# Large poisson game: Wikis

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# Encyclopedia

In game theory the large poisson game is a game with a random number of players. More exactly, N, the number of players is a Poisson random variable. The type of each player is selected randomly independently of other players types from a given set T. Each player selects an action and then the 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:
$P(N=k)=e^{-n}\frac{n^{k}}{k!}$

Stategy -

Nash equilibrium -

## Simple probabilistic properties

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

Poisson distribution