CAMS
Choice under Social Influence:
the Curse of Coordination
Jean-Pierre Nadal
Laboratoire de Physique Statistique (LPS) de l’ENS
and
Centre d’Analyse et Mathématique Sociales (CAMS), EHESS
http://www.lps.ens.fr/~nadal/
nadal@lps.ens.fr
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Collaborators
Mirta B. Gordon
Denis Phan
TIMC-IMAG (umr CNRS UJF) - Grenoble
CREM – Université de Rennes I, and:
GEMAS - UMR CNRS-Paris Sorbonne (Paris IV)
Viktoriya Semeshenko TIMC-IMAG (umr CNRS UJF) - Grenoble
Jean Vannimenus
Laboratoire de Physique Statistique – ENS - Paris
Project ELICCIR supported by the program
« Complex Systems in Human and Social Sciences » (CNRS-MENR)
collective behavior
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•
From a « microscopic » level (description of the agents and their interactions)
to a « macroscopic » level (description of collective behavior).
Examples from economics, theoretical neuroscience, statistical physics:
Elementary units
Interactions
Collective level
agents
preferences
social influences
( externalities )
market :
equilibrium price
neurons
synaptic weights
psychophysics:
associative memory
spins
(magnetic moments)
interactions
thermodynamics:
ferromagnetism
activation rule
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•
The dying seminar
T. C. Schelling
« Micromotives and Macrobehavior », Norton & Cy, 1978)
(« La tyrannie des petites décisions », PUF, 1980)
« critical mass model » :
• N scientists are asked to participate to a seminar/working group, meeting
every Saturday
•each week every one knows what was the attendance of the last Saturday
• every one has his own willingness to participate: scientist i wants to
participate if the number who attend is larger than ni
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The dying seminar
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F(n) = fraction of agents who want to attend if: ni ≤ n
η=n/N
n(1)/N = F(n(0))
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The dying seminar
F(n) = fraction of agents who want to attend if: ni ≤ n
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The dying seminar
F(n) = fraction of agents who want to attend if: ni ≤ n
heterogeneous agents
+ social interactions
= multiple equilibria
η=
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Discrete choices
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plan
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Market (and non market) model with a single good and externalities
(~ T. C. Schelling, « the dying seminar » and RFIM at T=0)
•
equilibrium properties and collective states
ƒ customer’s phase diagram
ƒ monopolist’s phase diagram
• JPN, D. Phan, M.B. Gordon, J. Vannimenus (2005)
Multiple equilibria in a monopoly market with heterogeneous agents and externalities
Quantitative Finance Vol.5, No. 6, 557-568 - preprint 2003 arXiv cond-mat/0311096
• M. B. Gordon, JPN, D. Phan and J. Vannimenus
Seller's dilemma due to social interactions between customers , Physica A, 2005
• M. B. Gordon, JPN, D. Phan and V. Semeshenko
Discrete Choices under Social Influence: Generic Properties, preprint 2007
http://halshs.archives-ouvertes.fr/halshs-00135405
• M. B. Gordon, JPN, D. Phan and V. Semeshenko
The perplex monopolist: optimal pricing with customers under social influence, in prepapration
•
repeated game framework:
ƒ hysteresis
ƒ behavorial learning by the customers
ƒ V. Semeshenko, M B Gordon, JPN & D Phan, proceedings of ECCE1, Springer 2006
ƒ JPN, V. Semeshenko, M B Gordon, in preparation
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Market model
simplifying hypothesis
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strategic complementarity : Jik > 0
making the same choice as the others is advantageous
homogeneous social influence (Jik=J):
weight of neighbours'
choices
•
global neighbourhood and large N :
1
J
ϑi
∑ ωk = J ηi
k∈ϑi
fraction of i’s neighbours
that adopt
N
1
1 N
ηi =
ωk ≅ ∑ ωk ≡ η = fraction of buyers
∑
N − 1 k =1
N k =1
(k ≠i)
η insensitive to fluctuations:
single agents cannot influence individually the collective term Jη
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Nash equilibria
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individual i’s choice : buy if
Vi − P = H + θi + J η − P > 0
fraction of buyers
fraction of buyers : if η = 0 ⇒ for H + θi > P : ωi = 1
if η = 1 ⇒ for H + θi < P − J : ωi = 0
f(H+θi)
non buyers
buyers
H
P-J
P-Jη
•
Nash equilibrium : η =
∞
∫ f(H + θ) dθ
P − Jη
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P
Hi=H+θi
IWP: logistic distribution
CAMS
logistic pdf
0,5
β
2 cosh2 βθ
0,4
f(βθ)
f( θ) =
f(βθ ) =
β
2 cosh 2 βθ
0,3
customers
phase diagram
0,2
0,1
η = fraction of buyers
0,0
-3
j
-2
-1
0
1
2
3
under blue=low h, above red=high h
h - p 2= β(H - P)
βθ
h-p
H>P
1
gle on
n
i
s ut i
so l
η
h- p
0
-1
-2
-3
2
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H<P
two
sol stab
uti
