Strategic Manipulation of Internet Opinion Forums:
Implications for Consumers and Firms
Model Extensions
Chrysanthos Dellarocas
R. H. Smith School of Business
University of Maryland
College Park, MD 20742
cdell@rhsmith.umd.edu
April 2006
This appendix analyzes an example duopoly setting, which is a direct counterpart of the monopoly
setting analyzed in Section 2 of the paper. The analysis shows that the results of the paper are
robust with respect to several extension of my basic model. Specifically:
• Appendix A.2 studies a setting where, in addition to praising themselves, firms can bad-mouth
their competitors.
• Appendix A.3 studies a setting where there are multiple forums and each consumer only visits
a subset of them.
• Appendix A.4 studies a setting where firms can also imperfectly signal their quality through
prices.
A.1 A duopoly setting
Consider a setting where two competing firms, A and B, simultaneously introduce imperfect substitute products in a new market. Just as in the monopoly case analyzed in Section 2 of the paper, we
assume that the appeal of each product to consumers is the sum of two independent components:
a horizontal component (“location”) and a vertical component (“quality”). A product’s location can
1
be reliably communicated to consumers, whereas a product’s true quality qj (j = A, B) can only
become known after the good is bought and consumed.
Consumers are uniformly distributed in the unit interval [0, 1] and have quadratic transportation
costs. Standard theory (D’Aspremont et al. 1979) then predicts that the two firms will locate
their products at the corners of the [0, 1] unit interval. Assume that product A is located at 0 and
product B is located at 1. Variable costs are assumed to be zero or, alternatively, marginal costs
are constant and identical for both firms, and prices are defined net of marginal costs.
Consumers have unit demand. A consumer’s utility from consuming products A, B is given by:
uiA = qA −
i2
− pA
2
uiB = qB −
(1 − i)2
− pB
2
(1)
where i ∈ [0, 1] is the consumer’s location in the unit interval. The distribution of consumer utility
functions is common knowledge.
Qualities qA , qB are known to both firms, but not to consumers. Consumers share common and
identical priors regarding each product’s quality. Prior beliefs are normally distributed with mean m
and precision τ . In addition, consumers have access to exogenously generated, normally distributed
signals xj of each product’s true quality with means qj , precision ρx , and full support. As before,
these signals can be thought of as the arithmetic means of online ratings posted by consumers who
have already tried the products.
All consumers and firms have access to the same realizations of signals xj . Let q = qA − qB
denote the quality differential between goods A and B and let θ = E(q|xA , xB ) be the mean
of consumer posterior beliefs regarding q (the perceived quality differential) after observing signals
xA , xB . Expected utility maximization and price-taking behavior imply the following linear demand
functions:
DA = P r[uiA ≥ uiB ] =
DB =
P r[uiA
<
uiB ]
=
1
2
1
2
+ θ − pA + pB
− θ − pB + pA
(2)
If firms set prices after the forum publishes quality ratings, signaling of qualities through prices is
not possible (see Appendix A.4). Maximization of expected sales revenues then implies:
prices
demand
sales revenues
1
θ
2 + 3
DA = 12 + 3θ
¡
¢2
wA = 12 + 3θ
pA =
1
θ
2 − 3
DB = 12 − 3θ
¡
¢2
wB = 12 − 3θ
pB =
(3)
Firms manipulate signals xj by posting fake anonymous ratings that praise their respective products.
This way firms can shift the mean of the distribution of average ratings from qj to qj + ηj at cost
2
c(ηj ) = ληj2 . Denote by yj the signals that result from manipulation of the original signals xj . As
before, we would like to understand how manipulation affects (i) the information quality of signal
y (relative to x), and (ii) the profits of the firms.
The analysis assumes that consumers are aware that firms may be attempting to manipulate online
ratings. Specifically, consumers understand that published ratings are the sum of three (indistinguishable) components:
yj = qj + ηj (qA , qB ) + εj
where qj are true qualities, ηj (qA , qB ) are the amounts by which each firm inflates its ratings and
εj are normally distributed error terms with mean zero and precision ρx .
