Some Analytical Properties of the Model for Stochastic Evolutionary Games in Finite Populations with Non-uniform Interaction Rate

Commun Theor Phys 60 (03) 37 47 Vol 60 No July 5 03 Some Analytical Properties of the Model for Stochastic Evolutionary Games in Finite Populations Non-uniform Interaction Rate QUAN Ji ( ) 3 and WANG Xian-Jia ( ) 34 GuangDong Electric Power Design Institute China Energy Engineering Group Co Ltd Guangzhou 50663 China Department of Management and Economics Tianjin University Tianjin 30007 China 3 Institute of Systems Engineering Wuhan University Wuhan 43007 China 4 School of Economics and Management Wuhan University Wuhan 43007 China (Received September 4 0; revised manuscript received January 0 03) Abstract Traditional evolutionary games assume uniform interaction rate which means that the rate at which individuals meet and interact is independent of their strategies But in some systems especially biological systems the players interact each other discriminately Taylor and Nowak (006) were the first to establish the corresponding non-uniform interaction rate model by allowing the interaction rates to depend on strategies Their model is based on replicator dynamics which assumes an infinite size population But in reality the number of individuals in the population is always finite and there will be some random interference in the individuals strategy selection process Therefore it is more practical to establish the corresponding stochastic evolutionary model in finite populations In fact the analysis of evolutionary games in a finite size population is more difficult Just as Taylor and Nowak said in the outlook section of their paper The analysis of non-uniform interaction rates should be extended to stochastic game dynamics of finite populations In this paper we are exactly doing this work We extend Taylor and Nowak s model from infinite to finite case especially focusing on the influence of non-uniform connection characteristics on the evolutionary stable state of the system We model the strategy evolutionary process of the population by a continuous ergodic Markov process Based on the limit distribution of the process we can give the evolutionary stable state of the system We make a complete classification of the symmetric games For each case game the corresponding limit distribution of the Markov-based process is given when noise intensity is small enough In contrast most literatures in evolutionary games using the simulation method all our results obtained are analytical Especially in the dominant-case game coexistence of the two strategies may become evolutionary stable states in our model This result can be used to explain the emergence of cooperation in the Prisoner is Dilemma Games to some extent Some specific examples are given to illustrate our results PACS numbers: 050Le 873Kg 0545Pq Key words: stochastic evolutionary games non-uniform interaction rate finite population evolutionary stable state Introduction Game theory has been widely recognized as an important tool in studying the social political and economic conflict and cooperation [] However the Nash equilibrium [] analysis as the analysis framework for traditional game theory has many limitations and defects First the Nash equilibrium is interpreted as the game player s best response to each other but the existing mathematical methods are not mature enough to solve it in some game models [3] Second the arrival of Nash equilibrium is not only impeded by the rationality degree of game players but also puzzled by multiple equilibriums [4 5] Finally the traditional analysis framework cannot explain the process of choosing and arriving of Nash equilibrium In contrast evolutionary game theory which is a theory frame suitable for bounded rationality can solve these problems to some extent Evolutionary game theory which is based on the population of individuals was pioneered by Maynard and Price [6] Unlike the static analysis methods of rational inference in the traditional game theory in the evolutionary game theory a learning and strategy adjustment process of individuals is introduced Theoretical