Epsilon-equilibrium

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Epsilon-equilibrium
A solution concept in game theory
Relationship
Superset of Nash Equilibrium
Significance
Used for stochastic games

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. [1]

Contents

Definition

There is more than one alternative definition.

The standard definition

Given a game and a real non-negative parameter , a strategy profile is said to be an -equilibrium if it is not possible for any player to gain more than in expected payoff by unilaterally deviating from his strategy. [2] :45 Every Nash Equilibrium is equivalent to an -equilibrium where .

Formally, let be an -player game with action sets for each player and utility function . Let denote the payoff to player when strategy profile is played. Let be the space of probability distributions over . A vector of strategies is an -Nash Equilibrium for if

for all

Well-supported approximate equilibrium

The following definition [3] imposes the stronger requirement that a player may only assign positive probability to a pure strategy if the payoff of has expected payoff at most less than the best response payoff. Let be the probability that strategy profile is played. For player let be strategy profiles of players other than ; for and a pure strategy of let be the strategy profile where plays and other players play . Let be the payoff to when strategy profile is used. The requirement can be expressed by the formula

Results

The existence of a polynomial-time approximation scheme (PTAS) for ε-Nash equilibria is equivalent to the question of whether there exists one for ε-well-supported approximate Nash equilibria, [4] but the existence of a PTAS remains an open problem. For constant values of ε, polynomial-time algorithms for approximate equilibria are known for lower values of ε than are known for well-supported approximate equilibria. For games with payoffs in the range [0,1] and ε=0.3393, ε-Nash equilibria can be computed in polynomial time [5] For games with payoffs in the range [0,1] and ε=2/3, ε-well-supported equilibria can be computed in polynomial time [6]

Example

The notion of ε-equilibria is important in the theory of stochastic games of potentially infinite duration. There are simple examples of stochastic games with no Nash equilibrium but with an ε-equilibrium for any ε strictly bigger than 0.

Perhaps the simplest such example is the following variant of Matching Pennies, suggested by Everett. Player 1 hides a penny and Player 2 must guess if it is heads up or tails up. If Player 2 guesses correctly, he wins the penny from Player 1 and the game ends. If Player 2 incorrectly guesses that the penny is heads up, the game ends with payoff zero to both players. If he incorrectly guesses that it is tails up, the game repeats. If the play continues forever, the payoff to both players is zero.

Given a parameter ε > 0, any strategy profile where Player 2 guesses heads up with probability ε and tails up with probability 1  ε (at every stage of the game, and independently from previous stages) is an ε-equilibrium for the game. The expected payoff of Player 2 in such a strategy profile is at least 1  ε. However, it is easy to see that there is no strategy for Player 2 that can guarantee an expected payoff of exactly 1. Therefore, the game has no Nash equilibrium.

Another simple example is the finitely repeated prisoner's dilemma for T periods, where the payoff is averaged over the T periods. The only Nash equilibrium of this game is to choose Defect in each period. Now consider the two strategies tit-for-tat and grim trigger. Although neither tit-for-tat nor grim trigger are Nash equilibria for the game, both of them are -equilibria for some positive . The acceptable values of depend on the payoffs of the constituent game and on the number T of periods.

In economics, the concept of a pure strategy epsilon-equilibrium is used when the mixed-strategy approach is seen as unrealistic. In a pure-strategy epsilon-equilibrium, each player chooses a pure-strategy that is within epsilon of its best pure-strategy. For example, in the Bertrand–Edgeworth model, where no pure-strategy equilibrium exists, a pure-strategy epsilon equilibrium may exist.

Related Research Articles

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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 game theory, the best response is the strategy which produces the most favorable outcome for a player, taking other players' strategies as given. The concept of a best response is central to John Nash's best-known contribution, the Nash equilibrium, the point at which each player in a game has selected the best response to the other players' strategies.

Matching pennies is the name for a simple game used in game theory. It is played between two players, Even and Odd. Each player has a penny and must secretly turn the penny to heads or tails. The players then reveal their choices simultaneously. If the pennies match, then Even keeps both pennies, so wins one from Odd. If the pennies do not match Odd keeps both pennies, so receives one from Even.

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, a Bayesian game is a game in which players have incomplete information about the other players. For example, a player may not know the exact payoff functions of the other players, but instead have beliefs about these payoff functions. These beliefs are represented by a probability distribution over the possible payoff functions.

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, a repeated game is an extensive form game that consists of a number of repetitions of some base game. The stage game is usually one of the well-studied 2-person games. Repeated games capture the idea that a player will have to take into account the impact of his or her current action on the future actions of other players; this impact is sometimes called his or her reputation. Single stage game or single shot game are names for non-repeated games.

