AMMCS-CAIMS 2015
Special Session (WDSEE)
Wealth Distribution and Statistical Equilibrium in Economics
The Equilibrium-Seeking Behaviour
of a Very Simple Model of Wealth
Garvin H Boyle
08 June 2015
Email: [email protected]
URL: Orrery-software.webs.com
URL: www.openabm.org/model/3860/version/3
X00
EiLab Project Goals
UNDERLYING THESES:
1. Entropy is, fundamentally,
neither a thermodynamic
phenomenon nor an
informational phenomenon,
but, rather, it is
fundamentally a
mathematical phenomenon
that finds expression in
thermodynamic data, in
communications data, and
also in many other types of
data such as economic data.
2. The best place to study
entropy is in simple ABMs.
X01
A
A.
B.
C.
D.
E.
F.
G.
H.
Outline of Presentation
Introduction of “Model I”
Definition of “Entropic Index”
The space H(4,8) and its H(4,8,20) Sub-Space
Transition Pairs and Asymmetric Probabilities
Transition Sets and WAECs.
e-curves and The E-curve
Q&A
The space H(4,100) and its H(4,100,250) Sub-Space
a.
b.
c.
d.
e-curves,
densities,
virtual equilibrium state,
tarry times
A00
History of “Model I” of EiLab
A Fully Bounded Capital Exchange Model
More complex variations of this model have
been discussed in received literature for at
least 26 years:
 Bennati – 1988, 1993
 Drăgulescu and Yakovenko – 2000
 Scalas, Garibaldi and Donadio – 2006
 Referred to as the BDY Model
EiLab is a software application in which models A through J have been
implemented, all being variations of the BDY model.
A01
EiLab, Model I, H(4,8,20) Instance
EiLab Application
K=4
A=8
W = 20
Parameter Selection
“IWiz” - MODEL
SELECTION
ABCDEFGH I J
Model Type/Design
Selection
Instance of Model I
Lots of “Empirical” Data
(MS Excel)
A02
Operational Description of “Model I”
A Fully-Bounded Capital Exchange Model
Variables of enumeration:
 i – amount of wealth held by agent, discrete integer, 1 <= i <= K
 ai – number of agents having wealth = $i
 A – total number of agents
 W – total number of dollars in an instance of the model
Process – Simple capital exchange:
 Randomly select 2 agents
 Randomly assign as 1 winner; 1 loser
 If loser has $1 or winner $K – transaction disallowed
 Otherwise loser pays winner $1
A = 8 Agents
Structure – There are three structural constants:
 K – number of allowed wealth levels – labeled $1 to $K
 Repeat
W = ($12)+ ($23)+ ($30)+ ($43)=$20
K = 4 wealth bins
A03
Initial and “Balanced” Configurations
of “Model I”, Version H(4,8,20)
K = 4 bins ($1, $2, $3 and $4); A = 8 agents; W = $20 total wealth.
Initial configuration.
Configuration Names:
“Balanced” configuration.
131( 0, 4, 4, 0 )
69( 2, 2, 2, 2 )
A04
Symbolic Notation for “Model I”
A Fully-Bounded Capital Exchange Model
Notation for Configurations or States:
 h – an ordered K-tuple of the form ( a1, …, aK ) represents a
configuration, or state, or histogram, of one instance of “Model I”
 An Excel process used to generate configurations, in order, assigns
a serial number h.
 h( a1, …, aK ) notation is used to link the serial number with the
configuration.
Examples: 131( 0, 4, 4, 0 ) or 69( 2, 2, 2, 2 )
Notation for State Spaces:
 H(K,A) = the set of all histograms with K bins and A agents.
 H(K,A,W) = the set of all histograms in H(K,A) having $W of total
wealth.
