Economic Harmony:

An Epistemic Theory of Economic Interactions

Ramzi Suleiman

Department of Psychology, University of Haifa , Israel Department of Philosophy , Al Quds University, Palestine

Keynote lecture presented at the International conference on:

Social Interaction and Society: Perspectives of Modern Sociological Science

ETH Zurich, May 26 – 28, 2016

1. Could fairness emerge in short interactions between rational players?

2. If yes, then under what conditions?

The main questions to be addressed in my talk are the following:

In Rabin’s “fairness equilibrium” model, fairness is loosely defined as a positive or negative sentiment or emotion which drives (costly) other-rewarding or other-punishing behavior.

What do we mean by fairness?

According to Rabin (AER, 1993) “If somebody is being nice to you, fairness dictates that you be nice to him. If somebody is being mean to you, fairness allows-and vindictiveness dictates-that you be mean to him”.

In other words: Fairness dictates reciprocity

But what if the interaction does not allow for reciprocity?

Example 1: All non-repeated interactions Example 2: Interaction with a dictator

While we are able to evaluate the behaviors and outcomes in such interactions as more or less fair, Rabin’s conception of fairness (as emotion which drives a reciprocal behavior) des not apply, nor does his “fairness equilibrium” model.

In the proceedings I shall introduce a theory of economic interactions called “economic harmony theory”.

In the proposed theory fairness is an attribute of the players’ outcomes.

By a “fair outcome” we mean an outcome which the reasonable person would consider as just.

In other words we define a “fairness outcome” as an outcome which maintains distributive justice

Formally, we define a fairness outcome as the vector of outcomes 𝒓∗= (𝑟1

∗, 𝑟2∗, 𝑟3

∗…𝑟𝑛∗) for

which the subjective utilities of all players’ outcomes are equal or:

𝑢𝑖 (𝑟𝑖∗) = 𝑢𝑗 (𝑟𝑗

∗) For all i and j (1)

In psychological terms, a fairness outcome is the outcome at which the satisfaction levels of all players are equal

Given the aforementioned operational definition, the prediction of standard Game Theory is that in non-cooperative games played in the short term, fairness (and cooperation) between rational players could not be achieved. In zero-sum games (e.g., in ultimatum bargaining) rational players will strive to maximize their own portion of the goods, thus eliminating the possibility of a fair division. In mixed-motive games (e.g., PD, PG, CPR) the possibility of fairness is excluded by free-riding.

However, ample empirical evidence from field and experimental studies, show that in many types of economic interactions (including the aforementioned games) satisficing levels of fairness (and cooperation) are consistently achieved.

As examples: In the ultimatum game, the mean offer out of 1MU is about 0.40 and the modal offer is 0.50.

In the dictator game the mean transfer is 0.20- 0.30 and offers of less than 0.20 are frequently rejected (because they are considered unfair) In the single-trial PD game the average rate of cooperation is 0.5 or more.

Several Economic models were proposed to account

for the cooperation and fairness observed in

strategic interactions.

Inequality Aversion theory (Fehr & Schmidt, 1999)

assumes 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.

The theory of Equity, Reciprocity and Competition

(ERC) (Bolton & Ockenfels, 2000), posits that along

with pecuniary gain, people are motivated by their

own payoffs relative to the payoff of others.

The prediction of IA is nonspecific as it requires an estimation of the relative weight of the fairness component in the proposer's utility function.

With regard to the standard ultimatum game the prediction of ERC is uninformative, as it predicts that the proposer should offer any amount that is larger than zero and less or equal 0.50

While in general the aforementioned theories yield better predictions than the SPE of standard game theory, they fall short of predicting the 0.60-0.40 split.

More important, both theories makes assumptions which contradict the rationality principle (by introducing other regarding components to the players’ utility functions).

We shall demonstrate that a plausible modification of the players utility functions without breaking the rationality principle is sufficient for accounting successfully for the fairness and cooperation witnessed in the UG and in several other games, including the PD, PG, CPR, and Trust games.

Economic Harmony Theory

We modify the players utility functions by defining the utility of each player i as:

𝑢𝑖 (𝑟𝑖

𝑎𝑖) (2)

Where 𝑟𝑖 is player i’s actual payoff, 𝑎𝑖 is his or her maximal aspired payoff, and u (..) is a bounded non-decreasing utility function with its argument (u(0) =0 and u(1) =1).

