Strict Nash equilibrium
From Theory
Notes for CS 8803 - Game Theory and Computer Science. Spring 2008
A strict Nash equilibrium is where any change looses money.
Consider the Coordination Game:
| A | B | |
| A | 2, 2 | 1, 1 |
| B | 1, 1 | 2, 2 |
| Coordination Game | ||
A,A and B,B are both natural Nash equilibrium. However there is also a mixed NE of each action being chosen with probability 1/2. This third NE results in the suboptimal payoff of 1.5 for each player. Worse, if either player deviates from an exactly even probability distribution then the response will be one of the preferable pure NE. It doesn't seem reasonable that the two players would arrive at this NE. We would like an equilibrium refinement which doesn't consider such tenuous solutions. The strict NE solution concept is a naive attempt at this goal.
Formally: 
is a strict NE iff:
Notice that this is only for pure policies as all mixed policies are not strict. For mixed NE choosing any of the actions in σ with probability greater than 0 will yield equal payoff, thus making it not strict.
Strict NE don't always exist as pure NE don't always exist (e.g. the matching pennies game).
The production of this material was supported in part by NSF award SES-0734780.