research of the evolutionary game theory mainly focuses on this learning process or the strategy adjustment dynamics of the population and then the corresponding stable states of the system Many mathematical approaches describing this dynamics have been proposed such as models based on ordinary differential equations [7 0] partial differential equations [ ] stochastic differential equations [3 5] cellular automata [6 9] and stochastic processes [0 5] The applied research of the evolutionary games mainly focuses on exploring the evolution of cooperation [6 3] Traditional evolutionary games assume uniform interaction rate which means that the rate at which individuals Supported by the National Natural Science Foundation of China under Grant Nos 73007 7079 and 6057407 Corresponding author E-mail: quanji3@whueducn c 03 Chinese Physical Society and IOP Publishing Ltd http://wwwioporg/ej/journal/ctp http://ctpitpaccn

38 Communications in Theoretical Physics Vol 60 meet and interact is independent of their strategies But in some systems especially biological systems the players interact each other discriminately As we know Taylor and Nowak [33] were the first to study evolutionary game dynamics non-uniform interaction rate Especially Tang et al [34] found the relationship between the interaction rate of strategies and the structure coefficient of the population very recently Structured population is a hot topic in evolutionary games these years and many interesting results have been obtained in the area [35 40] This makes the research of non-uniform interaction rate model more meaningful However Taylor and Nowak s model is based on replicator dynamics which assumes an infinite population In reality the number of individuals in the population is always finite and there will be some random interference in the individuals strategy selection process Therefore it is more practical to establish the stochastic evolutionary game model non-uniform interaction rate in finite populations In fact the analysis of evolutionary games in a finite size population is more difficult Just as Taylor and Nowak said in the outlook section of their paper The analysis of non-uniform interaction rates should be extended to stochastic game dynamics of finite populations [33] This is our motivation for this paper In this paper we use the stochastic model of Amir and Berninghaus [] in symmetric games finite populations Instead of well-mixed interaction we use the non-uniform interaction which allows the interaction rates to depend on the strategies This extension leads to non-linear payoff functions which result in more subtle results and their conclusions are the special case of ours We make a complete classification of the symmetric games and give the corresponding evolutionary stable state in each case All of the results given are analytical Especially in the dominant-case game coexistence of the two strategies may become evolutionary stable states in our model To illustrate our results some examples are given and we plot the corresponding area of each case The rest of the paper is organized as follows In Sec we describe our model in details In Sec 3 we give some analytical results prepared for the model In Sec 4 we apply our theory to all cases of symmetric games and give the main results of this paper Some numerical examples to illustrate our results are given in Sec 5 And the paper is concluded in the last section The appendix in the end contains all the proof of the theorems in this paper Model Consider a finite population size N playing symmetric games The strategy set of the game is A B} and the payoff matrix is A B ( A B ) a b c d which means an A player will obtain a when playing against another A or b when playing against B Choosing strategy B results in either obtaining c (against A) or d (against B) Suppose that the probability of interaction between two players is not independent of their strategies Let r r r 3 denote the reaction rate of two A players an A player and a B player two B players respectively (r r r 3 > 0) The payoff of each individual is determined by the average payoff over a large number of