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Economic Harmony: A Theory of Cooperation between Egoists
Ramzi Suleiman
University of Haifa
Working Paper
March 30, 2013
Keywords: Ultimatum Game, Dictator Game, Public Goods, Cooperation, Fairness, Social Justice,
ERC, inequality aversion, Golden Ratio.
Please address all correspondence to:
Dr. Ramzi Suleiman
Department of Psychology
University of Haifa
Haifa 31509, Israel
Email: suleiman@psy.haifa.ac.il,
Mobiles: + 972-(0)50-474- 215, +31-(0)6-8616-4553
Homepage: http://suleiman.haifa.ac.il
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Economic Harmony: A Theory of Cooperation between Egoists
Abstract
I propose a theory of cooperation, called Economic Harmony Theory (EHT), which assumes that
individuals strive to maximize their own payoffs relative to subjective reference points. EHT further
assumes that (1) different players might adhere to different reference points; (2) they are aware of
the prevailing norms of equality and equity in wealth distribution; and (3) there exists an effective
mechanism by which sanctions could be applied to deviations from group norms. Based on these
assumptions, I derive testable point-predictions of proposers' offers in the standard ultimatum game
(Güth, Schmittberger & Schwartze, 1982), in a mini-ultimatum game with information about the
proposers' intentions (Falk, Fehr & Fischbacher, 2003), in a three-person ultimatum game (Kagel &
Wolfe, 2001), and a three-person common-pool resource-dilemma game (Budescu, Suleiman &
Rapoport, 1995; Suleiman & Budescu, 1999). For the standard ultimatum game, EHT yields a
symmetric solution at the 50-50 split, and an asymmetric solution at the Golden Ratio, according to
which the proposer keeps a portion of ≈62% of the entire amount and transfer ≈38% to the
responder. Computer simulations of the repeated ultimatum game, in which two automatons play
according to a simple reinforcement learning rule (Roth & Erev, 1995), show that the proposers'
demands converge to the Golden Ratio division.
EHT predicts the level of cooperation (≈ 60/40 split) observed in numerous ultimatum experiments
conducted in industrialized countries (Oosterbeek, Sloof & Van de Kuilen, 2004) and in small-scale
societies (Henrich, 2006). It also accounts for the cooperation observed in a class of three-person
games. For all the tested games, Although EHT does no assume other-regarding preferences or
hard-wired altruism (Fehr & Gachter, 2002), it outperforms existing theories of cooperation,
including ERC of Bolton & Ockenfels (2000) and Inequality Aversion of Fehr & Schmidt (1999).
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1. Introduction
The observation of human behavior in real life and in controlled laboratory experiments reveals that
individuals frequently cooperate with genetically unrelated strangers whom they will never meet
again, even when such cooperation is costly to them (Sober & Wilson, 1998). This behavior is
puzzling, since it contradicts the law of natural selection, which works against cooperators and in
favor of free-riding. Several theories have been proposed to explain cooperative behavior, among
which are the theory of kin selection (Hamilton, 1964) and theories of direct and indirect (Axelrod
& Hamilton, 1981; Nowak & Sigmund, 1998) reciprocity. These theories fail to explain why
humans cooperate with strangers, when interactions are not repeated and reputation effects are
absent. Costly punishment, which as neurobiological tests indicate derives from a hard-wired taste
for fairness (De Quervain, 2004), has been proposed as an answer. Laboratory experiments, using
the public-goods game (Fehr & Gachter, 2002; Fehr & Fischbacher, 2004; Gächter, Renner &
Sefton, 2008; Gintis, 2008) and the ultimatum game (Xiao & Houser, 2005, 2009; Ellingsen &
Johannesson, 2008; Yamagishi, 2009; Suleiman & Samid, 2010), show that individuals are willing
to incur costs in order to punish unfair or non-cooperative others. Punishment might be driven by
selfish motives, as suggested by findings from many ultimatum game experiments, or by altruistic
motives, i.e., when punishment is costly and yields no material benefits to the punisher, as
suggested by the findings of many experiments on altruistic punishment in public goods games.
Several economic theories have been proposed to account for the cooperation observed in strategic
interactions by incorporating a component of fairness into the standard economic model. Two
significant attempts in this direction are Bolton & Ockenfels (2000) - hereafter BO - theory of
Equity, Reciprocity and Competition (or ERC) and Fehr & Schmidt's (1999) - hereafter FS -
Inequality Aversion theory. The two theories assume that, in addition to the motivation for
maximizing own payoffs, individuals are motivated to reduce the difference in payoffs between
themselves and others, although with greater distaste for having lower, rather than higher, earnings
than others (Kagel & Wolfe, 2001) - hereafter KW. The two theories have proven to be successful
in organizing a large body of experimental data. For example, they can explain why behavior in
competitive market experiments with complete contracts converges to the prediction of the game
theoretic model, whereas in two-person bargaining games, like the ultimatum game, strong
deviations from the standard model prediction towards more equitable allocations are observed.
