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A general classification of the replication dynamics with a unique fixed point in the interior of simplex $S_N$
Hongju Daisy Chen, Bin Yi, Zhanshan Sam Ma
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Summary
This paper establishes a general classification framework for replicator dynamics in evolutionary game theory, specifically focusing on systems with a unique fixed point in the interior of the simplex S_N for Nโฅ2. It derives sufficient and necessary conditions for the existence of such fixed points and links the stability properties (Nash Equilibrium, Evolutionarily Stable Strategy, Lyapunov stability) to the definiteness of a derived symmetric matrix B. The framework generalizes known results for N=2 (Hawk-Dove) and N=3 (Rock-Scissors-Paper) games.
Entities (12)
Relation Signals (7)
Replicator Dynamics โ isfoundationof โ Evolutionary Game Theory
confidence 95% ยท The replication dynamics (differential equation system) is the foundation of evolutionary game theory.
Theorem 1 โ providesconditionsfor โ Unique Fixed Point in Int S_N
confidence 95% ยท Theorem 1... the sufficient and necessary conditions of the unique fixed point of replicator dynamics Eqn. (1) in Int S_N are...
Theorem 2 โ linksmatrixdefinitenessto โ Stability Classification
confidence 94% ยท Theorem 2... p* is an ESS if and only if B is a positive definite matrix... if B is a negative definite matrix, then x* is unstable...
Evolutionarily Stable Strategy โ implies โ Asymptotic Stability
confidence 93% ยท the fixed point corresponding to ESS is the asymptotically stable fixed point of the Eqn. (1).
Nash Equilibrium โ correspondsto โ Fixed Point
confidence 92% ยท The point corresponding to the NE strategy must be the fixed point of Eqn. (1)
Hawk-Dove Game โ isexampleof โ Replicator Dynamics
confidence 88% ยท We first discuss the well familiar hawk-dove game... According to replicator dynamics, there is: x' = x(1-x)(b/2 - cx)
Rock-Scissors-Paper Game โ isexampleof โ Replicator Dynamics
confidence 87% ยท Now let us discuss the N=3 case. Assume that A=... Rock-Scissors-Paper (RSP) game with N=3 strategies
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Abstract
Abstract:The replication dynamics (differential equation system) is the foundation of evolutionary game theory. When n=2, there are four possible types of replication dynamics. When n=3, there are 49 possible types of replication dynamics. However, when n>3, the classification of replication dynamics has not been solved. In this article, the sufficient and necessary conditions of the replication dynamics equation with a unique fixed point in the interior of simplex $S_n$(Int$S_n$) for $n\geq 2$ are presented. Furthermore, the different types of replication dynamics equations with a unique fixed point in IntSn is discussed.
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- Source: https://arxiv.org/abs/2605.13883v1
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1 A general classification of the replication dynamics with a unique fixed point in the interior of simplex S N Hongju (Daisy) Chen 1,2,3,4 Bin Yi 3 Zhanshan (Sam) Ma 3,4,5* 1 School of Mathematics and Statistics, Guilin University of Technology, Guilin, Guangxi, China 2 Guangxi Colleges and Universities Key Laboratory of Applied Statistics, Guilin, Guangxi, China 3 Computational Biology and Medical Ecology Lab Biostatistics and Image Genetics Lab Kunming Institute of Zoology, Chinese Academy of Sciences, Kunming, China 4 Kunming College of Life Sciences, University of Chinese Academy of Sciences, Kunming, China 5 Faculty of Arts and Sciences Harvard University Cambridge, MA, 02138, USA *For All Correspondence: ma@vandals.uidaho.edu 2 Abstract The replication dynamics (differential equation system) is the foundation of evolutionary game theory. When ํ=2, there are four possible types of replication dynamics. When ํ=3, there