interactions In order to simplify the situation to obtain the analytical results we consider an outside player playing against the population [] When the population size is small there will be some difference between the outsider and insider methods owing to the slight difference of the payoff functions But when the population size is large enough the results of the two methods are the same For such a player when the number of individuals in the population choosing A is i the expected payoff of choosing strategy A and B are respectively: π i A ar i + br (N i) r i + r (N i) π i B cr i + dr 3 (N i) r i + r 3 (N i) () Introduce stochastic process z(t) that denotes the number of the individuals choosing strategy A at time t S 0 N} is the state space of the process Players update strategy by their payoff The three hypotheses: inertia myopia and mutation to describe the bounded rational behavior in Ref [] are used in this paper Because of inertia we can suppose that there is no chance that two or more players change their strategies at the same time Thus we can model this evolutionary learning system as a birth-death process Let λ i ε + κ (π i A π i B) + for i S N} λ N 0 () µ i ε + κ (π i B π i A) + for i S 0} µ 0 0 (3) where ε is a small positive number κ > 0 is constant f if f 0 f + 0 if f < 0 When z(t) i (i S) a player switches from strategy B to strategy A rate λ i and from strategy A to strategy B rate µ i Thus given that π i A > πi B (in this case µ i ε) and a transfer has occurred the player has more incentives to shift from strategy B to strategy A; that is z(t) jumps upward to i + probability

No Communications in Theoretical Physics 39 λ i /(λ i + ε) and also downward to i probability ε/(λ i + ε) because of noise or some uncertain factors in the decision making process Here ε denotes noise intensity and κ denotes the speed at which the players react to the environment The evolutionary process z(t) is a homogenous Markov chain in continuous time state space S For any t > 0 we have an (N + ) (N + ) transition matrix P (t) p ij (t)} ij S p ij (t) is the probability that z(t) will be in state j after time t given that presently it is in state i In our model p ij (t) pz(s + t) j z(s) i} s 0 (4) p ii+ (t) λ i t + o(t) for i S N} p ii (t) µ i t + o(t) for i S 0} p ii (t) (λ i + µ i )t + o(t) for i S p ij (t) o(t) otherwise (5) As ε > 0 the Markov chain is ergodic Based on the results in the birth-death process for any i S lim p ij(t) t + exists and all of them are equal Let lim p ij(t) v ε ξ j j t + N k0 ξ i S k ξ 0 ξ k λ 0λ λ k µ µ µ k ( k N) (6) So v ε (v0 ε v ε vn ε ) is the limit distribution of the process when noise intensity is ε 3 Analytical Properties of the Limit Distribution 3 Classification of Cases Without loss of generality let r (otherwise divide the denominator and numerator of π i A and πi B by r then replace r /r by r r 3 /r by r 3 ) Let x i/n y x Then Denote Thus π A (i) ar i + b(n i) r i + (N i) π B (i) ci + dr 3(N i) i + r 3 (N i) ar x + by r x + y cx + dr 3y x + r 3 y (7) α r r 3 (a d) + (b c) β r (a c) γ r 3 (b d) (8) h(x) (β + γ α)x + (α γ)x + γ (9) π A (i) π B (i) ar x + by r x + y cx + dr 3y x + r 3 y h(x) (r x + y)(x + r 3 y) (0) As (r x + y)(x + r 3 y) > 0 for any values of x y r r 3 so sign(π i A π i B) sign h(x) () The following cases should be distinguished Case β + γ α 0 In this case h(x) is a linear function of x h(x) (α γ)x + γ 0 x γ/(γ α) (-i) x [0 ] let i maxi i S i N x } (-i-a) γ 0 (-i-b) γ > 0 (-ii) x / [0 ] so x [0 ]: (-ii-a) h(x) > 0 (-ii-b) h(x) < 0 (-iii) x /N /N } h(x) 0 in this case α β γ 0 Case β + γ α > 0 In this case h(x) is a parabola opening upward Let (α γ) 4(β + γ α)γ α 4βγ (-i) α 4βγ 0 In this case for any x h(x) 0 (-ii) α 4βγ > 0 the two roots of h(x) 0 are x (α γ) α 4βγ (β + γ α) x (α γ) + α 4βγ (β + γ α) (x < x ) (-ii-a) x 0 x In this case for any x (0 ) h(x) < 0 (-ii-b) x 0 0 < x < Let i maxi i S i N x } (-ii-c) 0 < x < x Let i maxi i S i N x } (-ii-d) 0 < x < 0 < x < Let i maxi i S i N x } i maxi i S i N x } i < i Case 3 β + γ α < 0 In this case h(x) is a parabola opening downward (3-i) α 4βγ 0 In this case for any x h(x) 0 (3-ii) α 4βγ > 0 the two roots of h(x) 0 are x (α γ) + α 4βγ (β + γ α) x (α γ) α 4βγ (x < x (β + γ α) ) (3-ii-a) x 0 x In this case for any x (0 ) h(x) > 0 (3-ii-b) 0 < x < x Let i maxi i S i N x } (3-ii-c) x 0 0 < x < Let i maxi i S i N x } (3-ii-d) 0 < x < 0 < x < Let i maxi i S i N x } i maxi i S i N x } i < i 3 Analysis Results of v ε When ε 0 + Let lim ε 0 + v ε (v 0 v v N ) v ()

