Pricing strategies of goods with externalities

Pricing strategies of goods with externalities
Mirta B. Gordon* — Jean-Pierre Nadal**
— Denis Phan *** — Viktoriya Semeshenko ****
* Laboratoire TIMC-IMAG, UMR 5525 CNRS-UJF, Université Joseph Fourier, Grenoble, ** Centre d’Analyse et Mathématique Sociales (CAMS, UMR 8557 CNRS-
EHESS), EHESS, Paris, and Laboratoire de Physique Statistique (LPS, UMR 8550
CNRS-ENS-Paris 6-Paris 7), ENS, Paris *** Groupe d’Etude des Méthodes de l’Analyse Sociologique (GEMAS, UMR 8598 CNRS - Université Paris IV - Sorbonne), Paris
**** Present address: Buenos Aires, Argentina
RÉSUMÉ. Cet article résume l’effet de l’influence sociale sur un marché de monopole avec des
agents hétérogénes. Les équilibres du marché sont présentés dans le cas limite de l’influence
globale. Considérant une maximisation statique du profit, il peut y avoir deux régimes : vendre
soit à un grand nombre d’acheteurs avec un prix bas, soit à un petit nombre à un prix plus
élevé. Ceci s’observe pour un grand nombre de distributions mono-modales de la disposition
idiosyncrasique à payer si l’influence sociale est suffisament forte. La stratégie optimale du
vendeur passe d’un régime à l’autre pour une valeur des paramètres ou la demande a deux
equilibres de Nash différents; mais la stratégie des prix bas peut échouer à cause d’un manque
de coordination des consommateurs.
ABSTRACT. This paper
summarizes the effects of social influences in a monopoly market with heterogeneous agents. The market equilibria are presented in the limiting case of global influence.
Considering static profit maximization there may exist two different regimes: to sell either to a
large fraction of customers at a low price, or to a small fraction of them at a higher price. This
arises for numerous mono-modal distributions of idiosyncratic willingness to pay if the social
influence is strong enough. The seller’s optimal strategy switches from one regime to the other
at parameter values where the demand has two different Nash equilibria; but the strategy of
posting low prices to attract large fractions of buyers may fail due to a lack of coordination.
Choix binaires, systémes complexes, agents hététogènes en interaction, effets de
voisinage, influence sociale, monopole, externalités
MOTS-CLÉS :
KEYWORDS: Binary Choice, Complex Systems, Heterogeneous
Effects, Social Influence, monopoly, externalities
Interacting Agents, Neighborhood
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Nom de la revue ou conférence (à définir par )
1. Introduction
Social systems where the individuals’ decisions depend on the decisions of other
individuals present interesting properties that cannot be understood without introducing explicitly the social influence in the models. Social influence plays an important
role on the collective outcomes because it may give raise to multiple equilibria. The
decision of leaving a neighborhood (Schelling, (1971)), to attend a seminar (Schelling, (1973), Schelling, (1978)) or a crowded bar (Arthur, (1994), Arthur, (1999)), to
participate to collective actions such as strikes and riots (Granovetter, (1978)), are particular examples. It has been further suggested that social influence may explain the
school dropout (Crane, (1991)), the persistence in the educational level within some
neighborhoods (Durlauf, (1996)), etc. Which among the competing equilibria is actually realized in a given situation depends on the degree of coordination reached by
the population.
This paper summarizes a model with social influence that presents multiple equilibria. We assume that the individuals make rational decisions, i.e. that maximize their
surplus and that the latter depend on the decisions made by others.
In the model, customers have to decide whether or not to buy a good at a price
posted by a monopolist. The situation is close to the subscription to a telephone network, where the utility of adopting the telephone for each individual depends on how
many others have adopted it (see i.e. (Curien et al., (1987)) for a survey). We analyze the problem of a monopolist who has to choose the price without knowing which
equilibrium will actually be reached.
