strategy(in(contests(–an(introduc4on(faculty.haas.berkeley.edu/rjmorgan/phdba279b/contests -...

Strategy in Contests – An Introduc4on by Kai A. Konrad, Working Paper, January 2007* presenta4on for PHDBA 279B by Aisling ScoH and GianCarlo Moschini

* Now available as a book : Kai A. Konrad, Strategy and Dynamics in Contests, Oxford University Press, 2009.

1

Introduction • What is a contest?

•  Game where players have opportunity to expend resources in order to affect probabili4es of winning prizes

• A lot of theore4cal progress •  Less empirical work and open ques4ons

•  Importance: apply auc4on theory to many different contexts •  In all 3 contests

•  Players exert costly irreversible efforts while compe4ng for a prize •  A player’s probability of winning the prize depends on the players’ rela4ve expenditures

•  Each contest defines the exact probability differently •  Contest Success Func4on (CSF) maps vector of contestants’ expenditures to probability of winning

2

A contest

•  Strategic game among n contestants •  A prize is allocated, contestants value the prize •  Each player makes an “effort” at a cost •  The vector of all efforts determines contestants’ probability of winning •  Payoff to contestants: •  The “contest success func4on” can take alterna4ve forms

0ix ≥

1 2( , ,..., )nx x x x≡

1 2 1 2( , ,..., ) ( , ,..., ) ( )i n i n i i ix x x p x x x v C xπ = −

1 2( , ,..., )i np x x x

3

> =0 , 1,2,...,iv i n

( )i iC x

3 Contests •  3 Canonical Contests:

•  All-‐pay Auc4on •  Lobby and Military

•  Tournament •  Principal-‐Agent, Labor, and contract design

•  Tullock (or loHery) Contest •  R&D races, Poli4cal contests, and Rent-‐seeking compe44ons

•  Tullock and Tournaments usually have pure strategy equilibria •  Excep4ons exist

• All-‐pay usually has non-‐degenerate mixed strategy equilibria • Different cases seek to maximize or minimize contest effort expenditure

4

Examples •  Promo4onal compe44on (Friedman 1958)

•  Expend resources on adver4sing in order to compete •  CSF: share in total market •  Can change the total market size indica4ng that efforts increase public good

•  Internal labor markets •  Rela4ve performance schemes (Lazear and Rosen 1981 and Rosen 1986)

•  Li4ga4on •  Legal system determines rules of the game, but each group allocates effort •  Entry decision, rule of fees, informa4on assymetries

•  Beauty Contests •  Olympic games (Steward and Wu 1997)

•  Rent Seeking and influence ac4vi4es •  Bureaucrats and regulators (Tullock 1967, Kruger 1974, and Posner 1975) •  Status seeking (Congleton 1989, Konrad 1990 and Konrad 1992)

5

…and Examples • Campaigning and commiHee bribing

•  Different from adver4sing because payoffs are discon4nuous

•  Sports (Hoehn and Szymanski 1999 and Szymanski 2003) •  Role of entry fees, homogeneous par4cipants, and prize number and structure

•  Intra and Inter group contests • Military conflict

•  Rela4onship between numerical superiority and baHle success is loose •  Informa4onal assyme4res (Clark and Konrad 2006)

• R&D contests (Fullerton and McAfee 1999) •  Firms expend effort to get monopoly rent from patent or first developing product

6

First-Price All-Pay Auction (complete information)

•  Illustrate with special case of two players with , and

1

( , ) 1 2

0

i j

i i j i j

i j

if x x

p x x if x x

if x x

⎧ >⎪

= =⎨⎪ <⎩

1 2 0v v≥ > ( )i iC x x=

•  There is no equilibrium in pure strategies:

•  There is a mixed-‐strategy equilibrium described by cumula4ve distribu4on func4ons

and , on the support , with the property that

2v 1v

1 1( )F x 2 2( )F x [ ]20,v

[ ]1 1 2 1 1 1 1 2 1 2( ) ( ) , 0,x F x v x v v x vπ = − = − ∀ ∈

[ ]2 2 1 2 2 2 2 2( ) ( ) 0 , 0,x F x v x x vπ = − = ∀ ∈

0

7

•  Recall contestants’ payoff: 1 2 1 2( , ,..., ) ( , ,..., ) ( )i n i n i i ix x x p x x x v C xπ = −

