# fixedpoint.jp

## Playing the Poisson racing of Chicken (2022-11-05)

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.