Unit-sphere games

TL;DR

This paper introduces unit-sphere games where strategies are unit vectors, characterizes their Nash equilibria through eigenvalues, and shows existence and uniqueness results for non-negative payoff matrices, with efficient computation methods.

Contribution

It defines a new class of games on the unit sphere, characterizes Nash equilibria via eigenvalues, and establishes existence, uniqueness, and computational properties.

Findings

Nash equilibria characterized by eigenvalues and eigenvectors.

Unique equilibrium exists for non-negative payoff matrices.

Equilibrium can be efficiently computed via Cournot adjustment.

Abstract

This paper introduces a class of games, called unit-sphere games, where strategies are real vectors with unit 2-norms (or, on a unit-sphere). As a result, they can no longer be interpreted as probability distributions over actions, but rather be thought of as allocations of one unit of resource to actions and the multiplicative payoff effect on each action is proportional to square-root of the amount of resource allocated to that action. The new definition generates a number of interesting consequences. We first characterize sufficient and necessary conditions under which a two-player unit-sphere game has a Nash equilibrium. The characterization effectively reduces solving a unit-sphere game to finding all eigenvalues and eigenvectors of the product of individual payoff matrices. For any unit-sphere game with non-negative payoff matrices, there always exists a unique Nash equilibrium; furthermore, the unique equilibrium is efficiently reachable via Cournot adjustment. In addition, we show that any equilibrium in positive unit-sphere games corresponds to approximate equilibria in the corresponding normal-form games. Analogous but weaker results are extended to positive n-player unit-sphere games.