In game theory and economic modelling, a solution concept is a process via which equilibria of a game are identified. In this sense they are used as predictions of play, suggesting what the outcome of a particular game will be (i.e. what strategies will or might be adopted by players. Game theory is a branch of applied mathematics that uses models to study interactions with formalised incentive structures (games). It has applications in a variety of fields, including economics, international relations, evolutionary biology, political science, and military strategy. ...

Each of the following solution concepts (other than rationalisability) is a refinement of what precedes it that eliminates implausible equilibria in richer games.

Rationalisability

In this solution concept, players are assumed to be rational and so strictly dominated strategies are eliminated from the set of strategies that might feasibly be played. A strictly dominated strategy is one for which there is a strategy that a player is always better off playing and so a rational player would never play such a strategy. (Strictly dominated strategies are also important in minimax game-tree search). For example, in the (single period) prisoners' dilemma (shown below), cooperate is strictly dominated by defect for both players because either player is always better off playing defect, regardless of what his opponent does. Minimax is a method in decision theory for minimizing the expected maximum loss. ... Will the two prisoners cooperate to minimise total loss of liberty or will one of them, trusting the other to cooperate, betray him so as to go free? The prisoners dilemma is a type of non-zero-sum game (game in the sense of Game Theory). ...

Nash equilibrium

(Main article: Nash equilibrium) In game theory, the Nash equilibrium (named after John Nash) is a kind of optimal strategy for games involving two or more players, where no player has anything to gain by changing only ones own strategy. ...

A Nash equilibrium is a strategy profile (a strategy profile specifies a strategy for every player, e.g. in the above prisoners' dilemma game (cooperate, defect) specifies that prisoner 1 plays cooperate and player 2 plays defect) in which every strategy is a best response to every other strategy played. A strategy by a player is a best response to another player's strategy if there is no other strategy that could be played that would yield a higher pay-off in any situation in which the other player's strategy is played. In game theory, the best response is the strategy in a single period that creates the most favorable immediate outcome for the current player. ...

Backward induction

There are games that have multiple Nash equilibria, some of which are unrealistic. In the case of dynamic games, unrealistic Nash equilibria might be eliminated by applying backward induction, which assumes that future play will be rational. It therefore elimates noncredible (or incredible) threats because such threats would be irrational to carry out if a player was ever called upon to do so.

For example, consider a dynamic game in which the players are an incumbent firm in an industry and a potential entrant to that industry. As it stands, the incumbent has a monopoly over the industry and does not want to lose some of its market share to the entrant. If the entrant chooses not to enter, the payoff to the incumbent is high (it maintains its monopoly) and the entrant neither loses nor gains (its payoff is zero). If the entrant enters, the incumbent can fight or accommodate the entrant. It will fight by lowering its price, running the entrant out of business (and incurring exit costs – a negative payoff) and damaging its own profits. If it accommodates the entrant it will lose some of its sales, but a high price will be maintained and it will receive greater profits than by lowering its price (but lower than monopoly profits).

If the entrant enters, the best response of the incumbent is to accommodate. If the incumbent accommodates, the best response of the entrant is to enter (and gain profit). Hence the strategy profile in which the incumbent accommodates if the entrant enters and the entrant enters if the incumbent accommodates is a Nash equilibrium. However, if the incumbent is going to play fight, the best response of the entrant is to not enter. If the entrant does not enter, it does not matter what the incumbent chooses to do (since there is no other firm to do it to - note that if the entrant does not enter, fight and accommodate yield the same payoffs to both players; the incumbent will not lower its prices if the incumbent does not enter). Hence fight is a best response of the incumbent if the entrant does not enter. Hence the strategy profile in which the incumbent fights if the entrant does not enter and the entrant does not enter if the incumbent fights is a Nash equilibrium. Since the game is dynamic, any claim by the incumbent that it will fight is an incredible threat because by the time the decision node is reached where it can decide to fight (i.e. the entrant has entered), it would be irrational to do so. Therefore this Nash equilibrium can be eliminated by backward induction.

See also:

• Monetary policy theory
• Stackelberg competition

Monetary policy is the process of managing a nations money supply to achieve specific goals—such as constraining inflation, achieving full employment or more well-being. ... The Stackelberg leadership model is a model of duopoly in economics. ...

Subgame perfect Nash equilibrium

(Main article: Subgame perfect equilibrium) Subgame perfect equilibrium is an economics term used in game theory to describe an equilibrium such that players strategies constitute a Nash equilibrium in every subgame of the original game. ...

A generalisation of backward induction is subgame perfection. Backward induction assumes that all future play will be rational. In subgame perfect equilibria, play in every subgame is rational (specifically a Nash equilibrium). Backward induction can only be used in terminating (finite) games of definite length and cannot be applied to games with imperfect information. In these cases, subgame perfection can be used. The eliminated Nash equilibrium described above is subgame imperfect because it is not a Nash equilibrium of the subgame that starts at the node reached once the entrant has entered. A minigame is a (usually short) segment of a video game that uses a different style of gameplay than the rest of the game. ...

Perfect Bayesian equilibrium

Sometimes subgame perfection does not impose large enough restriction on unreasonable outcomes. For example, since subgames cannot cut through information sets, a game of imperfect information may have only one subgame – itself – and hence subgame perfection cannot be used to eliminate any Nash equilibria. A perfect Bayesian equilibrium is a specification of players’ strategies and beliefs about which node in the information set has been reached by the play of the game. A belief about a decision node is the probability that a particular player thinks that that node is or will be in play (on the equilibrium path). In particular, the intuition of PBE is that it specifies player strategies that are rational given the player beliefs it specifies and the beliefs it specifies are consistent with the strategies it specifies.

In a Bayesian game a strategy determines what a player plays at every information set controlled by that player. The requirement that beliefs are consistent with strategies is something not specified by subgame perfection. Hence, PBE is a consistency condition on players’ beliefs. Just as in a Nash equilibrium no player’s strategy is strictly dominated, in a PBE, for any information set no player’s strategy is strictly dominated beginning at that information set. That is, for every belief that the player could hold at that information set there is no strategy that yields a greater expected payoff for that player. Unlike the above solution concepts, no player’s strategy is strictly dominated beginning at any information set even if it is off the equilibrium path. Thus in PBE, players cannot threaten to play strategies that are strictly dominated beginning at any information set off the equilibrium path.

The Bayesian in the name of this solution concept alludes to the fact that players update their beliefs according to Bayes' theorem. They calculate probabilities given what has already taken place in the game. Bayes theorem is a result in probability theory, which gives the conditional probability distribution of a random variable A given B in terms of the conditional probability distribution of variable B given A and the marginal probability distribution of A alone. ...

Forward induction

Forward induction is so called because just as backward induction assumes future play will be rational, forward induction assumes past play was rational. Where a player does not know what type another player is (i.e. there is imperfect and asymmetric information), that player may form a belief of what type that player is by observing that player's past actions. Hence the belief formed by that player of what the probability of the opponent being a certain type is based on the past play of that opponent being rational.

See also

• "The Intuitive Criterion" (Cho and Kreps 1987)

See also

• Extensive form game