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. In the terms of games of timing can be formulated the auctions, games of war of attrition, competitive prediction of a random variable, etc. Using the symmetry we derive the equation to determine the equilibrium in these games.