… First-Price All-Pay Auction

2v

1

2

11 vv

−1 1( )F x

2 2( )F x

ix

•  Equilibrium mixed strategies:

[ ]11 2

21 1

1 2

0,( )

1

x for x vvF x

for x v

⎧ ∈⎪= ⎨⎪ >⎩

[ ]2 22 2

1 12 2

2 2

1 0,( )

1

v x for x vv vF x

for x v

⎧ ⎡ ⎤− + ∈⎪ ⎢ ⎥= ⎣ ⎦⎨⎪ >⎩

•  NOTE: ◊ Inefficiency: highest valua4on contestant does not always get the prize

◊ Expected efforts are and à 21 2vEx =

22

212vExv

= 1 2 2Ex Ex v+ <

•  More than 2 contestants: à same as case of

à con4nuum of equilibria

1 2 3 ... 0nv v v v≥ > ≥ ≥ > 2n =

1 2 1... ... 0j j nv v v v v+= = = > ≥ ≥ > 8

All-Pay Auction with liquidity constraints

•  Contestants might be constrained in their efforts (e.g., limits on campaign contribu4ons)

•  Applica4on to R&D contests with liquidity constraints

n contestant all value the prize equally at

each contestant’s effort limited by the fund availability . Assume

v

w 1 2 3 ... nw w w w> > ≥ ≥

Equilibrium mixed strategies:

[ )11 2

1 1

1 2

0,( )1

x for x wF x vfor x w

⎧ ∈⎪= ⎨⎪ ≥⎩

[ )2 22 2

2 2

2 2

1 0,( )

1

w x for x wv vF x

for x w

⎧ ⎡ ⎤− + ∈⎪ ⎢ ⎥= ⎣ ⎦⎨⎪ ≥⎩

v

1

21 wv

−1 1( )F x

2 2( )F x

ix2w

2wv

9

•  Contestants know their own valua4on and the distribu4on from which all valua4ons are drawn. Example: case of and .

With bid func4ons , the contestant’s payoff is

FOC:

All-Pay Auction with incomplete information

2n = [ ]0,1iv ∈

( )x vξ=

•  If is the uniform distribu4on,

the symmetric equilibrium is:

( )1( ) ( )i i i i ix F x v xπ ξ −= −

( )F v

( )1

1( ) 0 ( ) 1 0i i i ii

dx F x vdxξ

π ξ−

−ʹ′ ʹ′= → − =

2

2vx =

( )F v

1

1 2

ix

iv

( )vξ

•  Note: here contestant with highest valua4on

always wins prize (inefficiency noted earlier disappears)

10

•  Can nest alterna4ve legal fee shining rules in the Contest framework. Example:

•  American system: and (standard contest)

•  Bri4sh rule (loser pays all costs): and

All-Pay Auction and models of litigation

if wins

if wins( , , ) i i ji i j i

i j

v bx dx ix x v

ax tx jπ

− −⎧⎪= ⎨

− −⎪⎩

1a b= = 0d t= =

1a t= = 0b d= =

•  In an all-‐pay auc4on with incomplete informa4on, the expected total effort by

contestants is higher under the Bri4sh system than the American system

-‐-‐ see prof. Morgan’s lecture notes on Auc4ons

11

All-Pay Auction with additive noise (tournament)

•  Includes a random element (addi4ve noise) to the contest success func4on Effort produces a “performance” ; success func4on (for n=2) is 1

( , ) 1 2

0

i i j j

i i j i i j j

i i j j

if x x

p x x if x x

if x x

ε ε

ε ε

ε ε

⎧ + > +⎪

= + = +⎨⎪ + < +⎩

ε= +i i iy x

•  Define and assume it has distribu4on on

Suppose both contenders value winning equally ( ) and loser gets

payoff to contestants:

op4mality: where

2 1ε ε ε≡ − ( )G ε [ ],e e−

i Wv b= Lb

( , ) 1 ( , ) ( )i i i j W i i j L ip x x b p x x b C xπ ⎡ ⎤= + − −⎣ ⎦

( )( , )

