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.
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. 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 :. 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. 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.
For a more detailed discussion, see the Supplementary Information. Inspecting the observations reported by Henrich and coworkers 19 in more detail shows that the mean value of offers observed with the ethnical group of Gnau in Papua New Guinea, 0.
Also very close are the average offers observed with the Kazakh 0. Although it is speculative at this point to draw a conclusion, the coincidence of the average offer of 0. Suleiman 40 showed by a Two One-Sided Test TOST that the agreement of the theoretical prediction in terms of the GR is statistically significant both with the empirical data compiled by Henrich and coworkers 19 and with the meta-analysis by Oosterbeek and coworkers Seven out of 19 accepted offers were 0.
One proposer requested 5. Performing a precise statistical test with the data on the UG is difficult because of the rather wide and complex distribution of data. Thus, it is of interest to look not only at the average values from the experimental studies but also at the distribution of offers. A considerable fraction of people indeed make the equipartition offer of 0. This is well explained by the second convergent as given in Eq.
Figure 2 in ref. An observation reported by many authors is that offers below 0. Here we have presented a new solution concept for explaining the hypothesis that a division according to the Golden Ratio GR would characterize the relevant Nash equilibrium of the Ultimatum Game UG. That equilibrium implies that the proposer offers about Our solution concept is based on an optimization criterion, which can be interpreted by epistemic arguments or by a bargaining perspective.
To disguise the disadvantage for the responder best, it is favourable for the proposer to make an offer that is farthest away from any ratio of integer numbers, that is, an offer near the inverse GR. Such an offer is hardest to evaluate by the responder in view of whether it is still near equipartition. It is the natural boundary above which an unjustified inequity is no longer accepted In the light of the observation that people have learned rules of behaviour from iterated games that they apply also to one-shot games 14 , we here use arguments from both perspectives: that of one-shot games and that of iterated games.
Our above arguments are alternative and complementary to the explanation given by Suleiman 39 , In game theory in general, several alternative solution concepts 50 such as correlated equilibrium 60 , Kantian equilibrium 61 , and cooperative equilibrium 49 have been proposed, which even imply solutions that differ from Nash equilibria 50 , 60 , Correlated equilibrium means that in games where mixed strategies are relevant such as in a coordination game or Hawk-dove game in the case where information exchange between players is excluded , the average payoff is increased for both players when the decision probabilities are correlated In the concept of Kantian equilibrium 61 and the related co-action concept 62 , each agent maximizes her payoff assuming that all other agents in a symmetric situation will make the same decision.
These concepts were devised for non-sequential games, so that they are not immediately applicable to the UG which is, in addition, asymmetric. The key idea of the cooperative equilibrium concept 49 is that players form coalitions see Introduction. Justice theory predicts a single point of perfect justice equipartition in the case of the UG and a continuum of justice evaluation magnitudes Along those lines, one may argue that the set of Nash equilibria should be narrowed down to the region between 0.
Or the proposer may follow an alternating iteration in her mind in terms of predicted upper and lower bounds of what would probably be accepted by the responder. This is usually made in terms of ratios of small integers. Persons with a strong intuition may perform that iteration almost instantaneously.
In future experiments, test persons may be asked e. It is certainly of interest to investigate, in theoretical and experimental studies, iterated versions of the UG see ref.
There are different possible versions of iteration: The players may keep the total payoff of all rounds, or only the payoff of the last round, the numbers of rounds may or may not be known a priori, proposers may meet new responders or always the same, players may have full or incomplete information about previous outcomes etc. As mentioned above, the GR is relevant in phyllotaxis 42 and can there be derived from an optimality principle as well There appears to be a relationship between that observation and our solution concept Fig.
In the UG, the division is made so as to avoid that the fraction claimed by the proposer is a clear multiple of the offer. Above, we have explained the frequently observed offers of ca.
The proposer needs to express her offer in the decimal system or a ratio of integers rather than by an intuitively felt ratio. Thus, it is unlikely that she would propose This is reminiscent of the various approximations of the GR by ratios of Fibonacci numbers realized in phyllotaxis 42 , Ratios of two consecutive Fibonacci numbers turns divided by number of leaves are realized by numerous plants, apparently with lower effort than the GR.
It can be assumed that the less effort is spent, the less accurate is the approximation of the Golden Ratio. The Fibonacci series is also relevant in many other applications, for example, the enumeration of polyphenanthrenes in chemistry 63 and fatty acids The GR also arises in the solution of some problems in electric and fluidic circuits 53 , 54 and nonlinear dynamics in physics 55 , 59 see Supplementary Information and in dynamic optimization of biochemical pathways Another explanation follows from a bargaining perspective Thomas Pfeiffer, personal communication : When the responder asks to get half of the money, the proposer asks to get half of that i.
Although I do not claim that the above results can directly be applied to more complex bargaining situations in which one side has a priority whatsoever, such as salary negotiations, there are obvious relationships to such situations 10 , 11 , 29 , Offered salaries can be accepted or rejected by employees.
