Difference between revisions of "Dynamic game"
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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 . | |
+ | 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 | + | 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, | + | 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 |
Dynamic game. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dynamic_game&oldid=18028