11, 19, 66].Another interesting angle is that the interconnecteddemes exclusively with the PD-players are subject to the (in)famous social dilemma of the tragedy of commons(TOC) [26]: Any common resource therein would be ex-ploited without restrain. Migration, even when random,goes a long way to develop groups of cooperators, atevery deme, competing with the defectors so that theTOC may be averted. All it requires is that in a tinyfraction of the demes, the cooperation is encouraged;and migration propagates the virtue across at everyother deme. Thus, while overcoming the migrationdilemma, the cooperators also simultaneously tacklethe dilemma pertaining to the TOC. In the light ofthe recent studies [67–69] on the TOC using feedbackincluded replicator equation with payoff matrices fromtwo different games (like in this letter), we believe thatchaotic synchronization’s hitherto unexplored utility inthe TOC could be a useful future avenue of research.
ACKNOWLEDGMENTS
The authors are grateful to Srihari Keshavamurthy,Archan Mukhopadhyay, and Supratim Sengupta for help-ful discussions and invaluable suggestions.
Appendix A: A note on the coupling strength
The coupled map lattice (CML) is a very useful Eule-rian description of dynamical systems to model a varietyof phenomena in nonlinear systems. A CML consists ofa collection of maps (discrete-time dynamics) that inter-acts with each other by means of a network structureamong them. This interaction was modelled by simplediffusive coupling in the very early of its history [70, 71].As the simplest nontrivial case, consider that a partic-ular map dynamics at a site of the lattice is given by x n +1 = f ( x n ) . Then the equation for the i th site that iscoupled to the two nearest neighbours through a diffusivecoupling is given by, x in +1 = f ( x in ) + (cid:15) x i +1 n − x in + x i − n ) , (A1)where, (cid:15) is the coupling parameter. While f ( x ) could beany map and x could be any relevant physical quantity,for the specific case of the replicator map used in ourpaper, x is the population fraction. We immediately notethat in the continuum limit, we can change our site index i to a continuous spatial variable s and the time index n to a continuous time variable t to get the continuousequation, ∂x ( s, t ) ∂t = f ( x ( s, t )) − x ( s, t ) + (cid:15) ∂ x ( s, t ) ∂s . (A2)From Eq. (A2), we can conclude that (cid:15) is a diffusion co-efficient which can, in general, take any value between to ∞ . Physically, larger the (cid:15) (i.e., diffusion coeffi-cient), faster the diffusion; mathematically, if one waits0for infinite time, x at a site will completely diffuse out toneighbouring sites irrespective of the value of the diffu-sion coefficient.Eq. (A1) however has a mathematical problem whencompared with its continuum version, Eq. (A2): Sincethe coupling part represents the average of the popula-tion flux to the site i from the connecting sites, it isreasonable to restrict (cid:15) between zero to unity. Still onecan have unbounded solutions [72] for reasonable valuesof (cid:15) . Although this drawback of the modelling is ignoredoften, but in doing so we lose touch with the physicalreality. One way [72] to recover the physical reality isto separate the diffusive processes from the reproductiveprocess ( f ( x ) ). First, the new population in each cell istaken to be x (cid:48) in = f ( x in ) and this is followed by the dif-fusion process between the sites, so that the at the nextgeneration we get x in +1 = x (cid:48) in + (cid:15) x (cid:48) i +1 n − x (cid:48) in + x (cid:48) i − n ) , = ⇒ x in +1 = (1 − (cid:15) ) f ( x in ) + (cid:15) f ( x i +1 n ) + f ( x i − n )] . (A3)Another way of enforcing physical solution is to slightlymodify Eq. (A1) as follows: x in +1 = (1 − (cid:15) ) f ( x in ) + (cid:15) x i +1 n + x i − n ) . (A4)We note that if (cid:15) lies between 0 to 1, the righthand sideof Eq. (A4) is a convex combination of f ( x in ) and ( x i +1 n + x i − n ) / both of which are constrained to be between 0and 1; and hence, x in +1 must also always remain between0 to 1 to help keep the solution always physical. Thus, thenecessity of keeping the coupling parameter between to is to enforce physical reality, although in the continuumlimit it can take any real value. It is worth pointing outthat in the limit of weak selection f ( x ) ≈ x and bothforms, Eq. (A3) and Eq. (A4), are approximately thesame. Appendix B: Details of the numerical simulations
This section describes the numerical methods used inproducing the results reported in this paper. We useda parallel in-house C++ code and several libraries like“blitz++” for array operations, “YAML” for inputs, and“MPI” for task parallelism. Each deme/node/lattice sitein the CML was an object of our user-defined class.