l
on s e
βJB
4
6
j
j=βJ
8
10
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customer’s phase diagram
smooth mono-modal distribution of the IWP
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customer’s phase diagram
alternative representation
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IWP: triangular distribution
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f(h i)
distribution :
2/(3b)
customers phase diagram :
0.4
0.2
h-p
0.0
-2
h-b
h-b
0
h
2
2
4
h+2b x
h+2b
b
η = fraction of buyers
η=1
0
-b/2
0<η<1
-2
-2b
η
η=0
-4
j
h-p
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0
1
2
b
3
jB 2b
4
3b
5
6
j
coexistence
of 2 solutions
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Customer’s phase diagram
triangular distribution of the IWP
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Customer’s phase diagram
multimodal distribution: exple, bi-modal case
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Customer’s phase diagram
bi-modal distribution
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plan
CAMS
•
market model with a single good and externalities
•
equilibrium properties and collective states
J.-P. Nadal, D. Phan, M.B. Gordon, J. Vannimenus (2003) Multiple equilibria in a monopoly
market with heterogeneous agents and externalities
ƒ multiple solutions
ƒ customer’s phase diagram
ƒ Monopolist’s phase diagram
•
repeated game framework :
•
•
empirical data
perspectives
ƒ hysteresis
ƒ learning by the customers
experience Weighted Attraction (EWA) learning scheme (Camerer, 2003)
results with different EWA learning models
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Monopolist’s phase diagram
Monopolist:
Dilemma:
sell at high price to a small number of customers
or at low price to a large number of customers?
Maximisation of the profit
P = price of one unit of the good
C = cost per unit
Choose the price P which maximises
Π = ( P - C ) η(P) N
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Monopolist’s dilemma
coexistence
of 2
solutions
This figure: logistic distribution for the IWP
But = generic phase diagram for any smooth monomodal distribution
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Monopolist’s phase diagram
transition
Price (η-)
η+
Profit (−)
η-
Price (η+)
Profit (+)
j=βJ
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monopolist’s phase diagram
curse of coordination
j
curse of coordination: the optimal seller’s strategy corresponds to a price for which
the demand is not unique; the optimal profit will not be obtained if the population
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do not
coordinate
plan
CAMS
•
market model with a single good and externalities
•
equilibrium properties and collective states
•
J.-P. Nadal, D. Phan, M.B. Gordon, J. Vannimenus (2003) Multiple equilibria in a monopoly
market with heterogeneous agents and externalities
ƒ multiple solutions
ƒ customer’s phase diagram
ƒ Monopolist’s phase diagram
repeated game framework :
ƒ hysteresis
ƒ learning by the customers
•
•
empirical data
perspectives
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Hysteresis
•
Myopic Best-Response Dynamics
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Evolution of the demand as the price increases or decreases
(or as the mean willingness to pay/adopt, h, decreases or increases)
•
Known from statistical physics:
hysteresis and avalanches
(ref. : Sethna et al, 1993)
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Numerical simulations (« multi-agents »)
Thomas C. Schelling
« Some vivid dynamics can be generated by any reader with a half-hour to
spare, a roll of pennies and a roll of dimes, a tabletop, a large sheet of paper,
a spirit of scientific inquiry, or, lacking that spirit, a fondness for games. »
in From micromotives to macrobehavior (1978)
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Myopic Best-Response Dynamics
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Myopic Best-Response Dynamics
= jB
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Myopic Best-Response Dynamics
finite size effect
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Myopic Best-Response Dynamics
CAMS
plan
•
•
market model with a single good and externalities
equilibrium properties and collective states
•
repeated game framework :
ƒ multiple solutions
ƒ customer’s phase diagram
ƒ monopolist’s phase diagram
ƒ hysteresis
ƒ behavorial learning by the customers learning_to_choose.ppt
experience Weighted Attraction (EWA) learning scheme (Camerer, 2003)
results with different learning models
V. Semeshenko, M B Gordon, JPN & D Phan, 2006, proceedings of ECCE1, Springer
JPN, V. Semeshenko, M B Gordon, in preparation
•
•
empirical data
perspectives
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hypothesis
the demand adapts to the price faster than the time scale of price
revision:
price is assumed to be fixed during the customers' learning.
