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

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/272799>.

@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\lbrace \sup \_\{y\in B\_\{\}(x)\} u (y) + \inf \_\{ y \in B\_\{\}(x)\} u (y) \right\rbrace + \beta \_\{ B\_\{\}(x)\} u(y) y, \end\{equation*\}u ( x ) = α 2 ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ sup y ∈ B ε ( x ) u ( y ) + inf y ∈ B ε ( x ) u ( y ) ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ + β ∫ B ε ( x ) u ( y ) d y, forx ∈ 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},
number = {1},
pages = {81-90},
publisher = {EDP-Sciences},
title = {Dynamic Programming Principle for tug-of-war games with noise},
url = {http://eudml.org/doc/272799},
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
PY - 2012
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\lbrace \sup _{y\in B_{}(x)} u (y) + \inf _{ y \in B_{}(x)} u (y) \right\rbrace + \beta _{ B_{}(x)} u(y) y, \end{equation*}u ( x ) = α 2 ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ sup y ∈ B ε ( x ) u ( y ) + inf y ∈ B ε ( x ) u ( y ) ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ + β ∫ B ε ( x ) u ( y ) d y, forx ∈ 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/272799
ER -