November 28, 2012
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An alternative to Nash equilibrium
JDN 2456260 EDT 11:16
There is a problem with neoclassical game theory. Those who teach and use it would be the last to admit it, but still, the problem is there.
As I discussed a few days ago, the problem is that the “rationality” of neoclassical game theory can lead to results which are bad for everyone. In the extreme, yet all-too realistic example I mentioned, it could literally destroy human civilization.
It also leads to the paradoxical result that a group of “irrational” individuals can together fare much better than a group of “rational” individuals playing the same game. It can be advantageous to be “irrational.” “Rational.” You keep using that word… I do not think it means what you think it means.
Douglas Hofstadter proposed a solution, one which I would now like to extend using the same framework as Nash equilibrium. For that reason, I humbly propose it be called Julius-Hofstadter equilibrium.
Hofstadter’s proposal was called “superrationality”: The idea is that we know (by the Aumann Agreement Theorem) that two perfectly-rational agents will agree on all their beliefs about the world. If their preferences are the same, this means that their behaviors will be the same. (I will extend to non-symmetric games in a moment.)
Hence (this is the key insight), they must set their behaviors based on that knowledge. Payoffs that require two players to think differently simply aren’t valid; no superrational agent would ever get such a result.
Formally, this means we diagonalize the payoff matrix. All non-diagonal entries go to zero and get ignored. Then, we seek an equilibrium which is like a Nash equilibrium, but with one change. A Nash equilibrium asks whether each player has an incentive to change unilaterally. A Julius-Hofstadter equilibrium instead asks whether all players have an incentive to change multilaterally.
For example, consider this standard Prisoner’s Dilemma:
C
D
C
4,4
1,5
D
5,1
2,2
The neoclassical solution is to say that because defection is a dominant strategy for both places, they will both select it and stay there at Nash equilibrium. The result is that everyone gets a payoff of 2, when mutual cooperation would yield a payoff of 4 for everyone.
But when we diagonalize the matrix first, this is what we get instead:
C
D
C
4,4
0,0
D
0,0
2,2
Now this is a pure coordination game (indeed, all symmetric games reduce to pure coordination games), and there are now two Nash equilibria, one for mutual cooperation and one for mutual defection.
In fact, there is only one Julius-Hofstadter equilibrium, which is mutual cooperation. Why? Because if each player knows that the other player thinks as they do, then when they switch to cooperation, they know the other player will as well. This is strictly dominant, because it raises their payoff from 2 to 4.
There are games that have multiple Julius-Hofstadter equilibria, however. Any Nash equilibrium of a diagonalized game is a Julius-Hofstadter equilibrium as long as there is no other Nash equilibrium that has a higher payoff. All the Julius-Hofstadter equilibria of a game have equal payoffs.
For example, this coordination game has four equal Julius-Hofstadter equilibria:
A
B
C
D
A
1,1
0,0
0,0
0,0
B
0,0
1,1
0,0
0,0
C
0,0
0,0
1,1
0,0
D
0,0
0,0
0,0
1,1
What do we do if the payoffs aren’t symmetric? What then would a superrational agent do? Stay tuned for a later post.