40 Communications in Theoretical Physics Vol 60 Lemma 3 give the distribution of v for each case listed above They can be regarded as the preparation before giving the main results All the proofs are in the appendix Lemma In Case that is β + γ α 0 In case (-i-a) when i < (N )/ v puts probability on state N; when i > (N )/ v puts probability on state 0; when i (N )/ v puts probability /( + B N (0)) on 0 and B N (0)/( + B N (0)) on N B N (0) a (N )/+ a N a a (N )/ κ (πb (i) π A (i)) i (N )/ κ (π A (i) π B (i)) i > (N )/ In case (-i-b) v puts probability p on state i and p on state i + a i + p + + κ (πb (i) π A (i)) i i κ (π A (i) π B (i)) i > i In case (-ii-a) v puts probability on state N In case (-ii-b) v puts probability on state 0 In case (-iii) v puts the same probability on all the states in the state space 0 N} Lemma In Case that is β + γ α > 0 In case (-i) v puts probability on state N In case (-ii-a) v puts probability on state 0 In case (-ii-b) when i < (N )/ v puts probability on state N; when i > (N )/ v puts probability on state 0; when i (N )/ v puts probability /( + B N (0)) on 0 and B N (0)/( + B N (0)) on N B N (0) a (N )/+ a N a a (N )/ κ (πb (i) π A (i)) i (N )/ κ (π A (i) π B (i)) i > (N )/ In case (-ii-c) v puts probability p on state i and p on state i + + p + + κ (πa (i) π B (i)) i i or i > i κ (π B (i) π A (i)) i < i i In case (-ii-d) when i i > N v puts probability p on state i and p on state i + + p + + κ (πa (i) π B (i)) i i or i > i κ (π B (i) π A (i)) i < i i when i i < N v puts probability on state N; when i i N v puts probability p on state i p on state i + and p p on state N p + + + + + + + + a N p + + + + + + + a N Lemma 3 In Case 3 that is β + γ α < 0 In case (3-i) v puts probability on state 0 In case (3-ii-a) v puts probability on state N In case (3-ii-b) when i < (N )/ v puts probability on state N; when i > (N )/ v puts probability on state 0; when i (N )/ v puts probability /( + B N (0)) on 0 and B N (0)/( + B N (0)) on N B N (0) a (N )/+ a N a a (N )/ κ (πb (i) π A (i)) i (N )/ κ (π A (i) π B (i)) i > (N )/ In case (3-ii-c) v puts probability p on state i and p on state i + + p + + κ (πb (i) π A (i)) i i or i > i κ (π A (i) π B (i)) i < i i In case (3-ii-d) when i + < i v puts probability p on state i and p on state i + p +/( + + ); when i + > i v puts probability on state 0; when i + i v puts probability p on state 0 p on state i and p p on state i + a p + a + + + + + + p + + a + + + + + +

No Communications in Theoretical Physics 4 4 Main Results for all Cases of Symmetric Games symmetric games can be classified as dominantcase game coordinate-case game and coexistent-case game The following Theorem gives all the possible distribution of v for each case game The proof of the Theorem is in the appendix Theorem (i) In the dominant-case games (a) If strategy A is strictly dominant than strategy B that is a > c b > d only the following four cases can occur Case (-ii-a): v puts probability on state N Case (-i): v puts probability on state N Case (-ii-d): when i i > N v puts probability p on state i and p on state i + + p + + κ (πa (i) π B (i)) i i or i > i κ (π B (i) π A (i)) i < i i when i i < N v puts probability on state N; when i i N v puts probability p on state i p on state i + and p p on state N p + + + + + + + + a N p + + + + + + + a N Case (3-ii-a) v puts probability on state N (b) If strategy B is strictly dominant than strategy A that is a < c b < d only the following four cases can occur Case (-ii-b): v puts probability on state 0 Case (-ii-a): v puts probability on state 0 Case (3-i): v puts probability on state 0 Case (3-ii-d): when i + < i v puts probability p on state i and p on state i + p +/( + + ); when i + > i v puts probability on state 0; when i + i v puts