### The Trouble with Nash Equilibrium

2/10/2013 06:58:00 PM
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I've been dabbling in game theory lately, and was inspired to rant a bit about what a shame it is that John Nash came up with his game theory equilibrium concept before, say, the development of the "correlated equilibrium" concept.

At anyrate, this is a simplified version of the problem I have in mind: suppose that there are three committee members assigned to vote on whether to implement some proposal. A majority vote is required to approve the proposal, but the votes are cast by secret ballot. Lets suppose all three committee members prefer the proposal to pass.

This game turns out to have lots of Nash Equilibria. Lets identify some of them by listing out all possible pure strategies below. Each line represents a different strategy, and the format is this: (voter 1, voter 2, voter 3; outcome), where "Y" is a "yea" vote and "N" is a "nay" vote. I've crossed out the strategies that are not Nash Equilibria:
(Y, Y, Y; passes)
(Y, Y, N; passes)
(N, Y, Y; passes)
(Y, N, Y; passes)
(Y, N, N; fails)
(N, Y, N; fails)
(N, N, N; fails)
Each line that is not crossed out corresponds to an equilibrium. (There would be a lot more Nash equilibria if we allowed mixed strategies as well).  But look! There exists a Nash Equilibrium in which the proposal, which all three committee members support, fails to pass!

Why is this true? Keep in mind the definition of a Nash Equilibrium: it is a point at which all three committee members are playing strategies such that no single member would prefer the outcome if she changed her vote. No single member. Hence, the case where all three members vote no is a Nash equilibrium since if any single member changed her vote to "yea" the bill would still fail.

The Nash equilibrium concept turns out to be a terrible predictor of behaviors. In this case, not only is the Nash concept unable to tell us how any of the committee members will vote, it even fails to exclude the most obviously absurd possibilities.

I will however, say this in defense of the Nash equilibrium concept: sometimes reality really is that absurd. Just consider, for example, the "Vote Yes Hope No" caucus in the recent debt ceiling fiasco--the republicans who publicly voted no while privately lobbying for other Representatives to vote in favor of the extension. Considering that the bill's supporters almost caused the bill to fail, I guess it is possible in that case that there really was a possible equilibrium in which all GOP supporters of the bill would have voted no. But then, I attribute that not to the brilliance of the Nash concept, but rather the stupidity of the GOP when it comes to the debt ceiling.