Volume 18, Number 1, January-March 2012
|Page(s)||81 - 90|
|Published online||02 December 2010|
Dynamic Programming Principle for tug-of-war games with noise
Department of Mathematics, University of Pittsburgh,
2 Institute of Mathematics, Helsinki University of Technology, P.O. Box 1100, 02015 TKK, Finland
3 Departamento de Matemática, FCEyN UBA (1428), Buenos Aires, Argentina
Revised: 13 August 2010
We consider a two-player zero-sum-game in a bounded open domain Ω described as follows: at a point x ∈ Ω, Players I and II play an ε-step tug-of-war game with probability α, and with probability β (α + β = 1), a random point in the ball of radius ε centered at x is chosen. Once the game position reaches the boundary, Player II pays Player I the amount given by a fixed payoff function F. We give a detailed proof of the fact that the value functions of this game satisfy the Dynamic Programming Principle for x ∈ Ω with u(y) = F(y) when y ∉ Ω. This principle implies the existence of quasioptimal Markovian strategies.
Mathematics Subject Classification: 35J70 / 49N70 / 91A15 / 91A24
Key words: Dirichlet boundary conditions / Dynamic Programming Principle / p-Laplacian / stochastic games / two-player zero-sum games
© EDP Sciences, SMAI, 2010
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