Participation, Incentives and Efforts in
Contests:
A Maximization Perspective
Wolfgang Leininger, TU Dortmund
based on joint work with Jörg Franke, Christian Kanzow and Alexandra Schwartz
April 2011
Introduction
The Contest Game
The Maximization Problem
Conclusion
The efforts of men are utilized in two different ways: they are
directed to the production or transformation of economic
goods, or else to the appropriation of goods provided by
others.
Vilfredo Pareto, 1927
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Introduction
The Contest Game
The Maximization Problem
Conclusion
Introduction
Competitive situations among (heterogeneous) individuals are
frequently modeled as Tullock contest games.
Such situation include: marketing and advertising by firms,
litigation, relative- reward schemes (promotion), rent-seeking
(lobbying), patent races, sports, combat, war, sabotage etc.
Our focus: Contest organizer can design/choose/bias the
contest rules.
Research question: What is the optimal contest rule given
the organizer’s objective?
objective: Maximization of total contest efforts
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Introduction
The Contest Game
The Maximization Problem
Conclusion
The Tullock Contest
I n rent seekers compete for a prize/rent of size V . Each
contestant i expends effort xi to capture the rent.
I contest success: pi (x1 , . . . , xn ) = nxi
P
xj
j=1
I
I
I
i maximizes Θi (xi , x−i ) = pi (xi , x−i )V − xi
solution: unique Nash equilibrium :
xi∗ = n−1
V
n2
n−1
∗
rent dissipation:
n · xi = n V
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Introduction
The Contest Game
The Maximization Problem
Conclusion
I
General form of Tullock contest success function:
xir
pi (xi , x−i ) = P
n
xjr
j=1
I
Important property:
xir
lim P
n
r →∞
xjr
=
j=1
I

 1
1
m+1

0
xi > max{x−i }
xi = max{x−i }
xi < max{x−i }
All-pay auction:
I
I
highest bidder wins all bids,
all bids have to be paid.
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Introduction
The Contest Game
The Maximization Problem
Conclusion
Markets as Contests
i) A principal/employer has a ”wage budget” V in order to
incentivize n agents/workers to expend effort in output
production.
I The principal ’markets’ her budget : agents have to ”pay”
for wages with output they have produced.
I Suppose n agents supply x1 , . . . , xn units of output, then
”market clearing” requires:
px1 + . . . + pxn = V , p = price of output in wage units.
Hence:
V
p∗ = P
equilibrium price
n
xj
j=1
and each worker gets:
p ∗ xi =
V
n
P
j=1
xi =
xj
xi
n
P
V.
xj
j=1
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Introduction
The Contest Game
The Maximization Problem
Conclusion
ii) Suppose the budget of the principal is indivisible; i.e. a
”prize” of value V .
The principal now ”markets” lottery tickets, which the agents
can buy with their output; say, one ticket per unit of output.
Again, the equilibrium price of a ticket would have to be
p∗ =
V
n
P
, if i demands xi tickets.
xj
j=1
p ∗ xi =
V
n
P
j=1
xi with pi (xi , x−i ) =
xj
xi
n
P
xj
j=1
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Introduction
The Contest Game
The Maximization Problem
Conclusion
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Introduction
The Contest Game
The Maximization Problem
Conclusion
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Introduction
The Contest Game
The Maximization Problem
Conclusion
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Introduction
The Contest Game
The Maximization Problem
Conclusion
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Introduction
The Contest Game
The Maximization Problem
Conclusion
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Introduction
The Contest Game
The Maximization Problem
Conclusion
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Introduction
The Contest Game
The Maximization Problem
Conclusion
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Introduction
The Contest Game
The Maximization Problem
Conclusion
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Introduction
The Contest Game
The Maximization Problem
Conclusion
iii) Suppose the principal could use ”personalized” prices (e.g.
like a Lindahl-planner or a price-discriminating monopolist) for
tickets.
pi = αi · p̄, i = 1, . . . , n, p̄ > 0, αi > 0
Then market clearing requires:
α1 p̄x1 + . . . + αn p̄xn = V
→ p̄ ∗ =
V
n
P
and each agent spends:
αj x j
j=1
V
αi p̄xi = αi P
n
αj xj
j=1
xi =
αi xi
V
n
P
αj xj
j=1
→ biased Tullock Contest Success Function
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Introduction
The Contest Game
The Maximization Problem
Conclusion
A biased lottery is an aggregative game:
Suppose bidders have bidding cost functions according to
ci (xi ) = βi xi
θi (xi , x−i ) =
Then:
αi xi
V
n
P
αj xj
− βi x i
j=1
Set yi = αi xi and get
n
P
θi (xi , x−i ) = vi (yi , yj ) =
j=1
yi
n
P
V − γi yi with γi =
yj
βi
αi
j=1
Moreover:
A biased lottery is a well-behaved aggregative game
(Cornes/Hartley, 20??)