I will now show the existence of a symmetric PBE in linear strategies. Suppose that consumers
believe that the amounts by which firms inflate their online quality ratings are linear functions
ηA = g + hq, ηB = g − hq of the quality differential q = qA − qB , where g, h are real numbers that, at
an equilibrium, correspond to correct conjectures. From the above expressions for yA , yB it follows
that:
q = qA − qB =
εA − εB
yA − yB
−
2h + 1
2h + 1
(4)
in other words, the publicly observable statistic z = (yA − yB ) / (2h + 1) is a normally distributed
unbiased estimator of q with precision ρz = ρx (2h + 1)2 /2. If consumers update their beliefs
using Bayes rule, given the normality of prior beliefs and all observable signals, standard theory
(DeGroot, 1970) predicts that each consumer’s posterior beliefs about q will be normally distributed
with mean:1
θ=
ρz z
τ + ρz
(5)
Similarly to the monopoly case, the impact of manipulation on the precision of quality signal z
(relative to x) depends on the value of parameter h. In the absence of manipulation, the precision
of the difference of the two signals would be ρx /2. If h > 0, then ρz = ρx (2h + 1)2 /2 > ρx /2;
each firm’s manipulation strategy is a monotonically increasing function of the quality difference
1
From equations (3), prices and demand become negative if |θ| > 3/2. I implicitly assume that the precision of
each firm’s prior quality distribution is sufficiently high so that the probability that |θ| > 3/2. become negligible.
Since z is normally distributed with mean q and precision ρz and q is normally distributed with mean 0 and precision
ρz
1
5
zz
τ , θ = τρ+ρ
is normally distributed with mean 0 and variance σ 2 = (τ +ρ
2 + τ ≤ 4τ . The probability that θ falls
z
z)
outside the interval [-3/2, 3/2] becomes negligible if τ is sufficiently high.
3
between that firm and its competitor. In contrast, if −1/2 < h < 0 , then ρz < ρx /2; manipulation
strategies are then monotonically decreasing functions of the quality difference between each firm
and its competitor. If h = 0 both firms inflate their ratings by an identical amount, independently
of their relative qualities; manipulation then does not change the forum’s information value. Finally,
if h = −1/2, manipulation completely destroys the information of the original signal and renders
the resulting online ratings completely uninformative.2
Since sales revenues wj = ( 12 ± 3θ )2 are quadratic functions of θ, the marginal revenue ∂wj /∂θ =
2( 12 ± 3θ ) that a firm can obtain by inflating consumer perceptions of its quality is an increasing
function of their baseline perceptions. Firms that, in the absence of manipulation, would have been
perceived to possess higher quality, have more to gain from inflating their ratings. The intuitions of
Section 2.1 then predict that, in all symmetric linear PBE, higher quality firms would manipulate
more than lower quality firms (h > 0). Using an approach identical to the one we used to prove
Proposition 1, one can prove the following result:
Proposition A.1: Consider a duopoly setting where firm revenues are given by (3). There exist
symmetric linear PBE in which firm manipulation strategies have the form
ηA = g + h(qA − qB )
where
g=
ηB = g − h(qA − qB )
ρz
6λ(τ + ρz )(2h + 1)
and h is a positive real solution of the 5th degree polynomial equation:
λ=
ρ2z
9(τ + ρz )2 (2h + 1)h
Similarly to Proposition 1, Proposition A.1 shows that, in competitive settings where firm revenues
are convex with respect to perceived quality, manipulation strategies are monotonically increasing
in the quality difference between a firm and its competitor. Manipulation activity then increases the
informativeness of the online forum and becomes a form of quality signaling that benefits consumers.
