Strong Nash equilibrium
| Strong Nash equilibrium | |
|---|---|
| A non-cooperative games of more than 2 players |
In
Existence
Nessah and Tian[3] prove that an SNE exists if the following conditions are satisfied:
- The strategy space of each player is compact and convex;
- The payoff function of each player is concaveand continuous;
- The coalition consistency property: there exists a weight-vector-tuple w, assigning a weight-vector wS to each possible coalition S, such that for each strategy-profile x, there exists a strategy-profile z in which zS maximizes the weighted (by wS) social welfare to members of S, given x-S.
- Note that if x is itself an SNE, then z can be taken to be equal to x. If x is not an SNE, the condition requires that one can move to a different strategy-profile which is a social-welfare-best-response for all coalitions simultaneously.
For example, consider a game with two players, with strategy spaces [1/3, 2] and [3/4, 2], which are clearly compact and convex. The utility functions are:
- u1(x) = - x12 + x2 + 1
- u2(x) = x1 - x22 + 1
which are continuous and convex. It remains to check coalition consistency. For every strategy-tuple x, we check the weighted-best-response of each coalition:
- For the coalition {1}, we need to find, for every x2, maxy1 (-y12 + x2 + 1); it is clear that the maximum is attained at the smallest point of the strategy space, which is y1=1/3.
- For the coalition {2}, we similarly see that for every x1, the maximum payoff is attained at the smallest point, y2=3/4.
- For the coalition {1,2}, with weights w1,w2, we need to find maxy1,y2 (w1*(-y12 + y2 + 1)+w2*(y1 - y22 + 1)). Using the derivative test, we can find out that the maximum point is y1=w2/(2*w1) and y2=w1/(2*w2). By taking w1=0.6,w2=0.4 we get y1=1/3 and y2=3/4.
So, with w1=0.6,w2=0.4 the point (1/3,3/4) is a consistent social-welfare-best-response for all coalitions simultaneously. Therefore, an SNE exists, at the same point (1/3,3/4).
Here is an example in which the coalition consistency fails, and indeed there is no SNE.[3]: Example.3.1 There are two players, with strategy space [0,1]. Their utility functions are:
- u1(x) = -x1 + 2*x2;
- u2(x) = 2*x1 - x2.
There is a unique Nash equilibrium at (0,0), with payoff vector (0,0). However, it is not SNE as the coalition {1,2} can deviate to (1,1), with payoff vector (1,1). Indeed, coalition consistency is violated at x=(0,0): for the coalition {1,2}, for any weight-vector wS, the social-welfare-best-response is either on the line (1,0)--(1,1) or on the line (0,1)--(1,1); but any such point is not a best-response for the player playing 1.
Nessah and Tian[3] also present a necessary and sufficient condition for SNE existence, along with an algorithm that finds an SNE if and only if it exists.
Properties
Every SNE is a Nash equilibrium. This can be seen by considering a deviation of the n singleton coalitions.
Every SNE is weakly Pareto-efficient. This can be seen by considering a deviation of the grand coalition - the coalition of all players.
Every SNE is in the weak alpha-core and in the weak-beta core.[3]
Criticism
The strong Nash concept is criticized as too "strong" in that the environment allows for unlimited private communication. 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
A relatively weaker yet refined Nash stability concept is called
Confusingly, the concept of a strong Nash equilibrium is unrelated to that of a
References
- ^ 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..
- ^ .
- ^ ISSN 0022-247X.
- hdl:10016/4408.
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