probability p on state 0 p on state i and p p on state i + a p + a + + + + + + p + + a + + + + + + (ii) In the coordinate-case games that is a > c b < d only the following three cases can occur Case (-i-a): when i < (N )/ v puts probability on state N; when i > (N )/ v puts probability on state 0; when i (N )/ v puts probability /( + B N (0)) on 0 and B N (0)/( + B N (0)) on N B N (0) a (N )/+ a N /a a (N )/ κ (πb (i) π A (i)) i (N )/ κ (π B (i) π B (i)) i > (N )/ Case (-ii-b): when i < (N )/ v puts probability on state N; when i > (N )/ v puts probability on state 0; when i (N )/ v puts probability /( + B N (0)) on 0 and B N (0)/( + B N (0)) on N B N (0) a (N )/+ a N /a a (N )/ K (πb (i) π A (i)) i (N )/ K (π B (i) π B (i)) i > (N )/ Case (3-ii-b): when i < (N )/ v puts probability on state N; when i > (N )/ v puts probability on state 0; when i (N )/ v puts probability /( + B N (0)) on 0 and B N (0)/( + B N (0)) on N B N (0) a (N )/+ a N /a a (N )/ κ (πb (i) π A (i)) i (N )/ κ (π A (i) π B (i)) i > (N )/ (iii) In the coexistent-case games that is a < c b > d only the following three cases can occur Case (-i-b): v puts probability p on state i and p on state i + a i + p + + κ (πb (i) π A (i) i i κ (π A (i) π B (i) i > i Case (-ii-c): v puts probability p on state i and p on state i + + p a i + + κ (πa (i) π B (i)) i i or i > i κ (π B (i) π A (i)) i < i i Case (3-ii-c): v puts probability p on state i and p on state i + p + + +

Communications in Theoretical Physics 4 ( ai Vol 60 κ (πb (i) πa (i)) i i or i > i κ (πa (i) πb (i) i < i i Given the limit distribution of the process we can get the evolutionary stable state of the system It is necessary to emphasize that the evolutionary stable state we discussed in this paper is not the same but more stronger than the evolutionary stable strategy (ESS) proposed by Maynard Smith and Price in 973[6] According to Maynard Smith and Price a strategy is ESS if it cannot be invaded by a small number of individuals playing a different strategy In this paper what we discussed is a state on which the probability does not go to zero when the noise is vanishing This is equivalent the stochastic stable equilibrium (SSE) proposed by Dean Foster and Peyton Young[34] According to their theory a state P is an SSE if in the long run it is nearly certain that the system lies in every small neighborhood of P as the noise tends slowly to zero And it can be seen as a refinement of the ESS 5 Numerical Examples and Discussion Example Payoff matrix is ( 35 0 ) population size N 0 It is a dominant-case game and strategy B is strictly dominant than strategy A According to the Theorem the following three situations can occur: v puts probability on state 0; v puts probability p on state i and p on state i + ; v puts probability p on state 0 p on state i and p p on state i + Figure gives the corresponding areas about each situation when 0 < r r3 < 5 Fig The corresponding areas of the three situations when 0 < r r3 < 5 in the B-dominant game Fig The corresponding areas of the three situations when 0 < r r3 < 5 in the coordinate-case game It is a coordinate-case game According to the Theorem the following three situations can occur: v puts probability on state N ; v puts probability on state 0; v puts probability /( + BN (0)) on 0 and BN (0)/( + BN (0)) on N Figure gives the corresponding areas about each situation when 0 < r r3 < 5 3 Example 3 Payoff matrix is ( 5 ) population size 0 N 0 It is a coexistent-case game According to the Theorem the following three situations can occur: v puts probability p on state i and p on state i + ; v puts probability p on state i and p on state i + ; v puts probability p on state i and p on state i + Figure 3 gives the corresponding areas about each situation when 0 < r r3 < 5 Fig 3 The corresponding areas of the three situations when 0 < r r3 < 5 in the coexistent-case game 6 Conclusions Example Payoff matrix is ( 50 46 ) population size N In summary the motivation of this paper is based on the idea of non-uniform interaction rate model proposed

