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Game theory is the branch of mathematics in which games are studied: that is, models describing human behaviour. This is a glossary of some terms of the subject.

- Real numbers
- .
- The set of
**players** - .
- Strategy space
- , where
- Player i's strategy space
- is the space of all possible ways in which player
**i**can play the game. - A strategy for player
**i**

is an element of .

- Complements

an element of , is a tuple of strategies for all players other than **i**.

- Outcome space
- is in most textbooks identical to -
- Payoffs
- , describing how much gain (money, pleasure, etc.) the players are allocated by the end of the game.

A game in normal form is a function:

Given the *tuple* of *strategies* chosen by the players, one is given an allocation of *payments* (given as real numbers).

A further generalization can be achieved by splitting the **game** into a composition of two functions:

the **outcome function** of the game (some authors call this function "the game form"), and:

the allocation of **payoffs** (or **preferences**) to players, for each outcome of the game.

This is given by a tree, where at each vertex of the *tree* a different player has the choice of choosing an edge. The *outcome* set of an extensive form game is usually the set of tree leaves.

A game in which players are allowed to form coalitions (and to enforce coalitionary discipline). A cooperative game is given by stating a *value* for every coalition:

It is always assumed that the empty coalition gains nil. *Solution concepts* for cooperative games usually assume that the players are forming the *grand coalition*, whose value is then divided among the players to give an allocation.

A Simple game is a simplified form of a cooperative game, where the possible gain is assumed to be either '0' or '1'. A simple game is couple (**N**, **W**), where **W** is the list of "winning" **coalitions**, capable of gaining the loot ('1'), and **N** is the set of players.

- Acceptable game
- is a
**game form**such that for every possible**preference profiles**, the game has**pure nash equilibria**, all of which are**pareto efficient**.

- Allocation of goods
- is a function . The allocation is a
**cardinal**approach for determining the good (e.g. money) the players are granted under the different outcomes of the game.

- Best reply
- the best reply to a given complement is a strategy that maximizes player
**i'**s payment. Formally, we want:

.

- Coalition
- is any subset of the set of players: .

- Condorcet winner
- Given a
**preference***ν*on the**outcome space**, an outcome**a**is a condorcet winner if all non-dummy players prefer**a**to all other outcomes.

- Decidability
- In relation to game theory, refers to the question of the existence of an algorithm that can and will return an answer as to whether a game can be solved or not.
^{ [1] }

- Determinacy
- A subfield of set theory that examines the conditions under which one or the other player of a game has a winning strategy, and the consequences of the existence of such strategies. Games studied in set theory are Gale–Stewart games – two-player games of perfect information in which the players make an infinite sequence of moves and there are no draws.

- Determined game (or
**Strictly determined game**) - In game theory, a strictly determined game is a two-player zero-sum game that has at least one Nash equilibrium with both players using pure strategies.
^{ [2] }^{ [3] }

- Dictator
- A player is a
*strong dictator*if he can guarantee any outcome regardless of the other players. is a*weak dictator*if he can guarantee any outcome, but his strategies for doing so might depend on the complement strategy vector. Naturally, every strong dictator is a weak dictator. Formally:*m*is a*Strong dictator*if:*m*is a*Weak dictator*if: - Another way to put it is:
- a
*weak dictator*is -effective for every possible outcome. - A
*strong dictator*is -effective for every possible outcome. - A game can have no more than one
*strong dictator*. Some games have multiple*weak dictators*(in*rock-paper-scissors*both players are*weak dictators*but none is a*strong dictator*).

- a
- Also see
*Effectiveness*. Antonym:*dummy*.

- Dominated outcome
- Given a
**preference***ν*on the**outcome space**, we say that an outcome**a**is dominated by outcome**b**(hence,**b**is the*dominant*strategy) if it is preferred by all players. If, in addition, some player strictly prefers**b**over**a**, then we say that**a**is**strictly dominated**. Formally:

for domination, and

for strict domination.

An outcome**a**is (strictly)**dominated**if it is (strictly)**dominated**by some other**outcome**.

An outcome**a**is dominated for a**coalition****S**if all players in**S**prefer some other outcome to**a**. See also**Condorcet winner**.

- Dominated strategy
- we say that strategy is (strongly) dominated by strategy if for any complement strategies tuple , player
*i*benefits by playing . Formally speaking:

and

.

A strategy**σ**is (strictly)**dominated**if it is (strictly)**dominated**by some other**strategy**.

- Dummy
- A player
**i**is a dummy if he has no effect on the outcome of the game. I.e. if the outcome of the game is insensitive to player**i'**s strategy. - Antonyms:
*say*,*veto*,*dictator*.

- Effectiveness
- A coalition (or a single player)
**S**is*effective for***a**if it can force**a**to be the outcome of the game.**S**is α-effective if the members of**S**have strategies s.t. no matter what the complement of**S**does, the outcome will be**a**. **S**is β-effective if for any strategies of the complement of**S**, the members of**S**can answer with strategies that ensure outcome**a**.

