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Empirical Evidence for Evolutionary Game Theory : to connect with observable phenomena

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MIMS現象数理学ポスターセッション (先端数理科学研究科開設記念シンポジウム,2011/10/04)

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Empirical Evidence for Evolutionary Game Theory : to connect with observable phenomena

1. 1. Empirical Evidence for Evolutionary Game Theory : To Connect With Observable Phenomena Mitsuru KIKKAWA (Graduate School of Advanced Mathematical Sciences, Meiji Univ.) http://kikkawa.cyber-ninja.jp/ Development of mathematical sciences for practical use My Strategy : Modeling and Structural Estimation DEF. A Nash equilibrium of a strategic game n-person game G is a profile s*=(s* 1,…,s* n) with the property that for every position i=(1,…,n) we have the best response for another position’s strategy set s*-1. ⇒ 1. Rationality, 2. Statistical Population Focus on “Bounded Rationality (Large Population)”,“Nonequilibrium Dynamics” and formulate the game using “Statistical Mechanics”. Double Auction + main result 1, result 2 [REF.] [1] Kikkawa, M.: Statistical Mechanics of Games ― Evolutionary Game Theory ―, PTP Supplement, 179 (2009), 216-226. [2] Kikkawa, M.: Empirical Evidence for Evolutionary Game Theory, Submitted. [3] Kikkawa, M.: Convergence to Nash Equilibrium and Equilibrium Selection: A Bayesian Approach, Submitted. [4] Kikkawa, M.: Market Microstructure as a Double Auction, Mimeo. Modeling and Structural Estimation: Main Result 1. The probability of playing strategy πiα in the position i is obtained with the player’s payoff from the outcome is piα(s), ciα=Z-1 exp(γ piα(s)), γ: non-negative constant, Z: normalization parameter. Main Result 2. Under evolutionary approach, the following expression about the relationship between the payoff and the population size is achieved empirically : where is the average payoff of the entire population, is the position i ’s average payoff, Δ r is the variation in entire population size, and is the expected utilities’ variation by the population size changed.     , ˆ ˆ ' sp rpErp i ii      pˆ  spiˆ  ii rpE  Bayes’ Theorem Order Book Information/Sentiment ⇒ Text ming (Bayesian Estimation) Algorithmic Trading/High Frequency Trading (Cf. Replicator Eq. : )    niAxxAxx dt tdx ii i ,,1,  Fact 1. The volume distribution is proportional to the difference in reservation price between sellers and buyers. Fact 2. The volatility distribution is consistent with classical market microstructure’s results. Fact 3. The execution price and Walras equilibrium price are cointegration. Walras equilibrium has the price discovery role compared with the execution price. Fact 4. There are some cases that each investor does not choose the strategy rationally and hurries up selling and buying. @ MIMS 現象数理学ポスターセッション, 4th, Oct., 2011