A Problem of Calculus of Variations and Game Theory

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.

Figures (17)

• Figure 1: Labeled stream plot for $a=1$, where the neon line represents when $t=1$ and $f'(t)=0$, and the black dots represent $f(t)$ when $t=0$.
• Figure 2: Illustrates all $f(t)$ and $f'(t)$ values for $a=1$ and $t=0$.
• Figure 3: Stream plot of $f(t)$ from $t=0$ to $t=1$ for $a=102$.
• Figure 4: $f(t)$ when $t=0$ for the stream plot where $a=102$, with three solutions that satisfy the initial conditions. (*It may look like the purple curve is tangent to the $f'(t)$ axis, but if you zoom in, there are actually two solutions there.)
• Figure 5: All possible "startpoints" (when $t=0$) for the case where $a=1000$ (the straight lines in the middle of the graph are not supposed to be there, this was just how the datapoints were generated, and they show the jump in the $(f(t),f'(t))$ values at $t=0$).