Evolutionary Game Theory

“2008 Mathematical Economics” in RIMS, Kyoto Univ. (Kyodai-Kaikan 101): 2008/11/28 13:30- Statistical Mechanics of Games : Evolutionary Game Theory Graduate School of Economics, Kwansei Gakuin Univ. Mitsuru KIKKAWA (吉川 満) [email protected] This File is available at 1 http://kikkawa.cyber-ninja.jp/index.htm OUTLINE １．Introduction（Motivation, Purpose） ２．Related Literatures and Preliminaries ３．Our Model 3-1. Nearest neighbor （Ising TYPE） 3-2. Random Matching （SK MODEL） Annealed System, Quenched System ４．Implication : Cont- Bouchaud‟s Model ５．Summary and Future Works 2 １．INTRODUCTION 3 1. Introduction (Motivation, Purpose) 2. Related Literatures and Preliminaries 3. Our Model 3.1 Nearest neighbor （Ising TYPE） 3-2. Random Matching （SK MODEL） Annealed System, Quenched System 4. Implication : Cont- Bouchaud’s Model 5. Summary and Future Works OUR CONTRIBUTIONS 4 OUR CONTRIBUTIONS  This study formulates evolutionary game theory with a new concept using statistical mechanics. 5 OUR CONTRIBUTIONS  This study formulates evolutionary game theory with a new concept using statistical mechanics. →We add new parameter (γ: optimal choice behavior). 6 OUR CONTRIBUTIONS  This study formulates evolutionary game theory with a new concept using statistical mechanics. →We add new parameter (γ: optimal choice behavior).  Model: Ising model … nearest-neighbor interaction SK model … random matching interaction 7 OUR CONTRIBUTIONS  This study formulates evolutionary game theory with a new concept using statistical mechanics. →We add new parameter (γ: optimal choice behavior).  Model: Ising model … nearest-neighbor interaction SK model … random matching interaction  The emergence of the equilibrium using “Phase Transition(相 転移) “ 8 Research Fields (this study) Game Theory (Evolutionary Game Theory) 9 Research Fields (this study) Game Theory (Evolutionary Game Theory) Mathema tics (Probability) 10 Research Fields (this study) Game Theory (Evolutionary Game Theory) Mathema tics (Probability) 11 Theoretical Physics (Statistical Mechanics) Research Fields (this study) Game Theory (Evolutionary Game Theory) Mathema tics (Probability) 12 Theoretical Physics (Statistical Mechanics) Application Formulate a Market Situation 13 今までどのようにして高次元を 分析してきたのか？ 14 今までどのようにして高次元を 分析してきたのか？ →平均のみを考え、低次元系へ。 15 今までどのようにして高次元を 分析してきたのか？ →平均のみを考え、低次元系へ。 例）1. Micro : → Debreu and Scarf (IER,1963) →Replica Economy 2. Macro (Micro-foundation): →Representative man. 3. Game Theory : →1対1のゲームの束。Dynamics Matching and Bargaining Game, Evolutionary Game. 16 今までどのようにして高次元を 分析してきたのか？ →平均のみを考え、低次元系へ。 例）1. Micro : → Debreu and Scarf (IER,1963) →Replica Economy 2. Macro (Micro-foundation): →Representative man. 3. Game Theory : →1対1のゲームの束。Dynamics Matching and Bargaining Game, Evolutionary Game. 統計力学では、分布を考える。 17 MOVITATION  Numerous papers published have used statistical mechanics in game theory:  Blume[1], Diedeich and Opper [5], McKelvey and Palfrey [9, 10] 18 MOVITATION  Numerous papers published have used statistical mechanics in game theory:  Blume[1], Diedeich and Opper [5], McKelvey and Palfrey [9, 10]  These papers applied the Ising model and standard SK model in a straightforward manner. 19 MOVITATION  Numerous papers published have used statistical mechanics in game theory:  Blume[1], Diedeich and Opper [5], McKelvey and Palfrey [9, 10]  These papers applied the Ising model and standard SK model in a straightforward manner.  This study presents a novel model using statistical mechanics for evolutionary game theory with basic elements. 20 MOVITATION  Numerous papers published have used statistical mechanics in game theory:  Blume[1], Diedeich and Opper [5], McKelvey and Palfrey [9, 10]  These papers applied the Ising model and standard SK model in a straightforward manner.  This study presents a novel model using statistical mechanics for evolutionary game theory with basic elements.  Application : the description of the market 21 MOVITATION  Numerous papers published have used statistical mechanics in game theory:  Blume[1], Diedeich and Opper [5], McKelvey and Palfrey [9, 10]  These papers applied the Ising model and standard SK model in a straightforward manner.  This study presents a novel model using statistical mechanics for evolutionary game theory with basic elements.  Application : the description of the market 22 GENERAL EQUILIBRI UM ? ２． Related Literatures and Preliminaries 23 1. Introduction (Motivation, Purpose) 2. Related Literatures and Preliminaries 3. Our Model 3.1 Nearest neighbor （Ising TYPE） 3-2. Random Matching （SK MODEL） Annealed System, Quenched System 4. Implication : Cont- Bouchaud’s Model 5. Summary and Future Works Related Literatures Tom Siegfried A Beautiful Math: John Nash, Game Theory, And the Modern Quest for a Code of Nature (「世界で最も美 しい数学」) , Joseph Henry Press, 2006/09/25. → Ch. 11 24 Related Literatures Tom Siegfried A Beautiful Math: John Nash, Game Theory, And the Modern Quest for a Code of Nature (「世界で最も美 しい数学」) , Joseph Henry Press, 2006/09/25. → Ch. 11  Blume (GEB, 1993) , McKelvey and Palfrey (GEB, 1995, JER, 1996) → Ising model.  