The most exciting win-martingale among d+1 players is characterized by a Monge-Ampère equation, and its diffusion coefficient is sqrt(2(∇²g)^{-1}/(1-s)).
A ldous, What is the max-entropy win-probability martingale? https://www.stat.berkeley.edu/ al- dous/Research/OP/max ent mg.html
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Exciting games and Monge-Amp\`ere equations
The most exciting win-martingale among d+1 players is characterized by a Monge-Ampère equation, and its diffusion coefficient is sqrt(2(∇²g)^{-1}/(1-s)).