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Notwithstanding, both theories fail to account for the effect of intentionality observed in the mini-
ultimatum game by Falk et al. (2003) and, more importantly, they are strongly refuted by the
outcomes of three-person ultimatum experiments designed particularly to test their predictions
(Güth & Van Damme, 1998; Kagel & Wolfe, 2001). For example, in KW's experiment one
proposer allocates a sum of money between herself and two other players, one of which is randomly
chosen as responder and the other as non-responder. If the responder accepts, then each player
receives her allocation. If she rejects, then the proposer and the responder get zero and the non-
responder gets a consolation prize. The results show no reduction in rejection rates, holding offers
constant, with and without consolation prizes, contrary to both models.
The present article constitutes an effort to explain fairness based on a new theory of cooperation,
called economic harmony Theory, or EHT. The theory assumes that individuals base their decisions
solely on their self-interest. I demonstrate that EHT outperforms the previously mentioned theories
of cooperation in accounting for the existing two-person ultimatum data. It also succeeds in
predicting the results reported in several other games including the Falk et al. mini-ultimatum game
and the KW three-person ultimatum game.
2. Theory
EHT is based on the following four propositions:
1. Individuals are solely self-regarding players.
2. When making their decisions, individuals consider their payoffs relative to subjective reference
points (SRPs), rather than their absolute payoffs. A SRP can be social, when a player compares her
payoff to the payoff(s) of another member or members in her group (e.g., a co-worker's salary), or
non-social, when she compares her payoff with a neutral (non-social) reference point (e.g., her
expected expenditure).
3. Individuals are aware of the norms of equality and equity, as they are practiced in their social
group.
4. There exists a formal or informal sanctioning mechanism, by which sanctions are applied on
deviants from the group's norms and rules.
The assumption that players are cognizant of the norms of equality and equity does not mean that
they have a taste for fairness. Rather, it is assumed that individuals use the information about
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existing social norms of equality and equity and the sanctions for deviations from these norms for
their own self-interest.
The novelty of the theory lies in the plausible assumption that different players are motivated by
different reference points, and that adherence to a certain reference point is influenced by the social
structure of the interaction and by the position of the player in the social group. In real life, this
assumption is not hard to defend, not only because different individuals have different motivations
and intentions, but also because individual motivations and intentions are strongly affected by the
determinants of their position in a specific social setting. An employee who earns a monthly salary
of $x might compare her salary with the salary of another workmate, or with the average salary for
workers with similar expertise (social FPs), but she might also compare her salary with the salary
she could have received had she chosen another job offer, or with her monthly expenses (non-social
FPs). Her employer might compare the net income generated by the employee with the salary she
pays her employee, by the income generated by another equally paid employee (social FPs), or she
might compare her net income with a projected profit which could guarantee a minimum growth
rate of her workplace. The point to make here is that all intersections of employer-employee
reference points are plausible.
The present article focuses mainly on ultimatum bargaining and uses the proposed theory to derive
predictions of proposers' offers in class of games, including the standard ultimatum game (Güth,
Schmittberger & Schwartze, 1982), a mini-ultimatum games with intentionality (Falk et al., 2003),
a three-person ultimatum game (Kagel & Wolfe, 2001) and a three-person common pool resource
(CPR) game (Budescu, Suleiman & Rapoport, 1995). For the standard ultimatum game I report the
results of a computer simulation devised to model a repeated game between automatons who update
their behavior according to a simple learning rule. Comparison of the predictions of EHT with the
predictions of the SPE model, and with predictions of two major theories of economic cooperation,
namely, ERC (Bolton & Ockenfels, 2000) and Inequality Aversion (Fehr & Schmidt, 1999), shows
that that EHT outperforms all the aforementioned theories, and yields excellent point-predictions
for several experiments using the four above mentioned games. Generalization of the theory to
account for other (n-person) games, like the public good game with punishment (Fehr & Gachter,
2002; Fehr & Fischbacher, 2004), and possible applications, particularly in the area of corporate
efficiency and employee salaries, are briefly discussed.
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3. The Standard Ultimatum Game
In the one-stage ultimatum game, the proposer makes an offer (x, 1-x), for herself and a designated
recipient, respectively. The recipient responds by either accepting the offer, in which case both
players receive their offered shares, or by rejecting the offer, in which case the two players receive
nothing. The ultimatum game has proven to be a potent workhorse for studying selfishness,
fairness, cooperation, competition, and punishment (Kahneman, Knetsch & Thaler, 1986; Kagel,
Kim & Moser, 1996; Matthew, 1993; Prasnikar & Roth, 1992; Suleiman, 1996). It is well
documented that the modal and mean offers in the game are about 50% and 40%, respectively, and
that offers of 20% or less are rejected with high probability (Camerer, 2003; Suleiman, 1996).
