Economia Política - Cap 8

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INTRODUCTION TO CONTEST THEORY

1 Definition

A contest describes a situation where two or more agents ("con-

testants") spend resources in order to obtain a certain desirable out-

come ("win a prize").

Formally, a contest can be characterised by the following elements:

A set of contestantsN

= {1,...,n

} A prize B to be allocated among the contestants

Contestant is valuation of the prize: vi (B)

Each contestant i N can make an effort xi, at a cost Ci (xi)

Notice that a lower valuation of the prize is equivalent to

having a higher cost of making the effort, and vice versa

If effort is measured in monetary units, then Ci (xi) = xi

The probability that contestant i wins the prize is given by the

contest success function pi = pi (x1,...,xn) Thus, contestant i has an ex ante expected payoff of

i (x1,...,xn) = pi (x1,...,xn) vi (B) Ci (xi) . (1)

2 Examples

An illustrative specific example:

a certain number of cities competing for becoming the location

of the next Olympic Games

each city spends money on architects, marketing agencies, lobby-

ing, etc, in order to "improve" its proposal, increasing the prob-

ability of being chosen

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General political economy examples:

political lobbying

rent-seeking

beauty contests (e.g., the Olympic Games example)

political campaigning

bribery

Other examples:

promotional (advertising) competition

litigation

internal labour market tournaments (promotion contests)

R&D contests

military conflict

sports

3 Fully discriminatory contests

First-price all-pay auction

The contestant who expend the highest effort in the contest, wins the

prize for sure

More relevant for cases where

contest effort translates deterministically into some observable

quality or quantity variable for each contestant

the allocation of the contested prize is made on the basis of a

comparison of the value of this variable across the different con-

testants

Assume two contestants: 1 and 2

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Their valuations of the prize are given by v1 v2 > 0.

The effort cost function is C(xi) = xi, i = 1, 2.

The contest success function for contestant 1 is

p1 =

1 if x1 > x21

2if x1 = x2

0 if x1 < x2

, (2)

while the contest success function for contestant 2 is p2 = 1p1.

The players choose their efforts, xi 0, simultaneously to max-

imise expected profi

ts, given by (1).

This game has no Nash equilibrium in pure strategies

Mixed strategy equilibrium:

F1 (x1) =

(x1v

f or x1 [0, v2]

1 f or x1 > v2(3)

and

F2 (x2) = ( 1 v2

v1+ x2

v1f or x2 [0, v2]

1 f or x2 > v2, (4)

where Fi (xi) is the cumulative distribution function for the effort

choice of contestant i.

Expected payoffs in equilibrium are

E(1) = v1 v2

and

E(2) = 0.

Expected efforts in equilibrium are

E(x1) =v2

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and

E(x2) = (v2)

2

2v1.

Notice that the prize will not always be allocated to the contestant

with the higher valuation.

If there are more than two players, with

v1 v2 > v3 v4 ... vn,

then the unique equilibrium has xi = 0 for i > 2, while contestants 1

and 2 play (3) and (4), respectively.

4 Non-discriminatory contests

In many contests there are some noise and elements of randomness

In contests with noise, the contestant expending the highest effort does

not win the contest with certainty

typical example: political lobbying (rent-seeking) contests

The most commonly used imperfectly discriminating contest is the

Tullock contest

In a standard Tullock contest, the contest success function is given by

pi (x1,...,xn) =

(xrinj=1 x

rj

if max {x1,...,xn} > 0

1

notherwise

, (5)

where r > 0.

The parameter r represents the returns to contest effort (in terms of

success probability)

If r < 1, there are decreasing returns to effort

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If r > 1, there are increasing returns to effort

If r = 1, there are constant returns to effort

The special case of r = 1 is the "lottery contest".

the probability of winning corresponds to the relative shares of

"lottery tickets"

The Tullock contest success function converges to the contest success

function of a perfectly discriminatory contest, (2), when r

4.1 A general example of the Tullock contest

n players, with valuations v1 v2 ... vn

Effort is measured in monetary terms: C(xi) = xi

The contest success function is given by (5)

Contestants choose effort simultaneously

Expected profit for contestant i is

E(i) = xriPn

j=1 xrj

vi xi.