Setup and initial conditions:
The underlying game,e.g., the PD or the LG, within a node was specifiedduring the initialization of the system. We markedtwo games the LG and the PD by two numbers— and respectively. For a given fraction φ of the LG,we called a random number, r , between to usinga uniform random number generator—drand48(). If r ≤ φ , then we assigned the game type value of the nodeas , i.e., the LG; otherwise, game type was assigned , i.e., the PD. Subsequently, we set the values for the payoff matrix, which is the attribute of a node.This is how we distributed two game types among thenodes for a given φ . For initial cooperator fraction indifferent node was assigned randomly by choosing arandom number between to using a uniform randomnumber generator. To test the robustness of our results,we also used some special initial conditions, e.g., zerocooperator-fraction at nodes with the PD and x = 0 . at all other nodes. Dynamic random rewiring:
The CML used in thiswork is dynamic, i.e., it is a network such that its nodes’in- and out-degrees are stochastically changing overtime. We began with a simple linear chain networkwhere every node was connected to its two nearestneighbours with periodic boundary condition—in- andout-degree of each node was two—effectively creatinga simple ring network. The incoming edges correspondto the immigration from the neighbouring nodes. Atthe beginning of every time step, we did the followingrewiring of the simple ring network: For every node i , wegenerated a uniform random number r i between to .If r i ≤ p , then the corresponding i th node was selectedfor rewiring; otherwise, the node remained connectedto its nearest neighbours. If a node was selected forrewiring, then its incoming edges from its two nearestneighbours were deleted and new incoming edges weremade with two randomly chosen nodes other than itselfand its nearest neighbours. We repeated this process atevery time step starting from the simple ring network. Probability distribution function calculation:
In or-der to compute the probability distribution as presentedin Fig. 2(e), we saved the cooperator-fraction, x i , ateach node for time steps. Neglecting first transient time steps, we computed the normalizedprobability distribution using the data corresponding tothe last time steps using Python . For presentationpurpose, we chose two nodes randomly, one playing thePD and the other playing the LG. The probability dis-tribution function P( x ), where P( x ) dx is the probabilityof having the cooperator-fraction between x to x + dx for each node under study. Parameters in numerical simulations:
We used theCML with demes for every simulation and alsochecked for the robustness of the results with changein the CML’s size. We varied the game fraction φ andthe coupling strength (cid:15) from to in steps of . . Forthe PD, the variable payoff matrix elements were fixedat T = 1 . and S = − . , and for the LG we chose T = 8 and S = 7 . The later shows chaotic dynamics inisolation. To verify that, we computed the maximumLyapunov exponent for the replicator dynamics withthe payoff matrix of the LG and got a positive value asexpected. We simulated the system for time stepsin order to get a statistically steady state. Furthermore,as far as the cooperator-fraction and synchronization1parameters are concerned, we did an average over different realizations in parallel using the MPI. [1] M. A. Nowak, Evolution, games, and God : the princi-ple of cooperation (Harvard University Press, Cambridge,MA., 2013).[2] M. A. Nowak and R. M. May, Evolutionary games andspatial chaos, Nature , 826 (1992).[3] R. Axelrod,