are 49 possible types of replication dynamics. However, when ํ>3, the classification of replication dynamics has not been solved. In this article, the sufficient and necessary conditions of the replication dynamics equation with a unique fixed point in the interior of simplex S n (ํผํํกํ ) ) for ํโฅ2 are presented. Furthermore, the different types of replication dynamics equations with a unique fixed point in ํผํํกํ ) is discussed. Keywords: Replicator Dynamics; Evolutionary Game Theory (EGT); Nash Equilibrium (NE); Evolutionary Stable Strategy (ESS) Introduction Replicator dynamics have found wide application in computer science (Bloembergen et al., 2015; Tuyls et al., 2020; Ma & Krings 2011), economics (Sandholm, 2010), behavioural ecology and evolutionary biology (Maynard-Smith & Price 1972; Maynard-Smith 1982, McNamara & Leimar, 2020), and epidemiology (Nowak & May, 1994; Ma & Zhang 2025). A complete classification of the dynamics, especially for cases with a unique interior fixed point, is therefore of broad theoretical and practical interest. Consider an evolutionary game of single population of many individuals with N strategy. Let A=ํ ,- .ร. be the payoff matrix of the game, where N is the total number of pure strategies of the game, ํ ,- = ํํ , ,ํ - , is the payoff of a player (individual) with a pure strategy ํ , against another player (individual) with a pure strategy ํ - . The equation of replicator dynamics was first proposed by Taylor and Jonker (1978), and then named by Schuster and Sigmund (1983). It has become the most important equation in game dynamics (Cressman & Tao 2014). Suppose that ํฅ , is the frequency of choosing strategy ํ , in the population, then the replicator dynamics equations can be written as: 3 ํฅ 6 =ํฅ 6 ํดํฅ 6 โํฅํดํฅโฒ ํฅ 9 =ํฅ 9 ํดํฅ 9 โํฅํดํฅโฒ โฎ ํฅ . =ํฅ . ํดํฅ . โํฅํดํฅโฒ ๏ผ1๏ผ where ํฅ=(ํฅ 6 ,ํฅ 9 ,...,ํฅ . )โํ . , ํดํฅ , =ํฅ - ํ ,- . -@6 , and ํฅ A is the transpose of ํฅ and ํ . represent a ํโ1 dimension simplex, that is ํ . =(ํฅ 6 ,ํฅ 9 ,...,ํฅ . )|ํฅ , โฅ0,ํฅ , . ,@6 =1. For ํโฅ2, we divide the simplex ํ . into the following two parts: (i) โํโ1,2,...,ํ,ํฅ , >0,ํฅ , . ,@6 =1,and ํฅ=(ํฅ 6 ,ํฅ 9 ,...,ํฅ . )โํ . is the interior point of ํ . . The set of all such points is called the interior of ํ . and denoted by ํผํํกS M , which means that each strategy does not go extinct. (i) โํโ1,2,...,ํ,ํฅ , =0, and ํฅ=(ํฅ 6 ,ํฅ 9 ,...,ํฅ . )โํ . is a boundary point of ํ . . The set of all such points is called the boundary of ํ . and denoted by ํตํS M , which means at least one of the strategies goes extinct. The equations of replicated dynamics (1) can be divided into three cases (Hofbauer & Sigmund 1998; Nowak 2006): (i) There is a unique fixed point in ํผํํกS M . (i) There are infinitely many fixed points in ํผํํกS M . There is now a manifold in ํผํํกS M , and these fixed points are stable, but not asymptotically stable. (i) There is no fixed point in ํผํํกS M , and all orbits will tend to ํตํS M . In evolutionary game theory, two equilibria are very important (Huttegger & Zollman 2013): One is (strictly) Nash equilibrium (NE) and the other is evolutionarily stable strategy (ESS) (Maynard & Price 1973). The fixed point of the replication dynamics equation is closely related to the NE. We summarize the conclusions as follows: Theorem A (Hofbauer & Sigmund 2003; Cressman & Tao 2014), known as the Folk Theorem of Evolutionary Game Theory, states: (i) The point corresponding to the NE strategy must be the fixed point of Eqn. (1); (i) The strategy corresponding to the fixed point of Lyapunov stable in 4 Eqn. (1) must be NE; (i) The strategy corresponding to the ฯ limit points of the inner orbitals of Eqn. (1) must be NE. (iv) The point corresponding to the strictly NE strategy must be the asymptotically stable fixed point of Eqn. (1). Maynard & Price (1973) proposed the concept of ESS, and Taylor and Jonker (1978) proved that the fixed point corresponding to ESS is the asymptotically stable fixed point of the Eqn. (1). The study of the dynamic behavior (such as the fixed point and its stability) of the replication dynamics Eqn. (1) is essential for understanding the evolutionary game theory. In fact, when ํ=2, the payoff matrix is A= ํb cd , then it has four possible types of replicator dynamics (Nowak 2006; Sigmund 2010), with the following figures: Fig 1. The figures of replicator dynamics when n=2. Note: (1) for case 1 and 2, there is no fixed point in ํผํํกS 9 , and all the orbits tend to ํตํS 9 . 