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We extend the open games framework for compositional game theory to encompassalso mixed strategies, making essential use of the discrete probabilitydistribution monad. We show that the resulting games form a symmetric monoidalcategory, which can be used to compose probabilistic games in parallel andsequentially. We also consider morphisms between games, and show that intuitiveconstructions give rise to functors and adjunctions between pure andprobabilistic open games.

Compositional Game Theory with Mixed Strategies: Probabilistic Open Games Using a Distributive Law