Saturday, 25 July 2015

Nash equilibria in the ultimatum game

The ultimatum game is one of the most well known and well studied games. Yet there still seems much confusion and misunderstanding over the basic theory behind the game. This is worrying because it means one of the key lessons we can learn from the ultimatum game goes unrecognised.
       In the ultimatum game a proposer and responder need to decide how to split, say, $10. The proposer moves first by making a take-it-or-leave-it-offer. The responder then either accepts the offer or rejects it. If he accepts it the deal is done. If he rejects both get nothing. For example, the proposer could offer $3. If the offer is accepted the proposer gets $7 and the receiver $3. If it is rejected both get $0. 
      What is going to happen? In the experimental lab the almost universal outcome is that the proposer offers $5 and this is accepted. We also know, although the evidence for this is less unequivocal, that offers of less than $5 can be rejected.  
       What does theory say should happen? Here is the standard version you can find in the textbooks: The responder should accept any positive offer - because something is better than nothing. The proposer should, therefore, offer $0.01.
        The $0.01 'prediction' is clearly well wide of the mark. That leads many to the basic claim that 'game theory is wrong!'.
      Things, however, are far more complicated than that (even if we ignore social preferences). The standard prediction corresponds to the sub-game perfect Nash equilibrium of the game. There are though many other Nash equilibria. Indeed any offer is consistent with Nash equilibrium.
      To illustrate, suppose the responder will reject any offer less than, say, $5 and the proposer knows this. Then the proposer should offer $5. In other words, 'offer $5 and reject any offer less than $5' is a Nash equilibrium. This equilibrium is not sub-game perfect because it involves an incredible threat from the responder that he will reject an offer below $5. There is, though, no necessity that threats be credible.
       The behaviour we observe in the ultimatum game is, therefore, consistent with Nash equilibrium. Game theory is not wrong! We do need to question the relevance of sub-game perfection. But, sub-game perfection is only one of many equilibrium refinements that have been suggested by game theorists over the years. Sub-game perfection seems ill-suited to the ultimatum game. More suitable seems a refinement based on fairness principles or focal points (as advocated by Thomas Schelling in the The Strategy of Conflict way back in 1960).
       The ultimatum game is not the only challenge to sub-game perfection. Somewhat worryingly, however, sub-game perfection is still far and away the most common refinement concept used in economics. For instance, virtually every theoretical paper in industrial organization solves for the sub-game perfect Nash equilibrium. How reliable are the predictions from such an approach likely to be? Probably not very reliable at all.
          The way that game theory is applied may, therefore, need something of an overhaul. We certainly need to give consideration to Nash equilibria that are not sub-game perfect.