The first-order condition for optimal effort by contestant i is

E(i)

xi=

rxr1iP

j6=i xrjPn

j=1 xrj

2

vi 1 = 0. (6)

4.1.1 Special Case 1: Equal valuations (v1 = v2 = ... = vn = v)

Imposing symmetry, xi = x, in (6) yields

rxr1 (n 1) xr

(nxr)2v 1 = 0

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Solving for x yields

x = (n 1)n2

rv (7)

individual contest effort is

increasing in the valuation of the prize (v)

increasing in the parameter r

decreasing in the number of contestants (n)

Total contest effort is

X =n

Xi=1 xi = nx = n 1

n rv. (8) Total contest effort is increasing in the number of contestants

Notice that this pure strategy Nash equilibrium exists only if r nn1

Expected profits in the pure strategy equilibrium are

E() =1

nv

(n 1)

n2rv

=v

n 1

n 1

n r

Expected profits are non-negative only if r n

n1

For r > nn1

, expected profits from playing (7) is negative and

each contestant can do better by choosing zero effort.

For r > nn1

, only mixed strategy equilibria exist.

Rent dissipation

Rent dissipation is measured by the ratio of total effort and the

valuation of the prize: Xv

In the pure strategy equilibrium, the degree of rent dissipation is given

byX

v=

n 1

n

r (0, 1)

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Notice that, since r nn1

, there is never overdissipation in the pure

strategy equilibrium.

The contested rent is fully dissipated only in the extreme case ofr = 1

and free entry /(n )

If r < 1, there is always underdissipation of the rent, even with free

entry of contestants.

this means that rents survive even under perfect competition from

an infinite number of potential contestants!

4.1.2 Special Case 2: Two contestants with unequal valuations

Setting n = 2, the first-order condition (6) can be rewritten as

rxr1i xrj

(xr1

+ xr2

)2vi = 1; i, j = 1, 2; i 6= j

Taking the ratio of the two first-order conditions yields

x1

v1=

x2

v2,

implying

x1 x2 (since v1 v2 by assumption).

The player who value the prize more expends more effort in the contest

Both players dissipate the same (constant) fraction of their valuations

in rent-seeking activities, regardless of the parameter r.

4.1.3 Special Case 3: The lottery contest (r = 1)

Setting r = 1, the first-order condition (6) reduces toPj6=i xjPnj=1 xj

2

vi 1 = 0,

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which can be written as

(X xi) viX2

= 1,

where X =Pn

j=1 xj is total contest effort.

Solving for xi yields

xi = X

1

X

vi

(9)

By summing over n, we can derive total contest effort in equilibrium

X =

nXi=1

xi =

nXi=1

X1 Xvi = XnXnX

i=1

1vi! ,

which, when solving for X, yields

X =n 1Pni=1

1

vi

.

Alternatively, this expression can be written as

X =

n 1

n eVn, (10)

where eVn = nPni=1 1vi1 is the harmonic mean of the contestantsvaluations.

Notice from (9) that a contestant will participate in the contest (i.e.,

choosing xi > 0) if X < vi.

Total effort is always lower than the valuation of the active player

with the lowest valuation of the prize.

In general: More unequal valuations among the contestants will

reduce total effort

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5 Extensions to the standard contest set-up

Entry fees to participate in the contest

Multiple prizes

Budget constraints

Endogenous prizes

Contest effort affects the players valuations of the prize

Delegation

Sabotage

Contests for a public good

Strategic alliances

6 Contest design

How should a contest administrator optimally design a contest?

how many contestants?

which type of contestants?

entry fees or not?

optimal choice of prize

influence the cost-of-effort for the contestants

What is the objective function of a contest designer?

maximise total eff

ort?

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Economia Política - Cap 8