( )i i jW L i

i

p x xb b C x

x∂

ʹ′− =∂

( , ) ( )i i j i j

i i

p x x G x xx x

∂ ∂ −=

∂ ∂

12

… tournament

•  In a symmetric NE both contestant spend same effort and win with prob

the equilibrium solves

-‐-‐ note: increased dispersion of noise or reduc4on in win reward reduce effort

1 2

( ) *(0) ( )W LG b b C xʹ′ ʹ′× − =

13

•  Contest design problem: choose and to max obj func4on, e.g.,

•  Par4cipa4on constraint at symmetric NE requires

•  Hence designer problem is to max

•  Can implement the efficient solu4on sa4sfying

Wb Lb ( )2 W Lx b bϕ − −

*1 1 ( )2 2W Lb b C x+ ≥

( )2 2 ( )x C xϕ −

( )* *2 ( )x C xϕʹ′ ʹ′=

•  Note role of randomness in all the serngs considered so far

Standard Tullock

•  Contest success func4on: •  If r=1 loHery contest •  If r>0, allows for general types of contests •  As the above func4on converges to contest success func4on converges to no noise tournament

•  Imperfectly discrimina4ng

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14

Existence and Uniqueness

•  In a symmetric contest with n contestants there exists a pure strategy Nash Equilibrium if

•  Otherwise a mixed strategy Nash Equilibrium exists •  It is possible for total effort to exceed the valua4on of the prize •  Expected total effort cannot exceed the value of the prize

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15

Two Players

•  FOCs

•  Solve for the symmetric case:

x

x (x )

x (x )

x

1

1

1

2

2

2

10.50

0.5

1

2

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F'

Contests

Aisling Scott

Feb 2014

Standard Tullock

Solution for symetric:

xi = xj∀xi, xjNote: Vi = v ∀i and x1 = x2 = x∗ by symmetry

Thusr(x∗)(2r−1)

4(x∗)(2r)v = 1

x∗ = rv4 When does rent disaptaion exist? When 2x∗ = 2 rv

4

r = 2 all rent is disapated for symetric case

Let r = 1 so the best reaction functions are BR1 = x1 =√x2v − x2 and BR2 = x2 =

√x1v − x1

1

Contests

Aisling Scott

Feb 2014

Standard Tullock



Thusr(x∗)(2r−1)

4(x∗)(2r)v = 1


4



√x1v − x1

1

Contests

Aisling Scott

Feb 2014

Standard Tullock



Thusr(x∗)(2r−1)

4(x∗)(2r)v = 1


4



√x1v − x1

1

16

Two Players Tullock Example

Contests

Aisling Scott

Feb 2014

Standard Tullock



Thusr(x∗)(2r−1)

4(x∗)(2r)v = 1

x∗ = rv4

When does rent dissaptaion exist?

Since 2x∗ = 2 rv4

2v = 2x∗

r = 2

Thus when r = 2 all rent is dissapated.

Let r = 1 so the best reaction functions are:

BR1 = x1 =√x2v − x2

BR2 = x2 =√x1v − x1

1

Contests

Aisling Scott

Feb 2014

Standard Tullock



Thusr(x∗)(2r−1)

4(x∗)(2r)v = 1

x∗ = rv4


Since 2x∗ = 2 rv4

2v = 2x∗

r = 2



BR1 = x1 =√x2v − x2

BR2 = x2 =√x1v − x1

1

Contests

Aisling Scott

Feb 2014

Standard Tullock



Thusr(x∗)(2r−1)

4(x∗)(2r)v = 1

x∗ = rv4


Since 2x∗ = 2 rv4

2v = 2x∗

r = 2



BR1 = x1 =√x2v − x2

BR2 = x2 =√x1v − x1

1

Contests

Aisling Scott

Feb 2014

Standard Tullock



Thusr(x∗)(2r−1)

4(x∗)(2r)v = 1

x∗ = rv4


Since 2x∗ = 2 rv4

2v = 2x∗

r = 2



BR1 = x1 =√x2v − x2

BR2 = x2 =√x1v − x1

1

17

Best Reaction Functions

x

x (x )

x (x )

x

1

1

1

2

2

2

10.50

0.5

1

2

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3 6779


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F'

Contests

Aisling Scott

Feb 2014

Standard Tullock



Thusr(x∗)(2r−1)

