Betting on the Trend:
An Experimental Study on Network
Externalities with Continuous Demand
Vincent Mak and Rami Zwick
Department of Marketing
The Hong Kong University of Science and Technology
FUR XII 2006
LUISS, Rome, June 2006
Introduction
• Network externalities are said to exist when
the utility of a good or service to a consumer
increases with the number of consumers
already owning that good or using that
service.
• Often seen in technology adoption context:
Microsoft’s market dominance, VHS Vs
Betamax etc.
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Introduction
•
•
•
•
Theory: Leibenstein (1950), Rohlfs (1974), Katz &
Shapiro (1985, 1986, 1992) etc.
Empirical studies: Gandal (1994), Lopatka & Page
(1995), Park (2004) etc.
Experiments: relatively few – Chakravaty (2003a,
2003b), with seller(s) and buyers, offer mixed
support to Katz & Shapiro (1986)
Etziony & Weiss (2002) – experiment on binary
purchase decision with only consumer side and
exogenous price, closest in spirit to present study
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Introduction
• Previous network externalities research focus
on binary or small discrete choice sets for
consumers
• Continuous demand? “Fractional adoption”
within [0,1]?
• For example, commodity can be bought in
multiple units at price relatively small with
respective to endowment – demand is
approximately continuous
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Introduction
• How do individuals decide on their actions in
a market with network externalities, when
their aggregate actions exert endogenous
influence on individual benefits?
• How do they update their decisions based on
past information about the “trend”?
• Could coordination towards the Pareto
optimal outcome fail?
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Economic Model
• A setting for a repeated game
• Business telecommunication service,
targeted at large firms
• Monthly renewable/terminable subscription,
fixed per employee rate
• 1 month = 1 round
• Network externalities benefit derived from
underlying fully connected graph between all
employees subscribing to the service in the
market
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Economic Model
• Firms denoted by i = 1, 2, … N
• In a round, suppose firm i subscribes for ni
employees; assume “assigned budget” (endowment)
is constant across firm and this (rather than total no.
of employees) provides a constraint on ni
• Assume network externalities benefit is linear in:
• No. of internal links = ni (ni -1) / 2
• No. of external links = ni  n j
j i
• Additional assumption for simplification of payoff
expression: network “strength” per internal link =
2 * network “strength” per external link
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Economic Model
• After linear transformation of variables, obtain
the following payoff expression:
N
 i  w  xi 
xi  x j
j 1
kw
l w  xl  0
• k, N, w are exogenous parameters
• Define a stage game for an experimental
study of the repeated game
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Basic Properties of the Game
• A “voluntary contribution” game that does not allow
free-riding
• Game is “meaningful” only when 1 < k < N, focus on
this range of parameters
• Two pure-strategy equilibria: no-contribution, fullcontribution
• A mixed-strategy equilibrium (for risk neutral players
and possibly for slightly risk averse players) with
same expected utility as no contribution; predicted to
be unsustainable
• Purely risk neutral players’ best response to any
belief can only be 0 or w – expected not to be the
case in reality because of risk aversion, signalling
tendency and social factor
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Behavioural Model for Experiment
•
First-round contribution for player i, xi1, modelled as
combined influence of three factors:
1. Intrinsic propensity to contribute (an intercept term)
2. “Social factor” = kw/N
3. “Risk-dominance measure” = (kw-yi1)/(N-1), given a
candidate contribution yi1
•
Because of 3., decision becomes a fixed point
problem – more exactly, minimization of difference
between decided contribution and a “prescribed
contribution” which is a function with quantitative
representations of 1., 2. and 3. as arguments
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Behavioural Model for Experiment
•
•
Shall assume linear function and population-wise
homogeneity in parameters for prescribed
contribution in estimation
Solution is, if
x* 
•
a  bk / N  ck /( N  1)
w
1  c /( N  1)
with a, b, c constants to be determined, then:
xi1 = x* if 0 ≤ x*≤ 1, xi1 = 0 if x* < 0, xi1 = 1 if x* > 1.
Shall test model and estimate a, b, c from
experimental data
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Behavioural Model for Experiment
• An individual sequential adjustment model for dynamics
• Positive feedback on trend prediction and group imitation;
resistance of no-contribution equilibrium
• Let St be the sum of contributions in round t
 S t  kw 
 S t  kw  S t
 w   
  xit   

