# 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.^{[1]} 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.

## Contents

## Example[edit]

## Formal definitions[edit]

Large Poisson game - the collection , where:

- the average number of players in the game

- the set of all possible types for a player, (same for each player).

- the probability distribution over according to which the types are selected.

- the set of all possible pure choices, (same for each player, same for each type).

- the payoff (utility) function.

The total number of players, is a poisson distributed random variable:

Strategy -

Nash equilibrium -

## Simple probabilistic properties[edit]

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

Decomposition property for types - the number of players of the type is a Poisson random variable with mean .

Decomposition property for choices - the number of players who have chosen the choice is a Poisson random variable with mean

Pivotal probability ordering Every limit of the form 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 . This has a nice form: twice geometric mean minus arithmetic mean.

## Existence of equilibrium[edit]

Theorem 1. Nash equilibrium exists.

Theorem 2. Nash equilibrium in undominated strategies exists.

## Applications[edit]

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

## See also[edit]

## References[edit]

**^**Myerson, Roger (1998). "Population Uncertainty and Poisson games".*International Journal of Game Theory*.**27**(27): 375–392. CiteSeerX 10.1.1.21.9555. doi:10.1007/s001820050079.

- Myerson, Roger B. (2000). "Large Poisson Games".
*Journal of Economic Theory*.**94**(1): 7–45. doi:10.1006/jeth.1998.2453. - Myerson, Roger B. (1998). "Population Uncertainty and Poisson Games".
*International Journal of Game Theory*.**27**(3): 375–392. CiteSeerX 10.1.1.21.9555. doi:10.1007/s001820050079. - De Sinopoli, Francesco; Pimienta, Carlos G. (2009). "Undominated (and) perfect equilibria in Poisson games".
*Games and Economic Behavior*.**66**(2): 775–784. CiteSeerX 10.1.1.549.9282. doi:10.1016/j.geb.2008.09.029.