Betting on the Trend: An Experimental Study on Network Externalities with Continuous Demand Vincent Mak and Rami Zwick Department of Marketing The Hong.

Betting on the Trend:An Experimental Study on Network Externalities with Continuous Demand

Vincent Mak and Rami ZwickDepartment of Marketing

The Hong Kong University of Science and Technology

FUR XII 2006LUISS, Rome, June 2006

Betting on the Trend -- FUR 2006 2

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.


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


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


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?


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


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 =

• Additional assumption for simplification of payoff expression: network “strength” per internal link =

2 * network “strength” per external link

ij

ji nn


Economic Model

• After linear transformation of variables, obtain the following payoff expression:

• k, N, w are exogenous parameters• Define a stage game for an experimental

study of the repeated game

kw

xx

xw

N

jji

ii

1 0 lxwl


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, full-contribution

• 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


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



• Shall assume linear function and population-wise homogeneity in parameters for prescribed contribution in estimation

• Solution is, if

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

wNc

NckNbkax

)1/(1

)1/(/*



• 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

• If then• If then

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 for all -1 ≤ z ≤ 1

kwS t N

S

Nw

kwSx

Nw

kwSwx tt

itt

it

1

kwS t

N

Sw

Nw

kwSxw

Nw

kwSxw tt

itt

it )(1

1)()( zz



• Aggregate form of sequential adjustment:

• If , then ;

• If , then ;

where ν = λ + μ

• Predict that whether S1 ≥ kw is sufficient condition for reaching full-contribution equilibrium

kwS t t

tt S

Nw

kwSNwS

1

kwS t )(1 tt

t SNwNw

kwSSNw


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


Main Results


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.)

k = 2 k = 4 k = 8

N = 5 0.57 (0.38) 0.35 (0.40) NA

N = 10 0.48 (0.37) 0.46 (0.34) 0.45 (0.43)


First-Round Contribution

• Non-linear least squares estimation:

=

• All coefficients significant at p < 0.05• R-squared = 0.60

wN

NkNkx

)1/(60.11

)1/(60.1/73.142.0*1ix


Dynamics

• One-lag response plots at group level


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:

tt

t SNw

kwSNwS

58.080.002.01

)(14.057.01 tt

t SNwNw

kwSSNw

:kwS t

:kwS t


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 full-contribution 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


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


Thank You!

Betting on the Trend: An Experimental Study on Network Externalities with Continuous Demand Vincent Mak and Rami Zwick Department of Marketing The Hong.

Documents

n i n i

continuous slide

market slide

repeated game slide

technology fur

n i employees

network strength

present study slide