Strong Nash equilibrium | |
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A solution concept in game theory | |

Relationship | |

Subset of | Evolutionarily stable strategy (if the strong Nash equilibrium is not also weak) |

Significance | |

Used for | All non-cooperative games of more than 2 players |

In game theory a **strong Nash equilibrium** is a Nash equilibrium in which no coalition, taking the actions of its complements as given, can cooperatively deviate in a way that benefits all of its members.^{ [1] } While the Nash concept of stability defines equilibrium only in terms of unilateral deviations, strong Nash equilibrium allows for deviations by every conceivable coalition.^{ [2] } This equilibrium concept is particularly useful in areas such as the study of voting systems, in which there are typically many more players than possible outcomes, and so plain Nash equilibria are far too abundant.

The strong Nash concept is criticized as too "strong" in that the environment allows for unlimited private communication. In fact, strong Nash equilibrium has to be Pareto-efficient. As a result of these requirements, Strong Nash rarely exists in games interesting enough to deserve study. Nevertheless, it is possible for there to be multiple strong Nash equilibria. For instance, in Approval voting, there is always a strong Nash equilibrium for any Condorcet winner that exists, but this is only unique (apart from inconsequential changes) when there is a majority Condorcet winner.

A relatively weaker yet refined Nash stability concept is called coalition-proof Nash equilibrium (CPNE) ^{ [2] } in which the equilibria are immune to multilateral deviations that are self-enforcing. Every correlated strategy supported by iterated strict dominance and on the Pareto frontier is a CPNE.^{ [3] } Further, it is possible for a game to have a Nash equilibrium that is resilient against coalitions less than a specified size *k*. CPNE is related to the theory of the core.

Confusingly, the concept of a strong Nash equilibrium is unrelated to that of a weak Nash equilibrium. That is, a Nash equilibrium can be both strong and weak, either, or neither.

An **evolutionarily stable strategy** (**ESS**) is a strategy which, if adopted by a population in a given environment, is impenetrable, meaning that it cannot be invaded by any alternative strategy that are initially rare. It is relevant in game theory, behavioural ecology, and evolutionary psychology. An ESS is an equilibrium refinement of the Nash equilibrium. It is a Nash equilibrium that is "evolutionarily" stable: once it is fixed in a population, natural selection alone is sufficient to prevent alternative (mutant) strategies from invading successfully. The theory is not intended to deal with the possibility of gross external changes to the environment that bring new selective forces to bear.

In game theory, the **Nash equilibrium**, named after the mathematician John Forbes Nash Jr., is a proposed solution of a non-cooperative 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 their own strategy.

In economics, **general equilibrium theory** attempts to explain the behavior of supply, demand, and prices in a whole economy with several or many interacting markets, by seeking to prove that the interaction of demand and supply will result in an overall general equilibrium. General equilibrium theory contrasts to the theory of *partial* equilibrium, which only analyzes single markets.

In game theory, **coordination games** are a class of games with multiple pure strategy Nash equilibria in which players choose the same or corresponding strategies.

In game theory, the **centipede game**, first introduced by Robert Rosenthal in 1981, is an extensive form game in which two players take turns choosing either to take a slightly larger share of an increasing pot, or to pass the pot to the other player. The payoffs are arranged so that if one passes the pot to one's opponent and the opponent takes the pot on the next round, one receives slightly less than if one had taken the pot on this round. Although the traditional centipede game had a limit of 100 rounds, any game with this structure but a different number of rounds is called a centipede game.

In game theory, a **non-cooperative game** is a game with competition between individual players, as opposed to cooperative games, and in which alliances can only operate if self-enforcing.

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.

In game theory, the **core** is the set of feasible allocations that cannot be improved upon by a subset of the economy's agents. A coalition is said to *improve upon* or *block* a feasible allocation if the members of that coalition are better off under another feasible allocation that is identical to the first except that every member of the coalition has a different consumption bundle that is part of an aggregate consumption bundle that can be constructed from publicly available technology and the initial endowments of each consumer in the coalition.

In game theory, **trembling hand perfect equilibrium** is a refinement of Nash equilibrium due to Reinhard Selten. A trembling hand perfect equilibrium is an equilibrium that takes the possibility of off-the-equilibrium play into account by assuming that the players, through a "slip of the hand" or **tremble,** may choose unintended strategies, albeit with negligible probability.