If all employees reject the offer and do not work, neither side gets anything. At the scale of societies, too low wages lead to social unrest. Langen 10 , 11 suggested without proof that a fair salary system in a hierarchically structured company should imply that the income increases from level to level by the Golden Ratio cf.
As mentioned by Langen 10 , 11 , until two decades ago, there was an unwritten rule in large companies that the salary of a top manager should not exceed twentyfold the wage of a skilled worker.
On the basis of a Golden Ratio system, this corresponds to 6—7 levels in the hierarchy. Thus, the rule of the ratio of 20 might be based on an empirical application of the Golden Ratio. As mentioned in the Introduction, continuous frequency distributions of offers and of the acceptance probability as a function of the offered values can be determined empirically. It will be very interesting in future theoretical studies to compute optimal frequency distributions by variational calculus. This may be related to mixed Nash equilibria, which have scarcely been analysed for the UG so far 17 , It may be hypothesized, based on the above line of reasoning, that this characteristic value would be near the inverse GR.
The UG might even be relevant in microbiology. Several micro-organisms show the so-called sequential cross-feeding 67 , That is, a product excreted by one of them is taken up as a nutrient by the other. For example, some strain of the bacterium, Escherichia coli may take up glucose and convert it into acetate.
This can be taken up by another strain of the same species, which converts it further into CO 2 and water. During evolution, the two strains or species must come to an agreement about the metabolic intermediate to be excreted and taken up. It might have been ethanol or lactate as is the case for other pairs of species rather than acetate.
If the responder rejects the offered intermediate substance, the proposer will not survive either or at least be harmed because substances such as acetate and ethanol are toxic in higher concentrations, which is very reminiscent of an UG. On the other hand, they are not the poorest substrates possible 69 , so that they do not correspond to the subgame-perfect equilibrium.
This can be understood from a population perspective. If the responders only obtain very little energy, they cannot build up a population size sufficient to remove the harmful intermediates. The last offer from either side may be understood as an offer of a percentage of the dealer?
Bilateral trade negotiations: If the Bilateral trade negotiations break down, the gains from trade might get lost. Failure to pass legislation: Political coalition falls apart over the failure to agree on the distribution of economic rents. Negotiations for peace between two disputing countries: Such case can also be considered a pie-splitting game, take for example the dispute over the territory of Jerusalem between the Palestinian Liberation Organization and Israel.
Rejection of the last offer can often lead to an escalation of violence or war. These examples come from Dickinson []. Dickinson, David L.
Forsythe, Robert, Joel L. Horowitz, N. Charles Holt's website hosts a detailed bibliography on ultimatum game experiments. Alvin Roth's website has a bibliography of bargaining experiments and related papers.
Based on fMRI studies of the brain during decision-making, different brain regions activate dependent upon whether the participating subject "accepts" or "declines" an offer. Since to "decline" means that neither receives any money, the responder is actually "punishing" the player who makes a low offer. Punishing activates the part of the brain that is associated with the dopamine pathway — i. Hence, the subjects who refuse and punish in the process, possibly receive more pleasure from punishment than they would from accepting a low offer.
This is, therefore, an expected utility argument where the currency is in pleasures received rather than goods or their associated values in money. The classical explanation of the Ultimatum game as a well-formed experiment approximating general behaviour often leads to a conclusion that the Homo economicus model of economic self-interest is incomplete. However, several competing models suggest ways to bring the cultural preferences of the players within the optimized utility function of the players in such a way as to preserve the utility maximizing agent as a feature of microeconomics.
For example, researchers have found that Mongolian proposers tend to offer even splits despite knowing that very unequal splits are almost always accepted. Similar results from other small-scale societies players have led some researchers to conclude that " reputation " is seen as more important than any economic reward. This decay tends to be seen in other iterated games.
However, this explanation bounded rationality is less commonly offered now, in light of empirical evidence against it. It has been hypothesised e. The concept here is that if the amount to be split were ten million dollars a split would probably be accepted rather than spurning a million dollar offer.
Essentially, this explanation says that the absolute amount of the endowment is not significant enough to produce strategically optimal behaviour. However, many experiments have been performed where the amount offered was substantial: studies by Cameron and Hoffman et al.
Rejections are reportedly independent of the stakes as this level, with 30 USD offers being turned down in Indonesia, as in the United States , even though this equates to two week's wages in Indonesia. Other authors have used evolutionary game theory to explain behavior in the Ultimatum Game. These authors have attempted to provide increasingly complex models to explain fair behavior. The split dollar game is important from a sociological perspective, because it illustrates the human willingness to accept injustice and social inequality.
The extent to which people are willing to tolerate different distributions of the reward from " cooperative " ventures results in inequality that is, measurably, exponential across the strata of management within large corporations. See also: Inequity Aversion within companies.
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