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Dynamic Programming Principle for tug-of-war games with noise

Published online by Cambridge University Press:  02 December 2010

Juan J. Manfredi ,
Mikko Parviainen  and
Julio D. Rossi
Juan J. Manfredi
Affiliation:
Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA. manfredi@pitt.edu
Mikko Parviainen
Affiliation:
Institute of Mathematics, Helsinki University of Technology, P.O. Box 1100, 02015 TKK, Finland; Mikko.Parviainen@tkk.fi
Julio D. Rossi
Affiliation:
Departamento de Matemática, FCEyN UBA (1428), Buenos Aires, Argentina; jrossi@dm.uba.ar
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Abstract

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.

Keywords

Dirichlet boundary conditionsDynamic Programming Principlep-Laplacianstochastic gamestwo-player zero-sum games
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 18 , Issue 1 , January 2012 , pp. 81 - 90
Copyright
© EDP Sciences, SMAI, 2010

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