In game theory

Game theory

Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...

, an outcome is a set of moves or strategies

Strategy (game theory)

In game theory, a player's strategy in a game is a complete plan of action for whatever situation might arise; this fully determines the player's behaviour...

 taken by the players, or their payoffs resulting from the actions or strategies taken by all players. The two are complementary in that, given knowledge of the set of strategies of all players, the final state of the game is known, as are any relevant payoffs. In a game where chance or a random event is involved, the outcome is not known from only the set of strategies, but is only realized when the random event(s) are realized

Realization (probability)

In probability and statistics, a realization, or observed value, of a random variable is the value that is actually observed . The random variable itself should be thought of as the process how the observation comes about...

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A set of payoffs can be considered a set of N-tuples, where N is the number of players in the game, and the cardinality of the set is equal to the total number of possible outcomes when the strategies of the players are varied. The payoff set can thus be partially ordered

Partially ordered set

In mathematics, especially order theory, a partially ordered set formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the...

, where the partial ordering comes from the value of each entry in the N-tuple. How players interact to allocate the payoffs among themselves is a fundamental aspect of economics

Economics

Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...

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Choosing among outcomes

Many different concepts exist to express how players might interact. An optimal interaction might be one in which no player's payoff can be made greater, without making any other player's payoff lesser. Such a payoff is called Pareto efficient, and the set of such payoffs is called the Pareto frontier.

Many economists study the ways in which payoffs are in some sort of economic equilibrium

Economic equilibrium

In economics, economic equilibrium is a state of the world where economic forces are balanced and in the absence of external influences the values of economic variables will not change. It is the point at which quantity demanded and quantity supplied are equal...

. One example of such an equilibrium is the Nash equilibrium

Nash equilibrium

In game theory, Nash equilibrium is a solution concept of a 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 his own strategy unilaterally...

, where each player plays a strategy such that their payoff is maximized given the strategy of the other players.

Applications

Equilibria are not always Pareto efficient, and a number of game theorists design ways to enforce Pareto efficient play, or play that satisfies some other sort of social optimality. The theory of this is called implementation theory

Implementation theory

Implementation theory is an area of game theory closely related to mechanism design where an attempt is made to add into a game a mechanism such that the equilibrium of the game conforms to some concept of social optimality ....

. Other economists seek to design games based on a certain set of outcomes, an effort which goes under the name of mechanism design

Mechanism design

Mechanism design is a field in game theory studying solution concepts for a class of private information games...

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