Delayed Evolutionary Game Dynamics with Non-Uniform Interactions in Two Communities

Transcription

1 53rd IEEE Conference on Decision and Control December Los Angeles California USA Delayed Evolutionary Game Dynamics with on-uniform Interactions in Two Communities esrine Ben Khalifa Rachid El-Azouzi and Yezekael Hayel Abstract In this paper we study the evolution of a population composed of several communities where the interaction between them is non-uniform The main objective is to study the effect of slow time scale delays on the convergence of the replicator dynamics to the Evolutionary Stable Strategy (ESS) In particular we investigate the effect of time delays on the stability of the replicator dynamics We can distinguish two types of the time delay: strategic delay which is the delay associated with the strategies of players and spatial delay which is the delay associated with the communities of the players We prove that for large strategic delays the mixed intermediate ESS may become an unstable state for the replicator dynamics while for any value of the spatial delay the replicator dynamics converge to the intermediate ESS We also show that when both types of the time delay coexist the instability property is emphasized I ITRODUCTIO Evolutionary Game Theory was first used in biology as a way of understanding animal conflicts A key concept of evolutionary games is that of an evolutionary stable strategy (ESS) which investigates the stability against a small fraction of mutants in a population According to the original definition given by Maynard Smith [1] an ESS is a strategy that when adopted by almost all the population cannot be successfully invaded by a small fraction of mutants using an alternative strategy In a multi-population setting however various ESS definitions are proposed in the literature which differ in the level of stability For example an ESS definition given by Cressman [2] referred to as a weak ESS or Cressman ESS is a strategy in which at least one subpopulation stick to the ESS when a small whole fraction of mutants is introduced in the population An alternative ESS definition with a stronger stability condition is given by Taylor [3] in which the total payoff of the non-mutants in all sub-populations is higher than mutants total payoff While the ESS is a static concept that makes abstraction of how the equilibrium is reached several models of evolutionary dynamics are proposed in the literature in the aim of describing how the population evolves over time The replicator dynamics is the most studied one The classical replicator dynamics equation assumes an instantaneous effect of the population composition on the individual s fitness and so on the growth rate of a given strategy A more realistic model of the replicator dynamics would take into consideration some time delay in the evolution of frequencies of strategies in a population Such *This work has been partially supported by the European Commission within the framework of the COGAS project FP7-ICT and PHC-UTIQUE Project CERI/LIA University of Avignon France delays can have a non-negligible effect The authors in [4] studied the effect of a symmetric time delay in the replicator dynamics in a population model with two pure strategies More precisely they assumed that an individual s fitness at time t is equal to the expected payoff value of the strategy used by this individual τ time units ago considering the same delay for both strategies The authors proved that the unique interior stationary point becomes unstable for sufficiently large values of the delay ie the population profile persistently oscillates around the equilibrium point A similar result on the effect of the time delay was proved by the