The Evolution of Cooperation: Revised Edi-tion (Basic Books, New York, 2006).[4] P. Dasgupta, Trust and cooperation among economicagents, Philos. T. R. Soc. B , 3301 (2009).[5] B. Enke, Kinship, Cooperation, and the Evolution ofMoral Systems, Q. J. Econ. , 953 (2019).[6] O. Shehory, S. Kraus, and O. Yadgar, Emergent coop-erative goal-satisfaction in large-scale automated-agentsystems, Artif. Intell. , 1 (1999).[7] H. Celiker and J. Gore, Cellular cooperation: insightsfrom microbes, Trends Cell Biol. , 9 (2013).[8] M. A. Nowak, Five rules for the evolution of cooperation,Science , 1560 (2006).[9] S. A. West, C. E. Mouden, and A. Gardner, Sixteen com-mon misconceptions about the evolution of cooperationin humans, Evol. Hum. Behav. , 231 (2011).[10] D. Helbing and W. Yu, The outbreak of cooperationamong success-driven individuals under noisy conditions,Proc. Natl. Acad. Sci. U.S.A. , 3680 (2009).[11] J. Y. Wakano, M. A. Nowak, and C. Hauert, Spatial dy-namics of ecological public goods, Proc. Natl. Acad. Sci.U.S.A. , 7910 (2009).[12] H.-X. Yang, Z.-X. Wu, and B.-H. Wang, Role ofaspiration-induced migration in cooperation, Phys. Rev.E , 065101 (2010).[13] W. Yu, Mobility enhances cooperation in the presenceof decision-making mistakes on complex networks, Phys.Rev. E , 026105 (2011).[14] C. P. Roca and D. Helbing, Emergence of social cohesionin a model society of greedy, mobile individuals, Proc.Natl. Acad. Sci. U.S.A. , 11370 (2011).[15] F. Fu and M. A. Nowak, Global migration can lead tostronger spatial selection than local migration, J. Stat.Phys. , 637 (2012).[16] T. Wu, F. Fu, Y. Zhang, and L. Wang, Expectation-driven migration promotes cooperation by group interac-tions, Phys. Rev. E , 066104 (2012).[17] X. Chen, A. Szolnoki, and M. Perc, Risk-driven migrationand the collective-risk social dilemma, Phys. Rev. E ,036101 (2012).[18] P. Buesser, M. Tomassini, and A. Antonioni, Opportunis-tic migration in spatial evolutionary games, Phys. Rev.E , 042806 (2013).[19] A. Limdi, A. Pérez-Escudero, A. Li, and J. Gore, Asym-metric migration decreases stability but increases re-silience in a heterogeneous metapopulation, Nat. Com-mun. , 2969 (2018).[20] L. A. Dugatkin and D. S. Wilson, Rover: A strategyfor exploiting cooperators in a patchy environment, Am.Nat. , 687 (1991).[21] M. Enquist and O. Leimar, The evolution of cooperationin mobile organisms, Anim. Behav. , 747 (1993). [22] A. Rapoport and A. Chammah, Prisoner’s Dilemma (University of Michigan Press, Ann Arbor, MI., 1965).[23] E. Fehr and U. Fischbacher, The nature of human altru-ism, Nature , 785 (2003).[24] A. Rapoport, Exploiter, leader, hero, and martyr: Thefour archetypes of the 2 × , 81(1967).[25] B. Skyrms, The Stag Hunt and the Evolution of So-cial Structure (Cambridge University Press, Cambridge,U.K., 2003).[26] G. Hardin, The tragedy of the commons, Science ,1243 (1968).[27] V. Pareto,
Cours d’Économie Politique. Professé al’Université de Lausanne. (Lausanne: F. Rouge, Éditeur,Librairie de l’Université, 1896-97, 1896).[28] J. F. Nash, Equilibrium points in n-person games, Proc.Natl. Acad. Sci. U.S.A. , 48 (1950).[29] R. Cressman, Evolutionary dynamics and extensive formgames (MIT Press, Cambridge, MA., 2003).[30] R. Cressman and Y. Tao, The replicator equation andother game dynamics, Proc. Natl. Acad. Sci. U.S.A. ,10810 (2014).[31] P. D. Taylor and L. B. Jonker, Evolutionary stable strate-gies and game dynamics, Math. Biosci. , 145 (1978).[32] P. Schuster and K. Sigmund, Replicator dynamics, J.Theor. Biol. , 533 (1983).[33] J. Hofbauer and K. Sigmund, Evolutionary Gamesand Population Dynamics (Cambridge University Press,Cambridge, U.K., 1998).[34] K. M. Page and M. A. Nowak, Unifying evolutionary dy-namics, J. Theor. Biol , 93 (2002).[35] A. Traulsen, J. C. Claussen, and C. Hauert, Coevolution-ary dynamics: From finite to infinite populations, Phys.Rev. Lett. , 238701 (2005).[36] B. Skyrms, Chaos and the explanatory significance ofequilibrium: Strange attractors in evolutionary game dy-namics, PSA: Proceedings of the Biennial Meeting of thePhilosophy of Science Association , 374 (1992).[37] M. Nowak and K. Sigmund, Chaos and the evolutionof cooperation, Proc. Natl. Acad. Sci. U.S.A. , 5091(1993).[38] Y. Sato, E. Akiyama, and J. D. Farmer, Chaos in learninga simple two-person game, Proc. Natl. Acad. Sci. U.S.A. , 4748 (2002).[39] D. Vilone, A. Robledo, and A. Sánchez, Chaos and un-predictability in evolutionary dynamics in discrete time,Phys. Rev. Lett. , 038101 (2011).[40] M. Perc, Chaos promotes cooperation in the spatial pris-oner’s dilemma game, EPL , 841 (2006).[41] M. S. Krieger, S. Sinai, and M. A. Nowak, Turbulentcoherent structures and early life below the kolmogorovscale, Nat. Commun. , 2192 (2020).