# Poisson games

In game theory a Poisson game is a game with a random number of players, where the distribution of the number of players follows a Poisson random process. An extension of games of imperfect information, Poisson games have mostly seen application to models of voting.

A Poisson games consists of a random population of possible players of various types; every player in the game has some probability of being of some type. The type of the player affects their payoffs in the game; each type chooses an action and payoffs are determined.

## Formal definitions

Large Poisson game - the collection $(n,T,r,C,u)$ , where:
$n$ - the average number of players in the game
$T$ - the set of all possible types for a player, (same for each player).
$r$ - the probability distribution over $T$ according to which the types are selected.
$C$ - the set of all possible pure choices, (same for each player, same for each type).
$u$ - the payoff (utility) function.

The total number of players, $N$ is a poisson distributed random variable:
$P(N=k)=e^{-n}{\frac {n^{k}}{k!}}$ Strategy -

Nash equilibrium -

## Simple probabilistic properties

Environmental equivalence - from the perspective of each player the number of other players is a Poisson random variable with mean $n$ .

Decomposition property for types - the number of players of the type $t$ is a Poisson random variable with mean $nr(t)$ .

Decomposition property for choices - the number of players who have chosen the choice $c$ is a Poisson random variable with mean $...$ Pivotal probability ordering Every limit of the form $\lim _{n\to \infty }{\frac {P}{P}}$ is equal to 0 or to infinity. This means that all pivotal probability may be ordered from the most important to the least important.

Magnitude $2({\sqrt {xy}}-{\frac {x+y}{2}})$ . This has a nice form: twice geometric mean minus arithmetic mean.

## Existence of equilibrium

Theorem 1. Nash equilibrium exists.

Theorem 2. Nash equilibrium in undominated strategies exists.

## Applications

Mainly large poisson games are used as models for voting procedures.