denis phan 1 , stephane pajot 1 , jean pierre nadal 2 ,

1

The Monopolist's Market with Discrete Choices and Network Externality

Revisited:Small-Worlds, Phase Transition and Avalanches in an ACE Framework .

Denis Phan1, Stephane Pajot1, Jean Pierre Nadal2,

1 ENST de Bretagne, Département ESH & ICI - Université de Bretagne Occidentale, Brest1 Laboratoire de Physique Statistique, Ecole Normale Supérieure, Paris.

[email protected] - [email protected]

Ninth annual meeting of the Society of Computational Economics University of Washington, Seattle, USA, July 11 - 13, 2003

9th Society of Computational Economics, [email protected] 2

In this paper, we use Agent-based Computational

Economics

and mathematical theorising as complementary tools

Outline of this paper (first investigations)

1 - Modelling the individual choice in a social context Discrete choice with social influence: idiosyncratic and interactive

heterogeneity

2 - Local dynamics and the network structure (basic features)

Direct vs indirect adoption, chain effect and avalanche process From regular network towards small world : structure matters

3 - « Classical » issues in the « global » externality case Analytical results in the simplest case (mean field) « Classical » supply and demand curves static equilibrium

4 - Exploration of more complex dynamics at the global level « Phase transition », demand hysteresis, and Sethna’s inner

hysteresis Long range (static) monopolist’s optimal position and the network’s

structure


The demand side: I - modelling the individual choice in a social context

Discrete choice model with social influence :

(1) Idiosyncratic heterogeneity

ii i i i t i t

0,1max V H (t) S ( ) P

Agents make a discrete (binary) choice i in the set : {0, 1}

Surplus : Vi(i) = willingness to pay – price repeated buying

willingness to pay (1) Idiosyncratic heterogeneity : Hi(t) Two special cases (Anderson, de Palma, Thisse 1992) : « McFaden » (econometric) : Hi(t) = H + i for all t ; i ~ Logistic(0,)

Physicist’s quenched disorder (e.g. Random Field ) used in this paper

« Thurstone » (psychological): Hi(t) = H + i (t) for all t ; i (t) ~

Logistic(0,) Physicist’s annealed disorder (+ad. Assumptions : Markov Random

Field ) Also used by Durlauf, Blume, Brock among others…

Properties of this two cases generally differ (except in mean field for this model )


t i ik kk

S ( ) J . (t)

Myopic agents (reactive) : no expectations :

each agent observes his neighbourhood Jik measures the effect of the agent k ’s choice on

the agent i ’s willingness to pay: 0 (if k = 0 ) or Jik (if k = 1 )

Jik are non-equivocal parameters of social influence

(several possible interpretations)

The demand side: I - modelling the individual choice in a social context

Discrete choice model with social influence

(2) Interactive (social) heterogeneityWillingness to pay (2) Interactive (social) heterogeneity :

St(-i)

ik kiJ

J J J 0N

In this paper, social influence is assumed to be positive, homogeneous, symmetric and normalized across the neighbourhood)


The demand side: II - Local dynamics and the network structure 1 - Direct versus indirect adoption,chain effect and avalanche process

Indirect effect of prices: « chain » or « dominoes »

effectVariation in price

( P1 P2 )

Change of agent i

Change of agent k

k t i 1 2

k t i 2 2

H S ( P ) P

H S ( P ) P

i t i 1 2H S ( P ) P

Variation in price

( P1 P2 )

Change of agent i

Change of agent j

Direct effect of prices

An avalanche carry on as long as:

k t 1 i 2 2

k t i 2 2

k / H S ( P ) P

H S ( P ) P


The demand side: II - Local dynamics and the network structure 2 - From regular network towards small

world : structure matters (a)

Total connectivity

Regular network (lattice)

Small world 1(Watts Strogatz)

Random network

• Milgram (1967)“ six degrees of separation”

• Watts and Strogatz (1998)• Barabasi and Albert, (1999)

“ scale free ” (all connectivity) multiplicative process power law blue agent is “hub ” or “gourou ”


The demand side: II - Local dynamics and the network structure 2 - From regular network towards small

world : structure matters (b)

0

5

10

15

20

25

30

35

40

0,650,750,850,951,051,151,251,351,451,55

Price

Nu

mb

er

of

cu

sto

mers

Empty

Neighb2

Neighb4

World

Neighb2 + SW

Neighb4 + SW

World Empty

Neighb2

Neighb4

Neighb2+ SW

Neighb4 + SW


III - « Classical » issues in the « global » externality case

1 - Analytical results in the simplest case:global externality / full connectivity (main

field)

• H > 0 : only one solution• H < 0 : two solutions ; results depends on .J

P

max (P) P. 1 F P H J. (P)

Supply SideOptimal pricing by a monopolist

in situation of risk

m

1 F(z);

with : z P H J.

Demand SideIn this case, each agent observes only

the aggregate rate of adoption, Let m the marginal consumer: Vm= 0

1

1 exp( .z)

for large populations. With F logistic :

Aggregate demandmay have two fixedpoint for high low ; (here = 20)

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

Optimum / implicit derivation gives (inverse) supply curve :

d f (z)

dP 1 J.f (z) P

s 1

p ( ) J..(1 )


0.2 0.4 0.6 0.8 1

1

2

3

4

5

6

J = 4

J = 0

H = 0

Ps

Pd


2 - Inverse curve of supply and demand: comparative static

s 1p ( ) J.

