The Golden Ratio as a proposed solution of the Ultimatum Game: An explanation by continued fractions

TL;DR

This paper explores the use of continued fractions to explain why the Golden Ratio might be a fair division in the Ultimatum Game, offering an alternative to existing psychological theories.

Contribution

It introduces a novel explanation based on infinite continued fractions for the Golden Ratio as a fair split in the Ultimatum Game, complementing previous psychological models.

Findings

Proposes continued fractions as an explanation for the Golden Ratio in fairness

Provides a mathematical justification for the Golden Ratio as a fair division

Offers an alternative to psychological theories of fairness in the game

Abstract

The Ultimatum Game is a famous sequential, two-player game intensely studied in Game Theory. A proposer can offer a certain fraction of some amount of a valuable good, for example, money. A responder can either accept, in which case the money is shared accordingly, or reject the offer, in which case the two players receive nothing. While most authors suggest that the fairest split of 50 % vs. 50 % would be the equilibrium solution, recently R. Suleiman (An aspirations-homeostasis theory of interactive decisions (2014)) suggested the Golden Ratio, 0.618, to be the solution and argued that such a partitioning would be considered fair by both sides. He provided a justification in terms of an approach termed aspirations-homeostasis theory. The main idea is that responders tend to accept the minor fraction of the Golden Ratio because they feel that this fraction equals, in comparison to the larger fraction obtained by the proposer, the ratio of the larger fraction and the whole amount. Here we give an alternative explanation to that suggested solution, which complements the reasoning by Suleiman (2014) and is based on infinite continued fractions.