Optimal Decision Strategies for the Generalized Cuckoo Card Game

Cuckoo is a popular card game, which originated in France during the 15th century and then spread throughout Europe, where it is currently well-known under distinct names and with different variants. Cuckoo is an imperfect information game-of-chance, which makes the research regarding its optimal strategies determination interesting. The rules are simple: each player receives a covered card from the dealer; starting from the player at the dealer's left, each player looks at its own card and decides whether to exchange it with the player to their left, or keep it; the dealer plays at last and, if it decides to exchange card, it draws a random one from the remaining deck; the player(s) with the lowest valued card lose(s) the round. We formulate the gameplay mathematically and provide an analysis of the optimal decision policies. Different card decks can be used for this game, e.g., the standard 52-card deck or the Italian 40-card deck. We generalize the decision model for an arbitrary number decks' cards, suites, and players. Lastly, through numerical simulations, we compare the determined optimal decision strategy against different benchmarks, showing that the strategy outperforms the random and naive policies and approaches the performance of the ideal oracle.