Game of chance

A multi-stage game played by a single player. A game of chance $G$ is defined as a system

$$G=\langle F,f_0\in F,\{\Gamma(f)\}_{f\in F},u(f)\rangle,$$

where $F$ is the set of fortunes (capitals), $f_0$ is the initial fortune of the player, $\Gamma(f)$ is a set of finitely-additive measures defined on all subsets of $F$, and $u(f)$ is a utility function (cf. Utility theory) of the player, defined on the set of his fortunes. The player chooses $\sigma_0\in\Gamma(f_0)$, and his fortune $f_1$ will have a distribution according to the measure $\sigma_0$. The player then chooses $\sigma_1(f_1)\in\Gamma(f_1)$ and obtains a corresponding $f_2$, etc. The sequence $\sigma=\{\sigma_0,\sigma_1,\dots\}$ is the strategy (cf. Strategy (in game theory)) of the player. If the player terminates the game at the moment $t$, his gain is defined as the mathematical expectation $\sigma$ of the function $u(f_t)$. 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 $f$, may acquire $k$ lottery tickets of price $c$, $k=1,\dots,[f/c]$. To each $k$ corresponds a probability measure on the set of all fortunes and, after drawing, the fortune of the player becomes $f_1$. If $f_1<c$, the game is over; if $f_1\geq c$, the player may get out of the game or may again buy lottery tickets of a number in between one and $[f_1/c]$, 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)