authors in [5] in their social-type model The authors in [6] introduced an asymmetric time delay in a multiple access game in the context of a medium sharing problem Their dynamics model assumed random pairwise contests for the medium between players of the same interference range of each other The authors showed that the unique evolutionary stable strategy may be an unstable state when the transmission delay is large In all these papers time delay is shown to have a negative effect on the stability of the replicator dynamics Initially evolutionary game framework and replicator dynamics concept have been used to study biological systems and economic problems Recently this rich framework offers good capabilities to study large interaction networks of decision players and in this work we apply this framework to complex networks The idea is to study the propagation of strategies into groups not considering epidemic transmissions as in standard complex networks [7] but by the way of a strategic decision process in each individual The replicator dynamics aim to show the growth or death of a strategy into a population Some papers study particular noncooperative games on networks like coordination games [8] In this reference the authors consider an evolutionary game framework to study a coordination game between lots of individuals that are connected through a dynamic network Contrary to our formulation all the individuals are in the same group are symmetric in a sense They are particularly focus on the impact of the network dynamic on the ESS Another work presented in [9] presents mutation dynamics that converge to ESS when interactions occurs on a network The author applies their work to social networks Our paper focuses on different types of individuals such that any parities interaction does not lead to the same fitness depending on the type of individuals that are competing and not only the strategy used This type of group evolutionary game model has been proposed in theoretical contexts like in [10] We apply this concept to understand the deal of /14/$ IEEE 3809

2 strategies on the convergence of the replicator dynamics in the contact of group interactions In this paper we extend our work in [11] by investigating the effect of time delays on the stability of the replicator dynamics The paper is organized as follows In section II we present different ESS concepts related to mutations in the population composed of several communities In section III we describe the particular case with 2 interacting communities and we determine the different ESS We introduce the replicator dynamics equations and we analyze the impact of strategic delays and spatial delays in section IV and the impact of both delays in section V Finally we study evolutionary games in our context on random graphs in section VI and we conclude our paper in section VII II EVOLUTIOARY STABLE STRATEGIES We consider a large population of players or individuals divided into communities in which each community has its own set of strategy choices and payoff matrices Random matching occurs through pairwise interactions and may engage individuals from the same community or from different communities We denote by p ij the probability that an individual involved in an interaction interacts with an individual in community j with j p ij = 1 (see fig 1) We assume that there are n i pure strategies for each community i and a strategy of an individual is a probability distribution over the pure strategies We denote by A ij = (a ij kl ) k=1n il=1n j the payoff matrix where A ij = A ji If a player of community i using pure strategy k interacts with a player of community j using pure strategy l its payoff is a ij kl Let s = (s 1s ) with s i the