[42] K. Kaneko, Overview of coupled map lattices, Chaos ,279 (1992).[43] K. Kaneko and T. Yanagita, Coupled maps, Scholarpedia , 4085 (2014), revision Synchro-nization (Cambridge University Press, Cambridge, U.K., , 2259 (2020).[46] T. Börgers and R. Sarin, Learning through reinforcementand replicator dynamics, J. Econ. Theory , 1 (1997).[47] J. Hofbauer and K. H. Schlag, Sophisticated imitation incyclic games, J. Evol. Econ. , 523 (2000).[48] A. Bisin and T. Verdier, Beyond the melting pot: Cul-tural transmission, marriage, and the evolution of ethnicand religious traits, Q. J. Econ , 955 (2000).[49] J. D. Montgomery, Intergenerational cultural transmis-sion as an evolutionary game, Am. Econ. J. Microecon. , 115 (2010).[50] V. Pandit, A. Mukhopadhyay, and S. Chakraborty,Weight of fitness deviation governs strict physical chaosin replicator dynamics, Chaos , 033104 (2018).[51] A. Mukhopadhyay and S. Chakraborty, Periodic orbitcan be evolutionarily stable: Case study of discrete repli-cator dynamics, J. Theor. Biol. , 110288 (2020).[52] A. Mukhopadhyay and S. Chakraborty, Decipheringchaos in evolutionary games, Chaos , 121104 (2020).[53] S. Sinha, Random coupling of chaotic maps leads to spa-tiotemporal synchronization, Phys. Rev. E , 016209(2002).[54] R. Chattopadhyay, S. Sadhukhan, and S. Chakraborty,Effect of chaotic agent dynamics on coevolution of coop-eration and synchronization, Chaos , 113111 (2020).[55] S.-J. Wang, R.-H. Du, T. Jin, X.-S. Wu, and S.-X. Qu,Synchronous slowing down in coupled logistic maps viarandom network topology, Sci. Rep. , 23448 (2016).[56] C. Hilbe, Š. Šimsa, K. Chatterjee, and M. A. Nowak, Evo-lution of cooperation in stochastic games, Nature ,246 (2018).[57] Q. Su, A. McAvoy, L. Wang, and M. A. Nowak, Evo-lutionary dynamics with game transitions, Proc. Natl.Acad. Sci. U.S.A. , 25398 (2019).[58] S. Hummert, K. Bohl, D. Basanta, A. Deutsch,S. Werner, G. Theißen, A. Schroeter, and S. Schuster,Evolutionary game theory: cells as players, Mol. BioSyst. , 3044 (2014).[59] A. Cabrales and J. Sobel, On the limit points of discreteselection dynamics, Journal of Economic Theory , 407(1992). [60] E. Wagner, The explanatory relevance of nash equi-librium: One-dimensional chaos in boundedly rationallearning, Philosophy of Science , 783 (2013).[61] S. Wright, Isolation by distance, Genetics , 114 (1943).[62] M. Kimura and G. H. Weiss, The stepping stone modelof population structure and the decrease of genetic cor-relation with distance, Genetics , 561 (1964).[63] J. F. Crow and K. Aoki, Group selection for a polygenicbehavioral trait: a differential proliferation model, Proc.Natl. Acad. Sci. U.S.A. , 2628 (1982).[64] K. Aoki, A condition for group selection to prevail overcounteracting individual selection, Evolution , 832(1982).[65] B. Charlesworth, Effective population size and patternsof molecular evolution and variation, Nat. Rev. Genet. , 195 (2009).[66] M. S. Datta, K. S. Korolev, I. Cvijovic, C. Dudley, andJ. Gore, Range expansion promotes cooperation in an ex-perimental microbial metapopulation, Proc. Natl. Acad.Sci. U.S.A. , 7354 (2013).[67] J. S. Weitz, C. Eksin, K. Paarporn, S. P. Brown, andW. C. Ratcliff, An oscillating tragedy of the commonsin replicator dynamics with game-environment feedback,Proc. Natl. Acad. Sci. U.S.A. , E7518 (2016).[68] Y.-H. Lin and J. S. Weitz, Spatial interactions and oscil-latory tragedies of the commons, Phys. Rev. Lett. ,148102 (2019).[69] A. R. Tilman, J. B. Plotkin, and E. Akcay, Evolutionarygames with environmental feedbacks, Nat. Commun. ,915 (2020).[70] K. Kaneko, Period-Doubling of Kink-Antikink Patterns,Quasiperiodicity in Antiferro-Like Structures and SpatialIntermittency in Coupled Logistic Lattice*): Towards aPrelude of a “Field Theory of Chaos”, Progress of Theo-retical Physics , 480 (1984).[71] I. Waller and R. Kapral, Spatial and temporal structurein systems of coupled nonlinear oscillators, Phys. Rev. A , 2047 (1984).[72] T. Yamada and H. Fujisaka, Stability Theory of Syn-chronized Motion in Coupled-Oscillator Systems. II: TheMapping Approach, Progress of Theoretical Physics70