[Paper Review] A Game-Tree approach to discrete infinity Laplacian with running costs

Qing Liu, Armin Schikorra|arXiv (Cornell University)|May 31, 2013

Mathematical Biology Tumor Growth4 references3 citations

TL;DR

This paper presents a deterministic, game-theoretic approach to solving the discrete infinity Laplacian with running costs using game-trees and iterative convergence. It establishes the existence, uniqueness, and uniform boundedness of solutions to the dynamic programming principle (DPP) for biased tug-of-war games on general metric spaces, without requiring boundary mollification or semi-continuity assumptions.

ABSTRACT

We give a self-contained and elementary proof for boundedness, existence, and uniqueness of solutions to dynamic programming principles (DPP) for biased tug-of-war games with running costs. The domain we work in is very general, and as a special case contains metric spaces. Technically, we introduce game-trees and show that a discretized flow converges uniformly, from which we obtain not only the existence, but also the uniqueness. Our arguments are entirely deterministic, and also do not rely on (semi-)continuity in any way; in particular, we do not need to mollify the DPP at the boundary for well-posedness.

Motivation & Objective

• To establish the existence and uniqueness of solutions to the dynamic programming principle (DPP) for biased tug-of-war games with running costs on general metric spaces.
• To provide a deterministic, non-probabilistic proof of convergence for an iterative scheme without relying on boundary mollification or semi-continuity.
• To generalize the framework to include arbitrary metric spaces and non-uniform ball structures, beyond standard Euclidean domains.
• To analyze subsolutions and supersolutions of the DPP and derive uniform bounds independent of initial data.
• To demonstrate that the iterative scheme converges uniformly to the unique solution, even in the absence of positive $α$-weighting in the DPP as in prior works.

Proposed method

• Introduces a game-tree formalism to model the evolution of the value function over discrete time steps, where each node represents a state and moves correspond to choices in the sup-inf structure.
• Defines a discrete flow via iteration: $u_{k+1}(x) = \mu \sup_{B_\varepsilon(x)} u_k + (1-\mu)\inf_{B_\varepsilon(x)} u_k + f(x)$ for $x \in Y$, with fixed boundary values on $X \setminus Y$.
• Uses a recursive tree-based representation of the iteration to express $u_k(x)$ as a nested infsup over paths of length $k$, capturing the game-theoretic dynamics.
• Imposes admissibility constraints on trees to ensure only paths that preserve boundedness and convergence are considered, leading to a truncated game-tree with uniform bounds.
• Applies uniform convergence arguments by comparing $u_k$ to a reference sequence $v_L$ derived from a truncated game-tree, using geometric decay of path weights $\mu^{l(t)}(1-\mu)^{r(t)}$.
• Establishes convergence via a Cauchy-type argument: for sufficiently large $L$, the difference between $u_{k+L}$ and $v_L$ is uniformly small, implying uniform convergence of $u_k$.

Experimental results

Research questions

• RQ1 Does the dynamic programming principle (DPP) for biased tug-of-war with running costs admit a unique solution on general metric spaces?
• RQ2 Can the solution be obtained via a deterministic iterative scheme without requiring boundary mollification or semi-continuity?
• RQ3 What conditions ensure uniform boundedness of the iterates $u_k$ and their limit?
• RQ4 How does the game-tree structure help in proving convergence and uniqueness of the solution?
• RQ5 Can the convergence be quantified uniformly across all initial data and domains?

Key findings

• For any $\mu \in (0,1)$ and $\Lambda > 0$, there exists a constant $C = C(\mu, \Lambda)$ such that all subsolutions $\underline{u}$ with $\sup_X \underline{u} < \infty$ satisfy $\sup_X \underline{u} \leq C$, and all supersolutions $\bar{u}$ with $\inf_X \bar{u} > -\infty$ satisfy $\inf_X \bar{u} \geq -C$.
• The iterative scheme $u_{k+1}(x) = \mu \sup_{B_\varepsilon(x)} u_k + (1-\mu) \inf_{B_\varepsilon(x)} u_k + f(x)$ for $x \in Y$ converges uniformly to a unique solution $u$ on $X$, regardless of the initial bounded $u_0$.
• The solution $u$ to the DPP is the uniform limit of the iterates $u_k$, and this convergence holds even when $f \equiv 0$ or $f > 0$, without requiring $\alpha > 0$ as in prior works.
• The convergence is quantified: for any $\delta > 0$, there exist $L_0, K_0 > 0$ such that $\sup_X |u_{k+L}(x) - v_L(x)| \leq \delta/2$ and $\sup_X |v_{K+i}(x) - v_K(x)| \leq \delta/2$ for $L \geq L_0$, $K \geq K_0$, implying $\sup_X |u_{k+L+K}(x) - v_{L+K+i}(x)| \leq \delta$.
• The method avoids regularization or mollification at the boundary, and the proof relies solely on deterministic game-tree analysis and geometric decay of path weights.
• The framework applies to arbitrary metric spaces and more general sets $B(x)$ than open or closed balls, as formalized in Definition 1.5.