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A multi-stage game played by a single player. A game of chance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043220/g0432201.png" /> is defined as a system
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A multi-stage game played by a single player. A game of chance $G$ is defined as a system
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043220/g0432202.png" /></td> </tr></table>
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$$G=\langle F,f_0\in F,\{\Gamma(f)\}_{f\in F},u(f)\rangle,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043220/g0432203.png" /> is the set of fortunes (capitals), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043220/g0432204.png" /> is the initial fortune of the player, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043220/g0432205.png" /> is a set of finitely-additive measures defined on all subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043220/g0432206.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043220/g0432207.png" /> is a utility function (cf. [[Utility theory|Utility theory]]) of the player, defined on the set of his fortunes. The player chooses <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043220/g0432208.png" />, and his fortune <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043220/g0432209.png" /> will have a distribution according to the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043220/g04322010.png" />. The player then chooses <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043220/g04322011.png" /> and obtains a corresponding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043220/g04322012.png" />, etc. The sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043220/g04322013.png" /> is the strategy (cf. [[Strategy (in game theory)|Strategy (in game theory)]]) of the player. If the player terminates the game at the moment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043220/g04322014.png" />, his gain is defined as the mathematical expectation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043220/g04322015.png" /> of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043220/g04322016.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043220/g04322017.png" />, may acquire <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043220/g04322018.png" /> lottery tickets of price <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043220/g04322019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043220/g04322020.png" />. To each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043220/g04322021.png" /> corresponds a probability measure on the set of all fortunes and, after drawing, the fortune of the player becomes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043220/g04322022.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043220/g04322023.png" />, the game is over; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043220/g04322024.png" />, the player may get out of the game or may again buy lottery tickets of a number in between one and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043220/g04322025.png" />, 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.
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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|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)|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|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.
 
The theory of games of chance is part of the general theory of controlled stochastic processes (cf. [[Controlled stochastic process|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.

Latest revision as of 00:29, 25 November 2018

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)
How to Cite This Entry:
Game of chance. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Game_of_chance&oldid=11730
This article was adapted from an original article by E.B. Yanovskaya (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article