While the behavior of the recipient, and its emotional correlates, have been attracting a growing
research effort in the social and brain sciences (Pillutla & Murnighan, 1996; Sanfey et al., 2003;
Xiao & Houser, 2005; Yamagishi et al., 2009), interest in the proposer’s behavior has been
declining. There seems to be a consensus among researchers that what appear to be fair offers are
motivated mainly by self-regarding sentiments, and much less by other-regarding ones. By
proposing a reasonable offer, a rational proposer increases the probability that her offer will be
accepted. The fact that proposers care more about appearing fair (out of self-interest), and less
about being fair, has nicely been demonstrated by Kagel et al. (1996).
The question remains: Why do proposers offer on average of about 40% of the entire amount, and
not, say, 45% or 55% or maybe 35%? Despite hundreds of studies on the ultimatum game, which
replicate the 60/40-split result, the explanation of this finding remains elusive.
I show that application of the proposed EHT to the standard UG yields two points of balance: the
equal split (1/2, 1/2) and the split (Φ, 1-Φ) ≈ (0.38, 0.62), where Φ is the well-known Golden Ratio
equaling Φ = √5 1
2 ≈ 0.62) (Livio, 2002; Olsen, 2006).
To derive the solutions for the standard UG, consider all possible subjective reference points (RPs)
of both proposer (P) and responder (R). This yields two symmetric (P-social, R-social), (P-Non-
social, R-Non-social), and two asymmetric (P-social, R-Non-social), (P-Non-social, R-social),
intersection points. We consider these points in turn:
1. P-social, R-social: The only RP for each player is the expected payoff of his/her counterpart. If P
offers a division of (x, y), then social comparison implies that "balance" between the players'
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relative payoffs could be achieved if
= 1. Since y = 1- x, the point of balance is the equal split
(1/2, 1/2).
2. P-Non-social, R-Non-social: At this intersection point, balance requires that
=
, yielding the
equal split (1/2, 1/2).
3. P-Non-social, R-social: At this asymmetric point P compares her payoff to the entire amount that
was in her property at the initiation of the game, i.e.,
, while R compares his payoff to P's share;
i.e.,
. Balance is achieved if the R’s portion relative to the P's portion
1 x
x equals the P’s portion
relative to the entire amount (x
1 or
1 x
x =
x
1 …… (1)
Which could be rewritten as:
x2 x 1 0. …… (2)
Solving for , the harmony demand, we obtain = √
= (-
√
, √
), or
approximately: - 1.618 and 0.618, respectively. For positive amounts, balance is achieved for a
partition of (√
,
√
), or about (0.62, 0.38) shares for P and R, respectively.
4. P- social, R-Non-social: At this asymmetric point P compares her payoff to R's share; i.e., x/y; R
compares his payoff to the entire amount, i.e., y/1, and the point of balance in relative payoffs
should satisfy: x/y = y/1, (x +y =1), for which the positive solution x= √
and a partition of about
Interestingly, the asymmetric solution √
is the famous Golden Ratio,
φ
√
, where is the n
th term of the Fibonacci Series
(Posamentier & Lehmann, 2007): 0, 1, 1, 2, 3, 5, 8, 13, 21, …, in which each term is equal to the
sum of the two preceding terms ( While the equal split solution is in agreement
the prediction of the Equality Principle and by focal points considerations (Jost & Kay, 2010;
Messick & Sentis, 1983; Schelling, 1980), the asymmetric solution at the Golden ratio is a novel
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one. It emerges from the model's assumption that different players may compare their payoffs to
different reference points. Because the Golden Ratio is usually associated with harmony (Livio,
2002; Olsen, 2006), I refer hereafter to "balance" points also as "harmony" points.
To summarize, the analysis above yields two points of balance or harmony, one at the (symmetric)
equal division (1/2, 1/2) and the other at the asymmetric division at which the proposer receives a
payoff of √5 1
2 0.62 (62%) of the entire amount and the responder receives 1-
√5 1
2 0.38
(38%). It makes sense to argue that from the point of view of the proposer, the amount that she
could have received, had she retained the entire amount for herself, is most likely to be the
preferred reference point. Conversely, from the standpoint of the recipient comparison with the
proposer’s portion is more probable than other comparisons.
It is important to note that the balance or harmony point at the Golden Ratio ( is not in
equilibrium. For example the proposer will benefit from unilaterally deviating from the (62, 38)
split of 100 monetary units, say, to (70, 30). By doing so, she will increase her payoff, in equity
terms, from φ% (≈62%) to 70%, while reducing the responder's relative payoff from φ% ≈62%
(38/62), to 43% (30/70). Obviously, for proposers the best response is the SPE, offer (almost zero).