Diederich and Opper(PRA,1989) → SK model (Spin Glass) Contribution: SK model : Lyapunov function (fitness function) 25 Interpretation of Nash Equilibrium (J.F.Nash’s Ph D. Thesis)  1. “Rationality” ・・・ the players are perceived as rational and they have complete information about the structure of the game, including all of the players‟ preferences regarding possible outcomes, where this information about each other‟s strategic alternatives and preferences, they can also compute each other‟s optimal choice of strategy for each set of expectations. If all of the players expect the same Nash equilibrium, then there are no incentives for anyone to change his strategy. 26 26 (SOURCE :Press Release – The Royal Swedish Academy of Sciences, 11 October 1994) Nash has received a grant from the National Science Foundation to develop a new “evolutionar y” solution concept for cooperative games.(SOU RCE: the essential John Nash) Interpretation of Nash Equilibrium (J.F.Nash’s Ph D. Thesis)  1. “Rationality” ・・・ the players are perceived as rational and they have complete information about the structure of the game, including all of the players‟ preferences regarding possible outcomes, where this information about each other‟s strategic alternatives and preferences, they can also compute each other‟s optimal choice of strategy for each set of expectations. If all of the players expect the same Nash equilibrium, then there are no incentives for anyone to change his strategy.  2. “Statistical Populations” ・・・ is useful in so-called evolutionary games. This type of game has also been developed in biology in order to understand how the principles of natural selection operate in strategic interaction within among species.(→ Mass Action) 27 27 (SOURCE :Press Release – The Royal Swedish Academy of Sciences, 11 October 1994) Nash has received a grant from the National Science Foundation to develop a new “evolutionar y” solution concept for cooperative games.(SOU RCE: the essential John Nash) Interpretation of Nash Equilibrium (J.F.Nash’s Ph D. Thesis)  1. “Rationality” ・・・ the players are perceived as rational and they have complete information about the structure of the game, including all of the players‟ preferences regarding possible outcomes, where this information about each other‟s strategic alternatives and preferences, they can also compute each other‟s optimal choice of strategy for each set of expectations. If all of the players expect the same Nash equilibrium, then there are no incentives for anyone to change his strategy.  2. “Statistical Populations” ・・・ is useful in so-called evolutionary games. This type of game has also been developed in biology in order to understand how the principles of natural selection operate in strategic interaction within among species.(→ Mass Action) 28 28 (SOURCE :Press Release – The Royal Swedish Academy of Sciences, 11 October 1994) Nash has received a grant from the National Science Foundation to develop a new “evolutionar y” solution concept for cooperative games.(SOU RCE :the essential John Nash) RANDOM INTERACTION (SK MODEL) Diederich and Opper(1989) 29 RANDOM INTERACTION (SK MODEL) Diederich and Opper(1989)  Replicator Eq.: dx  x ( f  f ), dt 30 for   1,, N . RANDOM INTERACTION (SK MODEL) Diederich and Opper(1989)  Replicator Eq.: dx  x ( f  f ), for   1,, N . dt 1  Fitness Function: f   H   x c x , 2  where, f f  , c  c (    ) x This is a element of the Random Matrix , it is Gauss Distribution, Average is 0, Variance is 1/N. 31 We obtain the following Equations with Replica method in a Quenched System. 32 We obtain the following Equations with Replica method in a Quenched System. u v  q 2    1 (u  v)  2 2 33 dze    Z 2 / 2 dze ( z  ), Z 2 / 2 ( z  ) 2 , where   q (u  2v) We obtain the following Equations with Replica method in a Quenched System. u v  q 2    1 (u  v)  2 2 dze    Z 2 / 2 dze ( z  ), Z 2 / 2 ( z  ) 2 , where   q (u  2v) Competitive ↑↓、↓↑ S1 S2 ０．５ S1 A,A 0,0 S2 0,0 B,B Cooperative Parameter u ↑↑、↓↓ 34 2 We obtain the following Equations with Replica method in a Quenched System. u v  q 2    1 (u  v)  2 2 dze    Z 2 / 2 dze ( z  ), Z 2 / 2 ( z  ) 2 , where   q (u  2v) Competitive ↑↓、↓↑ S1 S2 ０．