vector describing the distribution of strategies in community i We denote by U i (k sp) the expected payoff of strategy k in community i when p =(p 1 p ) and the profile of all the population is s Then the expected payoff for a player in community i using z is given by: n i Ū i (zsp)= z k U i (k sp) k=1 where n i is the number of strategies in community i In the following we present different ESS characterisations that turn out to be interesting in analyzing the dynamical stability under the replicator dynamic These ESS characterizations differ mostly in the static stability condition Fig 1 Spatial interactions between 2 communities Definition 1: Strong ESS is a strategy that remains robust against invasion from a small group composed from all communities and using an alternative strategy More formally we can define the strong ESS as follows: A state s is a strong ESS if for all s = s there exists an (s) > 0 such that for all i =1 and (s) Ū i (s i s +(1 )s p) < Ūi(s i s +(1 )s p) (1) This strong ESS must in fact have a uniform invasion barrier or threshold where any proportion of invaders using an alternative strategy is repelled Equivalently an alternative definition can be established as follows: A state s is a strong ESS if and only if it meets two conditions for all i and for all s = s : Ūi(s i s p) Ūi(s i s p) (2) if Ūi(s i s p)=ūi(s i s p) then Ūi(sisp)<Ūi(s i sp)(3) The condition (3) states that all the communities have an incentive to remain at their ESS components when an alternative best-reply is used Indeed this is an immunity against a deviation from the ESS ow we introduce another ESS version with a weaker stability condition: the intermediate ESS In the intermediate ESS version [3] [12] the main focus is the total payoff of the whole population instead of the fitness of each community It guarantees that when the whole population adopts the strategy for any small group of deviant strategies the total fitness is worse than the intermediate ESS The formal definition of an intermediate ESS is given by: Definition 2: Intermediate ESS: A state s is an intermediate ESS if for all s = s there exists an (s) > 0 such that for all (s) Ū i (s i s +(1 )s p) < Ū i (s i s +(1 )s p) (4) Equivalently we have the following definition: Definition 3: A state s is an intermediate ESS if and only if for all s = s Ū i (s i s p) Ū i (s i s p) (5) If Ū i (s i s p)= Ū i (s i s p) then Ū i (s i sp) < Ū i (s i sp) (6) Proposition 1: The definitions 2 and 3 are equivalent Proof: see [11] It is fairly easy to check out that the strong ESS is an intermediate ESS The converse does not hold Indeed each population cannot earn a higher payoff by deviating from the strong ESS then the total payoff of all the populations ie the sum of the payoffs cannot be better and hence the requirement for an intermediate ESS is satisfied III AALYSIS OF PAIRWISE ITERACTIOS WITH TWO COMMUITIES AD TWO STRATEGIES We assume there are two communities each has a large number of individuals At each time step an individual from 3810

3 community i with i =1 2 may interact with an individual from the same community with probability p i or with an individual from the other community with probability 1 p i In addition each individual from community i has a twostrategy choice set {G i H i } The pairwise interactions are described by the matrices A B C and D Let s =(s 1 s 2 ) be the population profile with s i denotes the share of strategy G i in community i A = B = G 1 H 1 G 1 a 1 b 1 D= H 1 c 1 d 1 G 2 H 2 G 1 a 12 b 12 C= H 1 c 12 d 12 G 2 H 2 G 2 a 2 b 2 H 2 c 2 d 2 G 1 H 1 G 2 a 21 b 21 H 2 c 21 d 21 The payoff obtained by an individual in community 1 when he uses G 1 (resp H 1 ) writes: U 1 (G 1 sp)=p 1 (s 1 a 1 +(1 s 1 )b 1 )+(1 