game theory Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has appli ...

, an outcome is a situation which results from a combination of player's strategies. Formally, a path through the

game tree In the context of Combinatorial game theory, which typically studies sequential games with perfect information, a game tree is a graph representing all possible game states within such a game. Such games include well-known ones such as chess, ch ...

, or equivalently a terminal node of the game tree. A primary purpose of game theory is to determine the outcomes of games according to a solution concept (e.g.

Nash equilibrium In game theory, the Nash equilibrium, named after the mathematician John Nash, is the most common way to define the solution of a non-cooperative game involving two or more players. In a Nash equilibrium, each player is assumed to know the equili ...

). 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. 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 In mathematics, especially order theory, a partially ordered set (also poset) 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 r ...

, 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 () is the social science that studies the production, distribution, and consumption of goods and services. Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics anal ...

Choosing among outcomes

Many different concepts exist to express how players might interact. An optimal interaction may be one in which no player's payoff can be made greater, without making any other player's payoff lesser. Such a payoff is described as 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 In economics, economic equilibrium is a situation in which economic forces such as supply and demand are balanced and in the absence of external influences the ( equilibrium) values of economic variables will not change. For example, in the st ...

. One example of such an equilibrium is the

, where each player plays a strategy such that their payoff is maximized given the strategy of the other players. Players are persons who make logical economic decisions. It is assumed that human people make all of their economic decisions based only on the idea that they are irrational. A player's rewards (utilities, profits, income, or subjective advantages) are assumed to be maximised. The purpose of game-theoretic analysis, when applied to a rational approach, is to provide recommendations on how to make choices against other rational players. First, it reduces the possible outcomes; logical action is more predictable than irrational. Second, it provides a criterion for assessing an economic system's efficiency. In a Prisoner's Dilemma game between two players, player one and player two can choose the utilities that are the best response to maximise their outcomes. "A best response to a coplayer’s strategy is a strategy that yields the highest payoff against that particular strategy". A matrix is used to present the payoff of both players in the game. For example, the best response of player one is the highest payoff for player one’s move, and vice versa. For player one, they will pick the payoffs from the column strategies. For player two, they will choose their moves based on the two row strategies. Assuming both players do not know the opponents strategies. It is a a dominant strategy for the first player to choose a payoff of 5 rather than a payoff of 3 because strategy D is a better response than strategy C.

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. 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 is a field in economics and game theory that takes an objectives-first approach to designing economic mechanisms or incentives, toward desired objectives, in strategic settings, where players act rationally. Because it starts a ...

References

{{game theory Game theory