2. Customers’ model
The model assumes that individuals have to choose whether to buy or not a good at
a given price, i.e. they have to make binary decisions denoted s ∈ {1, 0}, where s = 1
stands for buying. If η is the fraction of individuals that choose s = 1, the utility U i of
individual i upon buying is
Ui (si = +1) = Hi + Jη − P
[1]
where Hi is the (prior) idiosyncratic willingness to pay, i.e. before taking into account
the others’ choices. We assume that this willingness to pay is distributed among the
population according to a probability density function (pdf) f (H i ) of variance σ. The
coefficient J > 0 represents the weight of the social component in the utility, η is the
fraction of the population that buy the good and P is its price. If s i = 0 the utility
is Ui (si = 0) = 0. Notice that the social component term may be written (to order
N −1 ) as follows
Jη =
N
J sk
N
k=1,
[2]
Titre abrégé de l’article (à définir par )
3
=h-p
0
B
L(j)
B
-2
U(j)
-4
0
jB
4
j
6
8
Figure 1. Customers’ phase diagram for a logistic pdf f (H i ) of mean H and variance
σ. The ordinate axis corresponds to the (normalized) gap between average willingness
to pay and posted price h − p = (H − P )/σ. Abscissa represent the (normalized)
social weight j = J/σ. The grey region indicates multiple equilibria. The critical
values are jB ≈ 2.21, hB ≈ −1.10.
meaning that the influence of each individual k on i’s decision has a weight J/N . In
this simple model all the individuals give the same weight to the decision of everybody else. A more realistic formalization would give more weight to friends than to
strangers, but we try to keep the model as simple as possible in order to put forward
the most important properties.
An interesting property of this model is that one can define a Lyapunov function for its dynamics. This is a function such has if the individuals make repeatedly
their choices with a myopic best response, this function either remains constant or increases. Notice however that this function is not the social utility, as one naively might
assume. The consequences are that
– under stochastic sequential dynamics (i.e. at each time step one randomly selected individual makes his choice) the system converges to fixed points,
– the Nash equilibria are the (relative) extrema of the social utility.
The same properties are shared by models with non uniform social weights provided that there is reciprocity, i.e. the weight J ik put by an individual i on the decision
made by k is the same as the one put by k on i’s decision. In other words, for symmetric interactions : Jik = Jki . If this is not the case, as one may expect in many social
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Nom de la revue ou conférence (à définir par d
(j;)
)
1.0
U
0.8
0.6
B
0.4
multiple
equilibria
j=1
j=jB
0.2
L
j=5
0.0
-8
-6
-4
U
-2 L B
0
2
4
Figure 2. Fractions of buyers for a logistic pdf f (H i ) of mean H and variance σ as
a function of the (normalized) gap between average willingness to pay and posted
price h − p = (H − P )/σ for three different values of the (normalized) social weight
j = J/σ.
systems, there is no longer a Lyapunov function and convergence to fixed points is not
guaranteed. Thus, without reciprocity Nash equilibria may not exist, and the system’s
dynamics may exhibit cycles and even chaotic behavior. To our knowledge, the only
works including non-symmetric interactions in the field of social sciences are a study
by Iori and Koulovassilopoulos (Iori et al., (2004)) of a model close to (1), and more
recently a model of cooperation and free-riding (Gordon et al., (2005b), Ma et al.,
(2009)).
Particular cases of the model (1) have already been published : (Gordon et al.,
(2005a), Nadal et al., (2005), Phan et al., (2008)). The generic properties for any type
of distribution of the idiosyncratic willingness to pay f (H i ) have been studied by
(Gordon et al., (2007)). If the social influence is large enough, the system exhibits
multiple equilibria. More precisely, there is a critical value J B , that depends on the
value of the H i pdf at its maximum ; for J > J B there are two Nash equilibria for
some range of the average willingness to pay of the population, H. The system exhibits multiple equilibria (there may exist as many as the number of modes of the
Hi distribution plus 1). A distribution with a single maximum (like a gaussian or a
logistic probability density function) may thus have two competing equilibria. This
result is summarized on Figures 1 and 2. The Nash equilibrium with the largest fraction of buyers is the optimal one, but it requires perfect coordination. If coordination
Titre abrégé de l’article (à définir par )
5
h
-0.5
hA
hB
hc(j)
A
h+(j)
h-(j)
B
U
-1.5
p=0
1.5
jA
2.0
jB
2.5
j
Figure 3. Optimal pricing strategies for a logistic pdf f (H i ) of mean H and variance
σ. The ordinate axis corresponds to the (normalized) average willingness to pay h =
H/σ. Abscissa represents the (normalized) social weight j = J/σ. In the white region
the optimal profit presents a single maximum at the optimal price. In the shaded region,
the profit presents two relative maxima as a function of the price. In the hatched region
the optimal strategy corresponds to high prices while in the grey region the optimal
strategy corresponds to low prices. Below the line p = 0 the willingness to pay of the
population is too low and there does not exist any viable strategy for the monopolist.
fails, the system converges on the equilibrium with a small fraction of buyers. Which
equilibrium is actually realized depends on the initial conditions and on details of the
decisions dynamics. It is interesting that the region with multiple equilibria occurs for
average willingness to pay far below the posted price. The customers’ utilities in that
case are positive thanks to the social interactions term in (1).
