untitled

Size: px

Start display at page:

Download "untitled"

めぐのこいたばし
5 years ago
Views:

1 c GM 1959 Lindsey [1] 1960 Howard [2] Howard 1 25 (Markov Decision Process) =25 9 Bellman [3] 1 Bellman 1 k D Esopo and Lefkowitz [4] 1 (SI) Cover and Keilers [5] 1 OERA (Offensive Earned- Run Average) OERA Howard Bellman 1997 Bukiet et al. [6] [7] Ano [8] [9] OERA [10] Bukiet et al. [6] Copyright c by ORSJ. Unauthorized reproduction of this article is prohibited.

2 SI OERA SIL OERAL 2008 [11] n Howard 2008 Turocy [12] Bellman 1 1 Kira and Inakawa [15] Turocy 2. (state) 1 1 (game in extensive form) 1 (move) 1 Shapley [13] Zachrisson [14] 1 2 (chance move) 1 (Markov property) 0 1 S (state space) s S s =(ι, τ, ω, λ, r, b,m) 1. ι {1, 2,...,12} ι =9 ι =12 2. τ {0, 1} (0) (1) 3. ω {0, 1, 2, 3} 4. λ Copyright c by ORSJ. Unauthorized reproduction of this article is prohibited

3 5. r =(r 3,r 2,r 1): r 1 {0, 1,...,9} 1 0, r i {0, 1} (i =2, 3) i 0, 1 6. b =(b 0,b 1): b i {1, 2,...,9} (i =0, 1) i 7. m {0, 1}: (0) (1) r r 1 S Q S s 0 =(ι, τ, ω, λ, r, b,m) 0 =(1, 0, 0, 0, (0, 0, 0), (1, 1), 1). S i i S i := {s =(ι, τ, ω, λ, r, b,m) S\S Q m = i}. S 0 S 1 s 0 4 (b 0 =4) 5 (r 1 =5) 6, 454, 296 (action space) A := { } s S 0 S 1 A(s) A (feasible action space) s =(ι, τ, ω, λ, r, b,m) S 0 S 1 A(s) τ m, A(s) τ = m, A(s) τ = m r 2 =0 r 1 1, A(s) τ = m ω 1 r i 1. i S Q := S 1 Q S 2 Q SQ, 5 S 1 Q := {s S ι 9, τ=1,ω=3,λ>0}, S 2 Q := {s S ι =9,τ=0,ω=3,λ<0}, S 3 Q := {s S ι 9, τ=1,λ<0}, S 4 Q := {s S ι =12,τ=1,ω=3,λ=0}, S 5 Q := {s S λ 30} {s S λ 30}. S 1 Q S 2 Q 9 S 3 Q S 4 Q 12 S 5 Q mercy-rule [12] mercy-rule 12 ω =3 s/ S Q 3 s 0 1 a A(s) X (a) { } a = { } a = { X (a) := a = } { } a =. X (a) 1 p(x s, a) [0, 1] s a A(s) x X(a) Copyright c by ORSJ. Unauthorized reproduction of this article is prohibited.

4 p(x s, a) =1, (s, a) s.t. a A(s). x X (a) X (a) s a A(s) x X(a) s s a x t s = t(s, a, x). s =(ι, τ, ω, λ, r, b,m) S Q i (payoff) ψ i(s) { { 1 λ>0, 0 λ 0, ψ 0(s) := ψ 1(s) := 0 λ 0, 1 λ< ( ). π i : S i A s S i π i(s) A(s) π i i (Markov policy) i Π i (i =0, 1). π 0 Π 0, π 1 Π 1 1 s 0 n X n {X n} (Markov chain) {X n} S Q T T := min{n X n S Q} <. 1 T i v i(s; π 0,π 1) v i(s; π 0,π 1):=E π 0,π 1 [ψ i(x T ) X 0 = s], i =0, 1. E π 0,π 1 π 0,π 1 Kira et al. [17] Copyright c by ORSJ. Unauthorized reproduction of this article is prohibited