A05
A – INTRODUCTION OF “MODEL I”
Parameters:
K = 4 bins in h
A = 8 agents
W = 20 dollars
Configurations:
h(a1, a2, a3, a4)
Configuration Spaces:
H(K,A)
H(K,A,W)
1. CHOOSE TWO RANDOM AGENTS
2. DESIGNATE ONE AS LOSER, RANDOMLY
3. CAPITAL EXCHANGE IS EFFECTED:
a. IF POSSIBLE, LOSER PAYS WINNER
b. ELSE, EXCHANGE IS DISALLOWED
Model I is a variation on the BDY Model (Bennati-Drăgulescu-Yakovenko) in which there
is both a lower bound ($1) and upper bound ($K) on the wealth of individual agents.
A01
B
A.
B.
C.
D.
E.
F.
G.
H.
Outline of Presentation
Introduction of “Model I”
Definition of “Entropic Index”
The space H(4,8) and its H(4,8,20) Sub-Space
Transition Pairs and Asymmetric Probabilities
Transition Sets and WAECs.
e-curves and The E-curve
Q&A
The space H(4,100) and its H(4,100,250) Sub-Space
a.
b.
c.
d.
e-curves,
densities,
virtual equilibrium state,
tarry times
B00
B – DEFINITION OF ENTROPIC INDEX
FOR ABMs
Clausius’
Physical
Thermodynamic
Entropy
Boltzmann’s
Statistical
Thermodynamic
Entropy
Jaynes’
Information
Theoretic
Concept of
Entropy
Shannon’s
Informational
Entropy
Yakovenko’s
Economic
Entropy(?)
ABM Entropy
B01
Pragmatic Approximations
PROBLEMS:
• A! is too large to
compute when A > 70
Formula as defined
Using Stirling’s approximation for ln(A!)
• Approximation to
ln(ai!) is poor when ai <
5
• Goldilocks formula
Using a variation on Lanczos’ approximation for GammaLn(A!)
B03
B04
Definition of Entropic Index For ABMs
Model I State
Model I Configuration
Histogram
Agents
Multiplicity of State h
Multinomial coefficient
Entropy of state h
f is a dimensionless scaling factor.
Super state space, configuration space
Entropic Index
Maximum entropy – for this
H(K,A) state space.
B02
C
A.
B.
C.
D.
E.
F.
G.
H.
Outline of Presentation
Introduction of “Model I”
Definition of “Entropic Index”
The space H(4,8) and its H(4,8,20) Sub-Space
Transition Pairs and Asymmetric Probabilities
Transition Sets and WAECs.
e-curves and The E-curve
Q&A
The space H(4,100) and its H(4,100,250) Sub-Space
a.
b.
c.
d.
e-curves,
densities,
virtual equilibrium state,
tarry times
C00
The H(4,8) Space, and its H(4,8,20) Sub-Space
W = 20
K=4
A=8
CONFIGURATION
GENERATOR
(MS Excel)
TRANSITION SET
GENERATOR
(MS Excel)
Lots of Analytical Data
(MS Excel)
C01
The H(K,A)=H(4,8) Space of Sub-Spaces
$32
C02
The H(K,A)=H(4,8) Space of Sub-Spaces
w(h)=$24
The subspace
H(4,8,24)
contains
only 10
states.
$32
C03
The H(K,A)=H(4,8) Space of Sub-Spaces
The sub-space
H(4,8,20) contains
13 states; 6 twin
states and one
singleton.
w(h)=$20
w(h)=$24
The subspace
H(4,8,24)
contains
only 10
states.
$32
C04
The H(K,A)=H(4,8) Space of Sub-Spaces
The entropic index in
space H(4,8) is
organized into
echelons.
S(h)0.4
C06
The H(K,A)=H(4,8) Space of Sub-Spaces
The sub-space
H(4,8,20) contains
13 states; 6 twin
states and one
singleton.
w(h)=$20
$32
C04
The
H(K,A,W)=H(4,8,20)
Sub-Space
With an initial
endowment of $20,
this space has only 13
possible states, with an
entropic index ranging
from 0.406 up to 1.000
for the “balanced”
state (#69).