2
To see why, for h = −1/2, manipulation strategies become ηA = g − (qA − qB )/2, ηB = g + (qA − qB )/2. Published
ratings then are equal to: yA = qA + ηA + εA = g + (qA + qB )/2 + εA , yB = qB + ηB + εB = g + (qA + qB )/2 + εB .
Observe that the distribution of both firms’ ratings becomes identical; published ratings then carry no information
about each firm’s quality relative to its competitor. In such settings ρz = 0: if consumers expect such firm behavior,
they will ignore online ratings.
4
A.2 Negative reviews
Assume now that, in addition to promotional reviews that praise their own product, firms A and
B can post negative reviews about their competitor’s product with the objective of reducing the
competitor’s average ratings. Denote by ηAA , ηBB the amounts by which firms A and B inflate
their own ratings and by ηAB , ηBA the amounts by which firms A and B deflate the ratings of their
respective competitor. Assume, further, that the costs of positive and negative manipulation are as
follows:
Positive manipulation of firm j’s ratings by itself :
2 + ξη η )
c(ηjj , ηij ) = λ(ηjj
ij jj
2 + ξη η )
Negative manipulation of firm j’s ratings by firm i : c0 (ηij , ηjj ) = αλ(ηij
ij jj
The new constant α intends to capture the fact that the marginal cost of negative manipulation
might have different parameters than the cost of positive manipulation. For example, it might be
cheaper (or more expensive) to compose a negative review than it is to compose a positive review
of a given level of credibility. The new factor ξηij ηjj , ξ > 0 models the fact that, by manipulating
its own ratings, firm j increases the negative manipulation costs of firm i (because, to reduce its
competitor’s average ratings, firm i must post enough fake negative ratings to compensate both the
presence of honest consumer ratings as well as the additional fake positive ratings posted by firm
j). For similar reasons, by manipulating firm j’s ratings, firm i increases the positive manipulation
costs of firm j.
The total manipulation costs of each firm are equal to:
2 + ξη
2
cA = λ(ηAA
BA ηAA ) + αλ(ηAB + ξηAB ηBB )
2 + ξη
2
cB = λ(ηBB
AB ηBB ) + αλ(ηBA + ξηBA ηAA )
As in the original model, I assume normally distributed common prior beliefs regarding the quality
difference q = qA − qB with mean zero and precision τ . I will now show existence of symmetric
linear PBE. Assume linear manipulation strategies ηAA = g + + h+ q, ηBB = g + − h+ q, ηAB =
g − + h− q, ηBA = g − − h− q. Then, published ratings are equal to yA = qA + ηAA − ηBA + ²A ,
yB = qB + ηBB − ηAB + ²B and the statistic z = (yA − yB )/(2(h+ + h− ) + 1) is an unbiased estimator
of q with precision ρz = ρx (2(h+ + h− ) + 1)2 /2. Substituting the above and following the steps of
the proof of Proposition 1 we can verify the existence of symmetric linear PBE where manipulation
strategies follow the postulated linear form and where h = h+ + h− must solve:
λ=
2(1 + α)
ρ2z
(2 − ξ)α 9(τ + ρz )2 (2h + 1)h
5
The relationship between h+ and h− is given by:
h− =
αξ + 2 +
h
2α + ξ
To better appreciate the intuition behind the above results, it is instructive to set ξ = 0. The above
expressions then simplify to:
λ
α
ρ2z
=
1+α
9(τ + ρz )2 (2h + 1)h
h− =
(6)
h+
α
Manipulation increases forum informativeness if and only if h = h+ + h− > 0. By (6), this condition
always holds. Comparison of the above equations with the results of Proposition A.1 reveals that
the presence of negative manipulation is mathematically equivalent to replacing the parameters λ of
α
the original model with λ0 = λ 1+α
. Observe that, in terms of affecting consumer perceptions about
the quality difference, posting negative reviews about one’s competitor is qualitatively equivalent
to posting positive reviews about one’s own product. The following points then describe the impact
of parameters α:
• If α is small then firms will predominantly use negative manipulation. The cumulative unit
cost parameter then becomes λ0 ' λα.