No Communications in Theoretical Physics 43 by Taylor and Nowak in 006 We have completed the work listed in the outlook part of their paper to extend the analysis of non-uniform interaction rates from deterministic dynamics of infinite populations to stochastic game dynamics of finite populations Especially in the stochastic framework we also get the following results for symmetric games: (i) In the dominant-case games ie strategy A is strictly dominant than strategy B the non-uniform interactive assumption may change the results of the evolutionary dynamics and coexistence of the two strategies may become the evolutionary stable state of the system However the nonuniform interactive assumption cannot change the results of the invasion dynamics That is strategy A can invade strategy B while strategy B cannot invade strategy A (ii) In the coordinate-case games the non-uniform interactive assumption cannot change the results of the evolution dynamics all A or all B are still the two evolutionary stable states of the system (iii) In the coexistentcase games the non-uniform interactive assumption cannot change the results of the evolution dynamics but only change the specific location of equilibrium point The coexistence of the two strategies is still the only evolutionary stable state of the system These results are the same as that of Taylor and Nowak s in the deterministic case The main discovery which is different from that of their deterministic framework can be summarized as follows: (i) The equilibrium discussed in our paper is stochastically stable which means that the equilibrium reached is independent of the initial state [34] But in the deterministic framework this is not the case More specifically in the dominant-case games ie strategy A is strictly dominant than strategy B when in the deterministic framework if most of the individuals choose strategy A in the initial state all individuals choose strategy A will be the evolutionary stable state of the system; However if most of the individuals choose strategy B in the initial state coexistence of strategy A and B will be the evolutionary stable state of the system [33] But in the stochastic framework the initial state of choosing the two strategies will not affect the final equilibrium at all (ii) In the stochastic framework the introduction of a vanishingly small noise term can cause the system to select among the ESS: some of the ESS may be stochastically stable while others are not But in our paper we prove all types of ESS in the deterministic framework can be preserved to be stochastically stable non-uniform interaction rate The results we obtained will be much more stronger than those of Taylor and Nowak s Specifically in the coordinate-case games both strategies are the ESS but only the risk dominant strategy is stochastically stable in the uniform interaction rate case [3] So the non-uniform interaction rate condition changes the stochastic stable equilibrium in the stochastic framework but not changes the ESS in the deterministic framework in this coordinate-case games Appendix Proof of Lemma In case (-i-a) when i i π B (i) π A (i); when i > i π B (i) < π A (i) Let κ (πb (i) π A (i)) i i thus λ i Denote ε i i κ (π A (i) π B (i)) i > i a i + ε i > i and µ i ai + ε i i ε i > i A j A j (ε) ξ j λ 0λ λ j µ µ µ j (a + ε) (a j + ε) B j B j (ε) ( + + ε) (a j + ε) (a + ε) ( + ε) ( j i ) (i + < j N) ε j (a +ε) (a j+ε) εj A j j i ε i + (a +ε) ( +ε) ε εi A i j i + ε i + ( + +ε) (a j +ε) (a +ε) ( +ε)ε j i εi j+ B j j > i + ξ k [ + εa + + ε i A i ] + [ε i A i ] + [ε i B i k0 + + + ε i N+ B N ] when i < (N )/ i N + < 0 v j lim vj ε ξ j j N lim ε 0 ε 0 k0 ξ k 0 otherwise v puts probability on state N; when i > (N )/ i N + > 0 v j lim ε 0 v ε j v puts probability on state 0; j 0 0 otherwise

