This is an introduction to a statistical game, called the Poisson racing of Chicken, that analyzes the failing process of the game of Chicken.
Two players \(A\) and \(B\) compete while sharing the knowledge of a real parameter, called distance, \(D > 0\). Each player decides a positive real number, say \(\lambda_A\) and \(\lambda_B\). Then, sample \(x_A\) and \(x_B\) from independent Poisson variables of mean \(\lambda_A\) and of mean \(\lambda_B\), respectively. If \(x_A + x_B \leq D\), then \(A\) wins if \(x_A > x_B\); \(B\) wins if \(x_A < x_B\); ties otherwise. However, both lose if \(x_A + x_B > D\).
Because of symmetry, both players should have the same best strategy \(\lambda^*\) of choice.
\(\lambda^* \approx D/2\) is a correct, intuitive, yet doomed answer.
We have invented this game for pondering over a peaceful future. Imagine that the players are nuclear-weapon states which budget a stockpile of nuclear ammunition under no information of each other's but evaluating the opponent's amount of investment. It is also relevant to think about a repeated game in which the base game is the above racing of Chicken.