５ S1 A,A 0,0 S2 0,0 B,B Cooperative Parameter u ↑↑、↓↓ 35 2 WHAT IS „‟ EVOLUTIONARY GAME THEORY‟‟ ? In 1973 Maynard Smith formalized a central concept in game theory called the evolutionary stable strategy (ESS), based on a verbal argument by G.R.Price. This area of research culminated in his 1982 book Evolution and the Theory of Games. The Hawk-Dove game is arguably his single most influential game theoretical model. ASSUMPTION: Large Number of Population (randomly matched) , Monotone (the strategy with higher payoff increases its shares) 36 Situation (Traditional Evolutionary Game Theory) 37 Situation (Traditional Evolutionary Game Theory) At Random (infinitely) Another players look at the game. 38 Situation (Traditional Evolutionary Game Theory) At Random (infinitely) Play a game Another players look at the game. 39 Situation (Traditional Evolutionary Game Theory) At Random (infinitely) Play a game Another players look at the game. 40 Replicator Equation REVIEW: Replicator Equation  REPLICATOR EQ. xi  xi  Ax i  x  Ax , i  1,, n. If the player's payoff from the outcome i is greater than the expected utility x Ax, the probability of the action i is higher than before. 41 REVIEW: Replicator Equation  REPLICATOR EQ. xi  xi  Ax i  x  Ax , i  1,, n. If the player's payoff from the outcome i is greater than the expected utility x Ax, the probability of the action i is higher than before. And this equation shows that the probability of the action i chosen by another players is also higher than before (externality). 42 REVIEW: Replicator Equation  REPLICATOR EQ. xi  xi  Ax i  x  Ax , i  1,, n. If the player's payoff from the outcome i is greater than the expected utility x Ax, the probability of the action i is higher than before. And this equation shows that the probability of the action i chosen by another players is also higher than before (externality). Furthermore, the equation is derived uniquely by the monotonic (that is if one type has increased its share in the population then all types with higher profit should also have increased their shares). 43 REVIEW: Replicator Equation  REPLICATOR EQ. xi  xi  Ax i  x  Ax , i  1,, n. If the player's payoff from the outcome i is greater than the expected utility x Ax, the probability of the action i is higher than before. And this equation shows that the probability of the action i chosen by another players is also higher than before (externality). Furthermore, the equation is derived uniquely by the monotonic (that is if one type has increased its share in the population then all types with higher profit should also have increased their shares). Two Strategies Classification：  x  x(1  x){b  (a  b) x} ・・・(＊) (I) Non-dilemma: a > 0. b < 0, ESS : one (II) Prisoner‟s dilemma : a < 0. b > 0, ESS :one S1 1 （III) Coordination : a>0,b>0, ESS two （IV) Hawk-Dove : a<0,b < 0, ESS one (mixed strategy) S 2 44 2 S 1 S 2 a,a 0,0 0,0 b,b Payoff Matrix REVIEW: Symmetric and Asymmetic Games  The difference between symmetric and asymmetric two person game is 45 REVIEW: Symmetric and Asymmetic Games  The difference between symmetric and asymmetric two person game is the payoff matrix . 46 REVIEW: Symmetric and Asymmetic Games  The difference between symmetric and asymmetric two person game is the payoff matrix . Type 2 Type 1 S1 S2 Type 2 S1 S2 A,A C,B B,C D,D Symmetric Two Person Game Replicator Equation： Situation： 47 Symmetric ： one Type 1 S1 S2 S1 A,E B,F S2 C,G D,H Asymmetric Two Person Game two Asymmetric ： seller and buyer etc. REVIEW: Ising Model, Spin Glass  Ising model ・・・  Spin Glass ・・・ 48 REVIEW: Ising Model, Spin Glass  Ising model ・・・相転移（異なる相へ移る）を記述する最も     簡単なモデル。 金属に外場から磁化をかけ、ある臨界値（Curie温度）を超え ると、磁石となる。 格子上にある（スピンの）状態 S_j ： {-1, +1}, j=1,…,N N個状態が「+1 or -1」 にすべて揃ったら「cooperative」、 「-1, 1」 の組ならば「competitive」、 Hamiltonian (Energy) H   J SS  i, j  Spin Glass ・・・ 49 i j REVIEW: Ising Model, Spin Glass  Ising model ・・・相転移（異なる相へ移る）を記述する最も     簡単なモデル。 金属に外場から磁化をかけ、ある臨界値（Curie温度）を超え ると、磁石となる。 格子上にある（スピンの）状態 S_j ： {-1, +1}, j=1,…,N N個状態が「+1 or -1」 にすべて揃ったら「cooperative」、 「-1, 1」 の組ならば「competitive」、 Hamiltonian (Energy) H   J SS  i j i, j  Spin Glass ・・・相互作用の符合が場所に一定ではないという ミクロ的な特徴を持っている。 例） CuMn・・・銅(強磁性体にならない)に微量のマンガン(磁 性原子)を混ぜ合わせて合金を作ると、マンガンの原子は銅の 結晶格子中でランダムな位置を占め、ガラスの性質に似たス ピン秩序を示すので、Spin Glass と呼ばれる。 50 REVIEW: PERCOLATION [d-dimensional Percolation] We examine each edge of Zd, and consider it to be open with probability p and closed otherwise, independent of all other edges. The edges of Zd represent the inner passageways of the stone, and the parameter p is the proportion of passages that are broad enough to allow water to pass along them. Suppose we immerse a large porous stone in a bucket of water. What is the probability that the center of the stone is wetted ? 51 REVIEW: PERCOLATION [d-dimensional Percolation] We examine each edge of Zd, and consider it to be open with probability p and closed otherwise, independent of all other edges. The edges of Zd represent the inner passageways of the stone, and the parameter p is the proportion of passages that are broad enough to allow water to pass along them. Suppose we immerse a large porous stone in a bucket of water. What is the probability that the center of the stone is wetted ? 52 ３．