00.10.20.30.40.50.60.70.80.91 x 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 dx/dt Case 1: a>c and b>d 00.10.20.30.40.50.60.70.80.91 x -0.04 -0.035 -0.03 -0.025 -0.02 -0.015 -0.01 -0.005 0 dx/dt Case 2: a<c and b<d 0.10.20.30.40.50.60.70.80.9 x -0.01 -0.005 0 0.005 0.01 0.015 0.02 0.025 dx/dt Case 3: a>c and b<d 0.10.20.30.40.50.60.70.80.9 x -0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 dx/dt Case 4: a<c and b>d 5 (2) for case 3, there is only one fixed point for equation in ํผํํกS 9 , and all the orbits (except for x ยบ x * ) tend to ํตํS 9 . (3) for case 4, there is only one fixed point for equation in ํผํํกS 9 , and all the inner orbits tend to x*. When ํ=3, the payoff matrix is ํด= ํ 66 ํ 69 ํ 6U ํ 96 ํ 99 ํ 9U ํ U6 ํ U9 ํ U , then it has 49 possible types of replicator dynamics (Bomze 1983, 1995). To the best of our knowledge, the classification of the replication dynamics equation for ํ>3 has not been solved (Sigmund 2010). The objective of this study is to solve part of the problem. The main results As introduced previously, the replication dynamics Eqn. (1) can be divided into three cases, i.e., infinitely many fixed points in ํผํํกS M , no fixed point in ํผํํกS M , and a unique fixed point in ํผํํกS M . This section includes the sufficient and necessary conditions of Eqn. (1), for the case that has a unique fixed point in Intํ . . The detailed proof for the sufficient and necessary condition is presented in this section. Theorem 1 Suppose that the payoff matrix of the single population with N strategies is ํด= (ํ XY ) .ร. , the sufficient and necessary conditions of the unique fixed point of replicator dynamics Eqn. (1) in ํผํํกS M are ํด .ร, ํด .ร(,Z6) >0,(ํ=1,2,...,ํโ1.), where ํด .ร, is the cofactor of the N-th row and the i-th column elements of the following determinant: ํ= ํ 66 โํ 96 ํ 96 โํ U6 โฎ ํ 69 โํ 99 ํ 99 โํ U9 โฎ ํ (.[6)6 โํ .6 1 ํ (.[6)9 โํ .9 1 โฏ โฏ โฑ ํ 6. โํ 9. ํ 9. โํ U. โฎ โฏ โฏ ํ (.[6). โํ .. 1 and the fixed point of Eqn. (1) in ํผํํกS . is ํฅ โ =ํฅ 6 โ ,ํฅ 9 โ ,...,ํฅ . โ ,ํฅ , โ = _ `รa b = _ `รa _ `รa ` acd . Notice that: ํฅ โ โํผํํกS . , thus ํ โ =ํฅ , โ . ,@6 ํ , is a Nash Equilibrium (NE), but it is not a strictly Nash Equilibrium (SNE). 6 Proof Consider that these replicator dynamics equations are given by: ํฅ 6 =ํฅ 6 ํดํฅ 6 โํฅํดํฅ A ํฅ 9 =ํฅ 9 ํดํฅ 9 โํฅํดํฅ A โฎ ํฅ . =ํฅ . ํดํฅ . โํฅํดํฅ A (1) where ํฅ=ํฅ 6 ,ํฅ 9 ,...,ํฅ . โํ .. It is easy to see that the fixed points in Intํ . if and only if ํดํฅ 6 =ํดํฅ 9 =โฏ=ํดํฅ . ํฅ 6 +ํฅ 9 +โฏ+ํฅ . =1. (2) That is, ํดํฅ , โํดํฅ ,Z6 =0๏ผํ=1,2,...,Nโ1 ํฅ 6 +ํฅ 9 +โฏ+ํฅ . =1 By expanding the above formula, we obtain: (ํ 66 โํ 96 )ํฅ 6 + ( ํ 69 โํ 99 )ํฅ 9 +โฏ+(ํ 6. โํ 9. )ํฅ . =0 (ํ 96 โํ U6 )ํฅ 6 + ( ํ 99 โํ U9 )ํฅ 9 +โฏ+(ํ 9. โํ U. )ํฅ . =0 โฎ (ํ .[66 โํ .6 )ํฅ 6 +(ํ .[69 โํ .9 )ํฅ 9 +โฏ+(ํ (.[6). โํ .. )ํฅ . =0 ํฅ 6 +ํฅ 9 +โฏ+ํฅ . =1 (3) This is an N-ary inhomogeneous linear equation for which the sufficient and necessary condition for a unique solution in ํ ) is the determinant of coefficients ํ= ํ 66 โํ 96 ํ 96 โํ U6 โฎ ํ 69 โํ 99 ํ 99 โํ U9 โฎ ํ (.[6)6 โํ .6 1 ํ (.[6)9 โํ .9 1 โฏ โฏ โฑ ํ 6. โํ 9. ํ 9. โํ U. โฎ โฏ โฏ ํ (.[6). โํ .. 1 โ 0 By expanding it on the N-th row, we discover that all the elements of the N-th row of the determinant ํ are 1, and we get ํ=1 . ,@6 รํด .ร, =ํด .ร, . ,@6 where ํด .