• If α is large then firms will predominantly use positive manipulation. The cumulative unit
cost parameter then becomes λ0 ' λ.
In conclusion, the possibility of negative manipulation does not qualitatively change the results of the
paper. In settings where negative manipulation is possible, symmetric linear PBE are characterized
by a mixture of positive and negative manipulation. The relative amounts of the two types of
manipulation are determined by the relative cost of posting a positive vs. a negative comment of a
given level of credibility: if credible negative comments are easier to generate, firms will focus most
of their resources on badmouthing each another. If the opposite is true, firms will focus most of
their energies on praising themselves.
A.3 Multiple online forums
The base model assumes that all consumers visit a single online forum and thus access (and contribute to) the same history of quality ratings. This set of assumptions describes settings where
6
large amounts of consumer opinions are concentrated in a small number of popular websites (such
as Amazon, Epinions, Citysearch, Yahoo, etc.). Although in some settings these assumptions correspond nicely to the current reality, there are other settings where a variety of smaller forums co-exist
and where each consumer visits and contributes only to a subset of them. If we assume that prices
are signal-free and that a consumer’s decision to visit a forum is uncorrelated with her decision to
visit any other forum as well as with her location in the taste interval, the presence of a set F of
forums (indexed by k = 1, ..., K) does not substantially change our model. In fact, if we make the
assumptions that, in addition to the total market size N , firms (i) know the number Nk of consumers who access each forum and the fraction rk of each forum’s visitors who contribute opinions,
and (ii) search all forums at the beginning of each period and, thus, have precise knowledge of the
aggregate ratings posted on each of them before they set prices, then a simple extension of the base
model applies to a setting with multiple forums.
The idea is that firms follow separate manipulation strategies ηjk for each individual forum and that
each of these strategies is qualitatively similar to the equilibrium strategies derived in Section A.1.
Let ρk = Nk rk denote the base precision of forum k ’s aggregate (honest) ratings. This expression
assumes that precision is proportional to the number of honest ratings. Assume linear manipulation
2 . Given published ratings y
strategies ηjk = gk ± hk q and cost functions ck (qj , ηjk ) = λk Nk rk ηjk
jk
the posterior beliefs (regarding the quality difference q) of a consumer who visits a subset S ⊆ F
P
P
of forums have mean value θS = k∈S zk ρzk /(τ + k∈S ρzk ) where ρzk = ρk (2hk + 1)2 /2 is the
precision of published ratings obtained from forum k and zk = (yAk − yBk )/(2hk + 1) is an unbiased
estimator of q based on ratings published in forum k. Given linear demand functions, maximization
of expected sales revenues implies:
prices
P
± 13 S∈2F nS θS
¡
¢2
P
wj = N 12 ± 31 S∈2F nS θS
pj =
sales revenues
1
2
where 2F denotes the powerset of F and nS = N −K
Q
k∈S
Nk
Q
l∈F −S (N
− Nl ) is the probability
that a random consumer will visit forum subset S. Sales revenues can be equivalently expressed as:
Ã
wj = N
K
1 1X
mk z k
±
2 3
!2
where mk = ρzk
k=1
X
S∈2F ∧k∈S
τ+
n
PS
l∈S
ρzl
Substituting the above and following the steps of the proof of Proposition 1 we can verify the
existence of linear PBE where manipulation strategies ηjk follow the postulated linear form.
A.4 Price signaling
This appendix shows that the main intuitions of the paper remain qualitatively robust in environments where sellers can also communicate their quality to consumers through prices. In such
7
environments, consumers pay attention both to online ratings and to prices and learn from both.