C05a
The
H(K,A,W)=H(4,8,20)
Sub-Space
With an initial
endowment of $20,
this space has only 13
possible states, with an
entropic index ranging
from 0.406 up to 1.000
for the “balanced”
state (#69).
C05b
The
H(K,A,W)=H(4,8,20)
Sub-Space
With an initial
endowment of $20,
this space has only 13
possible states, with an
entropic index ranging
from 0.406 up to 1.000
for the “balanced”
state (#69).
C05c
D
A.
B.
C.
D.
E.
F.
G.
H.
Outline of Presentation
Introduction of “Model I”
Definition of “Entropic Index”
The space H(4,8) and its H(4,8,20) Sub-Space
Transition Pairs and Asymmetric Probabilities
Transition Sets and WAECs.
e-curves and The E-curve
Q&A
The space H(4,100) and its H(4,100,250) Sub-Space
a.
b.
c.
d.
e-curves,
densities,
virtual equilibrium state,
tarry times
D00
Transition Pairs
• Two configurations are called a transition
pair if each is able to transition to the
other in one capital exchange.
• Configuration 100(1,2,5,0) can transition
to 96(1,3,3,1) if both the loser and the
winner are in bin 3.
• Configuration 96(1,3,3,1) can transition
to 100(1,2,5,0) if the winner is in bin 2
and the loser is in bin 4.
• Configurations 96 and 100 form a
transition pair.
100
(1,2,5,0)
96
(1,3,3,1)
D01a
Transition Pairs
• Two configurations are called a transition
pair if each is able to transition to the
other in one capital exchange.
• Configuration 100(1,2,5,0) can transition
to 96(1,3,3,1) if both the loser and the
winner are in bin 3.
• Configuration 96(1,3,3,1) can transition
to 100(1,2,5,0) if the winner is in bin 2
and the loser is in bin 4.
• Configurations 96 and 100 form a
transition pair.
100
(1,2,5,0)
96
(1,3,3,1)
96
(1,3,3,1)
100
(1,2,5,0)
D01b
Transitions Between Configurations
H(K,A,W) = H(4,8,20)
Entropic
Index
Levels
69
1.000000
96
0.905639
93
73
53
0.780639
100
0.649397
128
131
0.500000
0.405639
66
78
78 73 69
35
RATCHETS?
0.875000
49
126
Configuration Numbers – Global/Local Minima, Medial, Maximum
Transition paths – Asymmetric Probabilities
D02
Consider two
configurations
Transition Pairs
Ei =  0.26
96( 1, 3, 3, 1 )
Ei = 0.91
Ei = 0.65
100( 1, 2, 5, 0 )
D03a
96
Ei = 0.65
100( 1, 2, 5, 0 )
Ei = + 0.26
Ei =  0.26
96( 1, 3, 3, 1 )
Ei = 0.91
Transition Up
P(100 96) = 0.36
Consider two
configurations
Transition Pairs
100
Denote probability of transition up as P(100 96) = (5/8)(4/7) = 0.36.
D03b
Ei = 0.65
100( 1, 2, 5, 0 )
P(96 100) = 0.05
Ei = - 0.26
Ei = + 0.26
100
Transition Down
96
96
Ei =  0.26
96( 1, 3, 3, 1 )
Ei = 0.91
Transition Up
P(100 96) = 0.36
Consider two
configurations
Transition Pairs
100
Denote probability of transition up as P(100 96) = (5/8)(4/7)= 0.36.
Denote probability of transition down as P(96 100) = (3/8)(1/7) = 0.05.
D03c
Asymmetric Probabilities of Transition
Within a Transition Pair
Ratios: PUp / PDown
D04
Transitions Between Configurations
H(K,A,W) = H(4,8,20)
Entropic
Index
Levels
69
1.000000
96
0.905639
0.875000
93
73
53
30
0.780639
100
0.649397
66
128
131
0.500000
0.405639
49
78
78 73 69
35
126
Configuration Numbers – Global/Local Minima, Medial, Maximum
Transition paths – Asymmetric Probabilities
D05a(2b)
Expected: based on M100; Observed: based on 309,537 transitions.