- Finite game
- is a game with finitely many players, each of which has a finite set of
**strategies**.

- Grand coalition
- refers to the coalition containing all players. In cooperative games it is often assumed that the grand coalition forms and the purpose of the game is to find stable imputations.

- Mixed strategy
- for player
**i**is a probability distribution**P**on . It is understood that player**i**chooses a strategy randomly according to**P**.

- Mixed Nash Equilibrium
- Same as
**Pure Nash Equilibrium**, defined on the space of**mixed strategies**. Every finite game has**Mixed Nash Equilibria**.

- Pareto efficiency
- An
**outcome***a*of**game form***π*is (strongly)**pareto efficient**if it is**undominated**under all**preference profiles**.

- Preference profile
- is a function . This is the
**ordinal**approach at describing the outcome of the game. The preference describes how 'pleased' the players are with the possible outcomes of the game. See**allocation of goods**.

- Pure Nash Equilibrium
- An element of the strategy space of a game is a
*pure nash equilibrium point*if no player**i**can benefit by deviating from his strategy , given that the other players are playing in . Formally:

.

No equilibrium point is dominated.

- Say
- A player
**i**has a**Say**if he is not a*Dummy*, i.e. if there is some tuple of complement strategies s.t. π (σ_i) is not a constant function. - Antonym:
*Dummy*.

- Shannon number
- A conservative lower bound of the game-tree complexity of chess (10
^{120}).

- Solved game
- A game whose outcome (win, lose or draw) can be correctly predicted assuming perfect play from all players.

- Value
- A
**value**of a game is a rationally expected**outcome**. There are more than a few definitions of**value**, describing different methods of obtaining a solution to the game.

- Veto
- A veto denotes the ability (or right) of some player to prevent a specific alternative from being the outcome of the game. A player who has that ability is called
**a veto player**. - Antonym:
*Dummy*.

- Weakly acceptable game
- is a game that has
**pure nash equilibria**some of which are**pareto efficient**.

- Zero sum game
- is a game in which the allocation is constant over different
**outcomes**. Formally:

w.l.g. we can assume that constant to be zero. In a zero-sum game, one player's gain is another player's loss. Most classical board games (e.g. chess, checkers) are**zero sum**.

In probability theory, a **normal****distribution** is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is

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In mathematics, **Maass forms** or **Maass wave forms** are studied in the theory of automorphic forms. Maass forms are complex-valued smooth functions of the upper half plane, which transform in a similar way under the operation of a discrete subgroup of as modular forms. They are Eigenforms of the hyperbolic Laplace Operator defined on and satisfy certain growth conditions at the cusps of a fundamental domain of . In contrast to the modular forms the Maass forms need not be holomorphic. They were studied first by Hans Maass in 1949.

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The **table of chords**, created by the Greek astronomer, geometer, and geographer Ptolemy in Egypt during the 2nd century AD, is a trigonometric table in Book I, chapter 11 of Ptolemy's *Almagest*, a treatise on mathematical astronomy. It is essentially equivalent to a table of values of the sine function. It was the earliest trigonometric table extensive enough for many practical purposes, including those of astronomy. Centuries passed before more extensive trigonometric tables were created. One such table is the *Canon Sinuum* created at the end of the 16th century.

In the Newman–Penrose (NP) formalism of general relativity, independent components of the Ricci tensors of a four-dimensional spacetime are encoded into seven **Ricci scalars** which consist of three real scalars , three complex scalars and the NP curvature scalar . Physically, Ricci-NP scalars are related with the energy–momentum distribution of the spacetime due to Einstein's field equation.

**Stochastic portfolio theory (SPT)** is a mathematical theory for analyzing stock market structure and portfolio behavior introduced by E. Robert Fernholz in 2002. It is descriptive as opposed to normative, and is consistent with the observed behavior of actual markets. Normative assumptions, which serve as a basis for earlier theories like modern portfolio theory (MPT) and the capital asset pricing model (CAPM), are absent from SPT.

**Lagrangian field theory** is a formalism in classical field theory. It is the field-theoretic analogue of Lagrangian mechanics. Lagrangian mechanics is used to analyze the motion of a system of discrete particles each with a finite number of degrees of freedom. Lagrangian field theory applies to continua and fields, which have an infinite number of degrees of freedom.

- ↑ Mathoverflow.net/Decidability-of-chess-on-an-infinite-board Decidability-of-chess-on-an-infinite-board
- ↑ Saul Stahl (1999). "Solutions of zero-sum games".
*A gentle introduction to game theory*. AMS Bookstore. p. 54. ISBN 9780821813393. - ↑ Abraham M. Glicksman (2001). "Elementary aspects of the theory of games".
*An Introduction to Linear Programming and the Theory of Games*. Courier Dover Publications. p. 94. ISBN 9780486417103.

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