# Game involving the choice of the moment of time

A non-cooperative game in which the choice of a strategy by each player represents the choice of a time (from some fixed interval) to perform a specific action and for which the pay-off function of the players is continuous on the set of situations (with the exception of situations in which there are coincidences among the moments selected by the players) and is monotone increasing on the domain of continuity with respect to the strategies of the corresponding player. The most studied class of games of timing are the two-person zero-sum games (cf. Two-person zero-sum game) in which each player chooses a moment of time, that is, games on the unit square (cf. Game on the unit square) with discontinuous pay-off functions on the diagonal of the square, increasing in the first variable and decreasing in the second. In such games there exist a value for the game and optimal strategies for both players. Under certain additional assumptions, the solution of such games can be reduced to the solution of integral equations. Another class of games of timing are duels (cf. Duel), that is, two-person zero-sum games in which the actions of the players are directed towards the destruction of their opponents, so that in duels there are four possible different outcomes. For duels there are analytic methods for finding optimal (or $\epsilon$-optimal) strategies of the players. The theory of duels has military as well as economic applications (competition for markets, advertising campaigns, etc.).