p 1 )(s 2 a 12 +(1 s 2 )b 12 ) U 1 (H 1 sp)=p 1 (s 1 c 1 +(1 s 1 )d 1 )+(1 p 1 )(s 2 c 12 +(1 s 2 )d 12 ) Similarly the payoff obtained by an individual in community 2 when he uses G 2 (resp H 2 ) writes: U 2 (G 2 sp)=p 2 (s 2 a 2 +(1 s 2 )b 2 )+(1 p 2 )(s 1 a 21 +(1 s 1 )b 21 ) U 2 (H 2 sp)=p 2 (s 2 c 2 +(1 s 2 )d 2 )+(1 p 2 )(s 1 c 21 +(1 s 1 )d 21 ) In addition the expected payoff for any individual from community i is given by: Ū i (s i sp)=s i U i (G i sp)+(1 s i )U i (H i sp) In [11] we characterised the existence of pure ESSs and mixed ESSs under different stability conditions The following theorem summarises results on the existence of mixed ESS Theorem 1: Let and s 1 = (1 p 1)L 12 K 2 p 2 L 2 K 1 s 2 = (1 p 2)L 21 K 1 p 1 L 1 K 2 where =p 1 p 2 L 1 L 2 (1 p 1 )(1 p 2 )L 12 L 21 L 1 = a 1 b 1 c 1 +d 1 L 2 = a 2 b 2 c 2 +d 2 L 12 = a 12 b 12 c 12 +d 12 L 21 = a 21 b 21 c 21 +d 21 K 1 = p 1 (b 1 d 1 )+(1 p 1 )(b 12 d 12 ) K 2 = p 2 (b 2 d 2 )+(1 p 2 )(b 21 d 21 ) s =(s 1s 2) is an unique mixed ash equilibrium if 0 < 0 < (1 p 1)L 12K 2 p 2L 2K 1 (1 p 1)L 12K 2 p 2L 2K 1 < 0 < (1 p 2)L 21K 1 p 1L 1K 2 and (1 p 2)L 21K 1 p 1L 1K 2 < or < 0 (1 p 1)L 12K 2 p 2L 2K 1 < 0 < (1 p 1)L 12K 2 p 2L 2K 1 0 < (1 p 2)L 21K 1 p 1L 1K 2 < 0 and < (1 p 2)L 21K 1 p 1L 1K 2 Any mixed ash equilibrium with 0 <s i < 1 is not a strong ESS s is an intermediate ESS if and only if L 1 < 0 and 1 =4p 1 p 2 L 1 L 2 2 (1 p 1 )L 12 +(1 p 2 )L 21 > 0 Proof: see [11] Theorem 1 gives the conditions on the payoff matrices and the probabilities of interactions to be satisfied so that the ESSs exist IV REPLICATOR DYAMICS We introduce the replicator dynamics which describes the evolution of the various strategies in the communities Replicator dynamics is one of the most studied dynamics in evolutionary game theory In the replicator dynamics the share of a strategy G i in the population grows at a rate equal to the difference between the expected payoff of G i and the average payoff of community i [12] We assume that the probabilities of interaction p 1 and p 2 are stationary A Replicator Dynamics without delay The non delayed replicator dynamics equation writes for i =1 2 s i (t) =s i (t) U i (G i s(t)p) Ūi(s i (t) s(t)p) (7) with s(t) =(s 1 (t)s 2 (t)) which yields the following pair of non-linear ordinary differential equations: ṡ 1(t) =s 1(t)(1 s 1(t)) p 1s 1(t)L 1 +(1 p 1)s 2(t)L 12 + K 1 ṡ 2(t) =s 2(t)(1 s 2(t)) p 2s 2(t)L 2 +(1 p 2)s 1(t)L 21 + K 2) There are nine stationary points given by: (0 0) (1 1) (0 1) (1 0) (0 K2 p 2L 2 ) ( K1 p 1L 1 0) (1 (1 p2)l21+k2 p 2L 2 ) 1) and s =(s 1s 2) where ( (1 p1)l12+k1 p 1L 1 and s 1 = (1 p 1)L 12 K 2 p 2 L 2 K 1 s 2 = (1 p2)l21k1 p1l1k2 with =p 1 p 2 L 1 L 2 (1 p 1 )(1 p 2 )L 12 L 21 The interior stationary point corresponds to the mixed ash equilibrium given by theorem (1) and it is the only stationary point at which all the strategies co-exist Assuming that the state space is the unit square and that s exists the dynamic properties of this equilibrium point are brought out in the next theorem Theorem 2: 1 When L 1 < 0 L 2 < 0 and = p 1 p 2 L 1 L 2 (1 p 1 )(1 p 2 )L 12 L 21 > 0 then s is globally asymptotically stable for the replicator dynamics; 2 When =p 1 p 2 L 1 L 2 (1 p 1 )(1 p 2 )L 12 L 21 < 0 then the interior stationary point s is unstable for the replicator dynamics Proof: The first part has been shown in [11] The proof of the second part in the theorem is based on the linearization of the replicator dynamics equations around s We define x 1 (t) =s 1 (t) s 1 and x 2 (t) =s 2 (t) s 2 The linearized replicator dynamics writes: ẋ 1(t) γ 1 p1x 1(t)L 1 +(1 p 1)x 2(t)L 12 ẋ 2(t) γ 2 p2x 2(t)L 2 +(1 p 2)x 1(t)L 21 with γ 1 = s 1(1 s 1) and γ 2 = s 2(1 s 2) Which is of the form ẋ (t) =Ax(t) with γ A = 1p 1L 1 γ 1(1 p 1)L 12 γ 2(1 p 2)L 21 γ 2p 2L

