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Randomness and early termination: what makes a game exciting?

arXiv (Cornell University)(2023)

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摘要
In this short paper, we revisit an open problem posed by Aldous on the max-entropy win-probability martingale: given two players of equal strength, such that the win probability is a martingale diffusion, which of these processes has maximum entropy and hence gives the most excitement for the spectators? From a stochastic control perspective, the corresponding value function can be characterised by a log diffusion PDE, for which we show existence and uniqueness of classical solutions. We establish key qualitative properties of the solution including concavity, monotonicity, and convergence to a steady state. Moreover, we construct convergent numerical approximations, which allow us to highlight the behaviour of the win-probability process in the present case where the match may end early, in contrast to recent work by Backhoff-Veraguas and Beiglb\"ock where the match always runs the full length.
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early termination,game
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