The incentive to manipulate online ratings, thus, still remains. At equilibrium, the following analysis
finds that firms distort their prices upwards to signal their quality and engage in forum manipulation
at relative intensities that are qualitatively similar to those encountered in the baseline case: If firm
revenues are convex functions of consumer quality perceptions, the high quality firm manipulates
more, thus increasing the information value of the forum. Furthermore, for a broad range of parameter values, price signaling and ratings manipulation are substitutes: everything else being equal,
the higher (lower) the precision of information communicated through prices, the lower (higher) the
equilibrium intensity of manipulation.
The models of Section 2 of the paper as well as the baseline model of Appendix A.1 have been
carefully constructed to exclude the possibility of price signaling: Sellers set prices after they observe
average ratings. At that point there is no credible way for sellers to signal their quality through
prices.3
A simple perturbation of the model introduces opportunities for price signaling. Suppose that sellers
set prices at the same time that they make manipulation decisions, i.e. before they observe their
actual ratings. Profit maximizing prices then will reflect the sellers’ expectations regarding online
ratings; these expectations, in turn, are a function of each seller’s true quality. Anticipating such
behavior, consumers can then infer something about quality through prices.
A complication arises if we assume that prices are perfect signals of quality. Then, consumers
will ignore online ratings because they can perfectly infer quality from prices. However, it is the
anticipation of the impact of online ratings on beliefs that induces sellers to signal quality through
price in the first place. The argument is circular and leads to a contradiction.
The problem disappears if one assumes that prices carry imperfect information about quality (either
because firms themselves possess imperfect information or because a retailer introduces noise to the
final price).4 Suppose, for example, that retailers add random amounts to wholesale prices to reflect
their own competitive concerns. Suppose, further, that these amounts are uncorrelated to the
prices and qualities of the two products. Then, the retail prices that consumers observe are given
by p0j = pj + ζj , where ζi is a normally distributed noise term with mean zero and precision φ.
I will now establish existence of a symmetric linear PBE in prices and manipulation strategies. As
before, consumers conjecture linear manipulation strategies η j = g ± hq. Furthermore, they believe
that wholesale prices take the linear form:
3
To see why, suppose that consumers believe that prices carry some sort of information about quality. The mean
of their posterior beliefs regarding q = qA − qB will then be some function θ(yA , yB , pA , pB ). But then, equilibrium prices that maximize seller profits are simply the solution of the system of equations p∗A = arg maxpA pA ( 12 +
θ(yA , yB , pA , p∗B ) − pA + p∗B ), p∗B = arg maxpB pB ( 12 − θ(yA , yB , p∗A , pB ) − pB + p∗A ). Observe that true firm qualities
do not appear in the previous equations. Thus, no consumer beliefs that involve quality signaling through prices can
be consistent with seller actions.
4