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Introduction
The Contest Game
The Maximization Problem
Conclusion
Situations where contest organizer deviates from equal
treatment of competitors:
I Affirmative action: Disadvantaged participants are
favored by selection rules (e.g. university admission,
public procurement auctions etc.)
I Sports tournaments: Handicap rules in amateur golf,
NBA rookie draft (priority in drafting rookies for teams
with worst performance), etc.
I International trade: Import duties and tariffs favor
domestic firms.
I Rent-Seeking: Weak agents are encouraged by contest
organizer.
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Introduction
The Contest Game
The Maximization Problem
Conclusion
Our Model:
I Simple Tullock-contest game between n heterogeneous
players.
I Contest organizer biases contest success function (CSF)
by specifying individual weights → asymmetric lottery
contest.
I Contestants react to biased contest rule.
I Objective of contest organizer is maximal total effort
exertion which depends on weights.
In mathematical terms:
Bilevel mathematical program with equilibrium constraints.
max f (x ∗ (α)) subject to x ∗ (α) ∈ S(α).
α
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Introduction
The Contest Game
The Maximization Problem
Conclusion
Related Literature
I Theoretical contribution:
I
I
I
Balancing the playing field in competitive situations:
I
I
I
I
I
I
Complementary to: Esteban and Ray (1999)
Extension of: Fang (2002), Stein (2002), Nti (2004)
Auctions: Myerson (1981), McAfee & McMillan (1989)
Rank-order tournaments: Lazear & Rosen (1981)
All-pay auctions: Baye et al. (1993), Fu (1996), Clark &
Riis (2000)
Contests: Franke (2008), Runkel (2006).
Contest Design:
Dasgupta & Nti (1998), Nti (2004).
Bilevel progams with equilibrium constraints:
Luo et al. (1996), Outrata et al. (2002).
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Introduction
The Contest Game
The Maximization Problem
Conclusion
Baye, Kovenock and de Vries (1993, AER):
Exclusion Principle
2-Stage-Contest among n heterogeneous contestants
Stage I:
Contest Organizer selects participants of contest from the n
contestants.
Stage II:
Selected contestants (”finalists”) compete in a Tullock contest
with r = ∞; i.e. an all-pay auction.
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Introduction
The Contest Game
The Maximization Problem
Conclusion
Result:
If contestants are too heterogeneous (in a particular way) an
effort maximizing contest organizer may exclude some
contestants from going to Stage II; i.e.
{finalists } ⊂ {contestants}
Perversely, the most able contestants are excluded.
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Introduction
The Contest Game
The Maximization Problem
Conclusion
”Logic” behind result:
In an all-pay auction with heterogeneous bidders; i.e. bidders
have different valuations V1 , . . . , Vn for the prize; normally
only the bidders with the two highest valuations actually bid,
hence
I
Result 1:
Suppose: V1 = V2 > V 3 ≥ . . . ≥ Vn . Then exclusion of a
contestant does not pay for organizer.
I
Result 2:
Suppose V1 > V2 = V3 ≥ V4 ≥ . . . ≥ Vn . Then the
organizer maximizes total effort by excluding contestant 1.
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Introduction
The Contest Game
The Maximization Problem
Conclusion
Fang (2002): Inclusion Principle
2-Stage-Contest among n heterogeneous contestants
Stage I:
Contest Organizer selects participants of contest from the n
contestants.
Stage II:
Selected contestants (”finalists”) compete in a Tullock contest
with r = 1; i.e. a lottery.
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Introduction
The Contest Game
The Maximization Problem
Conclusion
Result:
The organizer cannot gain from excluding some contestants
from final lottery.
