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We show that the value function of an optimal stopping game driven by aone-dimensional diffusion can be characterised using a modification of theLegendre transformation if and only if the optimal stopping game exhibits aNash 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) ADuality Principle for the Legendre Transform. Journal of Convex Analysis,19(3), 609-630 where the `duality' between a concave-biconjugate which ismodified to remain below an upper barrier and a convex-biconjugate which ismodified to remain above a lower barrier is proven by appealing to theprobabilistic result in Peskir, G. (2008) Optimal stopping games and Nashequilibrium. Theory Probab. 53 (558-571). The main contribution of this paperis to show that, in this special case, the semi-harmonic characterisation ofthe value function may be proven using only results from convex analysis.

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