ร, represent the cofactor of the N-th row and the i-th column element of the determinant d. That is, ํด .ร, =โ1 .Z, รํ .ร, , where ํ .ร, represents the minor of the N-th row and the i-th column element of the determinant d. 7 By Cramer Rule, the unique solution of equation (2) can be written as: ํฅ , = b a b ,ํ=1,2,...,ํ. Where ํ , (ํ=1,2,...,ํ) is a new determinant, the i-th column element of the determinant d is replaced by the vector 0,0,...,0,1 A . If expand the determinant ํ , in terms of the i-th column, note that the first ํโ1 elements of the column are 0 and the last one is 1. Consequently, ํ , = ํด .ร, ,ํ=1,2,...,ํ, then we get ํฅ , = ํด .ร, ํ = ํด .ร, ํด .ร, . ,@6 ,ํ=1,2,...,ํ. However, for this solution to be in Intํ . , we must have ํฅ , = _ `รa b = _ `รa _ `รa ` acd >0,ํ=1,2,...,ํ. This implies the numerator and the denominator have to be both positive or negative, i.e., ํด .ร, ํด .ร(,Z6) >0,ํ=1,2,...,ํโ1. When the replicator dynamic equation (1) has a unique fixed point in Intํ . , and denote by ํฅ โ =ํฅ 6 โ ,ํฅ 9 โ ,...,ํฅ . โ ,ํฅ , โ = ํด .ร, ํ = ํด .ร, ํด .ร, . ,@6 >0,ํ=1,2,...,ํ. Denote ํ โ =ํฅ , โ R , M ,@6 , then by the above proof, we have ํดํฅ โ , =ํฅ โ ํดํฅ โ A ๏ผํ=1,2,...,N that is โํโ1,2,...,N,ฯR , ,ํ โ =ฯํ โ ,ํ โ . So, for any strategy ํ=ํฅ , R , M ,@6 ๏ผํฅ=(ํฅ 6 ,ํฅ 9 ,...,ํฅ . )โํ . we have ํํ,ํ โ =ฯํ โ ,ํ โ . Therefore, ํ โ =ํฅ , โ R , M ,@6 is a Nash Equilibrium. Theorem 1 has been proved. The following corollary can be drawn from the above Theorem 1: Corollary 1 If the determinant ํโ 0 in Theorem 1 and โํโ1,2,...,ํโ1๏ผsuch that ํด .ร, ํด .ร(,Z6) โค0, there is no solution for Eqn. (1) in ํผํํกS M , then every orbit tends to ํตํS M . 8 Proof Since the determinate ํโ 0 of theorem 1 and โํโ1,2,...,ํโ1, such that ํด .ร, ํด .ร(,Z6) โค0. By the proof of theorem 1, there is only one solution in ํ ) , but not in Intํ . . Therefore, there is no solution of equation (1) in Intํ . , and every orbit tends to ํตํํ . . As shown in figure 1, for case 3 and case 4, there is only one solution x * in Intํ 9 . For case 3, all the orbits (except for x ยบ x * ) tend to ํตํํ 9 , and for case 4, all inner orbits tend to x * . The general case for N ยณ 2, we obtain the following results: Theorem 2 Suppose that ํด=(ํ XY ) .ร. is the payoff matrix of a single population with N strategies that satisfied the condition of Theorem 1, B is a N-1 symmetric square matrix with ํ ,- = โ(ํ ,- +ํ -, โํ ., โํ ,. โํ .- โํ -. +2ํ .. ),1โคํ,ํโคํโ1, there are: (i) ํ โ is an ESS if and only if ํต is a positive definite matrix, particularly, ํฅ=ํฅ โ is globally asymptotically stable and all inner orbits converge to ํฅ โ . ํ โ is a NE, but it is not an ESS if and only if ํต is not a positive definite matrix. (i) If B is a positive semi-definite matrix, then ํฅ โ is Lyapunov stable, particularly, if ํต=0๏ผthen ํฅ , p a โ โกํ . ,@6 . (i) if B is a negative definite matrix, then ํฅ โ is unstable and every orbit tends to ํตํS M . To prove the Theorem 2, we first give the following two lemmas: Lemma 1 (Hofbauer et al. 1998, P64): The strategy ํ is an ESS in Intํ . if and only if ํAํ A =ํํ,ํ>ํํ,ํ=ํAํ A For all strategy ํโ ํ and ํโํ . . Lemma 2 Suppose that ํํฅ=ํํฅ 6 ,...,ํฅ ) =ํฅ 6 ,...,ํฅ ) ํ 66 โฏํ 6) โฎโฑโฎ ํ )6 โฏํ )) ํฅ 6 โฎ ํฅ ) , 9 where ํฅ=ํฅ 6 ,...,ํฅ ) โํ ) , ํด= ํ 66 โฏํ 6) โฎโฑโฎ ํ )6 โฏํ )) is a symmetric matrix and ํฟ is a positive number, then ํํฅ>0 in ํ x 0,ํฟ= ํฅ0<ํฅ=ํฅ 6 9 +โฏ+ํฅ ) 9 <ํฟ if and only if the matrix ํด is a positive definite matrix. Proof If ํด= ํ 66 โฏํ 6) โฎโฑโฎ ํ )6 โฏํ )) is a positive definite matrix, then ํํฅ>0 holds for all ํฅโ 0. Naturally, ํํฅ>0 ํํ ํ x 0,ํฟ= ํฅ0<ํฅ=ํฅ 6 9 +โฏ+ํฅ ) 9 <ํฟ. Assume ํํฅ>0 ํํ ํ x 0,ํฟ= ํฅ0<ํฅ=ํฅ 6 9 +โฏ+ํฅ ) 9 <ํฟ. By proof by contradiction, we suppose ํด= ํ 66 โฏํ 6) โฎโฑโฎ ํ )6 โฏํ )) is not a positive definite matrix, then there must exist ํฅ โ =ํฅ 6 โ ,...,ํฅ ) โ โํ ) ,ํฅ โ โ 0 such that ํํฅ โ โค0. Thus, there exist a large number ํ>0 such