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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.

**Game theory** is the study of mathematical models of strategic interaction between rational decision-makers. It has applications in all fields of social science, as well as in logic and computer science. Originally, it addressed zero-sum games, in which one person's gains result in losses for the other participants. Today, game theory applies to a wide range of behavioral relations, and is now an umbrella term for the science of logical decision making in humans, animals, and computers.

In economics, **perfect information** is a feature of perfect competition. With perfect information in a market, all consumers and producers have perfect and instantaneous knowledge of all market prices, their own utility, and own cost functions.

- Example
- Formal definitions
- Simple probabilistic properties
- Existence of equilibrium
- Applications
- See also
- References

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.

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 -

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.

Theorem 1. Nash equilibrium exists.

Theorem 2. Nash equilibrium in undominated strategies exists.

Mainly large poisson games are used as models for voting procedures.

In game theory, the **Nash equilibrium**, named after the mathematician John Forbes Nash Jr., is a proposed solution of a non-cooperative game involving two or more players in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only their own strategy.

**Mechanism design** is a field in economics and game theory that takes an engineering approach to designing economic mechanisms or incentives, toward desired objectives, in strategic settings, where players act rationally. Because it starts at the end of the game, then goes backwards, it is also called **reverse game theory**. It has broad applications, from economics and politics to networked-systems.

In game theory, the **best response** is the strategy which produces the most favorable outcome for a player, taking other players' strategies as given. The concept of a best response is central to John Nash's best-known contribution, the Nash equilibrium, the point at which each player in a game has selected the best response to the other players' strategies.

In probability theory, a **compound Poisson distribution** is the probability distribution of the sum of a number of independent identically-distributed random variables, where the number of terms to be added is itself a Poisson-distributed variable. In the simplest cases, the result can be either a continuous or a discrete distribution.

**Matching pennies** is the name for a simple game used in game theory. It is played between two players, Even and Odd. Each player has a penny and must secretly turn the penny to heads or tails. The players then reveal their choices simultaneously. If the pennies match, then Even keeps both pennies, so wins one from Odd. If the pennies do not match Odd keeps both pennies, so receives one from Even.

In game theory, a player's **strategy** is any of the options which he or she chooses in a setting where the outcome depends *not only* on their own actions *but* on the actions of others. A player's strategy will determine the action which the player will take at any stage of the game.

An **extensive-form game** is a specification of a game in game theory, allowing for the explicit representation of a number of key aspects, like the sequencing of players' possible moves, their choices at every decision point, the information each player has about the other player's moves when they make a decision, and their payoffs for all possible game outcomes. Extensive-form games also allow for the representation of incomplete information in the form of chance events modeled as "moves by nature".

In game theory, a **Bayesian game** is a game in which the players have incomplete information about the other players. For example, a player may not know the exact payoff functions of the other players, but instead have beliefs about these payoff functions. These beliefs are represented by a probability distribution over the possible payoff functions.

In game theory, **folk theorems** are a class of theorems about possible Nash equilibrium payoff profiles in repeated games. The original Folk Theorem concerned the payoffs of all the Nash equilibria of an infinitely repeated game. This result was called the Folk Theorem because it was widely known among game theorists in the 1950s, even though no one had published it. Friedman's (1971) Theorem concerns the payoffs of certain subgame-perfect Nash equilibria (SPE) of an infinitely repeated game, and so strengthens the original Folk Theorem by using a stronger equilibrium concept subgame-perfect Nash equilibria rather than Nash equilibrium.

In game theory, a **repeated game** is an extensive form game that consists of a number of repetitions of some base game. The stage game is usually one of the well-studied 2-person games. Repeated games capture the idea that a player will have to take into account the impact of his or her current action on the future actions of other players; this impact is sometimes called his or her reputation. *Single stage game* or *single shot game* are names for non-repeated games.

In game theory, a **correlated equilibrium** is a solution concept that is more general than the well known Nash equilibrium. It was first discussed by mathematician Robert Aumann in 1974. The idea is that each player chooses their action according to their observation of the value of the same public signal. A strategy assigns an action to every possible observation a player can make. If no player would want to deviate from the recommended strategy, the distribution is called a correlated equilibrium.