.(1 )

d 1 1

p ( ) H J. . ln

0.2 0.4 0.6 0.8 1

1

2

3

4

5

6

J = 4

J = 0

H = 2 PsPd = 1(one singleFixed point)

Dashed linesJ = 0no

externality

0.2 0.4 0.6 0.8 1

0.5

1

1.5

2

2.5

3

H = 1.9

J = 4

PsPd

Low / high P0.2 0.4 0.6 0.8 1

1

2

3

4

5

6

J = 4

H = 1

J = 0

Ps

Pd



3 - Phase diagram & profit regime transition

Full discussion of phasediagram in the plane

.J, .h, and numerically calculated solutions are

presented in:Nadal et al., 2003

+> -

+

-

-

+> -

+

P+

P -


IV - Exploration by ACE of more complex dynamics at the global level

1 - Chain effect, avalanches and hysteresis

0

10

20

30

40

50

60

70

80

90

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33

Chronology and sizes of induced adoptions in the avalanche when decrease from

1.2408 to 1.2407

0

200

400

600

800

1000

1200

1400

1 1,1 1,2 1,3 1,4 1,5

First order transition (strong connectivity)

i i i kk

V H J . P

P = H + JP = H

Homogeneous

population: Hi = H i

0

200

400

600

800

1000

1200

1400

1 1,1 1,2 1,3 1,4 1,5

= 5 = 20


IV - Exploration by ACE of more complex dynamics at the global

level 2 - hysteresis in the demand curve :

connectivity effectprices-customers hysteresis neighbours = 2

0

200

400

600

800

1000

1200

1400

0,9 1 1,1 1,2 1,3 1,4 1,5 1,6

prices

customers

prices-customers hysteresis neighbours = 4

0

200

400

600

800

1000

1200

1400

0,9 1 1,1 1,2 1,3 1,4 1,5 1,6

prices

customers

prices-customers hysteresis neighbours = 8

0

200

400

600

800

1000

1200

1400

1 1,1 1,2 1,3 1,4 1,5 1,6

prices

customers

prices-customers hysteresis neighbours = world

0

200

400

600

800

1000

1200

1400

1 1,1 1,2 1,3 1,4 1,5

prices

customers



(3) hysteresis in the demand curve :Sethna inner hystersis

(neighbourhood = 8, H = 1, J = 0.5, = 10) - Sub trajectory : [1,18-1,29]

0

200

400

600

800

1000

1200

1400

1,1 1,15 1,2 1,25 1,3 1,35 1,4

AB


1296 Agents optimal prices

adoptors profit Adoption rate

q

no externality 0,8087 1135 917,91 87,58%

Neighbour2 1,0259 1239 1 271,17 95,60%

Neighbour 4 1,0602 1254 1 329,06 96,76%

Neighbour 4_130x2 1,0725 1250 1 340,10 96,45% 5%

Neighbour 4_260x2 1,0810 1244 1 344,66 95,99% 10%

Neighbour 4_520x2 1,0935 1243 1 358,86 95,91% 20%

Neighbour 4_1296x2 1,1017 1237 1 362,35 95,45% 50%

Neighbour 6 1,0836 1257 1 361,48 96,99%

Neighbour 6_260x2 1,0997 1252 1 376,78 96,60% 7%

Neighbour 6_520x2 1,1144 1247 1 389,05 96,22% 13%

Neighbour 6_1296x2 1,1308 1241 1 403,03 95,76% 33%

Neighbour 6_1296x4 1,1319 1240 1 403,02 95,68% 66%

Neighbour 8 1,1009 1255 1 381,89 96,84%

Neighbour 8 260 x 2 1,1169 1249 1 395,43 96,37% 5%

Neighbour 8 520 x 2 1,1306 1245 1 407,20 96,06% 10%

Neighbour 8 1296x2 1,1461 1238 1 419,28 95,52% 25%

Neighbour 8 1296x4 1,1474 1239 1 421,97 95,60% 50%

Neighbour 8 1296x6 1,1498 1238 1 423,84 95,52% 75%

world 1,1952 1224 1 462,79 94,44%


Optimal long run (static) pricing by a monopolist: the influence of local network

structure

• optimal static (long run) monopoly prices increase with connectivity and small world parameter q ; higher with scale free than WS.

0.5 1 1.5 2

0.5

1

1.5

2

J=0

Global externality


Conclusion, extensions & future developments

Even with simplest assumptions (myopic customers, full connectivity, risky situation), complex dynamics may arise.

Actual extensions: long term equilibrium for scale free small world, and dynamic regimes with H<0.

In the future: looking for cognitive agents ….

Dynamic pricing & monopolist’s Bayesian learning process in the case of repeated buying

Dynamic pricing & agent’s learning process in the case of durable good (Coase conjecture)

Dynamic network and monopolist’s learning about the network ….


References Anderson S.P., DePalma A, Thisse J.-F.

(1992) Discrete Choice Theory of Product Differentiation, MIT Press, Cambridge MA.

Brock Durlauf (2001) “Interaction based models” in Heckman Leamer eds. Handbook of econometrics Vol 5 Elsevier, Amsterdam

Phan D. (2003) “From Agent-based Computational Economics towards Cognitive Economics”, in Bourgine, Nadal (eds.), Towards a Cognitive Economy, Springer Verlag, Forthcoming.www-eco.enst-bretagne.fr/~phan/moduleco

Phan D. Gordon M.B. Nadal J.P. (2003) “Social interactions in economic theory: a statistical mechanics insight”, in Bourgine, Nadal (eds.), Towards a Cognitive Economy, Springer Verlag, Forthcoming.

Nadal J.P. Phan D. Gordon M.B. Vannimenus J. (2003), "Monopoly Market with Externality: an Analysis with Statistical Physics and ACE", 8th Annual Workshop on Economics with Heterogeneous Interacting Agents, Kiel.

Any Questions ? (please speak slowly)

denis phan 1 , stephane pajot 1 , jean pierre nadal 2 ,

Documents

network externality

regular network

individual choice

h i t

social context discrete

demand hysteresis

s choice

local dynamics