Positional Games

a class of noncoalitional games in which decision-making by the players (that is, selection of strategies) is considered as a multistep or even continuous process. In other words, in a positional game, the player in the course of decision-making passes through a sequence of states, in each of which he must make some partial decision. The strategies of the players can therefore be considered as functions which establish correspondence between each informational state of the player, that is, what the player knows of the status of the game at a given moment—and the choice of a possible alternative in the state. (See discussion of chess in GAMES, THEORY OF.)

A player’s moves from one informational state to another may be accompanied by either acquisition or loss of information about previous informational states (both the player’s own states and those of the other players) and of the alternatives chosen in these states. The complete description of this process is said to be the player’s information in the positional game. The player’s information of himself (that is, of his former states and alternatives) is called his memory. The particular features of the information and memory of players in a game make it possible to simplify the description of its equilibria and to limit the search regions. Thus, if a position game with a finite number of informational states is a game with perfect information (that is, at any moment each player knows all the previous informational states and the choices made in them), it will contain equilibria of pure strategies and will not require recourse to mixed strategies.

These observations lose meaning if we consider position games with an infinite set of informational states, where mathematically highly complex, paradoxical phenomena are observed. An example of such a position game is one in which two players by turns call out the decimals a₁, a₂, a₃, a₄,. … If the resulting number 0. a₁, a₂, a₃, a₄, … belongs to a given set, then the first player gains a point; in the opposite case, the second player gains a point. If some player in a position game with a finite number of informational states has perfect memory (that is, he knows all his previous informational states and the choices made), he can limit his behavior strategically without harming his position, such that alternatives in different informational states can be chosen randomly, but stochastically independently.

Positional games (with a continuous set of informational states) include what are known as differential games. Dynamic programming may be thought of as theory of one class of one-person positional games. Problems that occur in multistep (sequential) statistical decisions can be naturally interpreted as position games. Since a player in a position game may acquire and lose information, game theory involves information theory.