Nash equilibirum and the Legendre transform in optimal stopping games with one dimensional diffusions

TL;DR

This paper demonstrates that the value function of a one-dimensional diffusion optimal stopping game can be characterized using a modified Legendre transform if and only if a Nash equilibrium exists, linking convex analysis to game theory.

Contribution

It provides an analytical proof connecting the value function's semi-harmonic property to convex analysis, complementing previous probabilistic approaches.

Findings

Characterization of value function via Legendre transform when Nash equilibrium exists

Analytical proof using convex analysis methods

Extension of duality principles in optimal stopping games

Abstract

We show that the value function of an optimal stopping game driven by a one-dimensional diffusion can be characterised using a modification of the Legendre transformation if and only if the optimal stopping game exhibits a Nash equilibrium (i.e. a saddle point of the optimal stopping game exists). This result is an analytical complement to the results in Peskir, G. (2012) A Duality Principle for the Legendre Transform. Journal of Convex Analysis, 19(3), 609-630 where the `duality' between a concave-biconjugate which is modified to remain below an upper barrier and a convex-biconjugate which is modified to remain above a lower barrier is proven by appealing to the probabilistic result in Peskir, G. (2008) Optimal stopping games and Nash equilibrium. Theory Probab. 53 (558-571). The main contribution of this paper is to show that, in this special case, the semi-harmonic characterisation of the value function may be proven using only results from convex analysis.