# Dynamic game

A variant of a positional game distinguished by the fact that in such a game the players control the "motion of a point" in the state space $X$. Let $I = \{ i \}$ be the set of players. To each point $x \in X$ corresponds a set $S _ {i} ^ {(} x)$ of elementary strategies of player $i \in I$ at this point, and hence, also, the set $S ^ {(} x) = \prod _ {i} S _ {i} ^ {(} x)$ of elementary situations at $x$. The periodic distribution functions

$$F ( x _ {k} \mid x _ {1} , s ^ {( x _ {1} ) } \dots x _ {k - 1 } , s ^ {( x _ {k-} 1 ) } ) ,\ x _ {i} \in X ,\ s ^ {( x _ {i} ) } \in S ^ {( x _ {i} ) } ,$$

representing the law of motion of the controlled point, which is known to all players, is defined on $X$. If $x _ {k}$ is fixed, the function $F$ is measurable with respect to all the remaining arguments. A sequence $P$ of successive states and elementary situations $x _ {1} , s ^ {( x _ {1} ) } \dots x _ {k} , s ^ {( x _ {k} ) } \dots$ is a play of a general dynamic game. It is inductively defined as follows: Let there be given a segment of the play (an opening) $x _ {1} , s ^ {( x _ {1} ) } \dots x _ {x-} 1$( $k \geq 2$), and let each player $i$ choose his elementary strategy $s _ {i} ^ {( x _ {k-} 1 ) } \in S _ {i} ^ {( x _ {k-} 1 ) }$ so that the elementary situation $s ^ {( x _ {k-} 1 ) }$ arises; the game then continues, at random, in accordance with the distribution $F ( \cdot \mid x _ {1} , s ^ {( x _ {1} ) } \dots x _ {k-} 1 , s ^ {( x _ {k-} 1 ) } )$, into the state $x _ {k}$. In each play $P$ the pay-off $h _ {i} ( P)$ of player $i$ is defined. If the set of all plays is denoted by $\mathfrak P$, the dynamic game is specified by the system

$$\Gamma = < I , X , \{ S _ {i} ^ {(} x) \} _ {i \in I , x \in X } , F , \{ h _ {i} ( P) \} _ {i \in I , P \in \mathfrak P } > .$$

In a dynamic game it is usually assumed that, at the successive moments of selection of an elementary strategy, the players know the preceding opening. In such a case a pure strategy $s _ {i}$ of player $i$ is a selection of functions $s _ {i} ^ {( x) } ( x _ {1} , s ^ {( x _ {1} ) } \dots s ^ {( x _ {k-} 1 ) } , x )$ which put the opening ending in $x$ into correspondence with the elementary strategy $s _ {i} ^ {(} x) \in S _ {i} ^ {(} x)$. Dynamic games in which the preceding opening is only known partly to the players — e.g. games with "information lag" — have also been studied.

For a game to be specified, each situation $s = \{ s _ {i} \}$ must induce a probability measure $\mu _ {s}$ on the set of all plays, and the mathematical expectation ${\mathsf E} h _ {i} ( P)$ with respect to the measure $\mu _ {s}$ must exist. This mathematical expectation is also the pay-off of player $i$ in situation $s$.

In general, the functions $h _ {i} ( P)$ are arbitrary, but the most frequently studied dynamic games are those with terminal pay-off (the game is terminated as soon as $x _ {k}$ appears in a terminal set $X ^ {T} \subset X$, and $h _ {i} ( P) = h _ {i} ( x _ {k} )$ where $x _ {k}$ is the last situation in the game), and those with integral pay-off ( $h _ {i} ( P) = \sum _ {k= 1 } ^ \infty h _ {i} ( x _ {k} , s ^ {( x _ {k} ) })$).

Dynamic games are regarded as the game-like variant of a problem of optimal control with discrete time. It is in fact reduced to such a problem if the number of players is one. If, in a dynamic game, $X \subset \mathbf R ^ {n}$, continuous time is substituted for discrete time and the random factors are eliminated, a differential game is obtained, which may thus be regarded as a variant of a dynamic game (see also Differential games).

Stochastic games, recursive games and survival games are special classes of dynamic games (cf. also Stochastic game; Recursive game; Game of survival).

How to Cite This Entry:
Dynamic game. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dynamic_game&oldid=46783
This article was adapted from an original article by V.K. Domanskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article