4(x∗)(2r)v = 1

x∗ = rv4


Since 2x∗ = 2 rv4

2v = 2x∗

r = 2



BR1 = x1 =√x2v − x2

BR2 = x2 =√x1v − x1

Note v1 = 2, v2 = 1, r = 1

1

18

Nash Equilibrium

x

x (x )

x (x )

x

1

1

1

2

2

2

10.50

0.5

1

2

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F'

19

Many Participants

• General Contest Success Func4on • Denote and let •  Yields FOC •  Thus we can characterize total efforts of par4cipants that expend nonzero efforts as:

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20

Why is Tullock (Lottery) model so popular? • Axioma4c Reasoning

•  Individual payoffs depend only on own preferences and aggregated values of other players’ efforts

•  For r=1 for a given aggregate effort of all contestants, the probability of winning is linear in contestant i’s own effort

•  Belle, Keeney, and LiHle (1975) prove that this is the only contest success func4on that has this property

•  Informa4on aspects •  Tullock (1980) assumed contestants know everything about each other

• Microeconomic underpinnings •  Simple search model •  Ra4o of efforts as a probability for winning the contest

21

Experimental • Most experiment find support for contest theory • Overbidding occurs (aggregate effort exceeds Nash equilibrium predic4ons)

•  Excep4on: rank-‐order tournaments • Why is this the case?

•  Open ques4on: What contributes to overbidding?

22

Robust Results •  1. One player’s increased efforts creates a nega4ve externality for another

•  More heterogeneous contestants, usually reduces aggregate equilibrium effort and increases net payoffs

•  2. Contestant with higher valua4on is usually more likely to win •  Also expends more effort •  Close one-‐to-‐one rela4onship for differences in valua4on and differences in individual cost of produc4on given effort levels

23

Sequential choices •  Stage 1: contestant simultaneously choose whether to exert effort “early” (E) or “late” (L)

•  Stage 2: simultaneous contest if 1st stage equilibrium is (E,E) or (L,L) sequen4al contest if 1st stage equilibrium is (E,L) or (L,E)

2x

1x

•

• •

2BR

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1S

2S

N

N

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•  The SPNE is (L,E) and •  The “weaker” player moves first and commits to a lower effort

2S

24

•  Contests may require an entry fee (or, contestant have an opportunity cost) •  If contest fully dissipates rent, why should contestants want to incur an entry fee? •  2-‐stage game with n possible contestants with same valua4on ; entry fee of

Stage 1: contestants decide whether or not to par4cipate

Stage 2: All-‐pay auc4on among contestants who entered

•  Asymmetric equilibria in pure strategy with only one contestant who enters •  Symmetric mixed strategy equilibrium: each enters with probability

•  Does all this cons4tute an “entry puzzle”? •  Possible resolu4on: players’ cost in any given contest may be a random draw

Voluntary participation

> 0v D

( ) −= −

1 ( 1)* 1 nq D v

25

•  Contest designer’s objec4ve func4on assumed to care about aggregate effort •  In general, high contestants’ valua4on (low costs) increases aggregate effort •  Also, homogeneity of contestants tend to increase aggregate effort

•  Counter-‐intui4ve result: it might be desirable to exclude contestant who value the prize

most highly

•  Illustrate with all-‐pay auc4on, n=3, and •  Equilibria where expected aggregate effort X is highest have •  But if contestant 1 is excluded,

Exclusion

26

⎛ ⎞= + <⎜ ⎟⎝ ⎠

2 22

11

2v vX v

v= 2X v

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•  Contestants act as “principals” and hire “agents” to do their bidding •  Illustrate with all-‐pay auc4on and two contestants •  Contract is , where down payment, and is the “delegated value”

•  Payoff to agent •  Payoff to principal à (with by IR)

•  Two pure strategy equilibria: and •  Contestants use delega4on to generate asymmetry between the two agents, thereby leading to a smaller dissipa4on of the prize

•  Note the role of “homogeneity” vs “heterogeneity” of players •  However, the delega4on contract is not renego4a4on-‐proof

Delegation

27

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Externalities •  Joint Ownership

•  Firms are no longer indifferent about who wins the prize •  Minority Shareholdings can reduce rents generated by the alloca4on of the prize and the firms’ aggregate equilibrium efforts

