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Construction by Jean-François Mertens and Zamir implementing with John Harsanyi's proposal to model games with incomplete information by supposing that each player is characterized by a privately known type that describes his feasible strategies and payoffs as well as a probability distribution over other players' types.^{ [1] }

**Jean-François Mertens** was a Belgian game theorist and mathematical economist.

**John Charles Harsanyi** was a Hungarian-American economist.

Such probability distribution at the first level can be interpreted as a low level belief of a player. One level up the probability on the belief of other players is interpreted as beliefs on beliefs. A recursive universal construct is built—in which player have beliefs on their beliefs at different level—this construct is called the hierarchy of beliefs.

The result is a universal space of types in which, subject to specified consistency conditions, each type corresponds to the infinite hierarchy of his probabilistic beliefs about others' probabilistic beliefs. They also showed that any subspace can be approximated arbitrarily closely by a finite subspace.

Another popular example of the usage of the construction is the Prisoners and hats puzzle. And so is Robert Aumann's construction of common knowledge.^{ [2] }

**Robert John Aumann** is an Israeli-American mathematician, and a member of the United States National Academy of Sciences. He is a professor at the Center for the Study of Rationality in the Hebrew University of Jerusalem in Israel. He also holds a visiting position at Stony Brook University, and is one of the founding members of the Stony Brook Center for Game Theory.

**Common knowledge** is a special kind of knowledge for a group of agents. There is *common knowledge* of *p* in a group of agents *G* when all the agents in *G* know *p*, they all know that they know *p*, they all know that they all know that they know *p*, and so on *ad infinitum*.

A **Bayesian network**, **Bayes network**, **belief network**, **decision network**, **Bayes(ian) model** or **probabilistic directed acyclic graphical model** is a probabilistic graphical model that represents a set of variables and their conditional dependencies via a directed acyclic graph (DAG). Bayesian networks are ideal for taking an event that occurred and predicting the likelihood that any one of several possible known causes was the contributing factor. For example, a Bayesian network could represent the probabilistic relationships between diseases and symptoms. Given symptoms, the network can be used to compute the probabilities of the presence of various diseases.

**Ray Solomonoff** was the inventor of algorithmic probability, his General Theory of Inductive Inference, and was a founder of algorithmic information theory. He was an originator of the branch of artificial intelligence based on machine learning, prediction and probability. He circulated the first report on non-semantic machine learning in 1956.

**Decision theory** is the study of the reasoning underlying an agent's choices against nature. Decision theory is where results depends on another and can be broken into two branches: normative decision theory, which gives advice on how to make the best decisions given a set of uncertain beliefs and a set of values, and descriptive decision theory which analyzes how existing, possibly irrational agents actually make decisions.

In game theory, a **signaling game** is a simple type of a dynamic Bayesian game.

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.

In game theory, a **solution concept** is a formal rule for predicting how a game will be played. These predictions are called "solutions", and describe which strategies will be adopted by players and, therefore, the result of the game. The most commonly used solution concepts are equilibrium concepts, most famously Nash equilibrium.

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 economics and game theory, **complete information** is an economic situation or game in which knowledge about other market participants or players is available to all participants. The utility functions, payoffs, strategies and "types" of players are thus common knowledge.

In game theory, a **Perfect Bayesian Equilibrium** (PBE) is an equilibrium concept relevant for dynamic games with incomplete information. A PBE is a refinement of both Bayesian Nash equilibrium (BNE) and subgame perfect equilibrium (SPE). A PBE has two components - *strategies* and *beliefs*:

In game theory, a **Bayesian game** is a game in which the players have incomplete information on the other players, but, they have beliefs with known probability distribution.

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, the **purification theorem** was contributed by Nobel laureate John Harsanyi in 1973. The theorem aims to justify a puzzling aspect of mixed strategy Nash equilibria: that each player is wholly indifferent amongst each of the actions he puts non-zero weight on, yet he mixes them so as to make every other player also indifferent.

In statistics, probability theory, and information theory, a **statistical distance** quantifies the distance between two statistical objects, which can be two random variables, or two probability distributions or samples, or the distance can be between an individual sample point and a population or a wider sample of points.

**Quasi-perfect equilibrium** is a refinement of Nash Equilibrium for extensive form games due to Eric van Damme.

**Auction theory** is an applied branch of economics which deals with how people act in auction markets and researches the properties of auction markets. There are many possible designs for an auction and typical issues studied by auction theorists include the efficiency of a given auction design, optimal and equilibrium bidding strategies, and revenue comparison. Auction theory is also used as a tool to inform the design of real-world auctions; most notably auctions for the privatization of public-sector companies or the sale of licenses for use of the electromagnetic spectrum.

**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.

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.

**Hierarchical temporal memory** (**HTM**) is a biologically constrained theory of intelligence, originally described in the 2004 book *On Intelligence* by Jeff Hawkins with Sandra Blakeslee. HTM is based on neuroscience and the physiology and interaction of pyramidal neurons in the neocortex of the mammalian brain.

**Mertens stability** is a solution concept used to predict the outcome of a non-cooperative game. A tentative definition of stability was proposed by Elon Kohlberg and Jean-François Mertens for games with finite numbers of players and strategies. Later, Mertens proposed a stronger definition that was elaborated further by Srihari Govindan and Mertens. This solution concept is now called Mertens stability, or just stability.

- ↑ Jean -François Mertens and Shmuel Zamir (1985-03-01). "Formulation of Bayesian analysis for games with incomplete information".
*INTERNATIONAL JOURNAL OF GAME THEORY*.**14**(1): 1–29. doi:10.1007/BF01770224. - ↑ Herbert Gintis (16 March 2009).
*The bounds of reason: game theory and the unification of the behavioral sciences*. Princeton University Press. p. 158. ISBN 978-0-691-14052-0 . Retrieved 3 March 2012.

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