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Contest Design

Part III: Another application

Paul Schweinzer

[email protected]

April 9, 2013.

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Contest Design, Part III

Intro The model Results Application

Programme for today

We’ve seen last time that it is possible to design contests such thatefficient allocations are obtained in situations where suchallocations are not typically expected. Today we will study aslightly different design objective based on

1. Dulleck, Kerschbamer & Sutter (2009): Inefficient provision ofcredence goods.

The corresponding contest paper is

1. Fleckinger, Roussillon & Schweinzer (2011): Endogenousexpert advice through labelling contests for credence goods.

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The main ideaWe model contests which endogenise the cost of assembling therelative ranking they are based on.

As an application we study a credence or experience goods market.

“A credence good is a good (or service) whose utility is difficult orimpossible to ascertain for the consumer. In contrast to experiencegoods, this utility is difficult to gauge even after consumption.”(wikipedia)

Sellers of these goods, by contrast, know the good’s quality.

Typical examples of goods with credence aspects include:◮ vitamin supplements,◮ car repairs,◮ many forms of medical treatment,◮ home maintenance services (e.g., plumbing and electricity).

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The labelling applicationWe consider a market for credence goods of varying butindividually fixed qualities produced by a set of competing firms.

Without add’l information, consumers expect a pooled quality.

Firms may choose to release information which allows ‘experts’ torank the products. This information is modelled as a public good.

Examples of such rankings employing labels include:◮ any product (EU flower, Warentest, Which?, FDA),◮ organic produce (Soil Assoc, Ökotest, Demeter, USDA),◮ firms, countries (S&P, Moody, Fitch), wine (WSJ, 20-11-09);◮ academic departments (REF), & research output (Journals),◮ novels (Pulitzer, Man Booker, Prix Goncourt, Bachmann),◮ movies (Oscar, Cannes, imdb.com, vcdq.com),◮ software, games (piratebay.org, gamerankings.com).

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Example: Ranking of UK economics departmentsAccording to the 2008 Research Assessment Exercise (RAE),UK economics departments can be ranked as follows:

# Department Staff% 4* 3* 2* Avg

1 London School of Economics 41.60 60 35 5 3.552 University College London 32.20 55 40 5 3.503 University of Warwick 49.63 40 55 5 3.354 University of Essex 34.31 40 55 5 3.355 University of Oxford 78.50 40 55 5 3.356 University of Bristol 19.45 30 55 15 3.157 University of Nottingham 47.25 30 55 15 3.158 Queen Mary, Uo London 23.00 30 55 15 3.159 University of Cambridge 38.00 30 45 25 3.0510 University of Manchester 34.80 25 55 20 3.05

The next Research Excellence Framework is due for 2014. 5 / 22



Literature: Credence goods / labelling / advertisingCredence goods

◮ Darby & Karni (1973), Pitchik & Schotter (1987)◮ Taylor (1995), Emons (1997), Feddersen & Gilligan (2001)◮ Dulleck, Kerschbamer (2006) [& Sutter (2010)]◮ Milgrom & Roberts (1986), Hahn (2004).

Labelling◮ Harbaugh, Maxwell & Roussillon (2011)◮ Lerner & Tirole (2006), Baski & Bose (2007)◮ Roe & Sheldon (2007), Lerner, Fahri & Tirole (2010).

(Comparative) Advertising◮ Barigozzi, Garella & Peitz (2009)◮ Bagwell (2007).

(P2P) File sharing, downloading◮ Alexander (2004), Peters (2005).

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The model of the credence market◮ There is a set N of n risk neutral firms, each producing a

single credence good of iid quality

θi ∼ F[0,s], i ∈ N , s ∈ N.

Wlog, we re-index firms such that θ1 ≥ θ2 ≥ · · · θn.We mainly use of the ratio of these qualities xi = θi/θ−i . Forn = 2 we write x = θ1/θ2 if there is no danger of confusion.

◮ Once a good is produced, there is nothing a firm can do toalter its quality. Marginal cost of (re)production is zero.(The paper also models (sequential) costly quality choice.)

◮ There is a large number of consumers with iid tastesµ ∼ G[0,s], g > 0 everywhere, and consumption utility

v(µ, θ) = µθ − p.

◮ Product qualities are mutually known among firms.7 / 22



The ranking model for exogenous qualities θi◮ Assume that firm i ∈ N can choose an amount of information

εi ∈ (−∞,∞) which it releases on its product.

◮ Assume that there exists a (full) noisy ranking of productsq(·). For this presentation, q(·) is the generalised Tullock csf

q1i (θ, r) =θri

θr1 + · · · + θrn

, with precision r = ε1 + · · · + εn.

Recall that qualities θi are fixed here; firms choose εi .

