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.^[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.

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:
$P(N=k)=e^{{-n}}{\frac {n^{{k}}}{k!}}$

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 $n$ .

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.

References

↑ Myerson, Roger (1998). "Population Uncertainty and Poisson games". International Journal of Game Theory (27): 375–392.

Myerson, Roger B. (2000). "Large Poisson Games". Journal of Economic Theory. 94 (1): 7–45. doi:10.1006/jeth.1998.2453.
Myerson, Roger B. (1998). "Population Uncertainty and Poisson Games". International Journal of Game Theory. 27 (3): 375–392. doi:10.1007/s001820050079.
De Sinopoli, Francesco; Pimienta, Carlos G. (2009). "Undominated (and) perfect equilibria in Poisson games". Games and Economic Behavior. 66 (2): 775–784. doi:10.1016/j.geb.2008.09.029.

Topics in game theory

Definitions	Normal-form game Extensive-form game Escalation of commitment Graphical game Cooperative game Succinct game Information set Hierarchy of beliefs Preference

Equilibrium concepts	Nash equilibrium Subgame perfection Mertens-stable equilibrium Bayesian Nash equilibrium Perfect Bayesian equilibrium Trembling hand Proper equilibrium Epsilon-equilibrium Correlated equilibrium Sequential equilibrium Quasi-perfect equilibrium Evolutionarily stable strategy Risk dominance Core Shapley value Pareto efficiency Gibbs equilibrium Quantal response equilibrium Self-confirming equilibrium Strong Nash equilibrium Markov perfect equilibrium

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

Classes of games	Symmetric game Perfect information Simultaneous game Sequential game Repeated game Signaling game Screening game Cheap talk Zero-sum game Mechanism design Bargaining problem Stochastic game n-player game Large Poisson game Nontransitive game Global games Strictly determined game Potential game

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 Rock-paper-scissors Pirate game Dictator game Public goods game Blotto games War of attrition El Farol Bar problem Fair division Fair cake-cutting Cournot game Deadlock Diner's dilemma Guess 2/3 of the average Kuhn poker Nash bargaining game Prisoners and hats puzzle Trust game Princess and monster game Monty Hall problem Rendezvous problem

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

Key figures	Albert W. Tucker Amos Tversky Ariel Rubinstein Daniel Kahneman David K. Levine David M. Kreps Donald B. Gillies Drew Fudenberg Eric Maskin Harold W. Kuhn Herbert Simon Hervé Moulin Jean Tirole Jean-François Mertens John Harsanyi John Maynard Smith John Nash John von Neumann Kenneth Arrow Kenneth Binmore Leonid Hurwicz Lloyd Shapley Melvin Dresher Merrill M. Flood Oskar Morgenstern Paul Milgrom Peyton Young Reinhard Selten Robert Aumann Robert B. Wilson Roger Myerson Samuel Bowles Thomas Schelling William Vickrey

See also	All-pay auction Alpha–beta pruning Bertrand paradox Bounded rationality Combinatorial game theory Confrontation analysis Coopetition List of game theorists List of games in game theory No-win situation Topological game Tragedy of the commons Tyranny of small decisions

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