For the sake of simplicity, we assume linearity, such that:

𝑢𝑖 (..) = 𝑟𝑖

𝑎𝑖 (3)

The aforementioned specification of ui implies a soft relaxation of the rationality principle, according to which 𝐞𝐚𝐜𝐡 𝐩𝐥𝐚𝐲𝐞𝐫 𝐩𝐮𝐭𝐬 𝐚𝐧 𝐮𝐩𝐩𝐞𝐫 𝐥𝐢𝐦𝐢𝐭 𝐭𝐨 𝐡𝐢𝐬 𝐨𝐫 𝐡𝐞𝐫 𝐠𝐫𝐞𝐞𝐝.

Following the adopted definition of fairness as distributive justice, a fairness solution of an economic interaction will be achieved if and only if:

Substituting: 𝑢𝑖 = 𝑟𝑖

𝑎𝑖 we get:

𝑟𝑖∗

𝑎𝑖 = 𝑟𝑗∗

𝑎𝑗 For all i and j

For reasons to be clarified in the proceedings we refer to the solution 𝒓∗ as a state of harmony

(4)

(5)

We shall demonstrate that assuming the utility function defined above, the theory prescribes reasonably fair solutions for a variety of economic interactions and that the prescribed solutions are in good agreement with experimental data.

We begin with the ultimatum game

In the Ultimatum game one player (the proposer) receives an amount of monetary units, and must decide how much to keep and how much to transfer to another player (the responder). The responder replies either by accepting the proposed offer, in which case both players receive their shares, or by rejecting the offer, in which case the two players receive nothing.

In ultimatum experiments conducted in many countries on participants from different cultures and socio-economic levels, using different stakes, and types of currency, the mean offer is around 0.40. In the first experimental study by Güth, et al. (1982), the mean offer was 0.419 Kahneman, et al. (1986), reported a mean offer of 0.421% (for commerce students in an American university) Suleiman (1996) reported a mean of 0.418 for Israeli students.

A meta-analysis performed on findings of ultimatum experiments conducted in twenty six countries with different cultural backgrounds (Oosterbeek, Sloof, & Van de Kuilen, 2004), reported a mean offer of 0.405

A large-scale cross-cultural study conducted in 15 small-scale societies (Henrich et al., 2005, 2006), reported a mean offer of 0.395.

Distributions of offers in two large-scale ultimatum studies

0.25 0.3 0.35 0.40 0.45 0.5 0.55 0.60

14.3

0

19.1

33.3

23.8

4.8 4.8

0

11.4

6.8

25 27.3

18.2

9.1

0

2.3

0

5

10

15

20

25

30

35

Fre

qu

en

cy (

in %

)

Offer

Henrich et al.(mean=39.5,std=8.3)

Oosterbeek etal.(mean=40.5,std=5.7)

Economic harmony solution

If player 1 (the proposer ) keeps 𝑥 MUs out of 1 MU (and player 2 -the responder gets 1- 𝑥 MUs), then:

𝑟1 = 𝑥, and 𝑟2 = 1- 𝑥 A harmony solution requires that:

𝑟1

𝑎1 = 𝑟2

𝑎2,

Or: 𝑥

𝑎1 = 1−𝑥

𝑎2 (6)

Solving for x we get: 𝑥 = 𝑎1

𝑎1+𝑎2 (7)

Which yields: 𝑟1∗ =

𝑎1

𝑎1+ 𝑎2 and 𝑟2

∗ =

(8)

In the absence of any constraints on the proposer’s decision, a self-interested proposer would aspire for the entire sum. i. e. , the maximal aspired payoff by th proposer is:

𝑎1= 1 Hypothesizing about the responder’s aspired payoff is trickier. We consider two plausible possibilities:

(1) He or she might aspire to receive half of the “cake”; (2) he or she might aspire to receive a sum that equals the sum the proposer wishes to keep for himself or herself.

Under the first assumption, we can set 𝑎1= 1 and 𝑎2= 1

2.

Substitution in equations 3 and 4 yields:

𝑟1∗=

𝑎1

𝑎1+ 𝑎2 =

1

1+ 1

2 = 2

3 (9)

And:

= 1 - 2

3 = 1

3 (10)

Under the second assumption, we have 𝑎1= 1 and 𝑎2= x Substitution in equations 3 yields:

𝑥

1 = 1−𝑥

𝑥 …… (11)

Solving for 𝑥 we get:

𝑥2 + 𝑥 - 1 = 0 … (12)

Which solves for:

𝑥 = 5−1

2 ≈ 0.618 … (13)

The solution 5−1

2 (≈ 0.618) is striking, since it

is the famous Golden Ratio (ϕ) known for its key role in the sciences and the arts. The corresponding portion for the recipient is: 𝑥𝑟 = (1- ϕ) ≈ 0.38, a very close result to the empirical one.