that p โ z <ํฟ, i.e., p โ z โํ x 0,ํฟ. However, ํ p โ z = 6 z ํํฅ โ โค 0. This is a contradiction. Thus, ํด= ํ 66 โฏํ 6) โฎโฑโฎ ํ )6 โฏํ )) is a positive definite matrix. This completes the proof. The proof of theorem 2 is presented based on the above two Lemmas. Proof: 1) Let ํฅ=(ํฅ 6 ,ํฅ 9 ,...,ํฅ . )โํ . , ํ=ํฅ , R , M ,@6 , by lemma 1, the strategy ํ โ is an ESS if and only if ํฅ โ Aํฅโฒ=ํํ โ ,ํ>ํํ,ํ=ํฅAํฅโฒ holds for all ํฅโํ . ,ํฅโ ํฅ โ . Let ํํฅ=ํํฅ 6 ,ํฅ 9 ,...,ํฅ . =ํฅ โ Aํฅ A โํฅAํฅ A , it is easy to see that ํํฅ โ =0, where ํํฅ is an N-ary quadratic function. So, the strategy ํ โ is an ESS if and only if 10 ํํฅ=ํํฅ 6 ,ํฅ 9 ,...,ํฅ . =ํฅ โ Aํฅ A โํฅAํฅ A >0=ํํฅ โ holds for all ํฅโํ . ,ํฅโ ํฅ โ . That is, strategy ํ โ is an ESS if and only if ํฅ โ is the strict minimum point of ํํฅ on the ํฅ= ํฅ 6 ,ํฅ 9 ,...,ํฅ . ํฅ , โฅ0,ํฅ , . ,@6 =1, where ํํฅ=ํํฅ 6 ,ํฅ 9 ,...,ํฅ . =ํฅ โ Aํฅ A โํฅAํฅ A . Note that ํฅ . =1โํฅ 6 โโฏโํฅ .[6 ๏ผํฅ . โ =1โํฅ 6 โ โโฏโํฅ .[6 โ ๏ผthereby, ํํฅ=ํํฅ 6 ,ํฅ 9 ,...,ํฅ . =ํฅ โ Aํฅ A โํฅAํฅ A =ํฅ 6 โ ,...,ํฅ .[6 โ ,1โํฅ 6 โ โโฏโํฅ .[6 โ ํด ํฅ 6 โฎ ํฅ .[6 1โํฅ 6 โโฏโํฅ .[6 โํฅ 6 ,...,ํฅ .[6 ,1โํฅ 6 โโฏโํฅ .[6 ํด ํฅ 6 โฎ ํฅ .[6 1โํฅ 6 โโฏโํฅ .[6 . It is a quadratic function of the N-1 element with respect to ํฅ 6 ,...,ํฅ M[6 , which is denoted by ํํฅ 6 ,ํฅ 9 ,...,ํฅ .[6 =ํํฅ 6 ,ํฅ 9 ,...,ํฅ . . Note that, ํฅ , โ = _ `รa b = _ `รa _ `รa ` acd >0 holds for any ํโ1,2,...,ํ. Set ํฟ=min p a โ . ,1โคํโคํ, we have ํฟ>0. Denote ํฆ โ =(ํฅ 6 โ ,...,ํฅ .[6 โ )โR .[6 ,ํฆ=ํฅ 6 ,ํฅ 9 ,...,ํฅ .[6 โR .[6 ; then we have ํํฆ= ํํฅ 6 ,ํฅ 9 ,...,ํฅ .[6 is defined on the set ํํฆ โ ,ํฟ=ํฆโํฆ โ =ํฅ , โํฅ , โ9 .[6 ,@6 <ํฟโ ํ .[6 .In other words, when ํฆ=ํฅ 6 ,...,ํฅ .[6 โํํฆ โ ,ํฟ, and let ํฅ . =1โํฅ 6 โโฏโํฅ .[6 , then we have ํฅ 6 ,...,ํฅ .[6 ,ํฅ . โํ . hold. However, ํํฅ 6 ,ํฅ 9 ,...,ํฅ .[6 =ํํฅ 6 ,ํฅ 9 ,...,ํฅ . 11 =ํฅ 6 โ ,...,ํฅ .[6 โ ,ํฅ . โ ํด ํฅ 6 โฎ ํฅ .[6 ํฅ . โํฅ 6 ,...,ํฅ .[6 ,ํฅ . ํด ํฅ 6 โฎ ํฅ .[6 ํฅ . =ํฅ , โ . ,@6 (ํดํฅ) , โํฅ , . ,@6 (ํดํฅ) , where ํฅ . =1โํฅ 6 โโฏโํฅ .[6 ๏ผํฅ . โ =1โํฅ 6 โ โโฏโํฅ .[6 โ ๏ผํดํฅ , =ํฅ - ํ ,- . -@6 . Note that it is an ํโ1-ary quadratic function in terms of ํฅ 6 ,...,ํฅ .[6 ; therefore, there are continuous second-order partial derivatives, and all the second-order partial derivatives are constant. We next prove that ํฆ โ =(ํฅ 6 โ ,...,ํฅ .[6 โ )is a stationary point of ํํฅ 6 ,ํฅ 9 ,...,ํฅ .[6 . By calculation, we have: ํํ ํํฅ 6 =ํฅ , โ . ,@6 ํ(ํดํฅ) , ํํฅ 6 โ( ํํฅ , ํํฅ 6 . ,@6 ํดํฅ , +ํฅ , ํํดํฅ , ํํฅ 6 ) =ํฅ , โ . ,@6 ํ ,6 โํ ,. โ(ํดํฅ 6 โํดํฅ . )โํฅ , ํ ,6 โํ ,. . ,@6 =(ํฅ , โ โํฅ , ) . ,@6 ํ ,6 โํ ,. โ(ํดํฅ 6 โํดํฅ . ) It is worth noting that when ํฅ , =ํฅ , โ ํ=1,2,...,ํโ1, there must be ํฅ . =1โํฅ 6 โโฏโํฅ .[6 =1โํฅ 6 โ โโฏโํฅ .[6 โ =ํฅ . โ . Note that ํฅ โ =(ํฅ 6 โ ,ํฅ 9 โ ,...,ํฅ . โ ) is a solution of equation (2), thus we have Aํฅ โ 6 =Aํฅ โ M . Hence, ยย ยp d | (p d โ ,...,p `ย d โ ) =0 holds. Similarly, ยย ยp a | (p d โ ,...,p `ย d โ ) =0(ํ=2,3,...,ํโ1). Thus, ํฆ โ =(ํฅ 6 โ ,...,ํฅ .[6 โ ) is a stationary point of ํํฅ 6 ,ํฅ 9 ,...,ํฅ .[6 . Because ํํฅ 6 ,ํฅ 9 ,...,ํฅ .[6 is an N-1-ary quadratic function with respect to ํฅ 6 ,...,ํฅ .[6 , which has continuous second-order partial derivatives, third-order partial derivatives and above on ํy โ ,ฮดโR M[6 are equal to 0, and ํฆ โ =(ํฅ 6 โ ,...,ํฅ .[6 โ ) is a stationary point of ํํฅ 6 ,ํฅ 9 ,...,ํฅ .[6 . Therefore, by Taylor Formula expansion of the multivariate function, we have: 12 ํํฅ 6 ,...,ํฅ .[6 =ํํฅ 6 โ ,...,ํฅ .[6 โ + ํํ ํํฅ , | (p d โ ,...,p `ย d โ ) .[6 ,@6 ร(ํฅ , โํฅ , โ ) + 1 2 ํฅ 6 โํฅ 6 โ ,...,ํฅ .[6 โํฅ .[6 โ ํปgํฆ โ ํฅ 6 โํฅ 6 โ โฎ ํฅ .[6 โํฅ .[6 โ = 6 9 ํฅ 6 โํฅ 6 โ ,...,ํฅ .[6 โํฅ .[6 โ ํปgํฆ โ ํฅ 6 โํฅ 6 โ โฎ ํฅ .[6 โํฅ .[6 โ ๏ผ4๏ผ where Hgํฆ โ =ํ ,- .[6.[6 ,ํ ,- = ํ 9 ํ ํํฅ , ํํฅ - | (p d โ ,...,p `ย d โ ) = ํ 9 ํ ํํฅ - ํํฅ , | (p d โ ,...,p `ย d โ ) =ํ -, , 1โคํ,ํโคํโ1. It is a Hessian matrix of ํํฅ 6 ,ํฅ 9 ,...,ํฅ .[6 at the point ํฅ 6 โ ,...,ํฅ .