• each agent must decide whether to buy or not
ƒunder imperfect information (he doesn't know the decisions of the others)
ƒand incomplete information (he doesn't know his actual payoff)
• based on the "attraction" of buying or not buying
• attraction values are updated (learned) based on the actual fractions of
buyers
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learning dynamics
CAMS
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At each time step each agent i makes a binary decision :
Pr oba[ωi ( t ) = 1] = f(Δi ( t ) ),
ƒ best response:
ƒ trembling hand:
relative attraction
for buying
with Δi ( t ) ≡ Ai1 ( t ) − Ai0 ( t )
⎧ 1 if Δi ( t ) − P > 0
ωi ( t ) = ⎨
⎩0 if Δi ( t ) − P < 0
P[ωi (t ) = 1] = 1 − ε (Δ i (t ) − P )
If logistic : P[ωi (t ) = 1] =
1
1 + exp− β [Δ i (t ) − P ]
• Learns his relative attraction [Camerer: Experience Weighted Attractions]
from the observation of the actual fraction of buyers η(t):
Δ i (t ) = H i + Jη̂i (t )
η̂i (t ) = agent i’s estimate of η
ηˆi (t + 1) = ηˆi (t ) + μ (t + 1){ [δ + (1 − δ )ωi (t )] η (t ) − ηˆi (t )}
μ(t + 1) = (1 − κ )
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μ( t )
+ κ (1 − φ)
μ( t ) + φ
CAMS
Experience Weighted Attraction
• updating attractions
in the EWA cube: no
learning
(Camerer, 2003)
1
μ = 1−φ
ˆ
ηi (t + 1) = ˆ
ηi ( t ) + μ(t + 1) { [δ + (1 − δ ) ωi ( t )]η( t ) − ˆ
ηi ( t )}
μ( t )
μ(t + 1) = (1 − κ )
+ κ (1 − φ)
μ( t ) + φ
unlearning
myopic
reinforcement
type of
learning rate
κ
myopic
fictitious
play
time
ficti average
d
tiou
s pla
we
y
ig
bel hted
tim
ief
e-a
0
v
e
→1 −raφge
ry μ ⎯⎯⎯
t →∞
o
dr
emm
ein
m
1
f→
o
orc0t
e
n
μ ⎯⎯⎯
mo
t
→∞
ry
0
of
rei δ = 0
pa
nfo
st
rce
pa
yo
me
ffs
nt
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φ
1
δ
fict = 1
itio
us p
la y
t .
h
ig vs
e
w ed yed
ay la
pl t p
no
δ
simulations
CAMS
phase diagram
for the
triangular distribution :
h-p
2
b
P1
η=1
0
-b/2
0<η<1
-2
-2b
η=0
-4
0
1
2
b
3
jB 2b
4
3b
5
6
j
P2
J=4
• Simplest case :
μ ( t + 1) = 1
1
myopic
fictitious
play
κ
type of
learning rate
1
0
me
mo
ry
φ
1
of
pa
st
0
pa
yo
f fs
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δ ht
ig s.
