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In this paper, we consider a three-person silent marksmanship contest. Each of three players 1, 2 and 3 has a gun with exactly one bullet which may be fired at any time on (0, 11 aiming at his own target. The accuracy function Ai(X) for player i is strictly increasing and differentiable with Ai(O) = 0 and Ai(l) = 1. The first player hitting his target gets payoff +1 and other two players get payoff zero. For the game, we get a Nash equilibrium point and equilibrium payoff for each player. The form of the Nash equilibrium point differs whether At (x)/A 2 (x)A leX), A 2(x)/A t (X)Al(X) and A l (x)/A t (X)A2(X) decrease or one of these increases. Some examples are given to illustrate the results. An m-person marksmanship contest is a non-zero-sum game with the follow­ ing structure: There are m players (i.e., contestants) 1, 2, ••• , m and m targets, and each target is associated with one player. Every player has a gun with exactly one bullet which may be fired at any time in (0, 1) aiming at his own target. Starting at time 0, each player walks toward his own target; he will reach his target at time 1. If player i fires his bullet at time x, the probability that he hits his target is Ai(x). The function Ai(x) is called the accuracy function for player i and is strictly increasing and differenti­ able on (0, 1) with A. (0) = ° and A. (1) = 1. The accuracy functions are fixed 'l- 'l- and known beforehand to all players. As soon as one of the players hits his target, the contest ends and the first player hitting his target gets payoff +1 and other players get payoff zero. If none of the players hit their targets or some players hit their targets at the same time, then the payoff is zero for every player. In this situation, each player wishes to delay firing as long as possiLle to increase his accuracy, while at the same time he does not wish to delay so long that his opponents can precede him with effectiveness.

ON A THREE-PERSON SILENT MARKSMANSHIP CONTEST