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. An extension of games of imperfect information, Poisson games have mostly seen application to models of voting.
Contents
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.
Formal definitions
Large Poisson game - the collection
The total number of players,
Strategy -
Nash equilibrium -
Simple probabilistic properties
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
Decomposition property for choices - the number of players who have chosen the choice
Pivotal probability ordering Every limit of the form
Magnitude
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.