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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.

1 Example
2 Formal definitions
3 Simple probabilistic properties
4 Existence of equilibrium
5 Applications
6 See also
7 Referencies

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 $n r (t)$

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.

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.

Referencies

Roger B. Myerson (2000). "Large Poisson Games". Journal of Economic theory ? (94): 7–45. http://ideas.repec.org/p/nwu/cmsems/1189.html.
Roger B. Myerson (?). "Population Uncertainty and Poisson Games". ? ? (?). http://ideas.repec.org/p/nwu/cmsems/1102.html.
Francesco de Sinopoli, Carlos G. Pimienta (?). "Undominated perfect equilibria in Poisson games". ? ? (?): ?. http://ideas.repec.org/p/cte/werepe/we073117.html.

Topics in game theory

Definitions	Normal-form game · Extensive-form game · Cooperative game · Information set · Preference

Equilibrium concepts	Nash equilibrium · Subgame perfection · Bayesian-Nash · Perfect Bayesian · Trembling hand · Proper equilibrium · Epsilon-equilibrium · Correlated equilibrium · Sequential equilibrium · Quasi-perfect equilibrium · Evolutionarily stable strategy · Risk dominance · Pareto efficiency · Quantal response equilibrium · Self-confirming equilibrium

Strategies	Dominant strategies · Pure strategy · Mixed strategy · Tit for tat · Grim trigger · Collusion · Backward induction

Classes of games	Symmetric game · Perfect information · Simultaneous game · Sequential game · Repeated game · Signaling game · Cheap talk · Zero-sum game · Mechanism design · Bargaining problem · Stochastic game · Large poisson game · Nontransitive game · Global games

Games	Prisoner's dilemma · Traveler's dilemma · Coordination game · Chicken · Centipede game · Volunteer's dilemma · Dollar auction · Battle of the sexes · Stag hunt · Matching pennies · Ultimatum game · Minority game · Rock-paper-scissors · Pirate game · Dictator game · Public goods game · Blotto games · War of attrition · El Farol Bar problem · Cake cutting · Cournot game · Deadlock · Diner's dilemma · Guess 2/3 of the average · Kuhn poker · Nash bargaining game · Screening game · Trust game · Princess and monster game

Theorems	Minimax theorem · Purification theorem · Folk theorem · Revelation principle · Arrow's impossibility theorem

See also	Tragedy of the commons · All-pay auction · List of games in game theory

Categories: Game theory

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