Rockstars & Agents
To Each According to its Degree:
The Meritocracy an Topocracy of Embedded
Markets
Florentino Borondo
Dep. de Química, and
Instituto de Ciencias Matemáticas (ICMAT)
Universidad Autónoma de Madrid
COMSOTEC 2015
Santander, Septiembre 2015
September 10, 2015
Rockstars & Agents
Outline
Introduction
Model
Mathematical analysis
Sparse network limit
The meritocracy of networks
Payoff distribution
Meritocracy and Topocracy
Rockstars & Agents
Rockstars & Agents
Introduction
INEQUALITY is observed in society at many levels
and this always have need a JUSTIFICATION
Rockstars & Agents
Introduction
For example, in ancient times
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Power of the Kings was justified as coming from God
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Examples: James I of England and VI of Scotland (1566-1625)
(his son Charles I)
Louis XIV King of France and Navarra (1638-1715)
Rockstars & Agents
Introduction
It was similar to the power of the Pope!
Rockstars & Agents
Introduction
Theoretitians
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Jean Bodin (1530-1596), On Sovereignty, and
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Jacques Bénigne Bossuet (1627-1704)
Rockstars & Agents
Introduction
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Later, Monarchies disappeared ...
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INEQUALITY has remained
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Other supporting theories are needed
Rockstars & Agents
Introduction
(Explanation of the title:)
“Jeder nach seinen Fähigkeiten, jedem nach seinen Bedürfnissen!”
“From each according to his ability, to each according to his need!’’
is a famous phrase in Karl Marx’s 1875 Critique of the Gotha Program
(draft program of the United Worker’s Party of Germany)
Rockstars & Agents
Introduction
(Explanation of the title:)
“To each according to what he and the instruments he owns produce’’
Milton FRIEDMAN
Rockstars & Agents
Introduction
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Other supporting theories
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I
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Marginal Productivity theory:
Markets reward people differently, because their
contributions to society are different
Society is unequal but not (necessarily) unfair. Meritocracy
Mechanisms, Individual Factors
(Talent, Industriousness, ...)
Rockstars & Agents
Introduction
But any new revision of these theories
in the 21st century
MUST INCLUDE Networks, since we are all connected
M. Granovetter, “Economic action and social structure: the
problem of embeddedness”, Am. J. Sociol. 91, 481 (1985)
Two decades ago !!!
contagion of diseases (Vespignani), social behavior(Centolla),
political participation (J. Borondo), ...
Position in th network: TOPOCRACY
Reason: sparsity of social networks
(1 individual is connected to 3,000)
Rockstars & Agents
Model
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Simple agent-based model of a market of cultural goods
(books, films, music):
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I
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Produce content: Rockstar (R) behavior
Help distribute: Agent (A) or Middleman behavior
Buy content; Consumer (C)
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Starting point: Exogenous network of N nodes, each one
characterized by its talent, 0 ≤ Ti ≤ 1
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Value generation: R’s produce new content with cost c = 0.
Sell it to Ti (N − 1) ' Ti N nodes at a fixed price s = 1,
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Value distribution: If R is not directly connected to C, the
sale is done through a chain of A’s, which share the
benefit, s.
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Question: By which behavior a node makes more money?:
Producing goods (R) or acting as intermediaries (A)?
Meritocracy vs. Topocracy
Rockstars & Agents
Model
Rockstars & Agents
Model
Rockstars & Agents
Mathematical analysis
Sparse network limit
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Sparse Network Limit
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Loops are negligible
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The average payoff of an individual can be split intro the
contributions from each of its two behaviors:
πi = πRi + πAi
Rockstars & Agents
Mathematical analysis
Sparse network limit
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R’s payoff:
hki2 hki3
+
+ ...
= Ti hki +
2
3
πRi
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= Ti
`
X
hkii
i=1
i
Finite network cutoff (when all nodes are reached):
`
X
hkii = N,
i=0
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Behavior: πRi scales linearly with Ti
`=
ln N
lnhki
Rockstars & Agents
Mathematical analysis
Sparse network limit
Numerical simulations
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hTi = 1/2
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Random (Erdös-Renyi) network
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N = 1000
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50 randomly chosen realization of the network
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10 random choices of the buyers
Rockstars & Agents
Mathematical analysis
Sparse network limit
Kinks takes place at the ` jumps:
Rockstars & Agents
Mathematical analysis
Sparse network limit
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Agents’ payoff:
hki2 hki3
+
+ ...