I
Total effort =
k ∗ −1
n
P
1
i=1
I
I
=
k ∗ −1
k∗
· HM(V1 , . . . , Vk ∗ )
Vi
{1, . . . , k ∗ } ⊆ {1, . . . , n} set of (voluntarily) active
participants (V1 ≥ . . . ≥ Vn )
k ∗ is determined as:
i−2
i−2
k ∗ = max{i Vi > i−1
· HM(V1 , . . . , V−i ) = i−1
}
P
j=1
⇔
largest i s.th. (i − 2) ·
1
Vi
<
i−1
P
j=1
1
Vj
1
;
Vj
i.e. at least two players participate.
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Introduction
The Contest Game
The Maximization Problem
Conclusion
Example:
V1 = 50
V2 = 40
i = 1X i = 3 : 38 >
i = 2X i = 4 : 38 >
V3 = V4 = 38
1
1
1
+ 40
50
≈ 22X
2
1
1
1
+ 40
+ 38
50
≈ 28X
All four contestants participate.
But, if V4 < 28 then n∗ = 3 and the last (least efficient)
contestant would submit a bid of 0; i.e. 4 would not compete.
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Introduction
The Contest Game
The Maximization Problem
Conclusion
The Underlying Contest Game
I n heterogeneous contestants compete for prize v = 1.
I Exerting effort xi induces costs: ci (xi ) = βi xi .
I Standard asymmetric lottery contest with individually
specified weights α = (α1 , . . . , αn ) ∈ (0, ∞)n yields
asymmetric CSF (Clark & Riis 1996):
αi xi
pi (xi , x−i ) = Pn
.
j=1 αj xj
I
Contestants are ordered:
β1
α1
≤ ... ≤
Expected payoff function: θi (xi , x−i ) =
βn
.
αn
Pnαi xi
j=1 αj xj
− βi x i .
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Introduction
The Contest Game
The Maximization Problem
Conclusion
Result from previous literature:
(Cornes/Hartley, 2005; Stein, 2002)
There exists a unique equilibrium in this contest game.
Characterization of equilibrium:
I Not all players are active → Subset K of active players:
K = {1, . . . , k} ⊆ N where 2 ≤ k(α, β) ≤ n.
I Closed form equilibrium effort:

 α1 (1 − αβi P(k−1)βj ) P(k−1)βj > 0, for all i ∈ K ,
i
i
j∈K αj
j∈K αj
xi∗ =
0,
for all i ∈
/ K.
I
Total equilibrium effort: f (α, β) =
k
P
xi∗ (α, β, k).
i=1
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Introduction
The Contest Game
The Maximization Problem
Conclusion
Problem: Solution is only implicitly determined (Stein, 2002)
k = k(α, β) = max{i |
i≤n
αi
βi
>
i−2
i−1
i−1
· HM( αβ11 , . . . , αβi−1
)}
This is equivalent to
k(α, β) = max{i |(i − 2) ·
i≤n
βi
αi
<
i−1
P
j=1
βj
}
αj
i.e. k ≥ 2
⇒ At least two players are active in equilibrium.
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Introduction
The Contest Game
The Maximization Problem
Conclusion
From the above characterization it also follows that
Heterogeneity Lemma:
k(α, β) ⊆ {i |
βi
αi
<
β1
α1
+
β2
}
α2
i.e. the two strongest players determine degree of
heterogeneity possible (→all-pay auction)
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Introduction
The Contest Game
The Maximization Problem
Conclusion
The Maximization Problem of the Contest Organizer
Contest Organizer specifies weighting factors in CSF such that:
P
xi∗ (α, β, K (α, β))
max n f (α) = max n
α∈(0,∞)
α∈(0,∞) i∈K (α,β)
Problematic issues:
1. Problem: Feasible set α neither closed nor bounded.
2. Problem: Active set K changes discretely with marginal
changes of α → Objective function f (α) not necessarily
continuous nor differentiable.
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Introduction
The Contest Game
The Maximization Problem
Conclusion
Addressing 1. Problem: Lemma 3.1, 3.2
I
Weights α can be rescaled to be bounded:
n
P
α̂i = 1.
i=1
I
Weights α can be varied for j ∈
/ K to some extent:
α is equivalent to α̂ = ({α̂}i∈K , {α̂j }j ∈K
/ ) with α̂j ∈ [0, ