Game of chance
A multi-stage game played by a single player. A game of chance is defined as a system
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where is the set of fortunes (capitals),
is the initial fortune of the player,
is a set of finitely-additive measures defined on all subsets of
, and
is a utility function (cf. Utility theory) of the player, defined on the set of his fortunes. The player chooses
, and his fortune
will have a distribution according to the measure
. The player then chooses
and obtains a corresponding
, etc. The sequence
is the strategy (cf. Strategy (in game theory)) of the player. If the player terminates the game at the moment
, his gain is defined as the mathematical expectation
of the function
. The aim of the player is to maximize his utility function. The simplest example of a game of chance is a lottery. The player, who possesses an initial fortune
, may acquire
lottery tickets of price
,
. To each
corresponds a probability measure on the set of all fortunes and, after drawing, the fortune of the player becomes
. If
, the game is over; if
, the player may get out of the game or may again buy lottery tickets of a number in between one and
, etc. His utility function may be, for example, the mathematical expectation of the fortune or the probability of gaining not less than a certain amount.
The theory of games of chance is part of the general theory of controlled stochastic processes (cf. Controlled stochastic process). A game of chance may be participated in by several persons, but from the theoretical point of view it is a single-player game, since the gain of a player does not depend on the strategy of his partners.
References
[1] | L.E. Dubins, L.J. Savage, "How to gamble if you must: inequalities for stochastic processes" , McGraw-Hill (1965) |
Game of chance. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Game_of_chance&oldid=43488