ΠR = NhTi hki +
2
3
hki2 2hki3 3hki4
ΠA = NhTi
+
+
+ ...
2
3
4
πAj
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= NhTi
`
X
hkii
i=1
= NhTi
i
`
X
i−1
i=2
i
hkii
`
`
X
kj2 X
i−1 i
i−1 i
hTi
2
= NhTi P 2
hki = kj
hki
2
i
hki
−
hki
i
i ki i=2
i=2
Prediction: πAj scales quadratically with kj
Rockstars & Agents
Mathematical analysis
Sparse network limit
Rockstars & Agents
Mathematical analysis
Sparse network limit
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Moreover:
Explanation:
Quadratic coeff:
hTi
hki2 −hki
πA ∼
k2
hki3/2
P`
i−1
i
i hki
i=2
∼
hki3
hki2
Rockstars & Agents
Mathematical analysis
Sparse network limit
Finally:
πi = πRi + πAi = CTi + Bki2
where
C=
`
X
hkii
i=1
i
`
Xi−1
hTi
and B =
hkii
hki2 + hki
i
i=2
Rockstars & Agents
Mathematical analysis
Sparse network limit
Income Map:
πA
πR +πA
Rockstars & Agents
Mathematical analysis
Sparse network limit
Rockstars & Agents
Mathematical analysis
The meritocracy of networks
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We have seen that:
1. A fully connected network is perfectly Meritocratic (M)
2. A sparse network tends to be Topocratic (T )
3. Or a star network is highly topocratic
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Question? When does the transition from M to T takes
place?
CNhTi
C
ΠR
=
=
Π
N(N − 1)hTi
N−1
1
N−1
ΠR
> =⇒ C >
Π
2
2
M =⇒
`
N − 1 X hkii
hki`
hkiln N/ lnhki
<
'
'
2
i
`
ln N/ lnhki
i
hki > N 1/2
Rockstars & Agents
Mathematical analysis
The meritocracy of networks
Rockstars & Agents
Mathematical analysis
Payoff distribution
Rockstars Payoff distribution:
πR = CT =⇒
`
X
P(πR ) = Uniform(0, C) = Uniform 0,
hkii
i=0
!
Rockstars & Agents
Mathematical analysis
Payoff distribution
Agents Payoff distribution:
πA =
Bk2
dk
1
P
=⇒ P(πA ) = P(k)
= √
dπA
2 BπA
r
πA
B
P(k) = Poisson(hki)
√
1
e−hki hki πA /B
p
P(πA ) = √
2 BπA ( πA /B)!
p
P(k) ' Gaussian(hki, hki)
1
P(πA ) = p
e
2 ΠhkiBπA
√
2
−
πA /B−hki /2hki
Rockstars & Agents
Mathematical analysis
Payoff distribution
Rockstars & Agents
Mathematical analysis
Meritocracy and Topocracy
Meritocracy
M = corr(T, πR + πA )
In a fully connected network (perfectly meritocratic)
π = (N − 1)T =⇒ M = corr(T, (N − 1)T) = 1
In general:
M = corr(T, πR + πA ) =
cov(T, πR ) + cov(T, πA )
σT σπR +πA
Using properties of the var and cov, and cov(T, πA ) = 0
CσT2
M=
σT
q
σπ2 R + σπ2 A
C
=q
C2 + 12σπ2 A
Rockstars & Agents
Mathematical analysis
Meritocracy and Topocracy
Topocracy
T = corr(k2 , πR + πA ) = q
σπA
c2 /12 + σπ2 A
σπA = B2 hki(1 + 6hki + 4hki2 )
Rockstars & Agents
Mathematical analysis
Meritocracy and Topocracy
Rockstars & Agents
Mathematical analysis
Meritocracy and Topocracy
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BIG CONCLUSION:
Internet makes the World more connected, and then more
MERITOCRATIC
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Complicar el modelo
Rockstars & Agents
Mathematical analysis
Meritocracy and Topocracy