Another interesting approach to the solution of the Ultimatum Game
I found interesting how in the analysis that you made above you make use of moral concerns about fairness or even also risk aversion to explain the outcomes of the UG in some populations. With my comment I don’t try to reject those theories but I recently read an article which gives another interpretation of certain outcomes of the UG and which I found really interesting since I think mathematical explanations can be really precise in comparison to social concerns such as morality.
A study conducted by Henrich, J. et al. analyzing 17 different ethnic groups all over the world showed that “the average offers ranged from 0.26-0.57 with a pronounced peak in the range 0.4-0.45 (9 ethnic groups). A meta-analysis of 37 papers with 75 results from UG experiments showed that, on average, the proposer offered 40% to the responder. Offers below 0.314, 18, 21, 22 or below 0.24, 12, 23 are usually rejected.”
The interesting part of these approach is that regardless of the underlying culture, such a percentage of offers can have another relation to a millenary number, the Golden Ratio.
In order to explain the relation we must first talk about what the GR is. It’s said in mathematics that two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. But how does this ratio relate with the UG?
The answer has been proposed by several economists but in my opinion, the most interesting one is the following and was proposed by Ramzi Suleiman. "the idea is that rational players strive to maximize utilities, which are functions of their actual payoffs relative to their aspired payoffs. Responders tend to accept an offer of the minor fraction of the GR, 1 − x*, because they feel that this fraction corresponds, in comparison to the larger fraction obtained by the proposer, to the ratio of the larger fraction to the whole amount. The latter is the proposer’s aspired payoff, that is, the maximum amount the proposer could get in principle. This equality of fractions can be felt by both players as a fair division.
Putting things simpler, the idea is that the proposer chooses an offer such that the responder cannot clearly approximate that the proposer gets a multiple of what he is getting. For such a strategy it can be proved that the GR is in fact the most irrational of all numbers, thus the proposer will choose a quantity which is approximately x=.618… (the reciprocal of the GR).
Since players are asked to choose a percentage of what their endowment is to offer to the responder, it can be quite difficult to offer a quantity such as the square root of x or simply choose a number as 38.57390… thus players approximate such a percentage to a ratio of 60:40. Since this ratio is commonly observed in the data, we can conclude that a solution to the UG can be related to the mystical GR.