 Nw 
 Nw  N
• If S t  kw then xit 1
• If S t  kw then
St 
 S  kw 
 S t  kw  
w  xit 1    t

(
w

x
)



w



 

it
N
 Nw 
 Nw  
where 0 ≤ α ≤ 1 is a constant; λ and μ are non-decreasing
functions over [-1,0), non-increasing over [0,1],
everywhere continuous except at 0
• Assume    ( z )   ( z )  1 for all -1 ≤ z ≤ 1
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Behavioural Model for Experiment
• Aggregate form of sequential adjustment:
S t  kw 
  St
 Nw 
• If S t  kw , then S t 1    Nw   
• If S t  kw , then Nw  S t 1
;
 S t  kw 

  ( Nw  S t ) ;
Nw


where ν = λ + μ
• Predict that whether S1 ≥ kw is sufficient condition for
reaching full-contribution equilibrium
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Experiment
• Total 240 subjects, HKUST undergraduates
• Five between-subject conditions with different (N, k): (5, 2), (5,
4), (10, 2), (10, 4), (10, 8)
• Six groups per condition, each play T = 20 rounds of repeated
game
• w fixed at 30 francs (experimental currency)
• Maximum per period payoff = HK$75 ((US$1=HK$7.8), kept
equal across conditions by varying exchange rate with real
currency
• Feedback at the end of each round: sum of contribution of that
round + individual contribution, payoff, etc.
• Choose two periods randomly at end of session and pay
subjects the average payoff from the two periods, plus HK$10
show-up fee
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Main Results
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Main Results
•
•
•
•
Convergence towards no-contribution equilibrium is
more noisy
Mixed-strategy equilibrium is not sustainable
Contribution is not limited to {0,w}
First-round contribution:
Mean xi1/w
(s.d.)
N=5
k=2
k=4
k=8
0.57 (0.38)
0.35 (0.40)
NA
N = 10
0.48 (0.37)
0.46 (0.34)
0.45 (0.43)
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First-Round Contribution
• Non-linear least squares estimation:
xi1 = x*  0.42  1.73k / N  1.60k /( N  1)  w
1  1.60 /( N  1)
• All coefficients significant at p < 0.05
• R-squared = 0.60
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Dynamics
• One-lag response plots at group level
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Dynamics
• ARCH(1) estimation of individual sequential
adjustment model assuming discount functions λ(z)
and μ(z) linear in z
• Include only parameters significant at p < 0.05,
aggregate to form estimation of group level
dynamics:

S  kw 
S t  kw : S t 1  0.02 Nw  0.80  0.58   t
  S t
 Nw 


 S  kw 
S t  kw : Nw  S t 1  0.57  0.14   t
  ( Nw  S t )
 Nw 

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Convergence Criterion
• From estimation of first-round contribution,
S1 ≥ kw iff k/N ≤ 0.48
• Assuming dynamics model is supported by data, this
suggests a criterion for convergence towards the fullcontribution equilibrium
• That is, critical mass (kw) needs be less than around
half of total population endowment (Nw) to ensure
convergence towards full-contribution equilibrium
• Experiment does support this criterion, and also that
whether S1 ≥ kw determines convergence towards
full-contribution equilibrium
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Conclusions
• Have suggested an economic model of
network externalities allowing for continuous
demand
• Payoff-dominant equilibrium not always
reached; have obtained a criterion for this
• First-round contribution and dynamics model
both supported by data
• Social factors are identified in the present
descriptive model which might have been
overlooked in more theoretical treatment
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Thank You!
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