•  Informa4on externali4es •  Revealing informa4on to uninformed voters •  Inverse campaigning dissipates rents (Konrad 2004)

28

Public Goods and Free Riding •  Homogeneous groups

•  Aggregate effort of a group and its win probability are completely independent of number of members in the two groups (Katz et al. 1990)

• Why? Free-‐riding might intensify as group gets larger •  Hetergeneous groups

•  Doesn’t effect aggregate equilibrium group effort •  Only highest group member valua4on maHers

29

Prizes •  Endogeneous Prizes

•  Suppose prize values are func4ons of the vector of efforts •  Theore4cally generates a different set of predic4ons •  In some case, all-‐pay auc4ons can have pure strategy equilibrium

• Mul4ple Prizes •  Sports tournaments onen award mul4ple prizes •  Nobel Prize exemplifies general non-‐op4mality of a single large prize •  How does prize structure influence contest outcome? •  Assump4on: Contest designer allocates a fixed amount of money either to one grand prize or split between mul4ple prizes

•  But how does the designer allocate these prizes? •  Sequen4al games or based on a CSF on basis of actual efforts

•  Many cases have been looked at but not all

30

“Handicapping” • More heterogenity between contestants may lead to lower aggregate equilibrium efforts

•  Contest designers may want to favor or bias towards the contestant who values the prize lower to increase aggregate efforts in equilibrium

•  These concepts have been used to study affirma4ve ac4on issues • Handicap effort of strong players or help the efforts of the weak

•  Clark and Riis (2000) show that this increases a bureaucrat's receipts

31

•  Introduce more detail on how contestants’ efforts translate into win probabili4es. Three structures are discussed:

•  Nested contests: contestants are actually groups, once the winning group is determined the prize needs to be allocated to members (another contest)

-‐-‐ Inter-‐group contest and intra-‐group contest

•  Alliances: the forma4on of groups in the inter-‐group contest may be endogenous • Mul0-‐ba3le contests: there are several stages to a contest

-‐-‐ Race

-‐-‐ Elimina4on tournament

-‐-‐ Tug of war

Grand Contests

32

•  Exogenous sharing rule (intra-‐group alloca4on): merit vs egalitarianism Example: 2 groups, prize allocated to a group based on the total effort of its members

Group win probabili4es

Sharing rule: the share of the prize to member j , if group wins, is A “merit” rule yields more group efforts than an “egalitarian” rule

•  Group choice of a sharing rule: possible 4me-‐consistency problem •  Hierarchical structures may be advantageous. Example: nm homogeneous contestants

•  One-‐stage Tullock contest: expected payoff of winner is •  Two-‐stage contest: 1st stage compe44on between n groups of m members each; 2nd stage compe44on between members of winning group. Expected payoff:

à two-‐stage contest dissipates less rent

Nested Contests

33

== ≡

+ ∑ 11 2

, jnii i ijj

xp v x xx x

− −⎛ ⎞− −⎜ ⎟⎝ ⎠2

1 11 n m vmnm

v

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11 nm vnm

α α= − +1(1 ) ij

iji i

xq

x nα =( 0) α =( 1)

•  “Race”: a contestant needs to accumulated a certain number of wins in sub-‐contests (baHles) in order to win the contest

Examples: Tennis, R&D, primary elec4ons

Illustra4on: need to win 2 baHles to win contest

Multi-battle contests

34

(2,2)

• • • (1,2)

• • •

• • (2,1) (1,1)

player 1 wins

player 2 wins

•  “Elimina4on tournament”: the field of contestants is reduced (e.g., halved) in each stage Examples: tennis tournaments; Jack Welch selec4on of his successor as CEO of GE

•  “Tug of War”: need a certain number of net stage successes (# wins minus # losses) to win the contest

• • • • • • • 0−1− −( 1)n−n n−( 1)n1. . . . . .

player 2 wins player 1 wins start

•  “Discouragement effect:” e.g., the fact that most of the prize is dissipated in the final of an elimina4on tournament discourages contestants in the semi-‐finals

strategy(in(contests(–an(introduc4on(faculty.haas.berkeley.edu/rjmorgan/phdba279b/contests -...

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