**Quantal response equilibrium** (**QRE**) is a solution concept in game theory. First introduced by Richard McKelvey and Thomas Palfrey, it provides an equilibrium notion with bounded rationality. QRE is not an equilibrium refinement, and it can give significantly different results from Nash equilibrium. QRE is only defined for games with discrete strategies, although there are continuous-strategy analogues.

**Risk dominance** and **payoff dominance** are two related refinements of the Nash equilibrium (NE) solution concept in game theory, defined by John Harsanyi and Reinhard Selten. A Nash equilibrium is considered **payoff dominant** if it is Pareto superior to all other Nash equilibria in the game. When faced with a choice among equilibria, all players would agree on the payoff dominant equilibrium since it offers to each player at least as much payoff as the other Nash equilibria. Conversely, a Nash equilibrium is considered **risk dominant** if it has the largest basin of attraction. This implies that the more uncertainty players have about the actions of the other player(s), the more likely they will choose the strategy corresponding to it.

**Proper equilibrium** is a refinement of Nash Equilibrium due to Roger B. Myerson. Proper equilibrium further refines Reinhard Selten's notion of a trembling hand perfect equilibrium by assuming that more costly trembles are made with significantly smaller probability than less costly ones.

In game theory, an **epsilon-equilibrium**, or near-Nash equilibrium, is a strategy profile that approximately satisfies the condition of Nash equilibrium. In a Nash equilibrium, no player has an incentive to change his behavior. In an approximate Nash equilibrium, this requirement is weakened to allow the possibility that a player may have a small incentive to do something different. This may still be considered an adequate solution concept, assuming for example status quo bias. This solution concept may be preferred to Nash equilibrium due to being easier to compute, or alternatively due to the possibility that in games of more than 2 players, the probabilities involved in an exact Nash equilibrium need not be rational numbers.

In game theory, a **stochastic game**, introduced by Lloyd Shapley in the early 1950s, is a dynamic game with **probabilistic transitions** played by one or more players. The game is played in a sequence of stages. At the beginning of each stage the game is in some **state**. The players select actions and each player receives a **payoff** that depends on the current state and the chosen actions. The game then moves to a new random state whose distribution depends on the previous state and the actions chosen by the players. The procedure is repeated at the new state and play continues for a finite or infinite number of stages. The total payoff to a player is often taken to be the discounted sum of the stage payoffs or the limit inferior of the averages of the stage payoffs.

In probability theory and statistics, the **Poisson distribution**, named after French mathematician Siméon Denis Poisson, is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant rate and independently of the time since the last event. The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.

In probability theory and statistics, the **Poisson binomial distribution** is the discrete probability distribution of a sum of independent Bernoulli trials that are not necessarily identically distributed. The concept is named after Siméon Denis Poisson.

The **Borel distribution** is a discrete probability distribution, arising in contexts including branching processes and queueing theory. It is named after the French mathematician Émile Borel.

**M equilibrium** is a set valued solution concept in game theory that relaxes the rational choice assumptions of perfect maximization and perfect beliefs. The concept can be applied to any normal-form game with finite and discrete strategies. M equilibrium was first introduced by Jacob K. Goeree and Philippos Louis.

- ↑ 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.

- 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. CiteSeerX 10.1.1.21.9555 . 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. CiteSeerX 10.1.1.549.9282 . doi:10.1016/j.geb.2008.09.029.

In computing, a **Digital Object Identifier** or **DOI** is a persistent identifier or handle used to identify objects uniquely, standardized by the International Organization for Standardization (ISO). An implementation of the Handle System, DOIs are in wide use mainly to identify academic, professional, and government information, such as journal articles, research reports and data sets, and official publications though they also have been used to identify other types of information resources, such as commercial videos.

**CiteSeer ^{x}** is a public search engine and digital library for scientific and academic papers, primarily in the fields of computer and information science. CiteSeer holds a United States patent # 6289342, titled "

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