A short solution to the many-player silent duel with arbitrary consolation prize

Abstract

The classical constant-sum ‘silent duel’ game had two antagonistic marksmen walking towards each other. A more friendly formulation has two equally skilled marksmen approaching targets at which they may silently fire at distances of their own choice. The winner, who gets a unit prize, is the marksman who hits his target at the greatest distance; if both miss, they share the prize (each gets a ‘consolation prize’ of one half). In another formulation, if they both miss they each get zero. More generally we can consider more than two marksmen and an arbitrary consolation prize. This non-constant sum game may be interpreted as a research tournament where the entrant who successfully solves the hardest problem wins the prize. We consider only the ‘symmetric’ case where all players are identical (having the same probability of missing at a given distance), and for this case we give the first complete solution to the many-player problem with arbitrary consolation prize. Moreover our theorem includes both the zero and non-zero-sum cases (by taking particular values for the consolation prize), and has a relatively simple proof.