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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034250/d0342501.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034250/d0342502.png" /> be the set of players. To each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034250/d0342503.png" /> corresponds a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034250/d0342504.png" /> of elementary strategies of player <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034250/d0342505.png" /> at this point, and hence, also, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034250/d0342506.png" /> of elementary situations at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034250/d0342507.png" />. The periodic distribution functions
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<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/d/d034/d034250/d0342508.png" /></td> </tr></table>
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representing the law of motion of the controlled point, which is known to all players, is defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034250/d0342509.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034250/d03425010.png" /> is fixed, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034250/d03425011.png" /> is measurable with respect to all the remaining arguments. A sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034250/d03425012.png" /> of successive states and elementary situations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034250/d03425013.png" /> 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) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034250/d03425014.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034250/d03425015.png" />), and let each player <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034250/d03425016.png" /> choose his elementary strategy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034250/d03425017.png" /> so that the elementary situation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034250/d03425018.png" /> arises; the game then continues, at random, in accordance with the distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034250/d03425019.png" />, into the state <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034250/d03425020.png" />. In each play <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034250/d03425021.png" /> the pay-off <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034250/d03425022.png" /> of player <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034250/d03425023.png" /> is defined. If the set of all plays is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034250/d03425024.png" />, the dynamic game is specified by the system
+
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
  
<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/d/d034/d034250/d03425025.png" /></td> </tr></table>
+
$$
 +
F ( x _ {k} \mid  x _ {1} , s ^ {( x _ {1} ) } \dots x _ {k - 1 }  , s ^ {( x _ {k-} 1 ) } ) ,\  x _ {i} \in X ,\  s ^
 +
{( x _ {i} ) } \in S ^ {( x _ {i} ) } ,
 +
$$
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034250/d03425026.png" /> of player <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034250/d03425027.png" /> is a selection of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034250/d03425028.png" /> which put the opening ending in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034250/d03425029.png" /> into correspondence with the elementary strategy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034250/d03425030.png" />. Dynamic games in which the preceding opening is only known partly to the players — e.g. games with "information lag" — have also been studied.
+
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
  
For a game to be specified, each situation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034250/d03425031.png" /> must induce a probability measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034250/d03425032.png" /> on the set of all plays, and the mathematical expectation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034250/d03425033.png" /> with respect to the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034250/d03425034.png" /> must exist. This mathematical expectation is also the pay-off of player <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034250/d03425035.png" /> in situation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034250/d03425036.png" />.
+
$$
 +
\Gamma  = < I , X , \{ S _ {i}  ^ {(} x) \} _ {i \in I ,
 +
x \in X }  , F , \{ h _ {i} ( P) \} _ {i \in I , P \in
 +
\mathfrak P }  > .
 +
$$
  
In general, the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034250/d03425037.png" /> are arbitrary, but the most frequently studied dynamic games are those with terminal pay-off (the game is terminated as soon as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034250/d03425038.png" /> appears in a terminal set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034250/d03425039.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034250/d03425040.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034250/d03425041.png" /> is the last situation in the game), and those with integral pay-off (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034250/d03425042.png" />).
+
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.
  
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, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034250/d03425043.png" />, 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]]).
+
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|Game of survival]]).
 
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|Game of survival]]).

Revision as of 19:36, 5 June 2020


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).

References

[1] 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
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