Two-Person Symmetric Whist

The game of two-person whist is played with a deck of cards. Each card belongs to a suit, and each suit is totally ordered. The cards are distributed between the two players so that each player receives the same number of cards. Hence both players have complete information about the deal. Play proceeds in tricks, with the obligation to follow suit, as in many real-world card games. We study the symmetric case of this game, that is, we assume that in each suit, the two players have the same number of cards. Under this assumption, the second player in each trick will always be able to follow suit. We show how to assign a value from a certain semigroup to each single-suit card distribution in such a way that the outcome of a multisuit deal under optimal play is determined by the sum of the values of the individual suit. This allows us to play a deal perfectly, provided that we can compute the values of its single-suit components. Although we do not have an efficient algorithm for doing this in general, we give methods that will allow us to find the value of a suit quickly in most cases. We also establish certain general facts about the game, for instance the nontrivial fact that a higher card is always at least as good as a smaller card in the same suit. 1. The game of two-person whist The game of two-person whist is played with a deck of cards. Each card belongs to a suit, and within each suit, the cards are ordered by rank. Realworld card packs sometimes have four suits with thirteen cards in each suit. We will not restrict ourselves to the “standard” deck, and in any case, the french suited 52 card pack too is just one of many variations. There does not even have to be the same number of cards in each suit. The cards are distributed between the two players, so that both players receive the same number of cards. We assume that both players have complete information about the situation. One of the players is said to have the lead. The player who has the lead plays, or leads, one of his cards. The other player, in response to this, plays one of his cards. If possible, he has to follow suit, that is, he has to play a card in the same suit as the card that was led. The player who played the highest card in the suit that was led wins the trick, and obtains the lead. The cards that have been played are removed, and play continues until all cards have been played. Each player tries to win as many tricks as possible. This game is a pure form of a common type of card game, trick taking games. Trick taking games exist in many different forms, and their history Date: February 21, 2005.