Nash equilibrium in the context of "Normal-form game"

An evolutionarily stable strategy (ESS) is a strategy (or set of strategies) that is impermeable when adopted by a population in adaptation to a specific environment, that is to say it cannot be displaced by an alternative strategy (or set of strategies) which may be novel or initially rare. Introduced by John Maynard Smith and George R. Price in 1972/3, it is an important concept in behavioural ecology, evolutionary psychology, mathematical game theory and economics, with applications in other fields such as anthropology, philosophy and political science.

In game-theoretical terms, an ESS is an equilibrium refinement of the Nash equilibrium, being a Nash equilibrium that is also "evolutionarily stable." Thus, once fixed in a population, natural selection alone is sufficient to prevent alternative (mutant) strategies from replacing it (although this does not preclude the possibility that a better strategy, or set of strategies, will emerge in response to selective pressures resulting from environmental change).

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.

Many solution concepts, for many games, will result in more than one solution. This puts any one of the solutions in doubt, so a game theorist may apply a refinement to narrow down the solutions. Each successive solution concept presented in the following improves on its predecessor by eliminating implausible equilibria in richer games.

In economics, market power refers to the ability of a firm to influence the price at which it sells a product or service by manipulating either the supply or demand of the product or service to increase economic profit. In other words, market power occurs if a firm does not face a perfectly elastic demand curve and can set its price (P) above marginal cost (MC) without losing revenue. This indicates that the magnitude of market power is associated with the gap between P and MC at a firm's profit maximising level of output. The size of the gap, which encapsulates the firm's level of market dominance, is determined by the residual demand curve's form. A steeper reverse demand indicates higher earnings and more dominance in the market. Such propensities contradict perfectly competitive markets, where market participants have no market power, P = MC and firms earn zero economic profit. Market participants in perfectly competitive markets are consequently referred to as 'price takers', whereas market participants that exhibit market power are referred to as 'price makers' or 'price setters'.

The market power of any individual firm is controlled by multiple factors, including but not limited to, their size, the structure of the market they are involved in, and the barriers to entry for the particular market. A firm with market power has the ability to individually affect either the total quantity or price in the market. This said, market power has been seen to exert more upward pressure on prices due to effects relating to Nash equilibria and profitable deviations that can be made by raising prices. Price makers face a downward-sloping demand curve and as a result, price increases lead to a lower quantity demanded. The decrease in supply creates an economic deadweight loss (DWL) and a decline in consumer surplus. This is viewed as socially undesirable and has implications for welfare and resource allocation as larger firms with high markups negatively effect labour markets by providing lower wages. Perfectly competitive markets do not exhibit such issues as firms set prices that reflect costs, which is to the benefit of the customer. As a result, many countries have antitrust or other legislation intended to limit the ability of firms to accrue market power. Such legislation often regulates mergers and sometimes introduces a judicial power to compel divestiture.

The Hobbesian trap (or Schelling's dilemma) is a theory that explains why preemptive strikes occur between two groups, out of bilateral fear of an imminent attack. Without outside influences, this situation will lead to a fear spiral (catch-22, vicious circle, Nash equilibrium) in which fear will lead to an arms race which in turn will lead to increasing fear. The Hobbesian trap can be explained in terms of game theory. Although cooperation would be the better outcome for both sides, mutual distrust leads to the adoption of strategies that have negative outcomes for both individual players and all players combined. The theory has been used to explain outbreaks of conflicts and violence, spanning from individuals to states.

Mutual assured destruction (MAD) is a doctrine of military strategy and national security policy which posits that a full-scale use of nuclear weapons by an attacker on a nuclear-armed defender with second-strike capabilities would result in the complete annihilation of both the attacker and the defender. It is based on the theory of rational deterrence, which holds that the threat of using strong weapons against the enemy prevents the enemy's use of those same weapons. The strategy is a form of Nash equilibrium in which, once armed, neither side has any incentive to initiate a conflict or to disarm.

The result may be a nuclear peace, in which the presence of nuclear weapons decreases the risk of crisis escalation, since parties will seek to avoid situations that could lead to the use of nuclear weapons. Proponents of nuclear peace theory therefore believe that controlled nuclear proliferation may be beneficial for global stability. Critics argue that nuclear proliferation increases the chance of nuclear war through either deliberate or inadvertent use of nuclear weapons, as well as the likelihood of nuclear material falling into the hands of violent non-state actors.

A coordination game is a type of simultaneous game found in game theory. It describes the situation where a player will earn a higher payoff when they select the same course of action as another player. The game is not one of pure conflict, which results in multiple pure strategy Nash equilibria in which players choose matching strategies. Figure 1 shows a 2-player example.

Both (Up, Left) and (Down, Right) are Nash equilibria. If the players expect (Up, Left) to be played, then player 1 thinks their payoff would fall from 2 to 1 if they deviated to Down, and player 2 thinks their payoff would fall from 4 to 3 if they chose Right. If the players expect (Down, Right), player 1 thinks their payoff would fall from 2 to 1 if they deviated to Up, and player 2 thinks their payoff would fall from 4 to 3 if they chose Left. A player's optimal move depends on what they expect the other player to do, and they both do better if they coordinate than if they played an off-equilibrium combination of actions. This setup can be extended to more than two strategies or two players.

A self-enforcing agreement is an agreement that is enforced only by the parties to it; no external party can enforce or interfere with the agreement. (In this respect it differs from an enforceable contract.) The agreement will stand so long as the parties believe it is mutually beneficial and it is not breached by any party.

In game theory, games in which cooperative behaviour can only be enforced through self-enforcing agreements are called non-cooperative games, whereas games allowing strategies relying on external enforcement are called cooperative games. Nash equilibrium is the most common kind of self-enforcing agreement.