For the harmony point to be in equilibrium, it must be supported by an effective punishment, such
that the proposer's negative payoff of such punishment, in relative terms, should exceed her benefit
from deviating from harmony. In mathematical terms, the punishment P(x) inflicted on a proposer
who increases her demand from the harmony point to x (x> ) should satisfy P(x) > x -
3.1 Computer Simulation of the Ultimatum Game
The third proposition, stated in section 2, prescribes that individuals are aware of their group's
norms regarding equality and equity. This assumption could be relaxed if individual players can
learn from experience, as a result of sanctions applied on norm deviations and rewards granted to
norms-abiders and keepers. I investigated this conjecture using a computer simulation of the
repeated ultimatum game. Specifically, we used the constructed simulation to test whether a simple
reinforcement learning model (Roth & Erev, 1995) would drive the dynamics of the interaction to
the harmony point, at which the proposer offers Φ=61.8% of the total amount. According to RE
learning model, in an ultimatum game played according to the strategy method, if player n (n = 1,
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2) plays his kth
pure strategy at time t and receives a payoff of x, then the propensity to play strategy
k is updated by setting:
1 n = q n + K x …………….. (3)
Where k is a learning parameter.
For all other pure strategies j,
1 n = n …………….. (4)
Thus, the probability P t n that player n plays his kth
pure strategy at time t is P n =
n / q t n where the sum is over all of player n’s pure strategies j.
To test whether simulated players, with no consciousness or TOM learn to converge to the
predicted point of balance, I ran 30 simulations of ultimatum games with a cake size of m=10
MUs, and learning coefficients of k = 3 for both players. Each simulation was conducted for t =
3000 rounds. Figure 1a depicts a typical run for a cake and Fig. 1b depicts the convergence
points of 30 simulations. As could be seen in the figure, most simulations approach the golden
ratio point. The mean demand by the simulated proposer is 61.14%, which is almost identical to
the model prediction of 61.80%, with a small standard deviation of 4.14%.
10
.
Figure 1 (a & b): A typical run for (Fig. a) and the convergence points of 30 simulations (Fig. b)
0
10
20
30
40
50
60
70
80
0 5 10 15 20 25 30
Pro
po
ser'
s D
eman
d (
in %
)
Simulation (n)
Φ (x 100) = 61.80%
Simulation Parameters:
k1=k2=3, t=3000
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4. Empirical Validation
4.1. The Standard UG
The predictive power of EHT was tested by comparing it with implications of the Equality Principle
(Jost & Kay, 2010; Messick & Sentis, 1983) and of two well-founded economic theories: Inequality
Aversion Theory (Fehr & Schmidt, 1999) and the Theory of Equity, Reciprocity and Competition
(Bolton & Ockenfels, 2000). For this purpose, I used two large-scale datasets: (1) data from a Meta-
analysis which integrated 37 studies conducted in 25 different countries, representing different
cultures and social-political systems (Oosterbeek, Sloof & Van de Kuilen, 2004), and (2) data
collected in a comprehensive study in 15 small communities, exhibiting a wide variety of economic
and cultural conditions (Henrich, 2006). Figures 2a & 2b depict the distribution of proposers’
offers, recipients’ re ections, and final payoffs in the two investigated data-sets. As shown in the
figure, the mean offers reported by the two studies are almost identical (40.5% and 39.5%,
respectively), and quite close to the predicted Divine Ratio equilibrium (38.2%). However, as
expected (Henrich, 2006), the behavioral variability of proposers in the small communities study
(adults) is considerably larger than the variability of the large industrial societies sample (university
students, standard deviations of 8.3 and 5.7 for the two studies, respectively).
Table 1 compares the predicted proposers’ offers by the EHT with the predictions of the Equality
principle (EQ), predicting a 50/50 split, the sub-game perfect equilibrium (SPE), predicting an ε →
0 offer, the Inequality Aversion (IA) theory, predicting ≈ 50/50 split, and the Equity-Reciprocity-
Conflict theory (ERC), which predicts that the proposer should offer any portion between an
infinitesimally small positive portion and 50% (Bolton & Ockenfels, 2000).
The bottom row in the table depicts the various predictions’ errors. It demonstrates that the
prediction of the proposed Φ-Fairness theory is superior to all the other theories, including the
equality principle which predicts the modal offer in most ultimatum experiments (Güth,
Schmittberger & Schwartze, 1982; Camerer & Thaler, 1995; Camerer, 2003; Kahneman, Knetsch &
Thaler, 1986; Kagel, Kim & Moser, 1996; Matthew, 1993; Prasnikar & Roth, 1992; Suleiman,
1996). To test the adequacy of the various theories in accounting for the experimental data, I used
the two one-sided test (TOST). This equivalence analysis determines whether a variable’s mean is
sufficiently close to a hypothesized parameter. If the confidence interval 100(1-2α) is within a
defined interval, we conclude that equivalence exists. For a pre-specified equivalence interval of
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±10% and a log-normal distribution of mean offers, the analysis yields a significant result (p<0.01)
for the large industrial societies (Oosterbeek, Sloof & Van de Kuilen, 2004) and a marginally
significant result for the small-scale communities (Henrich, 2006). Similar TOSTs performed for the
adequacy of all the other theories were statistically insignificant.
