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

• Volume: 18, Issue: 1, page 81-90
• ISSN: 1292-8119

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## Abstract

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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$u\left(x\right)=\frac{\alpha }{2}\left\{\underset{y\in {B}_{}\left(x\right)}{sup}u\left(y\right)+\underset{y\in {B}_{}\left(x\right)}{inf}u\left(y\right)\right\}+{\beta }_{{B}_{}\left(x\right)}u\left(y\right)y,$ for x ∈ Ω with u(y) = F(y) when y ∉ Ω. This principle implies the existence of quasioptimal Markovian strategies.

## How to cite

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Manfredi, Juan J., Parviainen, Mikko, and Rossi, Julio D.. "Dynamic Programming Principle for tug-of-war games with noise." ESAIM: Control, Optimisation and Calculus of Variations 18.1 (2012): 81-90. <http://eudml.org/doc/277814>.

@article{Manfredi2012,
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\begin\{equation*\} u(x) = \frac\{\alpha\}\{2\} \left\\{ \sup\_\{y\in \ol B\_\{\eps\}(x)\} u (y) + \inf\_\{ y \in \ol B\_\{\eps\}(x)\} u (y) \right\\} + \beta \kint\_\{ B\_\{\eps\}(x)\} u(y) \ud y, \end\{equation*\} for x ∈ Ω with u(y) = F(y) when y ∉ Ω. This principle implies the existence of quasioptimal Markovian strategies. },
author = {Manfredi, Juan J., Parviainen, Mikko, Rossi, Julio D.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Dirichlet boundary conditions; Dynamic Programming Principle; p-Laplacian; stochastic games; two-player zero-sum games; zero sum game; tug-of-war game; dynamic programming; quasioptimal strategies; Markovian strategies},
language = {eng},
month = {2},
number = {1},
pages = {81-90},
publisher = {EDP Sciences},
title = {Dynamic Programming Principle for tug-of-war games with noise},
url = {http://eudml.org/doc/277814},
volume = {18},
year = {2012},
}

TY - JOUR
AU - Manfredi, Juan J.
AU - Parviainen, Mikko
AU - Rossi, Julio D.
TI - Dynamic Programming Principle for tug-of-war games with noise
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2012/2//
PB - EDP Sciences
VL - 18
IS - 1
SP - 81
EP - 90
AB - 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\begin{equation*} u(x) = \frac{\alpha}{2} \left\{ \sup_{y\in \ol B_{\eps}(x)} u (y) + \inf_{ y \in \ol B_{\eps}(x)} u (y) \right\} + \beta \kint_{ B_{\eps}(x)} u(y) \ud y, \end{equation*} for x ∈ Ω with u(y) = F(y) when y ∉ Ω. This principle implies the existence of quasioptimal Markovian strategies.
LA - eng
KW - Dirichlet boundary conditions; Dynamic Programming Principle; p-Laplacian; stochastic games; two-player zero-sum games; zero sum game; tug-of-war game; dynamic programming; quasioptimal strategies; Markovian strategies
UR - http://eudml.org/doc/277814
ER -

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