[6 โ and a symmetric square matrix. By calculation, we get ํ ,- =โ(ํ ,- +ํ -, โํ ., โํ ,. โํ .- โํ -. +2ํ .. ),1โคํ,ํโคํโ1. If strategy ํ โ is an ESS, then ํฆ โ is a strict minimum point of ํํฆ=ํํฅ 6 ,...,ํฅ .[6 =ํฅ โ Aํฅ A โํฅAํฅ A on the set ํฅ=(ํฅ 6 ,ํฅ 9 ,...,ํฅ . )|ํฅ , โฅ0,ํฅ , . ,@6 =1. Thus, there must be a strict minimum point of ํํฆ on ํํฆ โ ,ํฟโํ .[6 . By lemma 2, Hgํฆ โ is a positive definite matrix. If Hgํฆ โ is a positive definite matrix, by Equation (4), ํํฅ 6 ,...,ํฅ .[6 >0 holds for all ํฅ 6 ,ํฅ 9 ,...,ํฅ .[6 โ ํฅ 6 โ ,...,ํฅ .[6 โ . Consequently, ํฆ โ is the strict minimum point of ํํฆ= ํํฅ 6 ,...,ํฅ .[6 =ํฅ โ Aํฅ A โํฅAํฅ A for any ํฅ=(ํฅ 6 ,ํฅ 9 ,...,ํฅ . )โํ . . Hence, the strategy ํ โ is an ESS. 2) Set ํํฅ=ํฅ , โ p a โ โ . ,@6 ํฅ , p a โ . ,@6 ํฅ=(ํฅ 6 ,ํฅ 9 ,...,ํฅ . )โํ . then, ํํฅโฅ0๏ผand ํํฅ=0โํฅ=ํฅ โ =ํฅ 6 โ ,ํฅ 9 โ ,...,ํฅ . โ . Thus, by the derivation process of 1) and calculation, we have: 13 ํํฅ= ํํ(ํฅ) ํํก =โ(ํฅ โ Aํฅ โฒ โํฅAํฅ โฒ )รํฅ , p a โ . ,@6 =โ 6 9 ํฅ 6 โํฅ 6 โ ,...,ํฅ .[6 โํฅ .[6 โ ํต ํฅ 6 โํฅ 6 โ โฎ ํฅ .[6 โํฅ .[6 โ รํฅ , p a โ . ,@6 If B is a positive semi-definite matrix, then โํฅ=ํฅ 6 ,ํฅ 9 ,...,ํฅ . โํ . ,ํฅโ ํฅ โ , we have ํํฅโค0 thus, ํฅ โ is a fixed point of Lyapunov stability of the equation (1). Particularly, if ํต=0, then ํํฅโก0. Thus, ํฅ , p a โ โกํ . ,@6 , where c is determined by its initial value. 3) if B is a negative definite matrix, then โํฅ=ํฅ 6 ,ํฅ 9 ,...,ํฅ . โํ . ,ํฅโ ํฅ โ ๏ผwe have ํํฅ=โ 6 9 ํฅ 6 โํฅ 6 โ ,...,ํฅ .[6 โํฅ .[6 โ ํต ํฅ 6 โํฅ 6 โ โฎ ํฅ .[6 โํฅ .[6 โ รํฅ , p a โ . ,@6 >0. That is, any orbits except ํฅโกํฅ โ , we obtain ํํฅ=ํฅ , โ p a โ โ . ,@6 ํฅ , p a โ . ,@6 is strictly monotonically increasing. It is note that max pโย ` ํํฅ=sup pโย)ยย ` ํํฅ=max pโยbย ` ํํฅ=ํฅ , โ p a โ . ,@6 , thus, every orbital except ํฅโกํฅ โ tends to ํตํํ . . Theorem 2 has been proved. Note that there are only five possibilities for matrix B, including a positive definite matrix, a positive semi-definite matrix, a negative definite matrix, a negative semi-definite matrix and an indefinite matrix. Based on the above theorem, ํ โ is a NE but not an ESS if and only if B is not a positive definite matrix; if B is a positive semi-definite matrix, then ํฅโกํฅ โ is Lyapunov stable; if B is a negative definite matrix, then except for ํฅโกํฅ โ , every orbit tends to ํตํํ . . In the next section, some examples are presented to illustrate that: if B is an indefinite matrix or a negative semi-definite matrix except for ํต=0, then some orbitals may tend to ํตํS M and some orbitals may tend to ํฅ โ . 14 Discussion In this section, we demonstrate that some existing results in the literature of the replication dynamics are the special cases of the results (theorems) presented in this article. Hawk-dove game (HD) with only one ESS in ํฐํํํบ ํต (ํโฅํ) We first discuss the well familiar hawk-dove game (Maynard-Smith & Price 1973; Maynard-Smith 1983). Conflicts between animals of same species over food, territory, mates, etc., in the animal kingdom, can be brutal and even deadly. However, under normal circumstances, conflicts between animals do not escalate, which is called conventional struggle by animal behaviorists. For example, in many species of snakes, males fight each other by wrestling rather than using their fangs. In growling contests, the males move parallel to each other and lock their antlers head forward, although the damage from antler attacks can be fatal, but this is rare. To date, most biologists have accepted the following view of the situation: conventional struggles are good for species, and fights that result in serious damages are bad for species. It turns out that in nature such conventional struggles are in great majority. However, this does not explain why sometimes it escalates the fight by breaking the rules (that is, inflicting serious damage on its opponent). In such exceptions, it can beat the competitors and achieves