Figure 2 (a &b): Distributions of offers, rejection rates and final payoffs in two large-scale studies
(a) Oosterbeek, Sloof & Van de Kuilen, (2004) and (b) Henrich et al. (2006)
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Mean Offer
(Experimental)
Subgame Perfect
Equilibrium
SPE
Inequality
Aversion
IA
Equity-Reciprocity Conflict
ERC
Equality
EQ
Economic Harmony
Theory
EHT
Oosterbeek et al.
(2004)
40.4 (5.7)
0
≈ 50
Any offer in [0, 50)
50
(1- √5 1
2) ≈38.2
Henrich (2006)
39.9 (8.3)
Error (in %)
≈ 40
≈ 10
≈ 15
≈ 10
1.3
Table 1. Comparison of the predictability of the proposed Φ-Fairness Theory with three well accepted theories
4.2 Mini-ultimatum game with information about proposers' feasible offers
I also tested the proposed model using data from the mini-ultimatum games studies by Falk et al.
(2003). In their study, they tested whether identical offers trigger different rejection rates depending on
the other offers available to the proposer. It was hypothesized that a certain offer with an unequal
distribution of material payoffs is much more likely to be rejected if the proposer could have proposed
a more equitable offer than if the proposer could have proposed only more unequal offers. The mini-
games used in their study are exhibited in Fig. 3. In all games the proposer P is asked to divide 10
points between himself and the responder R, who can either accept or reject the offer. Accepting the
offer leads to a payoff distribution according to the proposer’s offer. A re ection implies zero payoffs
for both players. As Figures 3a-3d indicate, P can choose between two allocations, x and y. In all four
games the allocation x is the same while the allocation y differs from game to game. If P chooses x and
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R accepts this offer, P gets 8 points while R receives 2 points. In game (a) the alternative offer y is
(5/5). In game (b) the alternative offer y is to keep 2 points and to give 8 points to R. In game (c) P has
in fact no alternative at all, i.e., he is forced to propose an offer (8/2). Finally, in game (d) the
alternative offer is (10/0).
Figure 3: Four Mini-Ultimatum Games (Source: Falk et al., 2003)
The standard game theoretic model predicts that in all games the allocation (8/2) is never rejected.
ERC and Inequality Aversion theories predict that the rejection rate of the (8/2)-offer is the same
across all games. Falk et al. argued that in the (5/5)-game a proposal of (8/2) is clearly perceived as
unfair because P could have proposed the egalitarian offer (5/5). In the (2/8)-game offering (8/2)
may still be perceived as unfair but probably less so than in the (5/5)-game because the only
alternative available to (8/2) gives P much less than R. In a certain sense, therefore, P has an excuse
for not choosing (2/8). Because one cannot unambiguously infer from his unwillingness to propose
an unfair offer to himself that he wanted to be unfair to the responder. Results showed that the
rejection rate of the (8/2)-offer in the (5/5)-game was highest (44.4%). 26.7% rejected the (8/2)-
offer in the (2/8)-game, 18% in the (8/2)-game and 8.9% in the (10/0)-game.
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Game Percentage of
8/2
(and the
alternative)
offer
Percentage of
rejections of
the 8/2 offer
Expected
payoff from
choosing the
8/2 offer
Expected
payoff from
choosing the
alternative
offer
EHT
predictions of
proposers'
offers, payoff
and
prediction
error in
payoff
5/5 31% (69%) 44.4% 4.44 5.00 Choose 5/5
5.00
(Error=3.4%)
2/8 73% (27%) 26.7% 5.87 1.96 Choose 8/2
8.00
(Error=39.9%
)
10/0 100% (0%) 8.9% 7.29 1.11 Choose 8/2
8.00
(Error= 8.9%)
Table 2: Proposer Expected Payoffs and Model's Predictions for the Tested Mini-Games
These results are inconsistent with the standard model as well as the two fairness models ERC and
Inequality Aversion. The proposer's data, shown in Table 2, clearly indicate that the percentage of
choosing the 8/2 offer increases dramatically as the alternative offer becomes more unfair, along
with a sharp decrease in the rejection rate of the 8/2 offers. As result, the 8/2 expected payoff
increases, and the alternative offer's expected payoff decreases, as the alternative offer becomes
more unfair, equaling only 1.1% for the 10/0 offer.
The EHT model prediction for the standard UG, defined for the continuous range of offers between
0 to 10, is {10 Φ, 10 (1-Φ)} ≈ (6.18, 3.82) for the proposer and the responder, respectively. To
derive the model's prediction for offers in the different binary, I used a minimum square difference
16
criterion. For the 5/5 condition the mean square differences (MS) between the predicted and
observed allocations for the 8/2 choice equals:
MS(8/2) 8 – 6 18 2 + 2 – 3 82 2 ≈ 6.63
The comparable MS associated with the alternative (5/5) offer is:
MS(5/5) 5 – 6 18 2 + 5 – 3 82 2 ≈ 2.79
Thus, for this condition, the minimal MS difference between the point of balance predicted by the
model and the feasible allocations in the mini-game is achieved by choosing the equal allocation.
