English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In this paper we revisit an open problem posed by Aldous on the max-entropywin-probability martingale: given two players of equal strength, such that thewin-probability is a martingale diffusion, which of these processes has maximumentropy and hence gives the most excitement for the spectators? We study aterminal-boundary value problem for the nonlinear parabolic PDE$2\partial_te(t,x)=\log(-\partial_{xx}e(t,x))$ derived by Aldous and prove itswellposedness and regularity of its solution by combining PDE analysis andprobabilistic tools, in particular the reformulation as a stochastic controlproblem with restricted control set, which allows us to deduce strictellipticity. We establish key qualitative properties of the solution includingconcavity, monotonicity, convergence to a steady state for long remaining timeand the asymptotic behaviour shortly before the terminal time. Moreover, weconstruct convergent numerical approximations. The analytical and numericalresults allow us to highlight the behaviour of the win-probability process inthe present case where the match may end early, in contrast to recent work byBackhoff-Veraguas and Beiglb\"ock where the match always runs the full length.

Randomness and early termination: what makes a game exciting?