Related papers: A Game-Tree approach to discrete infinity Laplacia…
We present a modified version of the two-player "tug-of-war" game introduced by Peres, Schramm, Sheffield, and Wilson. This new tug-of-war game is identical to the original except near the boundary of the domain $\partial \Omega$, but its…
We provide an alternative approach to the existence of solutions to dynamic programming equations arising in the discrete game-theoretic interpretations for various nonlinear partial differential equations including the infinity Laplacian,…
Motivated by tug-of-war games and asymptotic analysis of certain variational problems, we consider a gradient constraint problem involving the infinity Laplace operator. We prove that this problem always has a solution that is unique if a…
In tug-of-war, two players compete by moving a counter along edges of a graph, each winning the right to move at a given turn according to the flip of a possibly biased coin. The game ends when the counter reaches the boundary, a fixed…
We consider a stochastic differential game in the context of forward-backward stochastic differential equations, where one player implements an impulse control while the opponent controls the system continuously. Utilizing the notion of…
In two-player games on graphs, the players move a token through a graph to produce an infinite path, which determines the winner of the game. Such games are central in formal methods since they model the interaction between a…
We propose a new finite difference approximation to the Dirichlet problem for the homogeneous $\mathbf{p}$-Laplace equation posed on an $N$-dimensional domain, in connection with the Tug of War games with noise. Our game and the related…
This paper concerns value functions of time-dependent tug-of-war games. We first prove the existence and uniqueness of value functions and verify that these game values satisfy a dynamic programming principle. Using the arguments in the…
We study the Dirichlet problem of the following discrete infinity Laplace equation on a subgraph with finite width $$\Delta_{\infty} u(x) = \inf_{y \sim x}u(y)+\sup_{y \sim x}u(y)-2u(x) = f(x).$$ We say that a subgraph has finite width if…
We consider an autonomous navigation problem, whereby a traveler aims at traversing an environment in which an adversary tries to set an ambush. A two players zero sum game is introduced. Players' strategies are computed as random path…
An active line of research has considered games played on networks in which payoffs depend on both a player's individual decision and also the decisions of her neighbors. Such games have been used to model issues including the formation of…
Promoting behavioural diversity is critical for solving games with non-transitive dynamics where strategic cycles exist, and there is no consistent winner (e.g., Rock-Paper-Scissors). Yet, there is a lack of rigorous treatment for defining…
Nonlinear PDEs, mean value properties, and stochastic differential games are intrinsically connected. In this short expository note, we will describe how the solutions to certain PDEs (of $p$-Laplacian type) can be interpreted as limits of…
We develop methods to formally describe and compare games, in order to probe questions of game structure and design, and as a stepping stone to predicting player behavior from design patterns. We define a grammar-like formalism to describe…
Traditional solvable game theory and mean-field-type game theory (risk-aware games) predominantly focus on quadratic costs due to their analytical tractability. Nevertheless, they often fail to capture critical non-linearities inherent in…
In this paper, we investigate a class of tug-of-war games that incorporate a constant payoff discount rate at each turn. The associated model problems are $p$-Laplace type partial differential equations with zeroth-order terms. We establish…
In this paper, we resolve the computational complexity of a number of outstanding open problems with practical applications. Here is the list of problems we show to be PPAD-complete, along with the domains of practical significance:…
In this expository paper we illustrate the generality of game theoretic probability protocols of Shafer and Vovk (2001) in finite-horizon discrete games. By restricting ourselves to finite-horizon discrete games, we can explicitly describe…
Motivated by the "tug-of-war" game studied in [12], we consider a "non-local" version of the game which goes as follows: at every step two players pick respectively a direction and then, instead of flipping a coin in order to decide which…
In this paper, we focus our attention on the large capacities unsplittable flow problem in a game theoretic setting. In this setting, there are selfish agents, which control some of the requests characteristics, and may be dishonest about…