◮ Then the firm’s problem is to

maxεi

ui(θ, ε) =

n∑

k=1

qki (θ, r)Pk(θ, r , µ)− c(|εi |)

where Pk is the profit from the k th-ranked product and c(|εi |)is the strictly convex cost of information with c(0) = 0.

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A first property of precision contestsConsider the problem of firm i ∈ N = {1, 2}

maxεi

ui(θ, ε) = q(xi , r)P1(r) + (1− q(xi , r))P

2(r)− c(|εi |) (*)

where r = ε1 + ε2.

LemmaA necessary condition for a maximum of (*) is

c ′(|ε1|) + c′(|ε2|) = P

1′(r) + P2′

(r),

e.g., for quadratic cost ε1 + ε2 ≡ r∗ equals marginal profit.

Proof.

c ′(ε1)− P2′

(r) = q(r)P1′

(r) − q(r)P2′

(r) + P1(r)q′(r) − P2(r)q′(r),

−c ′(ε2) + P1′(r) = q(r)P1

′

(r) − q(r)P2′

(r) + P1(r)q′(r) − P2(r)q′(r).9 / 22



Ranking competition with a single exogenous prize P1

For x = θ1/θ2 and r = ε1 + ε2, player i ’s problem is

maxεi

ui (θ, ε) = q(xi , r)P1 − c(|εi |) =

1

1 + x−riP1 −

ε2i2

(*)

with focs

ε1 = P1 x

r log(x)

(1 + x r )2= −ε2.

Inserting r∗ = 0 from our lemma results in the eq’m bids

ε∗1 = P1 log(x)

4= −ε∗2. (**)

Proposition

Contest (*) with a constant prize P1 > 0 has the asymmetric, purestrategy equilibrium (**).

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Ranking competition with a single endogenous prize P1(r)Let P1(r) react to the amount of information produced in a linearfashion P1(r) = α(ε1 + ε2) = α r , α > 0. The players’ problem is

maxεi

ui(θ, ε) = q(xi , r)P1(r)− c(|εi |) =

1

1 + x−riαr −

ε2i2.

Taking the derivative wrt εi gives player i’s foc implicitly as

εi =αx ri (1 + x

ri + r log(xi ))

(1 + x ri

)2, r = ε1 + ε2.

Applying the insight from Lemma 1 that r∗ = P1′

= α, thistransforms into the equilibrium information dissemination functions

ε∗1 =αxα(1 + xα + α log(x))

(1 + xα)2, ε∗2 =

α(1 + xα − αxα log(x))

(1 + xα)2

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Information dissemination policies as fns of quality ratio

1 2 3 4 5

0.5

1.0

1.5

2.0

ε∗1, ε∗2

x =θ1

θ2

ε∗1 + ε∗2 = α

ε∗1(x)

ε∗2(x)

Application (see next slide):

The point here is that, since ε∗1 + ε∗2 = α,

every market quality combination sends the

same signal and a consumer cannot update

her prior information F on the distribution

of qualities.

In the paper we try to be more general.

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Application: The timing of the interaction

0 1 2 3

‘types’ aredrawn

firms choose(pki )

nk=1, εi(t)

rankingrealises

consumption& profits

time

Consumers observe prices & total r = ε1 + · · ·+ εn.

We analyse asymmetric, pure strategy eq’a ((pki )nk=1, εi )

ni=1.

In particular, we want to know about the (expected) welfare effectsof introducing a ranking into the credence market.

Since consumers’ valuations only depend on the ranking of thefirms—everything else being uninformative—each firm faces thesame optimisation problem for choosing pki and thus selects thesame equilibrium vector of conditional product prices. Nothingchanges when firms decide prices after the ranking realises.

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Illustrative example application setup

1. There are two firms N = {1, 2} of mutually known quality.

2. Consumers expect Uniform quality θ ∼ U[0,1].

3. Consumers have Uniform preferences µ ∼ U[0,1].

4. Labels are assigned using a generalised Tullock ranking tech

q(xi , r) =θri

θri + θrj

=1

1 + x−ri, i = {1, 2}, and j = 3− i

with precision r = ε1 + εn.

