<P> Games, as studied by economists and real - world game players, are generally finished in finitely many moves . Pure mathematicians are not so constrained, and set theorists in particular study games that last for infinitely many moves, with the winner (or other payoff) not known until after all those moves are completed . </P> <P> The focus of attention is usually not so much on the best way to play such a game, but whether one player has a winning strategy . (It can be proven, using the axiom of choice, that there are games--even with perfect information and where the only outcomes are "win" or "lose"--for which neither player has a winning strategy .) The existence of such strategies, for cleverly designed games, has important consequences in descriptive set theory . </P> <P> Much of game theory is concerned with finite, discrete games, that have a finite number of players, moves, events, outcomes, etc . Many concepts can be extended, however . Continuous games allow players to choose a strategy from a continuous strategy set . For instance, Cournot competition is typically modeled with players' strategies being any non-negative quantities, including fractional quantities . </P> <P> Differential games such as the continuous pursuit and evasion game are continuous games where the evolution of the players' state variables is governed by differential equations . The problem of finding an optimal strategy in a differential game is closely related to the optimal control theory . In particular, there are two types of strategies: the open - loop strategies are found using the Pontryagin maximum principle while the closed - loop strategies are found using Bellman's Dynamic Programming method . </P>

What are the different aspects of game theory