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Large poisson game: Wikis

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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.

Contents

Example

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.

See also

Poisson distribution

Referencies


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