The mixed temporal game 1×1 with uncertain actions

Abstract

Author's introduction: game solved in the present paper belongs to a class of temporal games whose general model was given by S. Karlin [Mathematical methods and theory in games, programming and economics, Addison-Wesley, Reading, Mass., 1959; MR0111634]. The rules are as follows. There are two opposing players, A and B. Each player has a certain number of actions which he can take in the time interval [0,1]. Neither player knows the number of actions he or his opposer has. Each player knows only the distribution of the number of actions for him and his opponent. More precisely, let k1 denote the number of actions of player A and k2 the number of actions of player B, and let k1 and k2 be independent random variables. The distributions of these random variables are as follows: P(k1=1)=p, P(k1=0)=1−p, P(k2=1)=q, P(k2=0)=1−q, p,q∈(0,1]. The opponent knows only both distributions. Thus player A [player B] knows beforehand that he as well as his opponent has zero or one action with the probability given above. If player A has an action then it is silent, i.e., his opponent does not know the moment when the action is taken. If player B has an action it is noisy, i.e. his opponent knows the moment when the action is taken. As a result of the action the player can win or, equivalently, his opponent can lose. A win can be achieved according to the probability given above. If player A [player B] takes an action at time t, then the probability of winning is equal to P(t) [to Q(t)]. Functions P(t) and Q(t) are called payoff functions. These functions satisfy the following conditions: (1) P(t) and Q(t) are differentiable and P′(t)>0 and Q′(t)>0 for t∈(0,1); (2) P(0)=Q(0)=0, P(1)=Q(1)=1. Both functions are known to the players before the game begins. The game ends when one player wins or when neither wins at time t=1. If only player A wins he obtains the payoff +1 from his opponent, and likewise if only B wins he obtains the payoff +1. In the remaining cases the payoff is equal to zero. Y. Teraoka solved a temporal game 1×1 with uncertain actions [Rep. Himeji Inst. Tech. 28 (1975), 1–8; RŽMat 1976:8 V662]. If in the game, played according to the rules given above, we put p=q=1, then this type of game is a particular case of the game solved by A. Styszynski [Zastos. Mat. 14 (1974), 205–225; MR0351487]. Here we present a solution to a more generalized game in which the first player has n actions. The game defined above can be interpreted as a duel. In the course of an action shots are fired and a win is achieved when the opponent is hit. The payoff function is then called an accuracy function. The duelists do not know beforehand whether the weapons are loaded or accurate. They know only the probability of drawing a loaded weapon. We shall use this interpretation in later sections of the paper. This game can also be viewed as an advertising Two enterprises are engaged in an advertising war to win a contract. Here an action can be viewed as setting in motion the advertising machine of the player's company. A win is achieved when the contract is awarded to the player's company as a result of his action. Before the campaign the players do not know whether they will receive funds from their central offices to engage in the advertising war.