The Golden Ratio ϕ, is defined as limn→ ∞𝑓𝑛

𝑓𝑛+1,

where fn is the nth term of the Fibonacci Series:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ….

in which each term is equal to the sum of the two preceding terms

𝒇𝒏 = 𝒇𝒏−𝟏 + 𝒇𝒏−𝟐

φ = 𝟓−𝟏

𝟐 ≈ 0.618

The golden ratio has fascinated intellectuals of diverse interests for at least 2,400 years.

“It is probably fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics” Livio M., (2002).The Golden Ratio: The Story of Phi, The World's Most Astonishing Number.

ϕ

is the only number that satisfies:

x + 1 = 𝟏

𝒙

Or:

𝒙𝟐 + x = 1

ϕ

The Golden Ratio is a significant number, not only due to its unique mathematical properties, but also due to its significant role in music, the arts and design, as well as in science and technology,

starting from physics and chemistry, to life

science, to humans' perception, cognition and

emotion

Quantum criticality

Phase Transition

Prediction of Information Relativity Theory (Suleiman, R.)

In quantum mechanics

Redshift 𝑧 = β

1−β

𝛽 = Recession velocity

In Cosmology

Big Bang Now

GZK cutoff

Is the point of harmony an equilibrium

or a steady state?

No!

Unless supported by an efficient external social or institutional mechanism.

Possible mechanisms for promoting fairness: 1. Moralizing (prosocial religiosity, civic norms of pro-sociality) 2. Sanctions (Punishment) second party (UG), Third Party (in the Dictator game), group punishment in PG games

3. Reputation (in repeated games) …

Shariff and Norenzayan (2007): priming religious or secular civic concepts in the dictator game

Sanctions (Punishment) as promoter of fairness

second party sanctioning (UG)

Third Party sanctioning in the Dictator game.

Group sanctioning in the PG game

I ran an experiment using a repeated δ-ultimatum game (Suleiman, 1996), with trial-to-trial feedback. In the δ-ultimatum game, acceptance of an offer of [x, S-x] entails its implementation, whereas its rejection results in an allocation of [δ x, δ (S-x)], where δ is a "reduction factor" known to both players (0≤ δ≤ 1). Varying the reduction factor, results in different recipients' punishment efficacy. For δ =0, the game reduces to the standard ultimatum game, in which the recipient has maximal punishment power, while for δ =1 the game reduces to the dictator game in which the recipient is powerless.

0

5

10

15

20

25

30

35

40

45

1 2 3 4 5 6 7 8 9 10

Off

er

(in

%)

Trial Block

Strong Punishment (no prime)

Weak Punishment (no prime)

Golden Ratio prediction = 38%

mean 0.39, sd = 0.09 acceptance rate = 73%

Mean = 0.19 , sd= 0.13 Acceptance rate 44%

Main Results

Predicting behaviors and outcomes in other games

CPR game

PG game

Trust Game

A Sequential step-level CPR Game A common pool of M MUs

Players make their requests from the common pool according to a predetermined order

Each player is informed about the sum of requests of the preceding players

If the sum of all requests is less or equal to the amount available in the pool then each player receives his or her requested MUs

If the sum of all requests is more than the amount available in the pool, then all Players receive zero MUs

Standard game theory predicts that the first mover will request the entire amount – a divisible ε

A sequential CPR game with n players could be viewed as an n-

player ultimatum game. It is easy to show that at the point

harmony, the requests vector 𝒓𝒉 = 𝑟1, 𝑟2, , . . 𝑟𝑛−1 , 𝑟𝑛 should

satisfy:

𝑟𝑘+1

𝑟𝑘= φ 𝑓𝑜𝑟 𝑘 = 1, 2, 3, …𝑛 ….(14)

Where φ is the Golden Ratio (φ ≈ 0.618).

