The Best Shot Game

1 Recap We have seen that there is a lot of “fairness” in Ultimatum games. While the dictator game shows that some is driven by the proposer, it also shows that lots is driven by the responder. The 4 countries paper and the Slonim and Roth paper showed that proposers seem to adapt their behavior to respond to different responder behavior: Proposer seem to best respond to the responders behavior. The next paper will show that this extends to games beyond the UG.

2 Best Shot Game: • Player 1 provides some quantity q1 • Player 2 learns q1, and provides some quantity q2. • This results in a quantity q of public good equal to q = max{q1, q2} • The players are paid based on a (common) payoff table, such that player j’s payoff (j=1,2) is of the form pj = R(q) – cqi. • That is, players are paid some (common) reward based on the amount of public good, but they are each charged for the amount qi they provided.

The name “best shot” reflects the maximum in the formula for the amount of public good q. When it is a minimum, it’s the “weak link” game, closely related to the minimum games we played in class. 3 Equilibrium

What is the form of the subgame perfect equilibrium?

4 Harrison, Glenn W. and Jack Hirshleifer, “An Experimental Evaluation of Weakest Link/Best Shot Models of Public Goods,” Journal of Political Economy, 97, 1989, 201- 225. They reported that two-player best shot games converged quickly to the perfect equilibrium (when players 1 and 2 moved sequentially). They further write:

“No subject was informed of the payoffs of any other subject in our experiments, and in particular the fact that all valuation schedules were the same was not revealed. Our theoretical analysis, in contrast, presumes that the payoffs are common public knowledge. This informational discrepancy made our experiments quite a severe test of the underlying theory.” (emphasis added)

5 Prasnikar and Roth (1992 QJE) designed an experiment • to see if considerations of fairness might play a more visible role in a complete information best shot game; and • if not, to understand the difference between best shot and ultimatum games.

6 7 8 9 10 11 • The strategic hypothesis: the difference between the ultimatum and best shot games lies in the incentives that player 1’s face away from the perfect equilibrium path, i.e. when player 1’s offer positive amounts in the ultimatum game, or provide positive amounts in the best shot game… • In the ultimatum game, we have seen that player 1’s who offer in the 40-50% range earn more than those who offer less (e.g. in all four countries of the four country experiment). • In the best shot game, if player 1 provides a positive quantity, player 2 free rides (and so 1 earns very little when he provides positive q1), but much more when he successfully free rides.

12 13 • So, we see games like the market games and best shot games in which players quickly learn to approach perfect equilibrium behavior. On the other hand, we see games like the ultimatum game, which is in some respects quite similar to the best shot game, in which there is no evidence at all that players are approaching perfect equilibrium behavior. • A number of theoretical approaches are being explored. None of them look as “finished” as standard game theory yet. We’ll talk more about fairness and some families of learning models.

14 Fairness papers

• Ernst Fehr and Klaus M. Schmidt "A Theory of Fairness, Competition and Cooperation", Quarterly Journal of Economics, Vol. 114, No. 3 (1999), 817-868. • Gary E. Bolton and Axel Ockenfels “ERC - A Theory of Equity, Reciprocity and Competition”, American Economic Review, March 2000, 90(1), 166-193

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