BASIC MODEL 53 1. Introduction (Motivation, Purpose) 2. Related Literatures and Preliminaries 3. Our Model 3.1 Nearest neighbor （Ising TYPE） 3-2. Random Matching （SK MODEL） Annealed System, Quenched System 4. Implication : Cont- Bouchaud’s Model 5. Summary and Future Works MODEL：  Each site on the lattice is the address of one player. ● ● SQUARE LATTICE 54 MODEL：  Each site on the lattice is the address of one player.  In Sec.2, player i and j play a game with nearest neighbor interaction. ● ● SQUARE LATTICE 55 MODEL：  Each site on the lattice is the address of one player.  In Sec.2, player i and j play a game with nearest neighbor interaction.  In Sec.3, the players are assumed to search at random for trading opportunities and when they meet the terms of game are started. ● ● SQUARE LATTICE 56 Situation (nearest neighbor interaction) 57 Situation (nearest neighbor interaction) 58 EXAMPLE S1(1) S2(2) S1(-1) S2(+1) S1(1) A,A 0,0 S1(-1) A,A 0,0 S2(2) 0,0 B,B S2(+1) 0,0 where A,B > 0 Ising Model 59 B,B PROBABILITY SPACE  Probability Space (Ω, F, P)   1, 1 Z2   exp[ H (S )]dS  F (Prop.１) μはそれ上の確率測度で, dSはΩ上の一様分布とす る。確率論的にはdS は密度1/2 のBernoulli 分布と 呼ぶものである。 60 60 ASSUMPTION , PROPOSITON ASSU.： All players are “rational”. 61 ASSUMPTION , PROPOSITON ASSU.： All players are “rational”. PROP.： Under Assu., we obtain the probability distributions of actions, {Si},i=1,…,N, and the palyer‟s payoff from the outcome is f (2.1) 1 i where {Si} is player i‟s action, γ is non-negative constant; for instance, γ is the optimal choice behavior f is the player‟s payoff from the outcome {Si}, and Z is the normalization parameter. P({S })  Z exp( f ) 62 ASSUMPTION , PROPOSITON ASSU.： All players are “rational”. PROP.： Under Assu., we obtain the probability distributions of actions, {Si},i=1,…,N, and the palyer‟s payoff from the outcome is f (2.1) 1 i where {Si} is player i‟s action, γ is non-negative constant; for instance, γ is the optimal choice behavior f is the player‟s payoff from the outcome {Si}, and Z is the normalization parameter. P({S })  Z exp( f )  INTERPRETATION： If payoff f is greater, then the probability of choosing the action is higher.  Distinction：STATICS, Non- Externality 63 Classical EVOLUTIONARY GAME ASSU.： All players are “rational”. 64 Classical EVOLUTIONARY GAME ASSU.： All players are “rational”.  Under this assumption, we obtain the unique solution: Selection Dy.→Replicator Eq.    xi  xi fi  f , i  1, N . 65 Classical EVOLUTIONARY GAME ASSU.： All players are “rational”.  Under this assumption, we obtain the unique solution: Selection Dy.→Replicator Eq.    xi  xi fi  f , i  1, N . INTERPRETATION：If the payoff fi is greater then the expected utility, the player choose the action with probability 1. 66 Distinction： DYNAMICS, EXTERNALITY REVIEW: Replicator Equation  x  xi  Ax i  x  Ax , i  1,, n. REPLICATOR EQ. If the player's payoff from the outcome i is greater than the expected utility x Ax, the probability of the action i is higher than before. And this equation shows that the probability of the action i chosen by another players is also higher than before (externality). Furthermore, the equation is derived uniquely by the monotonic (that is if one type has increased its share in the population then all types with higher profit should also have increased their shares). Two Strategies Classification：  x  x(1  x){b  (a  b) x} ・・・(＊) (I) Non-dilemma: a > 0. b < 0, ESS : one (II) Prisoner‟s dilemma : a < 0. b > 0, ESS :one S1 1 （III) Coordination : a>0,b>0, ESS two （IV) Hawk-Dove : a<0,b < 0, ESS one (mixed strategy) S 2 67 2 S 1 S 2 a,a 0,0 0,0 b,b Payoff Matrix DEFINITION DEF.：We define an order parameter, as how often a player has chosen an action in this game. (2.2) 68 DEFINITION DEF.：We define an order parameter, as how often a player has chosen an action in this game. (2.2) N m   Si PSi  i 1 where N is the number of the actions. 69 EXAMPLE  The actions‟ index {Si}={1,2},N=2, and the order parameter for each case is computed as follows. 70 S1(1) S2(2) S1(1) S2(2) A,A 0,0 0,0 B,B EXAMPLE  The actions‟ index {Si}={1,2},N=2, and the order parameter for each case is computed as follows. (i) If all the players' actions are {Action 1}, then we obtain m=1. 71 S1(1) S2(2) S1(1) S2(2) A,A 0,0 0,0 B,B EXAMPLE  The actions‟ index {Si}={1,2},N=2, and the order parameter for each case is computed as follows. (i) If all the players' actions are {Action 1}, then we obtain m=1. (ii) If all the players' actions are {Action 2}, then we obtain m=2. 72 S1(1) S2(2) S1(1) S2(2) A,A 0,0 0,0 B,B EXAMPLE  The actions‟ index {Si}={1,2},N=2, and the order parameter for each case is computed as follows. (i) If all the players' actions are {Action 1}, then we obtain m=1. (ii) If all the players' actions are {Action 2}, then we obtain m=2. (iii) If half of all the players‟ actions are {Action 1}, then we obtain m=3/2. 73 S1(1) S2(2) S1(1) S2(2) A,A 0,0 0,0 B,B EXAMPLE  The actions‟ index {Si}={1,2},N=2, and the order parameter for each case is computed as follows. (i) If all the players' actions are {Action 1}, then we obtain m=1. (ii) If all the players' actions are {Action 2}, then we obtain m=2. (iii) If half of all the players‟ actions are {Action 1}, then we obtain m=3/2. 