A Problem of Calculus of Variations and Game Theory

TL;DR

This paper explores a mathematical problem combining game theory and calculus of variations, analyzing optimal strategies and Nash equilibria through algebraic and computational methods, including polynomial approximation and linear programming.

Contribution

It introduces a novel approach to solving a combined calculus of variations and game theory problem, providing explicit strategies and equilibrium analysis.

Findings

Optimal strategies are derived algebraically.

Nash equilibria are computed via linear programming.

A variation of the game is also analyzed.

Abstract

In this paper, we study a theoretical math problem of game theory and calculus of variations in which we minimize a functional involving two players. A general relationship between the optimal strategies for both players is presented, followed by computer analysis as well as polynomial approximation. Nash equilibrium strategies are determined through algebraic manipulation and linear programming. Lastly, a variation of the game is also investigated.