4 det(a)= γ 1 γ 2 p1 p 2 L 1 L 2 (1 p 1 )(1 p 2 )L 12 L 21 If det(a) < 0 then the two eigenvalues of A are real and of opposite signs and the instability follows Corollary 1: The mixed intermediate ESS is globally asymptotically stable in the replicator dynamics Proof: From theorem (1) and theorem (2) we need to prove that if 1 =4p 1 p 2 L 1 L 2 ((1 p 1 )L 12 +(1 p 2 )L 21 ) 2 > then =p 1 p 2 L 1 L 2 (1 p 1 )(1 p 2 )L 12 L 21 > 0 (or equivalently 4 p 1 p 2 L 1 L 2 (1 p 1 )(1 p 2 )L 12 L 21 > 0) We have: 4p 1p 2L 1L 2 ((1 p 1)L 12 +(1 p 2)L 21) 2 4 p 1p 2L 1L 2 (1 p 1)(1 p 2)L 12L 21 = ((1 p 1)L 12 (1 p 2)L 21) 2 < 0 The proof follows B Replicator Dynamics with strategic delay In this section we focus on the impact of time delays of strategies on the dynamics An action taken today will have some effect after some time [6] We assume that the strategies G 1 G 2 H 1 and H 2 take a delay τ The replicator dynamics equation for the first community is then given by: Then we get: ṡ 1 (t) =s 1 (t)(1 s 1 (t)) p 1 L 1 s 1 (t τ)+(1 p 1 )L 12 s 2 (t τ)+k 1 Doing the same with the second group we get: ṡ 2 (t) =s 2 (t)(1 s 2 (t)) p 2 L 2 s 2 (t τ)+(1 p 2 )L 21 s 1 (t τ)+k 2 We introduce a small perturbation around s defined by x 1 (t) =s 1 (t) s 1 and x 2 (t) =s 2 (t) s 2 We then make a linearization of the two above equations around the interior equilibrium point s =(s 1s 2) we get the following system: ẋ 1 (t) = γ 1 p1 L 1 x 1 (t τ)+(1 p 1 )L 12 x 2 (t τ) ẋ 2 (t) = γ 2 p2 L 2 x 2 (t τ)+(1 p 2 )L 21 x 1 (t τ) with γ 1 = s 1(1 s 1) and γ 2 = s 2(1 s 2) Taking the Laplace transform of the system above we obtain the following characteristic equation λ 2 λ p 1γ 1L 1 + p 2γ 2L 2 e λτ + γ 1γ 2 p1p 2L 1L 2 (1 p 1) (1 p 2)L 12L 21 e 2λτ =0 (8) The zero solution is asymptotically stable if and only if all solutions of (8) have negative real parts [13] The equation (8) is typical for a linear system of two equations of the form ẋ(t) =Ax(t τ) which was extensively studied by the authors in [14] Based on their results we establish the following theorem on the asymptotic stability of the intermediate ESS in presence of symmetric strategic delays Theorem 3: The mixed intermediate ESS is asymptotically stable in the delayed replicator dynamics if and only if π τ< τ = min( 2 λ π + 2 λ ) with λ ± = p1γ1l1+p2γ2l2± β 2 and β = 2 p 1 γ 1 L 1 + p 2 γ 2 L 2 4γ1 γ 2 p1 p 2 L 1 L 2 (1 p 1 )(1 p 2 )L 12 L 21 Proof: It is previously shown that the conditions of existence of the mixed intermediate ESS are among others p 1 p 2 L 1 L 2 (1 p 1 )(1 p 2 )L 12 L 21 > 0 and L 1 < 0 and L 2 < 0 (corollary 1); which proves that the two solutions of λ 2 [p 1 γ 1 L 1 + p 2 γ 2 L 2 ]λ + γ 1 γ 2 [p 1 p 2 L 1 L 2 (1 p 1 )(1 p 2 )L 12 L 21 ]=0 given by λ ± = p1γ1l1+p2γ2l2± β 2 with β =[p 1 γ 1 L 1 + p 2 γ 2 L 2 ] 2 4γ 1 γ 2 [p 1 p 2 L 1 L 2 (1 p 1 )(1 p 2 )L 12 L 21 ] are real and negative For the remaining of the proof the reader should refer to [14] pp82 theorem (34) This theorem gives an upper bound on the delays for which the ESS remains an asymptotically stable state of the population Beyond this delay bound the mixed ESS loses its asymptotic stability property and persistent oscillations around the equilibrium state occur instead This result is similar to that given by the authors in [4] [6] which demonstrate that the unique mixed ESS becomes an unstable state for large delays in one-population two-strategy settings Simulations of the replicator dynamics equations are coherent with this theorem C Replicator Dynamics with spatial delay In this section we assume that delays are not associated ṡ 1(t) =s 1(t)(1 s 1(t)) U 1(G 1 s(t τ)p) U 1(H 1 s(t τ)p) with the