How do Tom and Jerry Play? A Simple Application of Convex Geometry in Game Theory
76 Pages Posted: 24 Jan 2022 Last revised: 30 Jan 2024
Date Written: January 18, 2022
Abstract
We propose a simultaneous-move hide-and-seek game, where one player wins by matching the other player, while the other player wins by mismatching in a connected space X in Euclidean space. We provide a complete characterization of Type I Nash Equilibrium where the seeker plays a pure strategy. We show that the center of mass of the hider’s strategy coincides with the seeker’s strategy at the center of the minimal cover ball of X. We also characterize Type II Nash Equilibrium where both players randomize their strategies. We show that the shape of X matters and the players only allocate their probability weights along a straight line. Some alternative settings are discussed and the equilibrium analysis is conducted. These results can be applied to a large number of scenarios, characterizing the behavior of two players in a zero-sum game, where one player aims to maximize the distance between them, while the other aims to minimize it.
Keywords: Hide and Seek, Simultaneous-move Game, Zero-sum Game, Convex Geometry, Nash Equilibrium
JEL Classification: C72
Suggested Citation: Suggested Citation