v
we ed yed
ay la
pl t p
no
ηˆi ( t + 1) = ηˆi ( t ) + μ ( t + 1) ⎡⎣η ( t ) − ηˆi ( t ) ⎤⎦
ηˆi ( t + 1) = η ( t )
myopic fictitious play
CAMS
j=4, κ=1, φ=0, δ=1
ωi(0)=0
ωi(0)=1
ωi(0) rational
consistent
1.0
η = fraction
of buyers
0.8
η
0.6
0.4
0.2
0.0
1
2
p1
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price
p2
3
4
discussion
CAMS
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•
asymptotics for φ < 1
ƒ if δ > 0
ƒ buyers : ηˆi → η
ηˆi
ƒ for non buyers:
ƒ discontinuity at Δ = P
→ δ η <η
: attractions are underestimated
asymptotics for φ > 1
ƒ the time decay of μ(t) may hinder the learning process
• more results:
ƒ decision through a trembling hand: interference between ε and μ
ƒ analytic results
ƒ the learning process as a special random walk
ƒ learning directly the attractions (estimations of Hi+Jη-P) leads to very
different results
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Trembling
hand
Adaptive
dynamics
IWP - P
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Trembling hand
smooth regime
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Stationary distribution of the attraction of a single agent
0 <(normalized attraction) < 1
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Ai(t)
(singular regime)
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Hi - P
singular regime
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Stationary distribution of the attraction of a single agent
0 <(normalized attraction) < 1
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CAMS
•
•
•
•
•
plan
market model with a single good and externalities
J.-P. Nadal, D. Phan, M.B. Gordon, J. Vannimenus (2003) Multiple equilibria in a
monopoly market with heterogeneous agents and externalities
equilibrium properties and collective states
ƒ multiple solutions
ƒ customer’s phase diagram
ƒ Monopolist’s phase diagram
repeated game framework :
ƒ hysteresis
ƒ learning by the customers
experience Weighted Attraction (EWA) learning scheme (Camerer, 2003)
results with different EWA learning models
empirical data
perspectives
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CAMS
Empirical Data
Number of cell phones
Q. Michard & J.-P. Bouchaud, preprint 2005
« Theory of collective opinion shifts: from smooth trends to abrupt swings »
http://arxiv.org/abs/cond-mat/0504079
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Empirical Data: scaling
CAMS
h ~ w-2/3
log(h)
log(w)
Q. Michard & J.-P. Bouchaud, preprint 2005
« Theory of collective opinion shifts: from smooth trends to abrupt swings »
http://arxiv.org/abs/cond-mat/0504079
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empirical data
« Of Songs and Men: a Model for Multiple Choice with Herding »
Christian Borghesi and Jean-Philippe Bouchaud
arXiv:physics/0606224 v1
(June 2006)
modelling/analysis of data:
web-based music market experiment,
M. J. Salganik, P. S. Dodds, D. J. Watts, Experimental Study of Inequality and
Unpredictability in an Artificial Cultural Market, Science 311, 854-856 (2006)
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CAMS
Current works and Perspectives
• Heterogeneous agents + social interactions + learning
- more on the monopolist model and its variants (joint
adaptive dynamics: seller and customers) ; more on
adaptive dynamics (market and non market contexts)
[+ M. B. Gordon, V. Semeshenko (TIMC, Grenoble), J. Vannimenus (LPS ENS); D.
Phan (GEMAS), R. Waldeck (ENST Bretagne)]
- diffusion of criminality - Project supported by the City of Paris
[+ L. Kebir, H. Berestycki (CAMS), V. Semeshenko, M. B. Gordon (TIMC,
Grenoble), S. Franz (ICTP, Trieste)]
- social interactions in medical care
[+ E. Tanimura (CAMS), J. Scheinkman (Princeton) ]
- adaptive dynamics and evolution of phoneme categories
[+ J. Pierrehumbert (Northwestern Univ.), Laurent Bonnasse-Gahot (CAMS)]
Orléans 15 mars 2007physics
- economics - mathematics - linguistics
CAMS