3. Monopoly pricing
Consider now the monopoly pricing problem (Gordon et al., (2005a), Nadal et al.,
(2005)), assuming that the distribution of the H i as well as the value of the parameter J
are known. The monopolist chooses the price that maximizes his profit Π ≡ N η(P −
C) where C is the unitary cost, assumed to be constant (without loss of generality we
put C = 0 hereafter). After a simple but cumbersome calculation, it turns out that the
optimal strategy is unique whenever j < j A , where jA depends on the pdf and satisfies
jA ≤ jB . Figure 3 shows the phase diagram of the monopoly pricing problem.
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-0.5
hM(j)
A
customers:
2 Nash equilibria
-1.0
h
hB
B
h+(j)
-1.5
-2.0
1.5
p(j)=0
[h=U(j)]
2.0
jB
2.5
3.0
3.5
4.0
j
Figure 4. Optimal pricing strategies for a logistic pdf f (H i ) of mean H and variance
σ. In the grey region the optimal pricing strategy requires customers’ coordination
For j > jA the optimal strategy changes abruptly along a line of discontinuity
hc (j). For h < hc (j) the profit is maximized by a high price strategy, targeting a
small fraction of buyers. For h > h c (j) the profit is maximized by a low price strategy and relatively large fractions of buyers. The inequality j A ≤ jB means that this
discontinuity occurs even in a region where customers have a single equilibrium. Therefore, whenever j < j B the monopolist can choose the optimal price (be it low or
high) and drive the customers to the expected Nash equilibrium, which is unique. But
if j > jB , the customers system’s response to price is not unique. In this region the
policy of posting low prices targeting large fractions of buyers is risky because customers may fail to coordinate. In that case, the good is sold to a smaller fraction of
buyers than expected and the seller does not get the expected utility. Surprisingly, the
curse of coordination exists in a very large region of the phase diagram, as shown in
Figure 4.
4. Conclusion
Endogenous externalities give raise to multiple equilibria, some of which require
coordination of large numbers of individuals. Pricing in systems with social influences
is risky, especially when the optimal strategy is to sell the good at a low price to large
fractions of buyers. If the latter fail to coordinate, only a small fraction of customers
buy at the (low) posted price ; the seller’s actual utility is much lower than expec-
Titre abrégé de l’article (à définir par )
7
ted. This mechanism was proposed by Becker (Becker, (1991)) long ago to explain
why popular restaurants do not increase their prices, by postulating a non-monotonic
demand curve. Our model provides a mathematical frame that provides individuallybased foundations for this theory, and allows to quantify the results.
Acknowledgements
This work is part of the project DyXi supported by the Programme Systèmes
Complexes et Modélisation Mathématique of the French Agence Nationale pour la
Recherche. Mirta B. Gordon, Jean-Pierre Nadal and Denis Phan are CNRS members.
5. Bibliographie
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Arthur W., « Complexity and the Economy », Science, vol. 284, p. 107-109, 1999.
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2. AUTEURS :
Mirta B. Gordon* — Jean-Pierre Nadal**
— Denis Phan *** — Viktoriya Semeshenko ****
3. T ITRE DE L’ ARTICLE :
Pricing strategies of goods with externalities
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Pricing with externalities
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* Laboratoire TIMC-IMAG, UMR 5525 CNRS-UJF, Université Joseph
Fourier, Grenoble, ** Centre d’Analyse et Mathématique Sociales (CAMS,
UMR 8557 CNRS-EHESS), EHESS, Paris, and Laboratoire de Physique
Statistique (LPS, UMR 8550 CNRS-ENS-Paris 6-Paris 7), ENS, Paris ***
Groupe d’Etude des Méthodes de l’Analyse Sociologique (GEMAS, UMR
8598 CNRS - Université Paris IV - Sorbonne), Paris **** Present address:
Buenos Aires, Argentina
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