5. Firms have quadratic information costs ci (|εi |) = ε2/2.

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Demand sideKnowing the distribution of qualities, the consumers’ expectationof the kth-ranked among n products is the order statistic

E[Θ(k:n)] =

∫ s

0θf(k:n)(θ)dθ ⇔ E[Θ(1:2)] = s

2

3, E[Θ(2:2)] = s

1

3

under Uniform qualities on [0, s] for n = 2.In eq’m, consumers observe a ranking of known precision r∗.In rank k = 1, 2, m = 3− k , a consumer expects quality

Λk(r) =E[Θ(k:2)]

r

E[Θ(k:2)]r + E[Θ(m:2)]r︸︷︷︸

=q̃k

E[Θ(1:2)] + (1− q̃k)E[Θ(2:2)],

A type-µ consumer is indifferent between the first & second rankedproducts iff µΛ1 − p1 = µΛ

2 − p2 defining the cutoff vector as

µ̂ = (µ̂01 = s, µ̂12(r) =

p1 − p2Λ1 − Λ2

, µ̂32(r) =p2Λ2

).15 / 22



Supply & demand (drawn for r ∗)

0

µ

Θ(2:2) =1/3

Θ(1:2) =2/3

µ̂12 =3×2r

2+7×2r

P1(r)

θ

µ̂23 =1

7+ 92r−1

P2(r)

µ̂01 = 1 = s

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Supply sideGiven these cutoffs, the first ranked firm maximises her profit bychoosing p1 (under Uniform consumer preferences)

arg maxp1

P1(r) = p1

∫µ̂01=s

µ̂12(r)

g(µ)dµ = (µ̂01 − µ̂12)p

∗

1 .

The second ranked firm maximises her profit by choosing p2

argmaxp2

P2(r) = p2

∫µ̂12(r)

µ̂23(r)

g(µ)dµ = (µ̂12 − µ̂23)p

∗

2 .

This gives prices (determining cut-offs µ) and thus the prizes

P1(r) = s34(2r−1)(1+21+r )

2

3(1+2r )(2+7×2r )2,

P2(r) = s3(1+21+r)(2r+4r−2)3(1+2r )(2+7×2r )2

. -0.5 0.5 1.0 1.5

-0.02

0.02

0.04

0.06

P1(r)

P2(r)

r

P i

‘maximum differentiation’

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Ranking competition with two endogenous prizesThese two prizes P1(r),P2(r) are non-linear in r ; firms’ choose

maxεi

1

1 + x−riP1(r) +

1

1 + x riP2(r)−

ε2i2

and eq’m r∗ is given by Lemma 1 as solution to

r∗ = P1

′

(r)+P2′

(r) =26+r

(

22 + 81× 2r + 27× 4r+1 + 59× 8r)

log(2)

(1 + 2r )2 (2 + 7× 2r )3≈ 1.759.

(This ‘stationary’ point equation cannot be solved analytically.)

Inserting this r∗ into the focs we obtain eq’m information policies

ε∗1 =P2

′

(r∗)+P1′

(r∗)x2r∗

+x r∗

(P1′

(r∗)+P2′

(r∗)+(P1(r∗)−P2(r∗)) log(x))

(1+x r∗)2 ,

ε∗2 =P1

′

(r∗)+P2′

(r∗)x2r∗

+x r∗

(P1′

(r∗)+P2′

(r∗)+(P2(r∗)−P1(r∗)) log(x))

(1+x r∗)2 .

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Endogenous prizes with demand effectUsing arbitrary example parameters of s = 4 and x = 3, these fnsgive firms’ optimal information dissemination choices as

ε∗1 = 1.847, ε∗

2 = −0.088 ⇔ r∗ = 1.759.

Redefining the previous prizes as

B1(r) = rγP1(r), B2(r) = rγP2(r), where γ ∈ {−1

2, 0,+

1

2}

we can derive market demand from the same model as above butwith a variable demand impact of the information provided.This gives (for the same parameters)

−1/2 : ε∗

1 = 1.157, ε∗

2 = −0.177, r∗ = 0.980 . . . decreasing,

0 : ε∗1 = 1.847, ε∗

2 = −0.088, r∗ = 1.759 . . . constant,

1/2 : ε∗

1 = 3.118, ε∗

2 = 0.089, r∗ = 3.207 . . . increasing.

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Further results & summary◮ The equilibria we just characterised actually do exist.◮ We extend the model to include the combined choice of

quality and precision, both simultaneous and sequential.◮ Welfare results:

◮ For any number of firms, if the distribution of consumer tastesG(µ) satisfies a weak condition on the lowest equilibriumcutoff µ̂nn+1, then the fully labelled and cartelised market isstrictly more efficient than its unlabelled equivalent.

◮ If only a single label is awarded per firm, then welfare in thefully labelled market is strictly higher than in the unlabelledmarket under a condition akin to MLRP.

————————————————————————–◮ We present the first model dealing with endogenous ranking

(or labelling) precisions that we are aware of.◮ As this ranking precision is endogenously chosen, we

effectively allow players to decide the ranking technology.21 / 22



That’s it!

In case you don’t have enough of this, I will present anenvironmental application ‘Efficient emissions reduction’ in theCESifo seminar on April 15, 12:00.

If you have any questions or you would like to discuss your ideaswith me you are welcome to come to my office hours in Room 211,later today.

Of course, you can also contact me through email [email protected].

Thank you for your attention!

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IntroThe modelResultsApplication

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