strategy(in(contests(–an(introduc4on(faculty.haas.berkeley.edu/rjmorgan/phdba279b/contests -...
TRANSCRIPT
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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WB
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
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
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
16
Two Players Tullock Example
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 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
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 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
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 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
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 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
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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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 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
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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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
1S2S
>1 2v v
• 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
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⎛ ⎞= + <⎜ ⎟⎝ ⎠
2 22
11
2v vX v
v= 2X v
> =1 2 3v v v
• 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
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( )ϕ ,i ib ϕ =i ∈ ⎡ ⎤⎣ ⎦0,ib bπ ϕ= − + −( , )Ai i i i j i ip x x b x
π ϕ= + −( , )( )Pi i i i j i ip x x v b π = −* *P
i i i ip v Ex π = 0AiE
( )= =* *1 22 ,b v b b ( )= =* *
1 2, 2b b b v
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)
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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
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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
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“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
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• 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
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• 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
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== ≡
+ ∑ 11 2
, jnii i ijj
xp v x xx x
− −⎛ ⎞− −⎜ ⎟⎝ ⎠2
1 11 n m vmnm
v
−⎛ ⎞−⎜ ⎟⎝ ⎠
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
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(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