44 Communications in Theoretical Physics Vol 60 p j 0 when i (N /) i N + 0 v j lim vj ε p j N p /( + B N (0)) ε 0 0 otherwise v puts probability /( + B N (0)) on 0 and B N (0)/( + B N (0)) on N In case (-i-b) when i i π A (i) π B (i); when i > i π A (i) < π B (i) κ (πa (i) π B (i)) i i ai + ε i i ε i i Let thus λ i and µ i κ (π B (i) π A (i)) i > i ε i > i a i + ε i > i Denote A j A j (ε) (a 0 + ε) (a j + ε) ( j i ) B j B j (ε) (a 0 + ε) ( + ε) ( + + ε) (a j + ε) (i + j N) ξ j λ 0λ λ j µ µ µ j k0 ξ k (a0+ε) (a j +ε) ε j [ + A ε + + A ] i ε i v j lim v ε ξ j j lim ε 0 ε 0 k0 ξ k (a 0+ε) ( +ε)ε j (i +) ε i ( + +ε) (a j+ε) Bj [ Bi + Aj ε j j i + + + ε i p j i p j i + 0 otherwise ε i j+ j i + B ] N ε i N+ + p a i + + v puts probability p on state i and p on state i + In case (-ii-a) π A (i) > π B (i) Let κ (π A (i) π B (i)) > 0 for any i so λ i a i + ε and µ i ε (0 i N) Denote A j A j (ε) (a 0 + ε) (a j + ε) ( j N) ξ j λ 0λ λ j (a 0 + ε) (a j + ε) µ µ µ j ε j A j ε j ( j N) ξ 0 k0 ξ k + A ε + + A N ε N v j lim v ε ξ j j lim ε 0 ε 0 k0 ξ k j N 0 otherwise v puts probability on state N In case (-ii-b) π A (i) < π B (i) Let κ (π B (i) π A (i)) > 0 for any i so λ i ε and µ i a i + ε (0 i N) Denote A j A j (ε) ( j N) (a + ε) (a j + ε) ξ j λ 0λ λ j µ µ µ j k0 ξ k + εa + + ε N A N v j lim ε 0 v ε j lim ε 0 ε j (a + ε) (a j + ε) εj A j ( j N) ξ 0 ξ j j 0 k0 ξ k 0 otherwise v puts probability on state 0 In case (-iii) π A (i) π B (i) for any i So λ i µ i ε ξ j v j /(N + ) (0 j N) v puts the same probability on all the states in the state space 0 N} Proof of Lemma We only give the proof of case (-ii-d) proof of other cases are similar but simpler When i i π A (i) π B (i); when i < i i π A (i) π B (i); when i > i π A (i) > π B (i) κ (πa (i) π B (i)) i i or i > i ai + ε i i or i > i Let thus λ i κ (π B (i) π A (i)) i < i i ε i < i i ε i i or i > i and µ i a i + ε i < i i Denote A j A j (ε) (a 0 + ε)(a + ε) (a j + ε) ( j i + ) B j B j (ε) (a 0 + ε) ( + ε) ( + + ε) (a j + ε) (i + < j i )

No Communications in Theoretical Physics 45 So C j C j (ε) (a 0 + ε) ( + ε) ( + + ε) (a j + ε) ( + + ε) ( + ε) (a 0+ε)(a +ε) (a j +ε) ε Aj j (a 0+ε)(a +ε) ( +ε) A i + ε i ( ++ε) (i + < j N) ε j j i j ε i ( ++ε) i + ξ j λ 0λ λ j µ µ µ j (a 0+ε) ( +ε)ε j (i +) ε i ( ++ε) (a j+ε) (a 0+ε) ( +ε)ε i i Bj ε i j+ i + < j i ε i ( ++ε) ( +ε)ε B i ε i i + j i + k0 ξ k [ + A [ C i + + (a 0+ε) ( +ε)ε i i ( ++ε) (a j +ε) ε + + Ai ε i ( ++ε) ( +ε)ε j i ε i ] [ A i ] [ + + Bi + + ε i ( + + ε) C ] N ε i i +N C j ε i i +j j > i + ε i + + B i ] ε i i + ε i i + + + When i i > N i > i i + N p j i v j lim vj ε ξ j lim ε 0 ε 0 k0 ξ p j i + p k 0 otherwise So v (v 0 v N ) puts probability p on state i and p on state i + j N When i i < N i < i i + N v j lim vj ε ε 0 + + + 0 otherwise So v puts probability on state N p j i When i i N i i i + N v j lim vj ε p j i + ε 0 p p j N 0 otherwise p + + + + + + + + a N p + + + + + + + a N So v puts probability p on state i p on state i + and p p on state N + [ B i ε i i + ] Proof of Lemma 3 We only give the proof of case (3-ii-4) proof of other cases are similar but simpler When i i π A (i) π B (i); when i < i i π A (i) π B (i); when i > i π A (i) < π B (i) Let κ (πb (i) π A (i)) i i or i > i κ (π A (i) π B (i)) i < i i ε i i or i > i ai + ε i i or i > i thus λ i and µ i a i + ε i < i i ε i < i i Denote A j A j (ε) (a + ε) (a j + ε) B j B j (ε) ( + + ε) (a j + ε) (a + ε) ( + ε) ( j i ) (i + < j i + ) ( C j C j (ε) + + ε) ( + ε) (a + ε) ( + ε) ( + + ε) (a j + ε) (j > i + )

46 Communications in Theoretical Physics Vol 60 So ε j (a +ε) (a j+ε) εj A j j i ε i + (a +ε) ( +ε) ε εi Ai j i + ξ j λ 0λ λ j µ µ µ j ε i + ( ++ε) (a j +ε) (a +ε) ( +ε)ε j i ε i j+ B j i + < j i ε i + ( ++ε) ( +ε) (a +ε) ( +ε)ε i i ( ++ε) εi i + Bi + ε i + ( ++ε) ( +ε)ε j i ++ε j i + (a +ε) ( +ε)ε i i ( ++ε) (a j+ε) εi i +j C j j > i + ξ k [ + εa + + ε i Ai k0 ] + [ε i Ai ] + [ε i B i + + + ε i i + B i ] [ + ε i i + B i ] + + [ε i i + C i + + ε + + + ε i i +N C N ] When i + < i i i + < 0 p j i v j lim vj ε ξ j lim ε 0 ε 0 k0 ξ p j i + k 0 otherwise p + + + So v (v 0 v N ) puts probability p on state i and p on state i + j 0 When i + > i i i + > 0 v j lim vj ε v puts probability on state 0 ε 0 0 otherwise p j 0 When i + i i i + 0 v j lim vj ε p j i ε 0 p p j i + 0 otherwise a p + a + + + + + + p + + a + + + + + + v puts probability p on state 0 p on state i and p p on state i + Proof of Theorem When β + γ α > 0 and α 4βγ > 0 x (α γ) α 4βγ (β + γ α) x (α γ) + α 4βγ (β + γ α) (x < x ) x < 0 α > γ or α < γ γ < 0 and x > α > β or α < β β < 0 0 < x x < α < 0 β > 0 γ > 0 0 < x < x > α < γ β < 0 γ > 0 x < 0 0 < x < α < β β > 0 γ < 0 When β + γ α < 0 and α βγ > 0 x (α γ) + α 4βγ (β + γ α) x (α γ) α 4βγ (x < x (β + γ α) ) x < 0 α < γ or α > γ γ > 0 and x > α < β or α > β β > 0 0 < x x < α > 0 β < 0 γ < 0 0 < x < x > α > γ β > 0 γ < 0 x < 0 0 < x < α > β β < 0 γ > 0 (i) In the dominant-case games (a) If strategy A is strictly dominant than strategy B β > 0 γ > 0 Case : β + γ α so α > γ > 0 x γ γ α > or < 0 h(0) γ > 0 h() α γ > 0 Case (-ii-a) satisfies the condition Case : β + γ > α When α 4βγ 0 case (-i) satisfies the condition When α 4βγ > 0 as β > 0 γ > 0 only case (-ii-d) satisfies the condition