D06
E
A.
B.
C.
D.
E.
F.
G.
H.
Outline of Presentation
Introduction of “Model I”
Definition of “Entropic Index”
The space H(4,8) and its H(4,8,20) Sub-Space
Transition Pairs and Asymmetric Probabilities
Transition Sets and WAECs.
e-curves and The E-curve
Q&A
The space H(4,100) and its H(4,100,250) Sub-Space
a.
b.
c.
d.
e-curves,
densities,
virtual equilibrium state,
tarry times
E00
Transitions Between Configurations
H(K,A,W) = H(4,8,20)
Entropic
Index
Levels
69
1.000000
96
0.905639
0.875000
93
73
53
30
0.780639
100
0.649397
66
128
131
0.500000
0.405639
49
78
78 73 69
35
126
Configuration Numbers – Global/Local Minima, Medial, Maximum
Transition paths – Asymmetric Probabilities
D05b(2b)
The transition set T100
96
73
100
131
78
Define the weighted average entropic
change (WAEC) on the transition set of h.
E02
The transition set T100
96
Reflexive Transition
100
(1,2,5,0)
100
(1,2,5,0)
73
100
131
78
Define the weighted average entropic
change (WAEC) on the transition set of h.
E02
WAEC of a
Configuration
• We can compute the WAEC
for each configuration.
• Note that it is negative for
those configurations with
the highest entropic index.
• The model will hover near
WAEC = 0.
Define the “virtual
equilibrium state” as that
virtual state for which
WAEC = 0.
E03
Transitions Between Configurations
H(K,A,W) = H(4,8,20)
Entropic
Index
Levels
69
1.000000
0.875000
93
73
53
0.780639
100
0.649397
66
128
131
0.500000
0.405639
49
78
78 73 69
VIRTUAL
EQUILIBRIUM
STATE?
96
0.905639
35
126
Configuration Numbers – Global/Local Minima, Medial, Maximum
Transition paths – Asymmetric Probabilities
E04
F
A.
B.
C.
D.
E.
F.
G.
H.
Outline of Presentation
Introduction of “Model I”
Definition of “Entropic Index”
The space H(4,8) and its H(4,8,20) Sub-Space
Transition Pairs and Asymmetric Probabilities
Transition Sets and WAECs.
e-curves and The E-curve
Q&A
The space H(4,100) and its H(4,100,250) Sub-Space
a.
b.
c.
d.
e-curves,
densities,
virtual equilibrium state,
tarry times
F00
e-curves and The E-curve
Enveloping
E-curve
Basal (inter-nodal)
e-curves
$32
F01
e-curves and The E-curve
Enveloping
E-curve
Basal (inter-nodal)
e-curves
CONSTRUCTION?
$32
F01
$16
$24
F02
e-curves and The E-curve
Basal (inter-nodal)
e-curve
$16
$24
$32
F03
e-curves and The E-curve
Basal (inter-nodal)
e-curve
EQUATIONS?
$16
$24
$32
F03
Equations of virtual e-curves & E-curves
A basal e-curve from node 2 (where W(h) = 16) to node 3 (where W(h) = 24) in the
space H(K,A)=H(4,8).
A generalized enveloping E-curve for the space H(K,A) identifies the maximal
entropy for each sub-space H(K,A,W).
Developed using Stirling’s approximation, due to ease of analytic resolution.
F04
$21
$19
Closed H(K,A,W) spaces:
• Virtual two-step transition path, as $1
is removed from loser, altering the
space to an H(K,A,W-1) space, then
added to the winner, restoring it to
the H(K,A,W) space.
Open H(K,A) spaces (not studied yet):
• Optional highways within H(K,A)
space along which a system which is
open with respect to wealth will find
a trajectory.
$20
Significance of e-curves
96
100
F06
Significance of The E-curve (?)