strategy used by an individual but rather with the opponent with which an individual interacts We consider that intra-community interactions take a delay τ in and intercommunity interactions take a delay τ o In this case the replicator dynamics writes: ṡ 1 (t) =s 1 (t)(1 s 1 (t))[u 1 (G 1 (s 1 (t τ in )s 2 (t τ o ))p) U 1 (H 1 (s 1 (t τ in )s 2 (t τ o ))p)] ṡ 2 (t) =s 2 (t)(1 s 2 (t))[u 2 (G 2 (s 1 (t τ o )s 2 (t τ in ))p) U 2 (H 2 (s 1 (t τ o )s 2 (t τ in ))p)] (9) Following the same procedure as in the previous sections we get the following characteristic equation: λ 2 λ(p 1 γ 1 L 1 + p 2 γ 2 L 2 )e τinλ + p 1 p 2 γ 1 γ 2 L 1 L 2 e 2τinλ (1 p 1 )(1 p 2 )γ 1 γ 2 L 12 L 21 e 2τoλ =0 (10) In order to simplify the previous equation we consider that intra-community interactions take no time delay but intercommunity interactions take a delay ie τ in =0and τ o > 0 The characteristic equation (10) then writes: λ 2 (p 1γ 1L 1 + p 2γ 2L 2)λ + γ 1γ 2p 1p 2L 1L 2 γ 1γ 2(1 p 1) (1 p 2)L 12L 21e 2λτo =0 or equivalently: λ 2 + αλ + β + δe λτ =0 (11) where τ = 2τ o α = (p 1 γ 1 L 1 + p 2 γ 2 L 2 ) β = γ 1 γ 2 p 1 p 2 L 1 L 2 δ = γ 1 γ 2 (1 p 1 )(1 p 2 )L 12 L 21 ow we summarize the stability properties of the delayed replicator dynamics in the following theorem which is based on the results of the authors in [15] related to the location of the roots of the characteristic equation (11) Theorem 4: The mixed intermediate ESS is asymptotically stable in the replicator dynamics with symmetric spatial delays for any τ o 0 Proof: The proof of this theorem is based on that given by Freedman and Kuang (theorem (41) page 202) related 3812

5 to the location of the solutions of the characteristic equation (11) which we state as it is given by its authors: If β 2 <δ 2 if s is unstable for τ =0then it is unstable for any τ 0; and if s is stable at τ =0 then it remains stable for τ inferior than some τ s 0 But if s is stable at τ =0 then =p 1 p 2 L 1 L 2 (1 p 1 )(1 p 2 )L 12 L 21 > 0 β 2 >δ 2 Therefore this case is excluded If β 2 >δ 2 2β α 2 > 0 and 2β α 2 2 > 4(β 2 δ 2 ) then the stability of the stationary point can change a finite number of times at most as τ is increased and eventually it becomes unstable But 2β α 2 = 2γ 1 γ 2 p 1 p 2 L 1 L 2 (p 1 γ 1 L 1 + p 2 γ 2 L 2 ) 2 = p 2 1γ 2 1L 2 1 p 2 2γ 2 2L 2 2<0 Therefore this case is excluded in our model Otherwise (this is the only case when s is stable at τ =0) the stability of the stationary point s does not change for any τ 0 s is asymptotically stable for any τ 0 Spatial delays do not alter the stability of the mixed ESS Indeed for any value of the delay τ o the frequency of strategies in the population converges to the mixed ESS after some damped oscillations Spatial delays preserve the asymptotic stability of the ESS and affect only the convergence rate umerical simulations are coherent with this theorem V REPLICATOR DYAMICS WITH SPATIAL-STRATEGIC DELAYS In this section we are interested in studying the stability of the replicator dynamics in presence of both strategic and spatial delays In particular we aim to study whether the spatial delay has a stabilizing effect on the replicator dynamics with strategic delay We define the delays as follows: τ 1 is the time delay that take the first strategies in both groups ie G 1 and G 2 when interacting with a strategy from the same group; τ 2 is the delay that take the second strategies in both groups ie H 1 and H 2 when interacting with a strategy from the same group; τ 1 is the delay that take the first strategies in both groups when interacting with a strategy from the other group; τ 2 is the delay that take the second strategies in both groups when interacting with a strategy from the other group In addition we