No Communications in Theoretical Physics 47 Case 3: β + γ < α so α > (β + γ) 4βγ As β > 0 γ > 0 only case (3-ii-a) satisfies the condition (b) If strategy B is strictly dominant than strategy A β < 0 γ < 0 Case : β + γ α so α < γ < 0 x γ/(γ α) > or < 0 h(0) γ < 0 h() α γ < 0 Case (-ii-b) satisfies the condition Case : β + γ > α so α < 0 α > (β + γ) 4βγ As β < 0 γ < 0 only case (-ii-a) satisfies the condition Case 3: β + γ < α When α 4βγ 0 case (3-i) satisfies the condition When α 4βγ > 0 as β < 0 γ < 0 only case (3-ii-d) satisfies the condition (ii) In the coordinate-case games β > 0 γ < 0 so α 4βγ > 0 Case : β + γ α so α > γ 0 < x γ/(γ α) < γ < 0 case (-i-a) satisfies the condition Case : β + γ > α so α < β As β > 0 γ < 0 only case (-ii-b) satisfies the condition Case 3: β + γ < α As β > 0 γ < 0 only case (3-ii-b) satisfies the condition (iii) In the coexistent-case games β < 0 γ > 0 so α 4βγ > 0 Case : β + γ α so α < γ 0 < x γ/(γ α) < γ < 0 case (-i-b) satisfies the condition Case : β + γ > α As β < 0 γ > 0 only case (-ii-c) satisfies the condition Case 3: β + γ < α As β < 0 γ > 0 only case (3-ii-c) satisfies the condition References [] J Von Neumann and O Morgenstern Theory of Games and Economic Behavior Princeton University Press Princeton (944) [] JF Nash Annals of Mathematics 54 (95) 89 [3] PJJ Herings and RJAP Peeters Econ Theory 8 (00) 59 [4] K Binmore Econ Philos 3 (987) 79 [5] K Binmore Econ Philos 4 (988) 9 [6] J Maynard Smith and GR Price Nature (London) 46 (973) 5 [7] PD Taylor and LB Jonker Math Biosci 40 (978) 45 [8] J Maynard Smith Evolution and the Theory of Games Cambridge University Cambridge (98) [9] JW Weibull Evolutionary Game Theory The MIT Cambridge (995) [0] J Hofbauer and K Sigmund Evolutionary Games and Population Dynamics Cambridge University Cambridge (998) [] VCL Hutson and GT Vickers J Math Biol 30 (99) 457 [] R Cressman and GT Vickers J Theor Biol 84 (997) 359 [3] D Foster and P Young Theor Popul Biol 38 (990) 9 [4] V Corradi and R Sarin J Econ Theory 94 (000) 63 [5] H Ohtsuki J Theor Biol 64 (00) 36 [6] MA Nowak and MM Robert Nature (London) 359 (99) 86 [7] G Szabo and T Czaran Phys Rev E 63 (00) 06904 [8] G Szabo J Vukov and A Szolnoki Phys Rev E 7 (005) 04707 [9] J Vukov and G Szabo Phys Rev E 7 (005) 03633 [0] M Kandori GJ Mailath and R Rob Econometrica 6 (993) 9 [] M Amir and SK Berninghaus Games Econ Behav 4 (996) 9 [] MA Nowak A Sasaki C Taylor and D Fudenberg Nature (London) 48 (004) 646 [3] C Taylor D Fudenberg A Sasaki and MA Nowak Bull Math Biol 66 (004) 6 [4] J Quan and XJ Wang Chin Phys B 0 (0) 03003 [5] J Quan and XJ Wang Commun Theor Phys 56 (0) 404 [6] E Pennisi Science 309 (005) 93 [7] MA Nowak and K Sigmund Nature (London) 437 (005) 9 [8] FC Santos and JM Pacheco Phys Rev Lett 95 (005) 09804 [9] H Ohtsuki C Hauert E Lieberman and MA Nowak Nature (London) 44 (006) 50 [30] MA Nowak Science 34 (006) 560 [3] PD Taylor T Day and G Wild Nature (London) 447 (007) 469 [3] XJ Wang J Quan and WB Liu Commun Theor Phys 57 (0) 897 [33] C Taylor and MA Nowak Theor Popul Biol 69 (006) 43 [34] C Tang X Li L Cao and J Zhan J Theor Biol 306 (0) [35] CG Nathanson CE Tarnita and MA Nowak PLoS Comp Biol 5 (009) e00065 [36] MA Nowak CE Tarnita and T Antal Philos T R Soc B 365 (00) 9 [37] CE Tarnita N Wage and MA Nowak Proc Natl Acad Sci 08 (0) 334 [38] J Wang B Wu DWC Ho and L Wang Europhys Lett 93 (0) 5800 [39] Z Wang A Szolnoki and M Perc Europhys Lett 97 (0) 4800 [40] C Hauert and LA Imhof J Theor Biol 99 (0) 06 [4] P Young Econometrica 6 (993) 57