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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.

[edit] Example

[edit] 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 -

[edit] 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 $n r (t)$

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.