Similar arguments were put forward by Caminal and Vives (1996) who investigated the role of quantities as a
signal of quality in the presence of possible price signaling.
8
pA = u + vq
pB = u − vq
u ∈ R+ , v ∈ R
Then p0A − p0B = 2vq + ζ1 − ζ2 and the observable statistic ω =
p0A −p0B
2v
ζ1 −ζ2
is a normally
2v
2
(2v) φ/2 = 2v 2 φ. If a
=q+
distributed unbiased estimator of the quality difference q with precision φω =
consumer now observes ratings yA , yB and retail prices p0A , p0B , her mean posterior beliefs regarding
q = qA − qB are given by:
θ(yA , yB , p0A , p0B )
p0 −p0B
A −y B
ρz y2h+1
+ φω A2v
ρz z + φω ω
=
=
τ + ρz + φω
τ + ρz + φω
Firm A’s expected revenues are then given by:
1
1
E[wA ] = E[p0A ( +θ(yA , yB , p0A , p0B )+p0A −p0B )] = pA +E[p0A θ(yA , yB , p0A , p0B )]+p2A +V [p0A ]−pA pB
2
2
where:
E[p0A θ(yA , yB , p0A , p0B )]
A −y B
ρz pA E[ y2h+1
] + φω E[p0A
=
τ + ρz + φω
p0A −p0B
2v ]
−pB
A −η B
ρz pA q+η
+ φω pA pA2v
+ φω
2h+1
=
τ + ρz + φω
V [p0A ]
2v
Similarly, firm B’s expected revenues are given by:
1
1
E[wB ] = E[p0B ( −θ(yA , yB , p0A , p0B )+p0B −p0A )] = pB −E[p0B θ(yA , yB , p0A , p0B )]+p2B +V [p0B ]−pA pB
2
2
where:
A −y B
ρz pB E[ y2h+1
] + φω E[p0B
0
0
0
E[pB θ(yA , yB , pA , pB )] =
τ + ρz + φω
p0A −p0B
2v ]
−pB
A −η B
ρz pB q+η
+ φω pB pA2v
− φω
2h+1
=
τ + ρz + φω
V [p0B ]
2v
Substituting into the equations that give period-1 expected payoffs:
vj = −cj + E [wj ] = −ληj2 + E [wj ] ,
q = qA − qB
and taking first order conditions with respect to both manipulation intensities ηA , ηB and wholesale
prices pA , pB we obtain:
9
µ
ηA =
1 3φω
+
2
4ρz
¶
µ
2ρz + 3φω
q,
6(τ + ρz + φω )
pB =
2ρz + 3φω
+ hq,
8(τ + ρz + φω )(2h + 1)λ
ηB =
pA =
+
1 3φω
+
2
4ρz
¶
−
2ρz + 3φω
q
6(τ + ρz + φω )
2ρz + 3φω
− hq
8(τ + ρz + φω )(2h + 1)λ
(7)
(8)
Consistency of beliefs requires that v and h solve the following system of constraints:
v=
2ρz + 3φω
(2ρz + 3φω )ρz
,λ =
6(τ + ρz + φω )
3(τ + ρz + φω )2 (2h + 1)4h
(9)
Substituting φω = (2v)2 φ/2 = 2v 2 φ and ρz = ρx (2h + 1)2 /2 the constraints become:
v=
ρx (2h + 1)2 + 6v 2 φ
(ρx (2h + 1)2 + 6v 2 φ)ρx (2h + 1)
,
λ
=
6(τ + ρx (2h + 1)2 /2 + 2v 2 φ)
24(τ + ρx (2h + 1)2 /2 + 2v 2 φ)2 h
(10)
Observe that v is always positive. Following a procedure similar to that used in the proof of
Proposition 1, we also find that admissible positive solutions h always exist.
z
Second-order conditions require ∂ 2 vj /∂ηj2 = −2λ < 0, ∂ 2 vj /∂p2j = − 2ρz4ρ
+3φω < 0. In addition, the
Hessian matrix:
¯
¯
¯
H=¯
¯
−2λ
ρz
(τ +ρz +φω )(2h+1)
must have a positive determinant det(H) =
ρz
(τ +ρz +φω )(2h+1)
z
− 2ρz4ρ
+3φω
(4h+8)ρ2z
.
3(τ +ρz +φω )2 (2h+1)2 (4h)
¯
¯
¯
¯
¯
The latter condition holds for
all h > 0.
Finally, from λ =
(2ρz +3φω )ρz
,
3(τ +ρz +φω )2 (2h+1)(4h)
substituting ρz = ρx (2h + 1)2 /2, setting h = h(φω ) and
differentiating the expression for λ with respect to φω , after a considerable amount of algebraic
manipulation we conclude that, if φω > τ , ∂h/∂φω < 0, that is, the higher (lower) the precision of information communicated through prices, the lower (higher) the equilibrium intensity of
manipulation.
References
Caminal, R. and Vives, X. (1996) Why Market Shares Matter: An Information-Based Theory.
RAND Journal of Economics, 27 (2), 221-239.
D’Aspremont, C. J. Jaskold, Gabszewicz, J. F. Thisse (1979) On Hotelling’s “Stability in Competition”. Econometrica 47, 1145-1150.
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