74 → If the order parameter m is near 1, then we know that there are many more players choosing {Action 1} than {Action 2}. S1(1) S2(2) S1(1) S2(2) A,A 0,0 0,0 B,B  EXAMPLE : Ising model  Si={-1,1} → m= -1,0(random),1 75  EXAMPLE : Ising model  Si={-1,1} → m= -1,0(random),1 m 1 O γc -1 76 γ  EXAMPLE : Ising model  Si={-1,1} → m= -1,0(random),1 m 1 O γc -1 Rational Choice Behavior Rational Player 77 γ Random behavior SIMULATION SKY BULUE＝Strategy 1, BLUE＝Strategy 2 m  0 ORDERED TYPE １ 78 NO ORDERED m  0 m 0 （s1, s1 ） Random  ORDERED TYPE 2 m  0 （s2,s2） Relation between order parameter and product of profit f and parameter γ γｆ：large γｆ：small m  tanh( f ) 79 79 Relation between order parameter and product of profit f and parameter γ  If the γf is large, order parameter approaches to 1. → We can find which action is occupied. γｆ：large γｆ：small m  tanh( f ) 80 80 Relation between order parameter and product of profit f and parameter γ  If the γf is large, order parameter approaches to 1. → We can find which action is occupied.  If the γf is small, order parameter approaches to 0. 81 81 γｆ：large γｆ：small m  tanh( f ) ORDERED PARAMETER IN REPLICATOR SYSTEM REPLICATOR Equation (symmetric two person game, the number of the strategy is two.)  x  x(1  x){b  (a  b) x} Stationary point (Nash equilibrium) b x  0,1, 0  1 ab  82 ORDERED PARAMETER IN REPLICATOR SYSTEM m ＋1  REPLICATOR Equation (symmetric two person game, the number of the strategy is two.)  x  x(1  x){b  (a  b) x} Stationary point (Nash equilibrium) 83 0  －１  b x  0,1, 0  1 ab  ORDERED PARAMETER IN REPLICATOR SYSTEM m ＋1  REPLICATOR Equation (symmetric two person game, the number of the strategy is two.)  x  x(1  x){b  (a  b) x} Stationary point (Nash equilibrium) 0 84 －１   b x  0,1, 0  1 ab  ORDERED PARAMETER has three points (corner point(-1,+1), interior point) in RE. SYS. EVOLUTIONARY STABLE STRATEGY （ESS) x   is an evolutionary DEF.：Weibull(1995): stable strategy (ESS) if for every strategy y  x there exists some  y  (0,1) such that the following inequality holds for all   (0,  y ) . u[ x,  y  (1   ) x]  u[ y,  y  (1  ) x]. 85 EVOLUTIONARY STABLE STRATEGY （ESS) x   is an evolutionary DEF.：Weibull(1995): stable strategy (ESS) if for every strategy y  x there exists some  y  (0,1) such that the following inequality holds for all   (0,  y ) . u[ x,  y  (1   ) x]  u[ y,  y  (1  ) x]. INTERPRETATION：incumbent payoff (fitness) is higher 86 than that of the post-entry strategy （ESS : ①the solution of the Replicator equation + ② asymptotic stable.） PROPOSITION PRO.(Bishop and Cannings (1978)): x is evolutionary stable strategy if and only if it meets these first-order and second-order best-reply : 87 PROPOSITION PRO.(Bishop and Cannings (1978)): x is evolutionary stable strategy if and only if it meets these first-order and second-order best-reply : Nash Eq. (2.4) u ( y, x)  u ( x, x), y, u ( y , x )  u ( x, x ) y  x, (2.5)  u ( y, y )  u ( x, y ), 88 PROPOSITION PRO.(Bishop and Cannings (1978)): x is evolutionary stable strategy if and only if it meets these first-order and second-order best-reply : Nash Eq. (2.4) u ( y, x)  u ( x, x), y, u ( y , x )  u ( x, x ) y  x, (2.5)  u ( y, y )  u ( x, y ), Asymptotic Stable Conditon 89 PROPOSITION PRO.: x is an evolutionary stable strategy in an evolutionary game with statistical mechanics, if there exists some m such that the inequality (2.6) holds for all m* 90 PROPOSITION PRO.: x is an evolutionary stable strategy in an evolutionary game with statistical mechanics, if there exists some m such that the inequality (2.6) holds for all m* (2.4) u ( y, x)  u ( x, x), y, (2.6) mm   , Lyapunov Stable Condition where, m* is the index of the equilibrium action. 91  EXAMPLE : Ising model  Si={-1,1} → m= -1,0(random),1 m 1 O γc -1 Rational Choice Behavior Rational Player 92 γ Random behavior ASYMMETRIC TWO PERSON GAME  Let this model add an order parameter; we can analyze an asymmetric two-person game in the same way.  Equilibrium Condition: 93 ASYMMETRIC TWO PERSON GAME  Let this model add an order parameter; we can analyze an asymmetric two-person game in the same way.  Equilibrium Condition: m '1  m  1 , m '2  m 2   2  1 94  PERCOLATION  The fundamental relationship between percolation and phase transition 95 PERCOLATION  The fundamental relationship between percolation and phase transition THE. (Coniglio, et al.(1976)) In the two-dimensional Ising model, we obtain     (i) If    c ,  ,0 C0    0,  ,0 C0    0. where  s , s  ,  is Gibbs measures. (ii) if μ is external to the set of all Gibbs states G ( , h)         C   C  0 96  0  0  PERCOLATION  The fundamental relationship between percolation and phase transition THE. (Coniglio, et al.