EH Solution for the Sequential CPR Game

A harmonious position effect

249

155

116

250

154

96

0

50

100

150

200

250

300

1 2 3

Re

qu

est

Position in the Sequence

ExpirementalTheoretical

Prediction of Requests in a Sequential CPR Dilemma with 3 players resource size = 500

Data Source: Budescu, Suleiman, & Rapoport (1995)

171

120

109

85

64

180

106

74 67

52

0

20

40

60

80

100

120

140

160

180

200

1 2 3 4 5

Re

qu

est

Position in the sequence

ExperimentalTheoretical

Prediction of Requests in a Sequential CPR Dilemma with 5 players resource size = 500

Data Source: Suleiman, Budescu, & Rapoport (1999)

Trust Game

In the single-trial Trust Game one player (the investor) is given an endowment of e MUs and is requested to transfer any amount x between 0 and e to a second player (the trustee). The amount transferred to the second player is multiplied by a factor α (α > 1). The second player is requested to transfer back any amount between 0 and α x.

Standard Game theory predicts one equilibrium solution: (transfer = 0, return =0)

In most experiments investors transfer substantial amounts of money and trustees transfer back significant amounts of money

EH solution of Trust Game Denote the amount transferred by x and the amount transferred back out of αx by y. Rewards: 𝑟1= e –x +y …. (15)

𝑟2= αx - y Maximal aspirations 𝑎1= 𝑎2= αe …. (16)

𝑟1

𝑎1 = 𝑟2

𝑎2

y = 𝛼+1 𝑥−𝑒

2 …. (17)

𝑟1= 𝑟2 = (𝑒+ 𝛼−1 𝑥)

2 …. (18)

For 𝛼 > 1, 𝑥 =e

𝑟1∗ = 𝑟1

∗ = 1

2 𝛼 e (equality) …. (19)

n= 7 Average payback $13.72

EH: $15 n=7

Average payback $7.3 EH: $7.5

Source: Berg, Dickhaut, & McCabe, GEB, 1995

EH: $7.61

Kosfeld et al. Nature, 2005

18

12

6

Source: Kosfeld et al. Nature, 2005

Public Goods Game

In a typical PG game, each of n players receives an endowment of e MUs and must chose how much to invest in a public project. For each 1MU invested in the project each player receives a positive payoff α (α <1) regardless of the amount of his or her contributions. The final reward for each subject is 𝑟𝑗= initial endowment – own contribution

+ α (Sum of all contributions)

EH solution to the PG game

The reward for each player j is given by: 𝑟𝑗 = e - 𝑥𝑗 + α 𝑥𝑖

𝑛𝑖=1 = e – (1- α) 𝑥𝑗 + α 𝑥𝑖𝑖≠𝑗 … (20)

And his/her maximal aspiration level is: 𝑎𝑗 = e + α (n-1) e …. (21)

Harmony is achieved when rj

aj = ri

ai For all i and j

Substitution and simplification yields: xi = xj (equal

contributions). Substitution in eq. 12 yields: 𝑟𝑗 = e + (α n -1) 𝑥𝑗 …. (22)

For α ≤ 𝟏

𝒏 𝒙𝒋 = 0 for all j (all-defect)

For α > 𝟏

𝒏 𝒙𝒋 = e for all j (full cooperation) … (23)

The Long-Run Benefits of Punishment

Gächter, Renner, & Sefton (2008),

Science, 322, 1510

n

EH prediction

α = 0.5, n = 3, 0.5 > 1

3 = 20 MU

Test of the theory on real-life data

In a recent study we looked at actual mean salaries of senior and junior employee in two high-tech professions and two non-high-tech professions, from 10 “developed” countries with high gross national income (GNI) and 10 “developing” countries with low GNI, representing different cultures around the world.

The high-tech professions were: computer programmer and electrical engineer, and the two non-high-tech (hereafter “low-tech”) professions were accountant and schoolteacher. The developed countries were: United States, England, Canada, Israel, Spain, New Zealand, Australia, Italy, Austria, and Japan. The developing countries were: Pakistan, Jordan, Lebanon, Oman, India, Bahrain, Egypt, Saudi Arabia, Brazil, and Thailand.

We computed the ratio of junior workers’ mean salaries out f the total (junior and senior) salaries.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4Ju

nio

r Sa

lary

/ T

ota

l Sal

arie

s

Developed Countries Developing Countries

High Tech

Low Tech

Mean ratios of junior salaries by country development and profession type

A field Study on salaries in 20 countries

1- Φ ≈ 0.38

(n = 10) (n = 10)

a

b

𝑎

𝑏 ≈0.618

𝑏

𝑎 ≈ 1.618

Golden Ratio, Fairness, & Beauty

In the languages I speak, the word “fair” is

used as synonym to “beautiful” and “good

looking” , but also for “equitable” and “just”

In Hebrew:

פנים יפות

גבר נאה

הכנסה יפה

סכום נאה

In Arabic:

