Poisson games

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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.

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]

  1. ^ 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.