(1976)) In the two-dimensional Ising model, we obtain     (i) If    c ,  ,0 C0    0,  ,0 C0    0. where  s , s  ,  is Gibbs measures. (ii) if μ is external to the set of all Gibbs states G ( , h)          C   C  0  0  0 (i) → there exists a.e. an infinite cluster of the corresponding sign and no infinite clusters of the opposite sign. 97 (ii) → there exists an infinite cluster for neither actions,  EXAMPLE : Ising model  Si={-1,1} → m= -1,0(random),1 m 1 O γc -1 Rational Choice Behavior Rational Player 98 γ Random behavior DEFINITION(CONNECTED) DEF. A subset A  B2 is called connected if and only if for every x, y  A , there exists a sequence b1 , b2 , bn  A such that (a) x  b1 and y  bn (b) For every , 1  i  n  1 there exists a point xi  Z 2 99 such that bi  bi 1  xi . DEFINITION(CONNECTED) DEF. A subset A  B2 is called connected if and only if for every x, y  A , there exists a sequence b1 , b2 , bn  A such that (a) x  b1 and y  bn (b) For every , 1  i  n  1 there exists a point xi  Z 2 such that bi  bi 1  xi . DEF . For , A  B , C  A is called A's connected component if and only if (a) C is connected, (b) For every b  A / C, C  b is not connected. 2  100 DEFINITION(CONNECTED) DEF. A subset A  B2 is called connected if and only if for every x, y  A , there exists a sequence b1 , b2 , bn  A such that (a) x  b1 and y  bn (b) For every , 1  i  n  1 there exists a point xi  Z 2 such that bi  bi 1  xi . DEF . For , A  B , C  A is called A's connected component if and only if (a) C is connected, (b) For every b  A / C, C  b is not connected. 2  DEF 2.11 A subset A  Z 2 is called (＊) connected if and only if for every x, y  A, there exists a sequence of points x1 , x2 ,, xn   A such that x0  x, xn1  y and for every ,1  i  n  1, xi  xi 1  1. 101     where , x  x1 , x 2  Z 2 , x  max x1 , x 2 . Concentric Circle Pattern and Chess Pattern  What kind of pattern do the actions‟ distribution on the lattice make ? 102 Concentric Circle Pattern and Chess Pattern  What kind of pattern do the actions‟ distribution on the lattice make ? Concentric Circle Pattern → red surrounded by a bigger blue, which is surrounded by a bigger red , …. 103 coexistence of finite (*) connected Chess Pattern → red and blue placed alternately coexistence of infinite (*) connected  EXAMPLE : Ising model  Si={-1,1} → m= -1,0(random),1 m 1 O γc -1 Rational Choice Behavior Rational Player 104 γ Random behavior Coexistence of infinite (*)-clusters TH. (Higuchi(1995) ) For every   0 is sufficiently small, pc 1 there exists h such that  ' h '  log  4 ', 1 1  pc 2 1  pc  4 , implies the coexistence of infinite (*) h  log 2 pc clusters with respect to the Gibbs state for  ,h . 105 105 Coexistence of infinite (*)-clusters TH. (Higuchi(1995) ) For every   0 is sufficiently small, pc 1 there exists h such that  ' h '  log  4 ', 1 1  pc 2 1  pc  4 , implies the coexistence of infinite (*) h  log 2 pc clusters with respect to the Gibbs state for  ,h . OUTLINE OF THE PROOF. Step 1. Lemma A.1 → 大小関係を表すための条件を得る. Step 2. ＋戦略がPercolationする確率(pc)と－戦略がPercolation する 確率(1- pc)を求める. 1- pc < p < pc であり, それらを同時に成り 立つ条件を求めと, 定理２の条件を導出することができる. (QED)  →無限＊クラスターの共存が存在 106 106  →このとき戦略の分布はチェス盤のパターン Ωに大小関係を入れる. 2 x  Z  任意の に対して  ( x)   ( x) となると きに,    とかくことにする. この大小関 係に対して Ω 上の関数 f が単調増加(減尐)と は,    なる ,   に対して常に f    f   となるときをいう. 107 Ωに大小関係を入れる. 2 x  Z  任意の に対して  ( x)   ( x) となると きに,    とかくことにする. この大小関 係に対して Ω 上の関数 f が単調増加(減尐)と は,    なる ,   に対して常に f    f   となるときをいう. DEF. Ω上の確率測度μとνに対して, μ≦ νとは, 任 意のΩ上の連続かつ単調増加関数 f に対して f ( ) (d )    となるときに言う.  108  f ( ) (d ) 定理A.1. (FKG-Holley Inequalities)   Z 2を有限集合とし て,   上の2つの確率測度μ, νが, 任意の 1 ,  2  に対して  (1   2 ) (1   2 )   (1 ) ( 2 ) を満たすならば, (  上の確率測度として) μ≦νであ (A.1) (1   2 )( x)  min 1 ( x),  2 ( x) , (1   2 )( x)  max 1 ( x),  2 ( x) とする. る. ただし 109 109 定理A.1. (FKG-Holley Inequalities)   Z 2を有限集合とし て,   上の2つの確率測度μ, νが, 任意の 1 ,  2  に対して  (1   2 ) (1   2 )   (1 ) ( 2 ) を満たすならば, (  上の確率測度として) μ≦νであ (A.1) (1   2 )( x)  min 1 ( x),  2 ( x) , (1   2 )( x)  max 1 ( x),  2 ( x) とする. る. ただし 系A1. Λを Z 2の有限部分集合とする. このとき以下の ことが成立する.   (i)  ,  が Ω≦η を満たすならば, q  q (ii) f,g を F可測な単調増加関数とすると任意の     に対して  fgdq    fdq    gdq . (iii)  h   ' h ' 4 |    ' | 0 ならば, 任意の    110 110     q   , h  q   ' , h ' .  に対して,  EX. :SPATIAL PRISONER‟S DILEMMA GAME, Nowak and May(Nature, 1992) Blue:C(cooperate), Red: D (defect), Yellow: D following a C, Green : C following a D 111 Coexistence of infinite (*)-clusters EXTENSION : Random Matching 112 1. Introduction (Motivation, Purpose) 2. Related Literatures and Preliminaries 3. Our Model 3.1 Nearest neighbor （Ising TYPE） 3-2. Random Matching （SK MODEL） Annealed System, Quenched System 4. Implication : Cont- Bouchaud’s Model 5. Summary and Future Works SK MODEL  Random Matching 113 SK MODEL  Random Matching  Payoff, Fitness H  J     J S S ij i j ij i j 2   ( ) J J  1  ij 0 exp  where P  J ij    2 2 2J 2 J   J0 : Average , J2 : Variance 114 Situation 115 Situation (random matching : annealed system) 116 Situation (random matching : Quenched System) At Random 117 ANEALED SYSTEM  Social Welfare Function, 分布関数の配位平均. F   log Z , Z    Si    Probability of Matching  dJ P J  exp( H J ), ij ij ij ( ij )    ( J ) 2 2   exp    J 0 Si S j  ( Si S j )   2  Si    (ij )  118 Fitness ANEALED SYSTEM  Social Welfare Function, 分布関数の配位平均. F   log Z , Z    Si    Probability of Matching  dJ P J  exp( H J ), ij ij ij ( ij )    ( J ) 2 2   exp    J 0 Si S j  ( Si S j )   2  Si    (ij )  Max F Solved Fitness m    1 1 2 2 4 2 2 4 F     J 0 ( Si )  ( J ) ( Si )   J 0 N  Si  ( J ) N  Si  2 2 i i i i    Si   119  m=<Si>, F 2 2 3 2 4 3  2 J 0 N m  2 J N m  0 m J0 m  0 or  2 2 J N As N →∞, m = 0 . 