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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

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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

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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).

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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.