Equilibrium in $n$-player competitive game of timing

Each player in the game of timing has to decide his time to shoot under the condition that he is not informed of the shooting times of his rivals. That is, we deal with silent games of timing. Games of timing are used to model auctions, games of war of attrition, competitive predictions of a random variable, etc. Using the symmetry of the model we derive the equation to determine the equilibrium of the game.