Game Theory Thoughtful Thursday

I just completed at Columbia a game theory course. Of course, I won't try and reproduce all I learned, let a lone all of game theory. There are plenty of good explanations such as Wikipedia, the first two chapters of our textbook by Osborne, and the entire text of A Course in Game Theory. But a basic problem is Prisoner's Dilemma, about which many of you have heard. For us, Dr Clarke reworded the problem as a public goods problem 1. Two neighbors share a yard. It costs $150.00 a year to plant and maintain a nice garden in the yard. Each would get $100.00 worth of enjoyment should the garden be planted. Each neighbor has a choice of contributing to what is a public good or reneging. The payoff matrix is

	Do Not Contribute	Contribute
Do not Contribute	0,0	$100,-$50.00
Contribute	-$50,$100	+$25.00,+$25.00

We assume that a person will continue planting once they started, even if the other person was not contributing their share. In a classic Nash equilibria, the parties would be stuck at both do not contribute. They just don't see the optimal solution Some look at it as a min-max solution or as Stahl put it, a no-regrets solution. I see it as both parties stuck at a local equilibrium. We studied the "Stag-Hare" problem. Here k out of n hunters can cooperate to get a Stag that they would share. It could just as easily be k people in a community building a community center or paying a share of any public good. Or they could all just do their own thing, hunt a "hare." The two Nash equilibria are everyone does their own thing or k people hunt a Stag. If everyone does their own thing, then one person isn't going to go off and hunt a Stag. They just won't catch it. Similarly if k hunters are happily searching for their Stag, there is reason for one to go off and do their own thing as they lose their chance to catch the Stag. The Nash equilibria theory simply does not provide for k people agreeing to hunt the Stag and does not say which equilibrium they would end up at. It just says they will get stuck at one like a hill climbing algorithm. (I asked my professor, aren't you just saying that the people will be stuck at a local equilibrium. She said yes.)

The mathematical theory says that the same thing would happen if both of our two homeowners knew they would share the yard for fifty years. The Subgame Perfect Equilibria of the finitely repeated game gives the same result. However, if the game is repeated indefintely, then both parties would contribute yielding the net benefit for both of them. The assumption is there is a discount factor that determines how much a garden in the future is worth compared to a garden this syear. If both parties are infinitely patient then the lower right result of both parties contributing to the garden prevails-- exactly opposite the Nash Equilibrium approach! If Party A decides to welch, not contributing while allowing the other party plant the garden, the other party welches the next time--the famous tit-for-tat. If Party A is somewhat impatient, it might take a few more seasons of "punishment" to get Party B to cooperate. The Nash Folk Theorem says that we can get the parties cooperating if the parties are sufficiently patient.

Hotelling used game theory to show that in a two-way election like we have with the two-party system, the parties will say about the same thing. If Candidate A is off to one side of the median position, then Candidate B will go to the median and win the election. And as Theiss-Morse showed, the American people on average think the government is at the right place on the left-right spectrum. (As many people think the government should be further to the left as further to the right.) But they would like to see more participatory democracy.

Reference

1. Clarke, Demand Revelation and the Provision of Public Goods Ballinger Press, Cambridge, 1980.