120  m=<Si>, F 2 2 3 2 4 3  2 J 0 N m  2 J N m  0 m J0 m  0 or  2 2 J N As N →∞, m = 0 . 1 . In Ising type, the order parameter is a tanh function; however, the order parameter is a point, like a replicator system. 121 2. If there are infinite players on this lattice, then the order parameter is 0. QUENCHIED SYSTEM  Quenched system : 122 QUENCHIED SYSTEM  Quenched system : Jij is chosen randomly, but then is fixed. 123 QUENCHIED SYSTEM  Quenched system : Jij is chosen randomly, but then is fixed. Social Welfare Function: 124 F   log Z QUENCHIED SYSTEM  Quenched system : Jij is chosen randomly, but then is fixed. Social Welfare Function : F   log Z Replica Method 125  log Z  lim Z n0 n  1 QUENCHIED SYSTEM  Quenched system : Jij is chosen randomly, but then is fixed. Social Welfare Function : Replica Method F   log Z  log Z  lim Z n0 n  a2  1 Hubbard-Stranovich Trans. exp    2 2 + saddle point method 126  1  x2   exp ax  2  dx  + replica symmetry QUENCHIED SYSTEM  Quenched system : Jij is chosen randomly, but then is fixed. Social Welfare Function : Replica Method F   log Z  log Z  lim Z n0 n  1  a2  1 Hubbard-Stranovich Trans. exp    2 2  x2   exp ax  2  dx  + saddle point method + replica symmetry   z2  1 ~ ~   m J qz J exp tanh      0 n dz Solved     2  2  q 127 1 2      z2  ~ ~ 2   exp tanh  J qz   J  0 n dz   2   Like a Ising Type TAP EQUATION  Let a model add another parameter hj (an effect of externality). H 128 J    J S S   h S ij i j ij i j i j j j TAP EQUATION  Let a model add another parameter hj (an effect of externality). H Annealed System →Solved 129 J    J S S   h S ij i j ij i j i j j j h j  2 m(1  N )( J 0  J m ) 2 2 TAP EQUATION  Let a model add another parameter hj (an effect of externality). H Annealed System →Solved J    J S S   h S ij i j ij i j i j j h j  2 m(1  N )( J 0  J m ) 2 Quenched System m  tanh   h  J m  , i  i  ij j  j   130 j 2 TAP EQUATION  Let a model add another parameter hj (an effect of externality). H Annealed System →Solved J    J S S   h S ij i j ij i j i j j h j  2 m(1  N )( J 0  J m ) 2 Quenched System m  tanh   h  J m  , i  i  ij j  j   1 . Weiss approximation : f [ s]  f  s , 2. Taylor expansion : 131 j 2 TAP EQUATION  Let a model add another parameter hj (an effect of externality). H Annealed System →Solved J    J S S   h S ij i j ij i j i j j h j  2 m(1  N )( J 0  J m ) 2 Quenched System m  tanh   h  J m  , i  i  ij j  j   1 . Weiss approximation : f [ s]  f  s , 2. Taylor expansion : 1 m  h , T  1 T  J  132 j 2 TAP EQUATION  Let a model add another parameter hj (an effect of externality). H Annealed System →Solved J    J S S   h S ij i j ij i j i j j j h j  2 m(1  N )( J 0  J m ) 2 2 Quenched System m  tanh   h  J m  , i  i  ij j  j   1 . Weiss approximation : f [ s]  f  s , 2. Taylor expansion : 1 m  h , T  1 T  J  133 →If the maximal eigenvalue of Jλ is 2J, the order parameter is discontinuous. →Multiple Equilibria MULTIPLE EQUIRIBRIA m O Externality, Random Matching, Quenched System γc Rational Player 134 γ Random behavior  EXAMPLE : Ising model  Si={-1,1} → m= -1,0(random),1 m Non-Externality, Nearest Neighbor Interaction 1 O γc -1 Rational Choice Behavior Rational Player 135 γ Random behavior 4. IMPLICATION Cont-Bouchaud‟s Model 136 1. Introduction (Motivation, Purpose) 2. Related Literatures and Preliminaries 3. Our Model 3.1 Nearest neighbor （Ising TYPE） 3-2. Random Matching （SK MODEL） Annealed System, Quenched System 4. Implication : Cont- Bouchaud’s Model 5. Summary and Future Works Cont-Bouchaud ‘s model ≒ §2 ‘s model.  This study discusses the simplified Cont and Bouchaud model through our models. IMAGE 137 Cont-Bouchaud ‘s model ≒ §2 ‘s model.  This study discusses the simplified Cont and Bouchaud model through our models. →We can understand the player‟s behavior in Cont and Bouchaud model. IMAGE 138 POINT! : Percolation Cluster ⇔ trading groups  A stock market with N AGENTS  Trading a SIGNLE asset 139  A stock market with N AGENTS  Trading a SIGNLE asset  The demand for stock of agent i is represented by a random variable Φi(t)( ∈ {-1,0,1}) Φi(t)>0 : BULL , <0 : BEAR, 0: not trade Pi  1  Pi  1  a, Pi  0  1  2a. 140  A stock market with N AGENTS  Trading a SIGNLE asset  The demand for stock of agent i is represented by a random variable Φi(t)( ∈ {-1,0,1}) Φi(t)>0 : BULL , <0 : BEAR, 0: not trade Pi  1  Pi  1  a, Pi  0  1  2a.  