Manfred Langen suggested that the GR would be a plausible solution to the UG and published it in online magazines on economics 10 , 11 , Without using mathematics, he argued that a ratio that occurs frequently in nature and is known to mankind since ancient times, is accepted most easily as well-proportionate and justified. Ramzi Suleiman 23 , 29 , 39 , 40 suggested the same ratio to be the solution. Suleiman had presented the idea in lectures in several talks back in , e. His idea is based on one out of several definitions for the GR:.
That equation says that the ratio between the smaller and larger fractions is the same as the ratio between the larger fraction and the whole. This leads to the solution. The GR is used frequently in architecture, painting and photography and occurs in phyllotaxis leaf positions on plants 42 , The evolution of plants has led to situations avoiding that leaves would be exactly above each other, that is, that integer numbers of leaves would fit into integer numbers of turns Fig.
Schematic representation of plant leaves positioned according to the Golden Ratio, seen from above. Suleiman termed his theory Economic Harmony 23 , 29 , 39 , The idea is that rational players strive to maximize utilities, which are functions of their actual payoffs relative to their aspired payoffs. This equality of fractions can be felt by both players as a fair division.
Interestingly, when the idea underlying Eq. That humans assume that the proposer has some priority is supported by the observation that a majority chooses the proposer role when they are allowed to choose See ref. I had the idea of the inverse GR being the solution in ca. I found that ratio intuitively appealing because it is one of the few numbers between 0. To the best of my knowledge, all four lines of work Jasso, Langen, Suleiman and myself started independently of each other. The GR plays a role also in other games analysed in game theory.
For example, in a cyclic four-player bargaining game, a mixed Nash equilibrium exists in which the probabilities of adopting certain strategies are related to the GR In a dyadic employer-employee interaction that is related to the UG 29 , in the cognitive hierarchy model of games proposed by Camerer and coworkers 47 and in some two-player extensive form games where the payoffs are drawn at random from a given feasible set 48 , the GR results for certain parameter settings.
Here the new solution concept to the UG is presented in a much more detailed way than in ref. The explanation is partly epistemic in that I analyse the reasoning of players in the game cf. On the mathematical side, I make use of continued fractions, a tool employed in several areas in mathematics 51 , 52 and physics 53 , 54 , The goal of the paper is to contribute to providing the theory asked for by Camerer 26 and Jasso We still assume that the proposer strives to maximize her payoff taken here as equivalent to utility but acknowledge that the responder may be willing to sacrifice some payoff, be it in the hope that the offer will be higher next time or for reasons of justice.
We assume that the proposer knows this attitude of the responder and considers it in maximizing her payoff. This entails that she will not offer too little because of fear of rejection. The restriction that all a i be positive is important for making the representation as unique as possible and for confining x to be less or equal to 1. Importantly, the continued fraction representation is unique and involves infinitely many levels for irrational numbers.
For rational numbers, for which the continued fractions are finite, implying that all a i are infinitely high from some index i on, there is a slight ambiguity in the last terms.
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Truncations of an infinite continued fraction yield approximations, called convergents, of the respective irrational number. The lower the a i , the slower the approximations converge 51 , When interpreting x as given by Eq. The following property of infinite fractions is worth noting.
In contrast, it is a monotonic increasing function of a 2 because that parameter is under the second fraction bar. Generally, x is a monotonic increasing decreasing function of all a i with an even odd subscript. Moreover, the proposer has to anticipate the wishes of the responder in order to avoid that the latter rejects the offer. Thus, she strives for compromise. A good compromise is found if all a i are minimized:. Besides the compromise argument, the optimality criterion can be substantiated by two arguments: First, the obvious unique solution. Usually, the numbers a i in continued fractions are integers.
Interestingly, the optimality principle 1 can even be written with real numbers, provided that the side constraint 6 is taken into account. Second, it is related to the mathematical property that the lower the a i , the slower the approximations converge 51 , It can be proved that the continued fractions do converge 51 , 52 , It will become clear below why slow convergence is meaningful in the context of the UG. Equations 5 and 8 lead to the infinite continued fraction. As any infinite part of the fraction is the same as the entire fraction, we can write.
Although the players cannot really bargain due to the ultimatum setup, the bargaining is implicit because the proposer has to anticipate the response. Page and Nowak 4 refer to the concept of empathy.
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This opens up an epistemic approach to explaining the new solution concept, in addition to the above axiomatic approach. We here present two ways of explanation based on bargaining arguments. Elaborating on this idea, our solution concept can be explained as follows. Most responders are likely to regret renouncing to take 0. In other words, they will approximate 0. The reasoning applies as long as x can still be considered as an approximation of 0.