the final victory. When the mutation for breaking the rule occurs, it may have a higher gene replication rate than any other members! (Hofbauer et al. 1998; Nowak 2006). Maynard-Smith and Price (1973) first explained the conventional struggle from the perspective of individual choice. Now suppose that there are two strategies, the hawk strategy (H) and the dove strategy (D). The hawk strategy is to escalate the fights, while the dove strategy is the opposite (such as retreating when the other side tries to escalate the fight). The payoff of the winner is b, and the payoff of the loser is c (b<c). If both players choose the hawk strategy, then the fight will escalate, and their final expected payoff is ย[ 9 . If one takes the hawk strategy and the other takes the dove strategy, then the former wins and gets the payoff is b, and the latter retreats and gets the payoff is 0. If both players take the dove strategy, neither player loses anything, and each player has a fifty-fifty chance of winning, then their expected payoff is ย 9 . Therefore, the payoff matrix of the Hawk-dove game can be derived as follows: 15 A= ํโํ 2 ํ 0 ํ 2 where ํ 6 =ํป, ํ 9 =ํท, are the hawk strategy and dove strategy respectively. Assume that the frequency of hawk strategy selection in this population be ํฅ; then the frequency of dove strategy selection is 1โํฅ. According to replicator dynamics, there is: ํฅ=ํฅ(1โํฅ)( ย 9 โ 9 ํฅ). Let ํฅ=0, and three fixed points can be derived as: ํฅ=0,1, ย , where ํฅ=0,1 is unstable, and ํฅ= ย is globally stable. In fact, it can be further shown that ํ= ย ํป+1โ ย ํท is an evolutionarily stable strategy (ESS) (Maynard-Smith 1982; Hofbauer et al. 1998; Nowak 2006). Particularly, when ํโซํ, it indicates that in the evolutionarily stable strategy, most individuals in the population will choose the dove strategy, which adequately explains the phenomenon of conventional struggles (i.e., no escalation). Thus, the ESS has two characteristics: Characteristics 1 The strategy ํ= ย ํป+1โ ย ํท is a mixed strategy, where ํ=( ย ,1โ ย )โํผํํกS 9 . It is an evolutionarily stable strategy (ESS) but is not a strictly Nash Equilibrium (SNE). Characteristics 2 From the point of view of evolutionary dynamics, this ESS is the only evolutionary endpoint. This naturally raises the following questions: Question 1 What are the sufficient and necessary conditions for any single population with N strategy game to have ESS in ํผํํกS M (ํโฅ2)? 16 In fact, when N=2, the answer is as follows (Maynard-Smith 1982; Hofbauer et al, 1998): Suppose the payoff matrix ํด= ํ 66 ํ 69 ํ 96 ํ 99 , then the sufficient and necessary conditions for ESS in ํผํํกS 9 are that ํ 66 <ํ 96 and ํ 99 <ํ 69 . By Theorems 1 and 2, the sufficient and necessary conditions for the equations (1) in ํผํํกS M ํโฅ2 to have ESS are: ํ ํด .ร, ํด .ร(,Z6) >0,(ํ=1,2,...,ํโ1.), where ํด .ร, is the cofactor of the N-th row and the i-th column elements of the following determinant: ํ= ํ 66 โํ 96 ํ 96 โํ U6 โฎ ํ 69 โํ 99 ํ 99 โํ U9 โฎ ํ (.[6)6 โํ .6 1 ํ (.[6)9 โํ .9 1 โฏ โฏ โฑ ํ 6. โํ 9. ํ 9. โํ U. โฎ โฏ โฏ ํ (.[6). โํ .. 1 . ํ ํต=(ํ ,- ) is a symmetric square matrix of order N-1 and a positive definite matrix, where ํ ,- =โ(ํ ,- +ํ -, โํ ., โํ ,. โํ .- โํ -. +2ํ .. ),1โคํ,ํโคํโ1. In summary, this completely solves Question 1. Particularly, if these two conditions are true for matrix A, then ํ โ =ํฅ , โ . ,@6 ํ , is the only endpoint of evolution, where ํฅ , โ = _ `รa b = _ `รa _ `รa ` acd . Rock- Scissors-Paper (RSP) game with N=3 strategies Now let us discuss the N=3 case. Assume that ํด= ํ 66 ํ 69 ํ 6U ํ 96 ํ 99 ํ 9U ํ U6 ํ U9 ํ U , without loss of generality, the elements in the diagonal are 0 (Nowak 2006, P58), i.e., ํด= 0ํ 69 ํ 6U ํ 96 0ํ 9U ํ U6 ํ U9 0 . By Theorem 1, it can be derived that the sufficient and necessary conditions for the Eqn. (1) to have a unique fixed point in ํผํํกS U are that the following conditions be satisfied: ํด U6 >0, ํด U9 >0,ํด U9 >0, or ํด U6 <0,ํด U9 <0,ํด U9 <0 where ํด U6 = ํ 69 ํ 6U โํ 9U โํ U9 ํ 9U , ํด U9 =โ โํ 96 ํ 6U โํ 9U ํ 96 โํ U6 ํ 9U , ํด U = โํ 96 ํ 69 ํ 96 โํ U6 โํ U9 17 In this case, the Eqn. (1) has a unique fixed point in ํผํํกS U and denoted by ํฅ โ =ํฅ 6 โ ,ํฅ 9 โ ,ํฅ U โ , where ํฅ , โ = _ ยฃa _ ยฃd Z_ ยฃ Z_ ยฃ . Thus, ํ โ = ํด 3ํ ํด 31 +ํด 32 +ํด 33 U ,@6 ํ , is a Nash Equilibrium (NE)., but it's not a strictly Nash Equilibrium (SNE). By Theorem 2, set ํต= 2ํ 6U +ํ U6 ํ 6U +ํ U6 +ํ 9U +ํ U9 โํ 69 โํ 96 ํ 6U +ํ U6 +ํ 9U +ํ U9 โํ 69 โํ 96 2ํ U9 +ํ 9U , thus ํ โ is an ESS if and only if B is positive definite matrix. B is a positive definite matrix if and only if all leading principal minors are greater than 0. That is, when 2ํ 6U +ํ U6 >0, and 2ํ 6U +ํ U6 ํ 6U +ํ U6 +ํ 9U +ํ U9 โํ 69 โํ 96 ํ 6U +ํ U6 +ํ 9U +ํ U9 โํ 69 โํ 96 2ํ U9 +ํ 9U >0, then ํ โ is an ESS. In other words, if B does not satisfy these conditions, then ํ โ is a Nash equilibrium, but not an ESS. When B is a positive semi-definite matrix (B is a positive semi-definite matrix if and only if all principal minors of B are non-negative), that is, when 2ํ 6U +ํ U6 โฅ0,2ํ U9 +ํ 9U โฅ0๏ผand 2ํ 6U +ํ U6 ํ 6U +ํ U6 +ํ 9U +ํ U9 โํ 69 โํ 96 ํ 6U +ํ U6 +ํ 9U +ํ U9 โํ 69 โํ 96 2ํ U9 +ํ 9U โฅ0๏ผ the fixed point ํฅ โ is Lyapunov stable. Particularly, if ํต=0, then ํฅ , p a โ โกํ . ,@6 . When B is a negative definite matrix (B is negative definite matrix if and only if all the odd-order leading principal minors of B are less than 0 and all the even-order leading principal minors of B are greater than 0), that is, when 2ํ 6U +ํ U6 <0 and 18 2ํ 6U +ํ U6 ํ 6U +ํ U6 +ํ 9U +ํ U9 โํ 69 โํ 96 ํ 6U +ํ U6 +ํ 9U +ํ U9 โํ 69 โํ 96 2ํ U9 +ํ 9U >0, all orbitals except ํฅโกํฅ โ tends to ํตํํ . . Now let us consider the Rock-Scissors-Paper game with matrix๏ผ ํด= 0ํโ1 โ10ํ ํโ10 where ํ>0, Eqn. (1) has a unique fixed point in ํผํํกS U , that is, ํฅ โ = 6 U , 6 U , 6 U and ํต= 2ํโ1ํโ1 ํโ12ํโ1 . (i) When ํ>1, B is a positive definite matrix and ํฅ โ is asymptotically stable, and ํ โ = 6 U ํ + 6 U ํ+ 6 U ํ is an ESS. All the inner orbits converge to ํฅ โ . (i) When ํ=1, B is a positive semi-definite matrix and ํต=0, ํฅ โ is Lyapunov stable, but not asymptotically stable, and ํฅ , d ยฃ โกํ U ,@6 , that is: ํฅ 6 ํฅ 9 ํฅ U โกํถ. (i) When ํ<1, B is a negative definite matrix and ํฅ โ is unstable, all the orbits except ํฅโกํฅ โ will tend to ํตํํ U . Obviously, most of the classical game literature are discussed base on the extension of the Rock- Scissors-Paper (RSP) game, which is a simple case of N=3 (Hofbauer & Sigmund 1998; Sigmund 2010; Nowak 2006). The objective of this study is to extend it to any ํโฅ2 cases. Other examples The matrix B has five possibilities: positive definite matrix, positive semi-definite matrix, negative definite matrix, negative semi-definite matrix and indefinite matrix. In this study, only specific theorems and conclusions are given for the first three cases. Let's discuss the last two cases (negative semi-definite matrix and indefinite matrix): ๏ผi๏ผ When ํ=3 and ํด= 03โ1 30โ1 110 , there is a unique fixed point of equation in ํผํํกS U , that is, 19 ํฅ โ = 1 3 , 1 3 , 1 3 which is a saddle point. In this case, ํต= 0โ6 โ60 and B is a negative semi-matrix. The corresponding phase diagram of the equation has been discussed in Figure 14 of Bomzeโs report (Bomze 1983), where some orbits tend to ํฅ โ , while some orbits tend to ํตํํ U . ๏ผi๏ผ When ํ=3 and ํด= 06โ4 โ305 โ130 , there is a unique fixed point of equation in ํผํํกํ U , that is, ํฅ โ = 1 3 , 1 3 , 1 3 which is locally asymptotically stable. At the time, ํต= โ100 016 is an indefinite matrix. The corresponding phase diagram of the equation has been discussed in Figure 12 of Bomzeโs report (Bomze 1983; Hofbauer& Sigmund 1998), where some orbits tend to ํฅ โ , while some orbits tend to the ํตํํ U . 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