have τ i >τ i i =1 2 Indeed τ i corresponds to a strategic delay to which is added a spatial delay due to the inter-community interaction The expected payoffs of strategies G 1 and H 1 in community 1 then write: U 1(G 1 (s 1(t τ 1)s 2(t τ 1))p)=p 1 (t τ 1)a 1 +(1 s 1(t τ 1))b 1 +(1 p1) s 2(t τ 1)a 12 +(1 s 2(t τ 1))b 12 U 1(H 1 (s 1(t τ 2)s 2(t τ 2))p)=p 1 (t τ 2)c 1 +(1 s 1(t τ 2))d 1 +(1 p1) s 2(t τ 2)c 12 +(1 s 2(t τ 2))d 12 Hence: ṡ 1 (t) =s 1 (t)(1 s 1 (t)) p 1 α 1 s 1 (t τ 1 ) p 1 α 2 s 1 (t τ 2 ) +(1 p 1 )α 3 s 2 (t τ 1 ) (1 p 1 )α 4 s 2 (t τ 2 )+K 1 (12) with α 1 = a 1 b 1 α 2 = c 1 d 1 α 3 = a 12 b 12 α 4 = c 12 d 12 Doing the same with the second community we obtain: ṡ 2 (t) =s 2 (t)(1 s 2 (t)) p 2 β 1 s 2 (t τ 1 ) p 2 β 2 s 2 (t τ 2 ) +(1 p 2 )β 3 s 1 (t τ 1 ) (1 p 2 )β 4 s 1 (t τ 2 )+K 2 (13) with β 1 = a 2 b 2 β 2 = c 2 d 2 β 3 = a 21 b 21 and β 4 = c 21 d 21 Following the same procedure in the previous sections we get the following characteristic equation: λ 2 λ(p 2γ 2β 1 + p 1γ 1α 1)e τ 1λ + λ(p 2γ 2β 2 + p 1γ 1α 2)e τ 2λ p 1p 2γ 1γ 2(α 1β 2 + α 2β 1)e (τ 2+τ 1 )λ + p 1p 2γ 1γ 2α 1β 1e 2τ 1λ +p 1p 2γ 1γ 2α 2β 2e 2τ 2λ (1 p 1)(1 p 2)γ 1γ 2α 3β 3e 2 τ 1λ + (1 p 1)(1 p 2)γ 1γ 2(α 3β 4 + β 3α 4)e ( τ 1+ τ 2 )λ (1 p 1) (1 p 2)γ 1γ 2α 4β 4e 2 τ 2λ =0 (14) When τ 1 = τ 2 = τ 1 = τ 2 = τ we find the characteristic equation (8) obtained when there is only symmetric strategic delay Under the approximation of small time delays we can expand the characteristic equation above and keeping only linear terms we get an equation of the form: where Aλ 2 + Bλ + C =0 (15) A =1+(p 1γ 1α 1 + p 2γ 2β 1)τ 1 (p 1γ 1α 2 + p 2γ 2β 2)τ 2 B = p 1γ 1L 1 p 2γ 2L 2 + p 1p 2γ 1γ 2(α 1β 2 + α 2β 1 2α 1β 1)τ 1 + p 1p 2γ 1γ 2(α 1β 2 + α 2β 1 2α 2β 2)τ 2 2γ 1γ 2(1 p 1)(1 p 2)(α 3β 3 + α 3β 4 + β 3α 4) τ 1 2(1 p 1)(1 p 2)γ 1γ 2(α 4β 4 + α 3β 4 + β 3α 4) τ 2 C = γ 1γ 2(p 1p 2L 1L 2 (1 p 1)(1 p 2)L 12L 21) Therefore for the small delays that satisfy the conditions A>0 B<0 and C>0 all roots of equation (15) admit negative real parts and the stability of the interior stationary points follows We make simulations of the system (12) and (13) to study its stability We take up the following illustrative numerical example inspired from the Hawk-Dove game: A = B = G1 H1 G H G2 H2 G H G D = H C = G2 G1 H2 H1 G H with p 1 = 04 and p 2 = 06 Then the values of the parameters are: L 1 = 2 L 2 = 2 L 12 = 09 L 21 = 32 K 1 =064 K 2 =124 s 1 =036 s 2 =064 τ =36 (theorem (3)) In the first case we keep only strategic delay and we fix it to τ 1 = τ 2 = 37 > τ = 36 time units The trajectories of the proportion of the population using G 1 and G 2 are given in Fig 2 Left and they oscillate around the equilibrium s 1 =036 and s 2 =064 In the second case we keep only spatial delay: τ 1 = τ 2 = 0 τ 1 = τ 2 =12 In Fig 2 Right We observe the convergence 3813

6 (t) (t) τ 1 time (t) (t) * * (t) (t) time * (t)(t) * Iterations (t)(t) * (t)(t) * Iterations time Fig 2 Left: Replicator dynamics with purely strategic delay Right:RD with purely spatial delay Bottom: RD with both strategic and spatial delays Iterations Fig 3 Stability of equilibria in a complete graph Left: Purely strategic delay Right: Purely spatial delay Bottom: both strategic and spatial delays to the equilibrium Indeed the replicator dynamics is stable for any value of the purely spatial delay (theorem (4)) In the last case (Fig 2 bottom) strategic and spatial delays are combined together Unlike our expectation spatial delay does not have a stabilizing effect Instead the oscillations are persistent with higher amplitude and lower frequency VI REPLICATOR DYAMICS O RADOM GRAPHS In this section we are interested in studying update rules that make the replicator dynamics converge