In Thermodynamics
In Economics
The maximum economic change that an
economic engine (e.g. a non-renewable
resource such as copper) is able to
produce in an economy varies directly
with the grade of the resource
endowment, as well as its size.
The maximum work that a heat engine is
able to do on the environment is the
Carnot Efficiency. The Carnot efficiency is
computable as 1 minus an entropic index.
𝑇
𝐶 = 1 − 𝑇𝐻 = Carnot Efficiency  Grade
𝐶
𝐸𝑥𝑒𝑟𝑔𝑦 =
U𝑠𝑒𝑓𝑢𝑙 𝑊𝑜𝑟𝑘 =
1−
𝐸𝑐𝑜𝑛𝑜𝑚𝑖𝑐 𝐺𝑟𝑎𝑑𝑒 ≡
𝑇𝐻
× 𝐸𝑛𝑒𝑟𝑔𝑦𝑖𝑛
𝑇𝐶
1−𝑆
𝑆
𝑚𝑎𝑥
𝑈𝑠𝑒𝑓𝑢𝑙
𝐸𝑐𝑜𝑛𝑜𝑚𝑖𝑐 =
𝐶ℎ𝑎𝑛𝑔𝑒
× 𝐸𝑛𝑒𝑟𝑔𝑦𝑖𝑛
1−
𝑆
𝑆𝑚𝑎𝑥
𝑠(ℎ)
1−
𝑠𝑚𝑎𝑥
×
𝐶𝑎𝑝𝑖𝑡𝑎𝑙
𝑉𝑎𝑙𝑢𝑒
𝑖𝑛
!
F07
F05
G
Garvin H Boyle
Email: [email protected]
URL: Orrery-software.webs.com
URL: www.openabm.org/model/3860/version/3
G01
H
A.
B.
C.
D.
E.
F.
G.
H.
Outline of Presentation
Introduction of “Model I”
Definition of “Entropic Index”
The space H(4,8) and its H(4,8,20) Sub-Space
Transition Pairs and Asymmetric Probabilities
Transition Sets and WAECs.
e-curves and The E-curve
Q&A
The space H(4,100) and its H(4,100,250) Sub-Space
a.
b.
c.
d.
e-curves,
densities,
virtual equilibrium state,
tarry times
H00
The Space H(4,100)
Configuration Generator --- K = 4; A = 100
H01
H02
(29,21,21,29)
(29,21,21,29)
H03
H09a
H09b
$20
H09c
$20
$250
H09c
H05
H12
H13
H14
Definition of Tarry Time: Number of contiguous ticks spent in a state or configuration
before exiting it.
Disallowed
Inceptive Transition
Reflexive
Inceptive Transition
Transitions
(entry to next state)
Transitions
(entry to state)
( = 1 tick each)
(= 1 tick)
( = 1 tick each)
( = 1 tick)
{
+
+
}{
H14b
H18a
H18b
H04
Cardinality of Entropic Level
Cardinality of Entropic Level vs. Entropic Level
H(K,A) = H(4,100); All 8,661 Levels
60
50
40
30
20
10
0
0.0
0.2
0.4
0.6
Entropic Level
0.8
1.0
H06
H07
H08
The Space H(4,100,250)
K = 4; A = 100; W = $250
G10
Transition Set Generator --- K = 4; A = 100; W = $250
H10
H11
H15
H16
H17
H19
H20
Transition Sets
Denote by Th the set of all configurations t which can be
assumed in exactly one time step. Call Th the “transition
set” of h. Note that h ϵ Th due to disallowed and reflexive
transitions.
E01
Plea for Related Research
I STRONGLY BELIEVE THAT
RESEARCH INTO THE ROLE OF ENTROPY
IN AGENT BASED MODELS
WILL PROVIDE DEEP INSIGHT INTO
THE ROLE OF ENTROPY IN
ECONOMIC PROCESSES.
X04
Hip Pocket Slides
X06