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
The main questions to be addressed in my talk
are the following:
1. Could fairness emerge in short interactions
between rational players?
2. If yes, then under what conditions?
What do we mean by fairness?
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.
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 0.200.30 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.
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
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.
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:
𝑟𝑖
𝑢𝑖 ( )
𝑎𝑖
(2)
Where 𝑟𝑖 is player i’s actual payoff, 𝑎𝑖 is his or her
maximal aspired payoff, and u (..) is a bounded nondecreasing 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:
(4)
Substituting: 𝑢𝑖 =
𝑟𝑖∗
𝑎𝑖
𝑟𝑖
𝑎𝑖
=
we get:
𝑟𝑗∗
𝑎𝑗
For all i and j
(5)
For reasons to be clarified in the proceedings
we refer to the solution 𝒓∗ as a state of harmony
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 socioeconomic 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
35
Oosterbeek et
al.
(mean=40.5,
std=5.7)
30
Frequency (in %)
Henrich et al.
(mean=39.5,
std=8.3)
33.3
27.3
25
25
19.1
20
15
23.8
18.2
14.3
11.4
9.1
10
6.8
4.8
5
0
0
0.25
0.3 0.35 0.40 0.45
Offer
0.5
4.8
0
2.3
0
0.55 0.60
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:
Or:
Solving for x we get:
Which yields:
𝑟1∗
𝑟1
𝑎1
=
𝑟2
,
𝑎2
𝑥
𝑎1
=
1−𝑥
𝑎2
(6)
𝑎1
𝑥=
𝑎1 +𝑎2
𝑎1
=
and
𝑎1 + 𝑎2
𝑟2∗ =
(7)
(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 =
Substitution in equations 3 and 4 yields:
𝑟1∗ =
𝑎1
1
2
=
1 =
3
𝑎1 + 𝑎2 1+
(9)
2
And:
2
=13
=
1
3
(10)
1
.
2
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)
5−1
2
… (13)
Which solves for:
𝑥=
≈ 0.618
5−1
2
The solution
(≈ 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
𝑓𝑛
as limn→ ∞
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
ϕ
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=
Or:
𝟏
𝒙
𝒙𝟐 + x = 1
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
Prediction of Information Relativity Theory (Suleiman, R.)
In quantum mechanics
Quantum criticality
Phase Transition
In Cosmology
Now
GZK cutoff
Big Bang
Redshift 𝑧 =
β
1−β
𝛽 = Recession velocity
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.
Main Results
45
Strong Punishment (no prime)
40
mean 0.39, sd = 0.09
acceptance rate = 73%
35
30
Offer (in %)
Weak Punishment (no prime)
Golden Ratio prediction = 38%
25
20
Mean = 0.19 , sd= 0.13
Acceptance rate 44%
15
10
5
0
1
2
3
4
5
6
Trial Block
7
8
9
10
Predicting behaviors and outcomes in other games
CPR game
Trust Game
PG 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
EH Solution for the Sequential CPR Game
A sequential CPR game with n players could be viewed as an nplayer 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
Where φ is the Golden Ratio (φ ≈ 0.618).
A harmonious position effect
….(14)
Prediction of Requests in a Sequential CPR Dilemma with 3 players
resource size = 500
300
250
249 250
Expiremental
Theoretical
Request
200
155 154
150
116
96
100
50
0
1
2
3
Position in the Sequence
Data Source: Budescu, Suleiman, & Rapoport (1995)
Prediction of Requests in a Sequential CPR Dilemma with 5 players
resource size = 500
200
180
180
171
Experimental
Theoretical
160
140
Request
120
120
106
109
100
85
74
80
67
64
60
52
40
20
0
1
2
3
4
Position in the sequence
5
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
𝑟2 = αx - y
Maximal aspirations
𝑎1 = 𝑎2 = αe
…. (15)
…. (16)
𝑟1
𝑎1
=
𝑟2
𝑎2
𝛼+1 𝑥−𝑒
2
y=
𝑟1 = 𝑟2 =
(𝑒+
…. (17)
𝛼−1 𝑥)
2
…. (18)
For 𝛼 > 1, 𝑥 =e
∗
∗
1
𝑟1 = 𝑟1 = 𝛼 e (equality)
…. (19)
n= 7
Average payback $13.72
EH: $15
n=7
Average payback $7.3 EH: $7.61
EH: $7.5
Source: Berg, Dickhaut, & McCabe, GEB, 1995
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
𝑛
𝑖=1 𝑥𝑖
= e – (1- α) 𝑥𝑗 + α
𝑖≠𝑗 𝑥𝑖
… (20)
And his/her maximal aspiration level is:
𝑎𝑗 = e + α (n-1) e
Harmony is achieved when
rj
aj
…. (21)
=
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 ≤
For α >
𝟏
𝒏
𝟏
𝒏
𝒙𝒋 = 0 for all j (all-defect)
𝒙𝒋 = e for all j (full cooperation)
… (23)
The Long-Run Benefits of Punishment
Gächter, Renner, & Sefton (2008),
Science, 322, 1510
EH prediction
n = 3, α = 0.5,
0.5
1
>n
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.
A field Study on salaries in 20 countries
Mean ratios of junior salaries by country development and profession type
Junior Salary / Total Salaries
0.4
1- Φ ≈ 0.38
0.35
0.3
High Tech
0.25
Low Tech
0.2
0.15
0.1
0.05
0
Developed Countries
(n = 10)
Developing Countries
(n = 10)
Golden Ratio, Fairness, & Beauty
𝑎
𝑏
≈0.618
𝑏
𝑎
≈ 1.618
b
a
In the languages I speak, the word “fair” is
used as synonym to “beautiful” and “good
looking” , but also for “equitable” and “just”
In Arabic:
‫وجه حسن‬
‫سلوك حسن‬
َ ‫حسن‬
‫الخلق وال ُخلق‬
In Hebrew:
‫פנים יפות‬
‫גבר נאה‬
‫הכנסה יפה‬
‫סכום נאה‬
On fairness and beauty
?
=
is
Lady Justice
Human Brain Waves
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)
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
An intriguing possibility
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?
Economic harmony and religious morals
treat others as you treat yourself
In three versions
 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).
Treating others as you treat yourself is also a valued civic moral
This moral is frequently misunderstood as reciprocity
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
𝑟𝑖
𝑢𝑖 (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 nperson 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
Ultimatum Game
Prediction
1-φ ≈ 0.382
𝑟𝑘+1
𝑟𝑘
Sequential CPR Game
Trust game
Supporting evidence
Aplenty
Budescu et a. ,1995 (n=3)
Suleiman et al. 1999 (n=5)
=φ
Equality
Did not test
1
2
𝑥 =e; 𝑟1 ∗ = 𝑟1 ∗ = 𝛼 e
Public goods
Equality
𝟏
For α ≤ 𝒏 𝒙𝒋 = 0 (all-defect)
Did not test
𝟏
For α > 𝒏 𝒙𝒋 = e (full cooperation)
Prisoner's Dilemma
Ultimatum game with
one-sided uncertainty
about the "pie" size
Three-person ultimatum
game
Equality
Rapoport & Chammah (1965)
mutual cooperation and mutual
Pruitt (1967),
defection (Same prediction as
Rabin's "fairness equilibrium model"
Rapoport, Sundali, & Seale
− 𝑎+𝑏 + (𝑎+𝑏)2 +16 𝑏(𝑎+𝑏)
x=
(1996)
8
𝑒
x = 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