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A variant of a [[Positional game|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 _ {k-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|Differential games]]).

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

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  N.N. Vorob'ev,   "The present state of the theory of games"  ''Russian Math. Surveys'' , '''25''' :  2  (1970)  pp. 77–136  ''Uspekhi Mat. Nauk'' , '''25''' :  2  (1970)  pp. 81–140</TD></TR>
</table>