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
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.
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.
H. Everett. "Recursive Games". In H.W. Kuhn and A.W. Tucker, editors. Contributions to the theory of games, vol. III, volume 39 of Annals of Mathematical Studies. Princeton University Press, 1957.
R. Radner. Collusive behavior in non-cooperative epsilon equilibria of oligopolies with long but finite lives, Journal of Economic Theory, 22, 121–157, 1980.