Similarly, for 2/8 and 10/0 conditions we get MS(2/8) 2 – 6 18 2 + 8 – 3 82 2 ≈ 31.95 and
MS(10/0) 10 – 6 18 2 + 0 – 3 82 2 ≈ 29.19. Since both MS values are much larger than
MS(8/2) ≈ 6.63 , for the two alternative offers, 2/8 and 10/0 the EHT predicts that proposers will
choose the 8/2 offer. These predictions are supported by Falk et al. results. In the 5/5 condition 69%
chose the 5/5 alternative, and in the other conditions 73% and 100% chose the 8/2 alternative. The
prediction errors for payoffs in each condition are:
1 For the 5/5 condition:
Error (5/5) = [│(predicted payoff – observed payoff) │/ predicted payoff ] x 100
= [ (4.44 x 0.31) + (5.00 x 0.69) – 5 ]/5 x 100 = [(5- 4.83)/5] x 100 = 3.4 %
2 For the 2/8 condition the prediction error is:
Error(2/8) = [ (5.87x 0.73) + (1.96 x 0.27) – 8 ]/ 8x 100 = [(4.81 – 8)/8] x 100= 39.9%,
3 For the 10/0 condition we get:
Error(10/0) =[7.29 x 1 + 10.00 x 0) – 8 ]/ 8 x 100 = 8.9%.
4.3 Three-person ultimatum Game
I also tested the EHT in the three-person ultimatum game (Kagel & Wolfe, 2001), designed to test
the ERC and Inequality Aversion theories. The experimental design was as follows: Player X offers
to split a sum of money (m-(y+z), y, z) between herself, player Y, and player Z, respectively. One
of the latter is chosen at random (with probability p) to accept or reject the offer. If the responder
accepts, then the proposed allocation is binding, as in the standard ultimatum game. However, if the
responder rejects, then both she and X receive zero payoffs. In one experiment the non-responder
receives a "consolation" payoff of c. The "consolation" payoff to the non-responder varied across
four conditions with c= 0, 1, 3, 12, the probability of designating Y or Z as responder was p= 1/2
17
and the amount to be allocated in all conditions was m=$15. The study findings rejected both the
strong and weak versions of the ERC and IA models. Contrary to the two theories' predictions,
frequent rejections of offers were detected, when both models call for acceptance. In addition, the
effect of the "consolation" payoff for the non-responder on the probability of responders accepting
offers was small, and did not increase monotonically with the size of the “consolation” payoff as a
weak version of both theories would suggest.
The study's findings reveal that the modal offer of the proposer was the equal distribution of ($5,
$5, $5), and the median was ($7, $4, $4). The proposers' mean keep was quite similar across all
treatments, including the one with high "consolation" payoff of c=12. It decreased only slightly
over the course of the repeated game (see Table 3 and Figure 4).The reported rejection rates were
lowest for the equal distribution ($5, $5, $5), and the ($6, $4.50, $4.50) distribution, 1% and 6%,
respectively, with corresponding expected keep of $4.96 and $5.67. The modal distribution, (7, 4,
4), yielded nearly maximum expected keep with relatively few rejections (of 11%).
To derive the EHT predictions, we take note of the fact that with each other player (Y or Z) the
proposer X faces a compound game in which she faces an ultimatum game or a dictator game with
probability of 1/2 for each game to be realized. For the ultimatum game with a cake size of m, the
EHT yields two predictions: a symmetric equal split (m/2, m/2) and the asymmetric Golden Ration
split of approximately (0.62m, 0.38m). Since in the dictator game no sanctions could be applied for
punishing an unfair allocator, the EHT predicts that the proposer keeps the cake and allocate (m, 0).
18
Figure 4: Average Keep for Proposers by Round in Four Prize Treatments (Source: Kagel & Wolfe, 2001)
Table 3: Average Keep for Proposers by Round in Four Prize Treatments (Source: Kagel & Wolfe, 2001)
The amount x kept by X could be written as:
{
……….. (5)
19
where Φ is the golden ratio, Φ≈ 0.62. Thus, the expected value is equal to:
= + = ( ……….. (6)
For p =
= $7.5 and Φ = 0.62 we obtain:
= (
≈ (1+ 0.62)
≈ $6.08.
