Poisson games
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. 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 formula_1, where: formula_2 - the average number of players in the game formula_3 - the set of all possible types for a player, (same for each player). formula_4 - the probability distribution over formula_3 according to which the types are selected. formula_6 - the set of all possible pure choices, (same for each player, same for each type). formula_7 - the payoff (utility) function. The total number of players, formula_8 is a poisson distributed random variable: formula_9 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 formula_2 . Decomposition property for types - the number of players of the type formula_11 is a Poisson random variable with mean formula_12. Decomposition property for choices - the number of players who have chosen the choice formula_13 is a Poisson random variable with mean formula_14 Pivotal probability ordering Every limit of the form formula_15 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 formula_16. 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.