وجه حسن

سلوك حسن

حسن الَخلق والُخلق

Lady Justice

? =

On fairness and beauty

is

Delta 1 (1.5 Hz) ; Delta 2 (2–3 Hz); Theta (6–8 Hz); Alpha (10 Hz); Beta1 (13–17 Hz); Beta2 (22–27 Hz); Gamma1 (30–50 Hz); Gamma2 (50–80 Hz)

Human Brain Waves

Source: Roopun A. K. et al . (2008), Frontiers in Neuroscience

𝒇𝟏

𝒇𝟐 = 0.60,

𝒇𝟐

𝒇𝟑 = 0.357,

𝒇𝟑

𝒇𝟒 = 0.70,

𝒇𝟒

𝒇𝟓=𝟐

𝟑,𝒇𝟓

𝒇𝟔 = 0.612,

𝒇𝟔

𝒇𝟕 = 0.613,

𝒇𝟕

𝒇𝟖 = 0.615

Is the use of the word “fair” to express evaluations of beauty as well as evaluations of justice, imply similar feelings aroused by beauty and justice? Do beauty and justice have similar brain correlates?

An intriguing possibility

Economic harmony and religious morals

The Christian Bible teaches that “the whole law is fulfilled in one word: ‘You shall love your neighbor as yourself.’ (Galatians 5:14).

In the Islamic “Hadith”, which contains the oral

teachings of Prophet Mohamed, the same rule appears as: “None of you truly believes until he loves for his brother what he loves for himself.”

And in the Bible of Judaism, the same rule appears as

“thou shalt love thy neighbor as thyself” (Leviticus 19:18).

treat others as you treat yourself In three versions

This moral is frequently misunderstood as reciprocity

Treating others as you treat yourself is also a valued civic moral

It is easy to show that abiding to the above stated moral as constraint for self-interest yield the same prediction of EH

Summary and concluding remarks

We proposed a theory of economic interaction termed “economic harmony theory”, in which the players’ standard utility functions 𝑢𝑖 (𝑟𝑖) are modified such that

𝑢𝑖 (𝑟𝑖) = 𝑟𝑖

𝑎𝑖

𝑢𝑖 (0) =0, 𝑢𝑖 (𝑎𝑖) =1

Where 𝑟𝑖 is player i’s actual payoff, 𝑎𝑖 is his or her maximal aspired payoff.

The assumption that rational players have a finite aspiration level 𝑎𝑖 such that u(𝑟𝑖 ≥ 𝑎𝑖) ≈ 1 is a plausible one.

The finite size of the collective goods

guarantees that such a limit exists (one could not aspire more than what exists!)

For risk averse players there always exists an

asymptotic limit for which u(𝑟𝑖 ≥ 𝑎𝑖) ≈1

The proposed modification is equivalent to requiring that players place an upper limit to their aspirations.

We demonstrated that the proposed theory is successful in predicting the levels of fairness and cooperation in several two-person and n-person games, as well as real-life data on employees salaries, without a gross violation of the rationality assumption (i.e., by introducing an other regarding component in the utility function).

Game Prediction Supporting evidence

Ultimatum Game 1-φ ≈ 0.382 Aplenty

Sequential CPR Game 𝑟𝑘+1

𝑟𝑘= φ

Budescu et a. ,1995 (n=3)

Suleiman et al. 1999 (n=5)

Trust game Equality

𝑥 =e; 𝑟1∗ = 𝑟1

∗ = 1

2 𝛼 e

Did not test

Public goods Equality

For α ≤ 𝟏

𝒏 𝒙𝒋 = 0 (all-defect)

For α > 𝟏

𝒏 𝒙𝒋 = e (full cooperation)

Did not test

Prisoner's Dilemma Equality

mutual cooperation and mutual

defection (Same prediction as

Rabin's "fairness equilibrium model"

Rapoport & Chammah (1965)

Pruitt (1967),

Ultimatum game with one-sided uncertainty about the "pie" size

x = − 𝑎+𝑏 + (𝑎+𝑏)2+16 𝑏(𝑎+𝑏)

8

Rapoport, Sundali, & Seale

(1996)

Three-person ultimatum game

x = 𝑒

4 (φ +1) ≈ 0.4045 e Kagel & Wolfe (2001)

Further investigations of the theory: Lab and Simulation studies on repeated and

evolutionary games

(hypothesis: Not an automatic reinforcement learning but an adaptation of players’ aspiration levels until a steady state is reached.

An fMRI study on the Neuro-correlates of fairness and beauty

Thank you for

your attention