The amount sent to each player (Y and Z) is equal to is equal to:
= $4.46
To summarize, the points of harmony for the discussed game are the equal distributions of (5, 5, 5)
and the Golden Ratio distribution of (6.08, 4.46, 4.46). These predictions are supported by the
reported results. The symmetric solution is equal to the observed modal one (5, 5, 5), which
generated almost no rejections (1%), while the asymmetric solution (6.08, 4.46, 4.46) is almost
identical to the second leas rejected (6%) distribution of (6, 4.5, 4.5). Note that the reported median
distribution of (7, 4, 4) is also very close to the Golden Ratio distribution, since the relative share of
the X vis a vis each one of the two player Y and Z is pro7
7 4 =
7
11 0 64.
Finally, note that the reported statistical independence of the portion kept by X on the size of the
consolation prize, or on whether it is positive or negative is also confirmed by the proposed model,
since its two solutions are independent on the value of c.
4.4 A Sequential Common Pool Resource Dilemma
I also tested the EHT with data from an experiment utilizing the sequential Common Pool Resource
(CPR) Dilemma (Budescu, Suleiman, & Rapoport, 1995; Suleiman & Budescu, 1999). When the
resource size is fixed and known to all players, the game can be viewed as a generalization of the
ultimatum game for any number of players n ≥ 2. In this game, a player occupying the j’th position
in the sequence (1 < ≤ n) is informed about the total requests of the preceding -1 players. He can
"reject" the offer by requesting an amount that exceeds the remaining portion of the resource, or
"accept" it by requesting a lesser amount. The sub-game perfect equilibrium for the game prescribes
that the first mover demands almost all the amount available in the common resource and leaves a
small portion for the others. Budescu et al. (1995) conducted an experiment using a three-person
20
CPR game. Their main objective was to test the effect of uncertainty regarding the CPR size and the
player's position in the sequence of play on her request. In the sequential protocol investigated by
Budescu et al., individual requests are made in an exogenously determined order, which is common
knowledge, such that each player knows his position in the sequence and the requests of the players
who precede him in the sequence. The size of the common pool was uniformly distributed in the
range (m-r/2, m+r/2), where m and r are the mean and the range of the distribution, respectively. In
Experiment 2 of Budescu et al., the parameters of the game were n= 3 players, m=500 MU, and r
varied according to three treatments r = 0 (no uncertainty), r = 250 (mid-range uncertainty), and r =
500 (maximum uncertainty) in a within-subject design. Results showed that the SPE, prescribing
that the first mover appropriates almost all the amount is strongly refuted. On the average, neither
the first mover, nor any of the following players, took advantage of their ultimatum position, as
SPE predicts; late movers frequently "rejected" excessive demands made by previous players. The
study's main result revealed two interesting findings that hold for several studies using different
group sizes and uncertainty range that: 1. Mean requests increase monotonically with r, causing
lower efficiency in the groups' benefit from the resource. 2. There exists a reverse positional order
effect exhibited in an inverse relationship between a player's request in the sequence and her
demand from the resource.
We focus here on the certainty condition. Under this condition, the first mover requested, on
average, 249 MUs, the second requested 155, and the third 116 points. The EHT prediction for the
game is that the relative payoff of a preceding player relative to the amount allocated between her
and the subsequent player should equal the golden ratio Φ≈ 0.62. Indeed, the first mover's portion
of the amount shared by her and the second player in the sequence is 249/(249+155) ≈ 0.62!. The
second player's portion relative to the amount shared between her and the third player in the
sequence is 155/(116+155) ≈ 0.57. These results not only predict the qualitative decline in requests
as function of the players' positions in the sequence, as a game theoretic prediction does (ref.), but
also yield good point predictions of the player's mean requests.
5. Summary and Concluding Remarks
The present article demonstrates that the decisions observed in the standard ultimatum game, as
well as in a class of two- and three-person games, could be accounted for without relaxing the
21
standard economic assumption of self-serving individuals who act in order to benefit themselves,
and who have no positive regard towards the benefit of others. The main departure of the proposed
EHT from the standard economic model lies in its assumption that individuals consider to their
payoffs relative to subjective reference points, which depend on the game structure of the
interaction and on the players' positions in the interaction. The points of balance or harmony in the
games studied above were derived from simultaneously considering the players' subjective focal
points, each of them is quite reasonable, and not from the best response argument of the equilibrium
concept. Like the theory of altruistic punishment, the proposed EHT posits that behaving unfairly
may be costly; hence, self-regarding individuals would behave in a way that is judged to be fair by
others.
For the two-person ultimatum game, the theory yields a symmetric solution at the 50-50 split, and
an asymmetric solution at the Golden Ratio, prescribing about a 62-38 split. The first solution does
not reflect the asymmetric structure of the game, which at the outset gives the proposer the
advantage of having the entire amount and the right to allocate it. It is most likely that she compares
her payoff with the entire amount, and for the responder to compare her payoff with the amount the
proposer keeps for herself. Framing the ultimatum situation as a symmetric game is bound to shift
the point of balance or harmony to the equal split. This conjecture is nicely demonstrated by Larrick
& Blount (1995) and Budescu et al. (1995, Expr. 2), who reported results showing that in the
ultimatum game subjects behave in a more self-interested manner than they do in a structurally
equivalent two-person sequential CPR game, in which players opted more for the equal split. This
result is explained by the difference in "framing" the game. While ultimatum bargaining games are
perceived in the context of "power" relations, CPR games are perceived in the context of "affinity"
relations (Larrick & Blount, 1995).