Between price changes and excess demand: x(t )  x(t  1)  x(t )  The number of clusters (coalitions) 141 N  (t )   i 1 CLUSTER x(t )  1 1 k The size of cluster W  (t )   1 i λ- Market Depth It measure the sensitivity of price to fluctuations in excess demand Aggregate Excess Demand Price Valuation 陳昱 パーコレーションと金融市場の価格変動 より転載 142 Heavily tails 陳昱 パーコレーションと金融市場の価格変動 より転載  For decrease in the activity parameter a showing its 143 similarity with real stock market phenomena: the heavily tails observed in the distribution of stock market. Random Matching (Cont-Bouchaud)  Annealed Sys. (+ externalitiy) →  Quenched Sys →  Quenched Sys. + externality → 144 Random Matching (Cont-Bouchaud)  Annealed Sys. (+ externalitiy) →One action occupied. The price is higher or lower than before.  Quenched Sys →  Quenched Sys. + externality → 145 Random Matching (Cont-Bouchaud)  Annealed Sys. (+ externalitiy) →One action occupied. The price is higher or lower than before.  Quenched Sys →like a Ising type.  Quenched Sys. + externality → 146 Random Matching (Cont-Bouchaud)  Annealed Sys. (+ externalitiy) →One action occupied. The price is higher or lower than before.  Quenched Sys →like a Ising type.  Quenched Sys. + externality →multiple equilibria (The rate of price change is dependent on the size of γ). 147 5．Summary and Future Works 148 1. Introduction (Motivation, Purpose) 2. Related Literatures and Preliminaries 3 Our Model 3.1 Nearest neighbor （Ising TYPE） 3-2. Random Matching （SK MODEL） Annealed System, Quenched System 4. Implication : Cont- Bouchaud’s Model 5. Summary and Future Works SUMMARY Add the parameter (optimal choice behavior). 149 SUMMARY Add the parameter (optimal choice behavior).  Sec. 2: We construct the nearest neighborhood model (Ising Type)  Sec. 3: We construct the randomly matched model (Annealed Sys., Quenched Sys.)  Sec.4: Quenched System + Externality → multiple equilibria  Sec.5: Apply to econo-physics‟ model 150 SUMMARY Add the parameter (optimal choice behavior).  Sec. 2: We construct the nearest neighborhood model (Ising Type)  Sec. 3: We construct the randomly matched model (Annealed Sys., Quenched Sys.)  Sec.4: Quenched System + Externality → multiple equilibria  Sec.5: Apply to econo-physics‟ model FUTURE WORKS： relation between this model and DMBG( Dynamic Matching and Bargaining Game), Simulation (Monte Calro Simulation) 151 Fin REFERENCE  Blume, Lawrence. E. :„‟ The Statistical Mechanics of Strategic Interaction,‟‟ Games and         152 Economic Behavior, Vol. 5 (1993), pp.387-424. Coniglio,A. N. , Chiara R., Peruggi, F. and Russo, L.: ''Percolation and phase transitions in the Ising model,'„ Communications in Mathematical Physics, Vol. 51, Number 3(October, 1976), pp. 315-323. Cont, R. and Bouchaud, J-P.: „‟Herd Behaivor and Aggregate Fluctuations in Financial Markets,‟‟ Macroeconomic Dynamics, 4 (2000), pp. 170-196. Diederich, S. and Opper, M.: ''Replicators with random interactions: A solvable model,'„ Physical Review A, Vol.39, Number 8 (1989), pp.4333-4336. 樋口保成: 「イジングモデルのパーコレーション」 数学, Vol. 47 (1995), pp.111-124. 吉川満: 「統計力学を用いた進化ゲーム理論」 『京都大学数理解析 研究所講究録』1597号(2008年5月), pp.220-224. . McKelvey, Richard D. and Palfrey, Thomas, R. (1996): „‟A Statistical Theory of Equilibrium in Games,‟‟ Japanese Economic Review, Vol. 47, pp. 186-209. Mezard, M., Parisi, G. and Virasoro, M. A.: Spin Glass Theory and Beyond, World Scientific, 1987. Weibull, J.W.: Evolutionary Game Theory, MIT Press,1995. FAQ. Q１）なぜ、1対1のゲームなのですか？ 153 数理生物学の分野でも格子モデルを使って、空 間構造のあるゲームを分析している研究があると 思います(例えば, Nowak and May (1992), Nowak (2006) など)が、これらの研究はノイマン近傍の相手と のゲームなどあると思いますが、なぜ隣の相手と のゲームなのでしょうか？ A1) まず理論としてきちんと定式化したかったので 、IsingモデルやSKモデルのアイデアを借り、定式 化するために、最近接の相手とのゲームとしまし た。数理生物学の分野では多くはシミュレーショ ンによるアプローチだと思います。そこが我々の ものとは異なります。相手が複数ある場合のゲー ムなどは今後の拡張となると思います。  Q2) このモデルを拡張するとしたら、どのような ことが考えられますか？  A2) まず考えられるのが、戦略が2以上の場合。 例えば、Celluer Automata からのアプローチ（ Domany and Kinzel (1984), PRL, vol. 53, number 4. pp. 311-314）などが考えられます。つまりIsingモデル では状態が2つでしたが、0から1までの実数とす れば、無限個の戦略がある場合に拡張することが 出来るわけです。 次にはゲームの相手が1対1ではなく、グループ でゲームをする場合。これは主に数理生物学で研 究されています。 154 さらには、このモデルで重要なパラメーター を内生化する研究（超統計(super statistics)) 。 などいろいろ考えることができます。  Q3) 統計力学という言葉には聞き馴染みがないのです が、経済学ではよく使われている概念なのでしょうか ？  A3) はい。ミクロ, マクロ経済学教科書レベルのもの では取り扱われていませんが、経済学の研究にも使わ れています。  取り上げた研究以外にも、Follmer (JME, 1974)では Isingモデルを。Grandmont(JET, 1992), Foley (JET, 1994) など多数あります。  元来統計力学は、古典力学では分析できない高次元 系を分析するために開発されたものです。もちろん 市場などの経済システムは大多数の人間が売買を繰 り返す複雑なシステムであるので、有効であると思 います。シミュレーションまで行うと、より現実に 迫れるのではないかと思います。 155 Acknowledgements この報告内容は「 2007年度 夏季研究会」(関西学 院大学経済学研究会),「第4回 生物数学の理論と応 用」（京都大学数理解析研究所) 」, 「経済物理学 III」 (京都大学基礎物理学研究所), 「第4回数学総合 若手研究集会」(北海道大学数学COE),「京都ゲーム理 論ワークショップ2008」, 「日本経済学会 2008年春季 大会」(東北大学) において報告したものを、加筆・ 訂正したものである。 筆者はこれらの研究会で報告することによって, 多 くのコメントや刺激を受けたことに感謝したします. 2008年11月 吉川 満。 http://kikkawa.cyber-ninja.jp/index.htm 156 156