Winning II

 

Step to the Rear

Will everyone here kindly step to the rear
And let a winner lead the way
Here's where we separate
The notes from the noise
The men from the boys
The rose from the poison ivy

So what does it take to be a winner?

The Ultimatum Game is an instrument in Economics and Game Theory. One player, the proposer, is given a sum of money, say $100. The proposer must split it with another player, the responder. Once the proposer makes his offer, the responder may accept it or reject it. If the responder accepts, the money is split according to the offer. If the responder rejects the offer, both players receive nothing.

A “fair” distribution would be a 50/50 split by the proposer.  That is not the User Optimal solution for the proposer.  The User Optimal solution of the proposer should be to offer the lowest non-zero amount and keep the balance. The User Optimal solution of the responder is to accept any non-zero offer by the proposer.  In the case of a $100 gift and integer offers, an offer of a $1 from the proposer to the responder should be accepted, since it would be the User Optimal solution of both players.

However in experiments, (Güth, Schmittberger, & Schwarze, 1982), responders typically rejected offers of less than 30%.  This was verified in different societies and cultural settings (Henrich, 2004) (Oosterbeek, 2004). Why did responders reject offers of less than 30% when that was not their User Optimal solution? With the User Optimal solution, a responder would at least have the offer, instead of both players having nothing.

The answer is that in that offer, the proposer is not only making an offer.  He is also conveying his expectation for repeating this game with the roles reversed, and if the game is repeated, then the proposer is conveying the discount rate that the proposer places on the future.  The System Optimal solution is the sum of the accepted offers in repeated games.  If the offer is 50/50 in each game, and the roles are reversed after each game, then after two games each player would have $100 and the System of all players would have $200.  If each player follows the User Optimal Solution in each game and makes an unfair offer and that offer is rejected, then after two games both players have $0 and the System has $0.

If the offer follows a User Optimal strategy and offers the bare minimum, then in addition to that offer the proposer is conveying information that he does not expect to ever repeat this game with the roles reversed OR if the roles are reversed and the game is repeated that he has a very high discount rate (i.e. places a very low value on the future).  The System Optimal solution requires that the game be repeated and that a reasonable discount rate is used to value that future.

If the offer is a 50/50 split then the offeror may be conveying that he does expect the game to be repeated with reversed roles or his discount rate is 0% CAGR.  If the offer is $30, then the proposer is also conveying that he expects the game to be repeated with roles reversed,but if it is repeated once per year then his discount rate is 40% CAGR, ( $50-$30)/$50.  When the responder rejects an offer, then the responder is responding not only for himself as a User, but for the System of all potential players.  By rejecting the offer, the responder is indicating that the proposer should be excluded from the System because the proposer either does not believe in a future, or if he does believe in the future he places a low value, i.e. a high discount rate, on that future.

Even though the System does not appear to have a role in the Ultimatum game, the responder is playing not only for themselves as a User,  but for the entire System. The winner is the System, not the player.

References

Güth, W., Schmittberger, R., & Schwarze, B. (1982). "An experimental analysis of ultimatum bargaining". Journal of Economic Behavior & Organization. 3 (4), 367–388.

Henrich, J. R. (2004). Foundations of Human Sociality: Economic Experiments and Ethnographic Evidence from Fifteen Small-Scale Societies. Oxford University Press.

Oosterbeek, H. R. (2004). "Cultural Differences in Ultimatum Game Experiments: Evidence from a Meta-Analysis". Experimental Economics. 7 (2), 171–188.