to the ESS in a complete random graph We assume that individuals from both communities are located randomly in a graph and each individual may interact in a non-uniform fashion with any other individual from any group in the graph At each iteration (or a time step) a randomly chosen individual from group i revises its strategy He chooses either G i or H i according to argmax A {G ih i} U i (A sp) ie an individual chooses a strategy that is a best response to the (delayed) population profile We consider the cases of strategic delay spatial delay and strategic-spatial delay We aim to observe the effect of this update rule and time delays on the convergence to the equilibrium We consider a numerical example in a random complete graph with 1 = 800 individuals from group 1 and 2 = 1200 individuals from group 2 Then the probabilities of intra-community interaction and inter-community interaction in the first (resp second) community are p 1 = 04 and 1 p 1 = 06 (resp p 2 = 06 and 1 p 2 = 04) The payoff matrices are the same as in the previous section We observe that using this updare rule the replicator dynamics converges to the equilibrium in presence of a spatial delay (figure (3) Right) When a strategic delay of a value τ = 37 time units > τ is introduced we observe persistent oscillations (figure (3) Left) Also when both types of the delay coexist oscillations are persisent (figure (3) Bottom) The effect of time delays is similar to that observed in the previous section on the replicator dynamics VII COCLUSIOS In this paper we focused our analysis on the replicator dynamics with time delays in the context of non-uniform interactions between two communities In such contexts time delays may be associated with the used strategies and with the communities of the interacting players as well We proved that the mixed intermediate ESS may be an unstable state for the replicator dynamics for large strategic delays while it remains a stable state for any value of the purely spatial delay In presence of both types of the delay the instability property is emphasized and the amplitude of oscillations significantly increases REFERECES [1] J Maynard Smith Evolution and the theory of Games Cambridge University Press UK 1982 [2] W H Sandholm Population Games and Evolutionary Dynamics MIT Press December 2010 [3] P D Taylor Evolutionarily stable strategies with two types of player J Appl Prob vol 16 no 1 pp Mars 1979 [4] T Yi and W Zuwang Effect of time delay and evolutionarily stable strategy Jour Theor Biol vol 187 no 1 pp July 1997 [5] J Alboszta and J Miekisz Stability of evolutionarily stable strategies in discrete replicator dynamics with time delay Jour Theor Biol vol 231 no 2 p ovembre 2004 [6] H Tembine E Altman and R El-Azouzi Delayed evolutionary game dynamics applied to medium access control in MASS Pisa Italy oct 2007 [7] M ewman The structure and function of complex networks SIAM Review vol 45 pp [8] M Tomassini and E Pestelacci Coordination games on dynamical networks Games vol 1 pp [9] R Olfati-Saber Evolutionary dynamics of behavior in social networks in CDC IEEE 2007 [10] G Szabo and G Fath Evolutionary games on graphs Physical Report vol 446 pp [11] Ben Khalifa R El-Azouzi Y Hayel H Sidi and I Mabrouki Evolutionary stable strategies in interacting communities in Valuetools Turin Italy dec 2013 [12] J W Weibull Evolutionary Game Theory Cambridge: MIT Press 1995 [13] R Bellman and K Cooke Differential Difference Equations ew York: Academic Press 1963 [14] T Hara and J Sugie Stability region for systems of differentialdifference equations Funkcialaj Ekvacioj vol 39 no 1 pp [15] H I Freedman and Y Kuang Stability switches in linear scalar neutral delay equations Funkcialaj Ekvacioj vol 34 no 1 pp