Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-28T16:27:45.131Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Le Gruyer, E., On absolutely minimizing Lipschitz extensions and PDE Δ ∞(u) = 0. NoDEA 14 (2007) 29–55. Google Scholar
Le Gruyer, E. and Archer, J.C., Harmonious extensions. SIAM J. Math. Anal. 29 (1998) 279–292. Google Scholar
Maitra, A.P. and Sudderth, W.D., Borel stochastic games with limsup payoff. Ann. Probab. 21 (1993) 861–885. Google Scholar
A.P. Maitra and W.D. Sudderth, Discrete gambling and stochastic games, Applications of Mathematics 32. Springer-Verlag (1996).
Manfredi, J.J., Parviainen, M. and Rossi, J.D., An asymptotic mean value property characterization of p-harmonic functions. Proc. Am. Math. Soc. 138 (2010) 881–889. Google Scholar
J.J. Manfredi, M. Parviainen and J.D. Rossi, On the definition and properties of p -harmonious functions. Preprint (2009).
Oberman, A., A convergent difference scheme for the infinity Laplacian : construction of absolutely minimizing Lipschitz extensions. Math. Comp. 74 (2005) 1217–1230. Google Scholar
Peres, Y. and Sheffield, S., Tug-of-war with noise : a game theoretic view of the p-Laplacian. Duke Math. J. 145 (2008) 91–120. Google Scholar
Peres, Y., Schramm, O., Sheffield, S. and Wilson, D., Tug-of-war and the infinity Laplacian. J. Am. Math. Soc. 22 (2009) 167–210. Google Scholar
S.R.S. Varadhan, Probability theory, Courant Lecture Notes in Mathematics 7. Courant Institute of Mathematical Sciences, New York University/AMS (2001).