Our solution concept is based on the idea that the offer should be such that the responder cannot clearly say that the proposer gets a multiple of what the responder obtains. The GR is hardest to approximate by any rational numbers in the sense that it is furthest away from any ratio of two small integers.
This is related to the above-mentioned observation that the lower the coefficients in the continued fraction 5 , the slower the approximations converge 51 , For each convergent, there is no fraction with a smaller denominator that approximates the GR better than the fraction under consideration. The infinite continued fraction 9 converges as slowly as possible because it only contains the lowest positive integer, unity, so that the partial denominators are small This is expressed by the minimization criterion 7 subject to 6.
The property of being most irrational can be understood by inspecting the optimal positions of plant leaves Fig. A second way of explanation in terms of bargaining corresponds very well with the convergents of the continued fraction 9. These are quotients of consecutive Fibonacci numbers 51 , 52 , A graphic representation is shown in Fig. The first convergents are. Scheme illustrating the convergence of continued fractions to the inverse Golden Ratio 0. The justice zone suggested by Vermunt 37 and Jasso 36 to be between equipartition and the inverse GR is indicated.
The Fibonacci numbers are defined by the recursive equation This leads to the series 1, 1, 2, 3, 5, 8, 13, …. It can easily be shown that the ratio of two consecutive Fibonacci numbers tends to the GR 57 :.
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In striking accordance with these properties of the convergents, the reasoning by many proposers can be subdivided into the following steps, thus representing an iterative bargaining. However, the answer by the responder is clear: She would certainly decline. That first estimate is equal to the first convergent 11a of the continued fraction 9.
It can be derived from a minimization principle applied to fraction 5 :. She is very likely to accept because this is the fairest division. This assumption is very plausible and is used by many authors in the field.
The responder could push up offers by rejecting offers that are equal to, or even higher than 0. Interestingly, the ethnical group of the Lamalera in Indonesia were found to make average offers larger than 0. However, the overwhelming majority of observations show that responders are not that hard-boiled to reject high offers, all the more as they do not have the means of communicating that threat in a one-shot game unlike in iterated versions of the UG.
Moreover, equipartition is the solution to many symmetric games.
The offer made in step ii corresponds to Eq. It can be derived from the minimization principle:. The two steps i and ii need not be gone in the given order. This does not, however, affect our line of reasoning. The question is by which factor the share of the responder can be diminished. The answer by the responder is more difficult to predict. Empirical observations show that some responders accept and some decline.
There is a case for her declining because she may be disappointed to get only half of what the proposer gets see above. It is too obvious that she has a pronounced disadvantage by this division. A clever proposer will anticipate that the responder tries to push up offers, but still leaving more than half to the proposer.
The fraction expressed by Eq. Then, the fraction kept by the proposer is given by Eq.
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We can see that the convergents truncated fractions alternate around the inverse GR Fig. This is related to the above-mentioned observation that the a i with even and odd subscripts contribute in different ways to x , notably increasing or decreasing it, respectively. The disadvantage in comparison to the share of the proposer can be accepted by considering it to be minor. This, however, may inspire the proposer to slightly lower the offer. Extending the reasoning in terms of mental bargaining in a straightforward way, the offer ideally converges to the expression given in Eq.
It is worth noting that Fibonacci numbers and the GR occur in various problems in physics 53 , 54 , 55 , A prominent example is provided by resistance ladders 53 , In fact, these can be used for illustrated the reasoning in terms of convergents Supplementary Information.
The GR solution also has the following interesting property for a proof, see Supplementary Information. Assume that the proposer will play the same game with a second responder, using her part x and offering the same fractional part of it as she offered before to the first responder. If the proposer divides according to the GR, the proposer keeps, after the second game, as much as the first responder received in the first. As the proposer has to express her offer in usual numbers rather than by mentioning the GR or the square root of 5, she will usually round the offer. This coincidental equality of the rounded value with a ratio of Fibonacci numbers makes this approximate solution very appealing.
It is in agreement with a frequently observed offer 19 , 20 , Another argument for the solution is that it is risky to offer the lowest amount she thinks the responder will accept. It certainly increases chances of acceptance if the offer is increased by a certain margin. Offers are often rounded to full or half units of the respective currency rather than to full 10 percent units. It is worth noting that there are other number series that lead to the GR as well, for example, the Lucas numbers These are less relevant for the solution concept presented because the convergents of the infinite continued fraction 9 involve Fibonacci numbers rather than any other numbers.
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