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)

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