A simple random walk game

Stefan Ankirchner (1) , Hugalf Bernburg (1) , Julian Wendt (1)

1 Friedrich-Schiller-Universität = Friedrich Schiller University Jena [Jena, Germany]

We consider a dynamic two player game where each player can control the step sizes of her own random walk. At any time, the players can choose a step size equal to 0 or 1. Moreover, whenever a player is ahead, i.e. her state is above the opponent's state, she receives a fixed reward. The player behind does not receive anything. We assume that both players aim at maximizing the expected sum of discounted rewards. We show that there exists a threshold that the correlation between the two random walks must not exceed in order for the game to possess an equilibrium in pure strategies. We explicitly compute the threshold level and show, provided the correlation is smaller than or equal to the treshold, that an equilibrium is given if the player ahead chooses step size 0 and the player behind step size 1.

Mots clés: Nash equilibrium; random walk; pure strategy; correlation; step size control; two player game

Domaines: Probabilités [math.PR]

Stefan Ankirchner, Hugalf Bernburg, Julian Wendt. A simple random walk game. 2022. ⟨hal-03607763⟩