5 3. (Nash equilibrium) 3.1 ( ). (π0,π 1 ) Π 0 Π 1 (Markov perfect equilibrium) (π0,π 1 ) v 0(s; π 0,π 1 ) v 0(s; π 0,π1), s S, π 0 Π 0, v 1(s; π0,π 1) v 1(s; π 0,π1), s S, π 1 Π ( ). (π0,π 1 ) (V 0(s),V 1(s)) := (v 0(s; π 0,π1),v 1(s; π0,π 1)) s (backward induction) Turocy [12] MLB Turocy (game in strategic form) OR 9 s Q S Q V i(s Q)=ψ i(s Q) 2 s V i(s ) V i(s ) i s V i(s ) i 3.1 (Bellman ). V i : S [0, 1] (π0,π 1) Π 0 Π 1 V i(s) = ψ i(s) s S Q, Max V i(t(s, a, x))p(x s, a) s S i, a A(s) x X (a) V i(t(s, π j (s),x))p(x s, π j (s)) s S j, x X (π j (s)) π i (s) arg max a A(s) x X (a) V i(t(s, a, x))p(x s, a), s S i. (i, j) =(0, 1), (1, 0) !? Copyright c by ORSJ. Unauthorized reproduction of this article is prohibited.

6 Turocy [12] ON OFF 8 1 ON OFF 64 3 ON Intel R Core TM i7-3770k 16GB PC ON ON ON ON 3 Turocy Turocy Bukiet et al. [6] 8! σ =(1, 2, 3, 4, 5, 6, 7, 8, 9) σ =(2, 3, 4, 5, 6, 7, 8, 9, 1) Copyright c by ORSJ. Unauthorized reproduction of this article is prohibited

4 1 mercy-rule 30 20 1 2.43 144 3.49 6. [1] Lindsey, G. R., Statistical Data Useful for the Operation of a Baseball Team, Operations Reserach, 7, 197 207, 1959. [2] Howard, R. A.

7 4 1 mercy-rule [1] Lindsey, G. R., Statistical Data Useful for the Operation of a Baseball Team, Operations Reserach, 7, , [2] Howard, R. A., Dynamic Programming and Markov Processes, M.I.T. Technology Press and Wiley, Cambridge, Mass, [3] Bellman, R., Dynamic Programming and Markovian Decision Processes, with Applicarion to Baseball, Optimal Strategies in Sports, S. P. Ladany and R. E. Macol (eds.), Elsevier-North Holland, New York, 77 85, [4] D Esopo, D. A. and Lefkowitz, B., The Distribution of Runs in the Game of Baseball, Optimal Strategies in Sports, S.P.LadanyandR.E.Macol(eds.),Elsevier North-Holland, 55 62, [5] Cover, T. M. and Keilers, C. W., An Offensive Earned-Run Average for Baseball, Operations Research, 25, , [6] Bukiet, B., Harold, E. R. and Palacios, J. L., A Markov Chain Approach to Baseball, Operations Research, 45, 14 23, [7] OR 24, , [8] Ano, K., Modified offensive earned-run average with steal effect for baseball, Applied Mathematics and Computation, 120, , [9] 47, , [10] 49, , [11] 18, , [12] Turocy, T. L., In Search of the Last-Ups Advantage in Baseball: A Game-Theoretic Approach, Journal of Quantitative Analysis in Sports, 4(2) Article 5, [13] Shapley, L. S., Stochastic games, Proceedings of the National Academy of Sciences of the United States of America, 39, , [14] Zachrisson, L. E., Markov games, Annals Mathematics Studies: Advances in Game Theory, M. Drescher, L. S. Shapley and A. W. Tucker (eds.), Princeton University Press, Princeton, 52, , [15] Kira, K. and Inakawa, K., On Markov perfect equilibria in baseball, Bulletin of Informatics and Cybernetics, toappear. [16] (Accessed 2014 Apr. 7). [17] Kira, K., Ueno, T. and Fujita, T., Threshold probability of non-terminal type in finite horizon Markov decision processes, Journal of Mathematical Analysis and Applications, 386, , [18] jp/nuldata/(accessed 2014 Apr. 7) Copyright c by ORSJ. Unauthorized reproduction of this article is prohibited.

WPA(Win Probability Added) 1 WPA WPA ( ) WPA WPA WPA WPA WPA

21 4 25 1 31 WPA(Win Probability Added) 1 WPA WPA ( ) WPA WPA WPA WPA WPA 1 1 2 WPA 3 2.1 WPA(Win Probability Added)................................. 3 2.2........................... 3 2.2.1...................................