Fundamental to EHT are two conditions. The first is that individuals internalize the norms of
equality and equity, and the second is the existence of an effective sanctioning mechanism for
punishing deviations from these norms. The first condition states that individuals are cognizant of
the norms of equality and equity; it does not imply that they follow them. Being selfish, they use the
information about their groups' norms in order to advance their self-interest. Such information is
vital for evading punishment by minimizing the difference between their payoffs (relative to
respective reference points) and achieving a balanced interaction. In fact, the computer simulation
22
(see section 2.1) demonstrates that the "awareness" condition could be relaxed if the game is
repeated, thereby allowing players to learn (the hard way) from experience.
The asymmetric solution at the Golden Ratio is appealing not only because it provides an answer to
the mean allocation of ≈60-40 frequently observed in ultimatum experiments, but also because it is
aesthetically pleasing (Pittard, Ewing & Jevons, 2007).It is commonly known that due to its
mathematical properties the Golden Ratio plays an important role in aesthetics, art, design, and
music, due to its visual and auditory pleasantness (Pittard et al., 2007; Hammel, 1987; Hammel &
Vaughan, 1995). Rectangular shapes with width to length ratio of ≈ 0.618 are widely used in the
design of TVs, computer screens, and credit cards, due to feelings of pleasantness and harmony that
the Golden Ratio is believed to induce (Livio, 2002; Olsen, 2006; Pittard, Ewing & Jevons, 2007).
In addition, together with the Fibonacci Series (which converges to the Golden Ratio), it plays a key
role in life sciences by determining the structure of plants and animals (Hammel, 1987; Klar, 2002),
the human body (Livio, 2002), human DNA, and brain waves (Weiss & Weiss, 2003; Conte et al.,
2009; Weiss & Weiss, 2010; Roopun et al., 2008; Merrick, 2010). Recently, the Golden Ratio also
has been discovered in physics (Coldea et al., 2010; Suleiman, 2012). To the best of my knowledge,
this is the first time that the Golden Ratio appears in social and economic interactions.
It is worth stressing that since the proposed theory is not specific with regard to the nature of the
sanctioning mechanism, or which player or agency has the power or authority to use it, it could be
applied to other situations in which the sanctioning agency could be a third party or a central
authority. Although I have not tested this feature of the theory, supporting evidence for the
applicability of the theory to such cases is provided by experimental results from the dictator game
with third party punishment played by adult participants from 15 small-scale societies (Henrich,
2006). This large-scale study reveals that that the option of a third party punishment raised the
dictator offer to about 30-45%, a proportion close to the mean offers usually detected in the
ultimatum game, and close to the Golden Ratio split. Reputation has a similar effect (Haley &
Fessler, 2005; Gallagher & Frith, 2003). For example, introducing reputation in the dictator game
increases the offers from 17% to 30.3% of the entire amount (Servátka, 2009).
Because the EHT has been tested only on a class of two- and three-person games, no claim for
generality is justified. A first step in this direction would be to develop the theory for accounting to
situations involving n players, like the public goods game and the n-person CPR dilemma game.
23
Nonetheless, we contend that the concept of harmony between relative payoffs, measured against
different subjective reference points, could prove to be useful not only for explaining proposers'
offers in controlled experimental situations, but also for understanding, and possibly making policy
recommendation for simultaneously improving productivity and enhancing fairness in
organizations. A straightforward application is to apply the concept of economic harmony for
assessing the levels of harmony or disharmony in the distribution of salaries in a given organization.
By modeling organizational structures as n-person games, it becomes possible to apply the theory in
order to specify the conditions required for achieving harmony in the distribution of wages in the
workplace, and consequently enhancing both profitability of the workplace and fairness in profit
allocation.
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Figure Captions
Figure 1 (a & b): A typical run for (Fig. a) and the convergence points of 30 simulations (Fig. b).
Figure 2 (a &b): Distributions of offers, rejection rates and final payoffs in two large-scale studies
(a) Oosterbeek, Sloof & Van de Kuilen, (2004) and (b) Henrich et al. (2006).
Figure 3: Four Mini-Ultimatum Games (Source: Falk et al., 2003).
Figure 4: Average Keep for Proposers by Round in Four Prize Treatments (Source: Kagel &
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Acknowledgments
This research was supported by the Israeli Science Foundation (grant No. 992/08). I wish to thank
Judith Avrahami, Ken Binmore, David V. Budescu, Werner Güth and Amnon Rapoport for very
helpful comments.
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