In game theory, **folk theorems** are a class of theorems about possible Nash equilibrium payoff profiles in repeated games. The original Folk Theorem concerned the payoffs of all the Nash equilibria of an infinitely repeated game. This result was called the Folk Theorem because it was widely known among game theorists in the 1950s, even though no one had published it. Friedman's (1971) Theorem concerns the payoffs of certain subgame-perfect Nash equilibria (SPE) of an infinitely repeated game, and so strengthens the original Folk Theorem by using a stronger equilibrium concept subgame-perfect Nash equilibria rather than Nash equilibrium.

In game theory, an **outcome** is a situation which results from a combination of player's strategies. Every combination of strategies is an outcome of the game. A primary purpose of game theory is to determine which outcomes are stable according to a solution concept.

**Hobart Peyton Young** is an American game theorist and economist known for his contributions to evolutionary game theory and its application to the study of institutional and technological change, as well as the theory of learning in games. He is currently centennial professor at the London School of Economics, James Meade Professor of Economics Emeritus at the University of Oxford, professorial fellow at Nuffield College Oxford, and research principal at the Office of Financial Research at the U.S. Department of the Treasury.

In game theory, a **subgame perfect equilibrium** is a refinement of a Nash equilibrium used in dynamic games. A strategy profile is a subgame perfect equilibrium if it represents a Nash equilibrium of every subgame of the original game. Informally, this means that if the players played any smaller game that consisted of only one part of the larger game, their behavior would represent a Nash equilibrium of that smaller game. Every finite extensive game with perfect recall has a subgame perfect equilibrium.

**Risk dominance** and **payoff dominance** are two related refinements of the Nash equilibrium (NE) solution concept in game theory, defined by John Harsanyi and Reinhard Selten. A Nash equilibrium is considered **payoff dominant** if it is Pareto superior to all other Nash equilibria in the game. When faced with a choice among equilibria, all players would agree on the payoff dominant equilibrium since it offers to each player at least as much payoff as the other Nash equilibria. Conversely, a Nash equilibrium is considered **risk dominant** if it has the largest basin of attraction. This implies that the more uncertainty players have about the actions of the other player(s), the more likely they will choose the strategy corresponding to it.

In game theory, an **epsilon-equilibrium**, or near-Nash equilibrium, is a strategy profile that approximately satisfies the condition of Nash equilibrium. In a Nash equilibrium, no player has an incentive to change his behavior. In an approximate Nash equilibrium, this requirement is weakened to allow the possibility that a player may have a small incentive to do something different. This may still be considered an adequate solution concept, assuming for example status quo bias. This solution concept may be preferred to Nash equilibrium due to being easier to compute, or alternatively due to the possibility that in games of more than 2 players, the probabilities involved in an exact Nash equilibrium need not be rational numbers.

**Algorithmic game theory** is an area in the intersection of game theory and computer science, with the objective of understanding and design of algorithms in strategic environments.

The concept of **coalition-proof Nash equilibrium** applies to certain "noncooperative" environments in which players can freely discuss their strategies but cannot make binding commitments. It emphasizes the immunization to deviations that are self-enforcing. While the best-response property in Nash equilibrium is necessary for self-enforceability, it is not generally sufficient when players can jointly deviate in a way that is mutually beneficial.

**Jean-François Mertens** was a Belgian game theorist and mathematical economist.

**Mertens stability** is a solution concept used to predict the outcome of a non-cooperative game. A tentative definition of stability was proposed by Elon Kohlberg and Jean-François Mertens for games with finite numbers of players and strategies. Later, Mertens proposed a stronger definition that was elaborated further by Srihari Govindan and Mertens. This solution concept is now called Mertens stability, or just stability.

**Maximal lotteries** refers to a probabilistic voting system first considered by the French mathematician and social scientist Germain Kreweras in 1965. The method uses preferential ballots and returns so-called maximal lotteries, i.e., probability distributions over the alternatives that are weakly preferred to any other probability distribution. Maximal lotteries satisfy the Condorcet criterion, the Smith criterion, reversal symmetry, polynomial runtime, and probabilistic versions of reinforcement, participation, and independence of clones.

- ↑ R. Aumann (1959),
*Acceptable points in general cooperative*n*-person games in "Contributions to the Theory of Games IV"*, Princeton Univ. Press, Princeton, N.J.. - 1 2 B. D. Bernheim; B. Peleg; M. D. Whinston (1987), "Coalition-Proof Equilibria I. Concepts",
*Journal of Economic Theory*,**42**: 1–12, doi:10.1016/0022-0531(87)90099-8. - ↑ D. Moreno; J. Wooders (1996), "Coalition-Proof Equilibrium",
*Games and Economic Behavior*,**17**: 80–112, doi:10.1006/game.1996.0095, hdl: 10016/4408 .

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