Drawing room game

A game with several players in which the outcome is determined not only by the skill of the players but also by random factors (shuffling cards, throwing a dice, etc.). Investigations of drawing room games (mainly card games) occupy an important position in the theory of games (cf. Games, theory of). Drawing room games are, on the one hand, an inexhaustible source of mathematically interesting models of games and, on the other, they are models of more serious conflicts — war, economics, etc. Poker, the first model of which was investigated by J. von Neumann in 1928 (see [1]), played a role at the beginning of the theory of games that is analogous to the role of dice games in probability theory. Later many special models of poker-type games were constructed and solved, and in the 1950s S. Karlin and R. Restrepo laid the foundations of the general theory of "poker" -type two-person zero-sum games (cf. Two-person zero-sum game). In models of them a player I (respectively, II) knows his card $ \xi $( respectively, $ \eta $), which is the realization of a random variable having a distribution $ F ( \xi ) $( respectively, $ G ( \eta ) $). The strategies (cf. Strategy (in game theory)) of the players I and II are vectors $ \phi ( \xi ) $ and $ \psi ( \eta ) $, and the pay-off function has the form

$$ K ( \phi , \psi ) = \ \int\limits \int\limits P [ \xi , \eta , \phi ( \xi ), \psi ( \eta )] \ dF ( \xi ) dG ( \eta ). $$

Karlin and Restrepo found sufficient effective methods for solving games like this (see [2]).

Many investigations have been devoted to bridge. Since in bridge two partners are, in fact, a single player, it is an example of a game without complete memory.

References

[1]	J. von Neumann, "Zur Theorie der Gesellschaftsspiele" Math. Ann , 100 (1928) pp. 295–320
[2]	S. Karlin, "Mathematical methods and theory in games, programming and economics" , Addison-Wesley (1959)
[3]	G.L. Thompson, "Signaling strategies in $n$-person games" , Contributions to the theory of games , 2 , Princeton Univ. Press (1953) pp. 267–277