In game theory, a correlated equilibrium is a solution concept that is more general than the well known Nash equilibrium. It was first discussed by mathematician Robert Aumann in 1974. The idea is that each player chooses their action according to their observation of the value of the same public signal. A strategy assigns an action to every possible observation a player can make. If no player would want to deviate from the recommended strategy, the distribution is called a correlated equilibrium.

In game theory, the purification theorem was contributed by Nobel laureate John Harsanyi in 1973. The theorem aims to justify a puzzling aspect of mixed strategy Nash equilibria: that each player is wholly indifferent amongst each of the actions he puts non-zero weight on, yet he mixes them so as to make every other player also indifferent.

Quantal response equilibrium (QRE) is a solution concept in game theory. First introduced by Richard McKelvey and Thomas Palfrey, it provides an equilibrium notion with bounded rationality. QRE is not an equilibrium refinement, and it can give significantly different results from Nash equilibrium. QRE is only defined for games with discrete strategies, although there are continuous-strategy analogues.

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Proper equilibrium is a refinement of Nash Equilibrium due to Roger B. Myerson. Proper equilibrium further refines Reinhard Selten's notion of a trembling hand perfect equilibrium by assuming that more costly trembles are made with significantly smaller probability than less costly ones.

In game theory, a stochastic game, introduced by Lloyd Shapley in the early 1950s, is a dynamic game with probabilistic transitions played by one or more players. The game is played in a sequence of stages. At the beginning of each stage the game is in some state. The players select actions and each player receives a payoff that depends on the current state and the chosen actions. The game then moves to a new random state whose distribution depends on the previous state and the actions chosen by the players. The procedure is repeated at the new state and play continues for a finite or infinite number of stages. The total payoff to a player is often taken to be the discounted sum of the stage payoffs or the limit inferior of the averages of the stage payoffs.

A continuous game is a mathematical concept, used in game theory, that generalizes the idea of an ordinary game like tic-tac-toe or checkers (draughts). In other words, it extends the notion of a discrete game, where the players choose from a finite set of pure strategies. The continuous game concepts allows games to include more general sets of pure strategies, which may be uncountably infinite.

In algorithmic game theory, a succinct game or a succinctly representable game is a game which may be represented in a size much smaller than its normal form representation. Without placing constraints on player utilities, describing a game of players, each facing strategies, requires listing utility values. Even trivial algorithms are capable of finding a Nash equilibrium in a time polynomial in the length of such a large input. A succinct game is of polynomial type if in a game represented by a string of length n the number of players, as well as the number of strategies of each player, is bounded by a polynomial in n.

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.

M equilibrium is a set valued solution concept in game theory that relaxes the rational choice assumptions of perfect maximization and perfect beliefs. The concept can be applied to any normal-form game with finite and discrete strategies. M equilibrium was first introduced by Jacob K. Goeree and Philippos Louis.

The Berge equilibrium is a game theory solution concept named after the mathematician Claude Berge. It is similar to the standard Nash equilibrium, except that it aims to capture a type of altruism rather than purely non-cooperative play. Whereas a Nash equilibrium is a situation in which each player of a strategic game ensures that they personally will receive the highest payoff given other players' strategies, in a Berge equilibrium every player ensures that all other players will receive the highest payoff possible. Although Berge introduced the intuition for this equilibrium notion in 1957, it was only formally defined by Vladislav Iosifovich Zhukovskii in 1985, and it was not in widespread use until half a century after Berge originally developed it.

References

Inline citations
  1. V. Bubelis (1979). "On equilibria in finite games". International Journal of Game Theory. 8 (2): 65–79. doi:10.1007/bf01768703.
  2. Vazirani, Vijay V.; Nisan, Noam; Roughgarden, Tim; Tardos, Éva (2007). Algorithmic Game Theory (PDF). Cambridge, UK: Cambridge University Press. ISBN   0-521-87282-0.
  3. P.W. Goldberg and C.H. Papadimitriou (2006). "Reducibility Among Equilibrium Problems". 38th Symposium on Theory of Computing. pp. 61–70. doi:10.1145/1132516.1132526.
  4. C. Daskalakis, P.W. Goldberg and C.H. Papadimitriou (2009). "The Complexity of Computing a Nash Equilibrium". SIAM Journal on Computing . 39 (3): 195–259. CiteSeerX   10.1.1.68.6111 . doi:10.1137/070699652.
  5. H. Tsaknakis and Paul G. Spirakis (2008). "An optimisation approach for approximate Nash equilibria". Internet Mathematics. 5 (4): 365–382. doi: 10.1080/15427951.2008.10129172 .
  6. Spyros C. Kontogiannis and Paul G. Spirakis (2010). "Well Supported Approximate Equilibria in Bimatrix Games". Algorithmica . 57 (4): 653–667. doi:10.1007/s00453-008-9227-6.
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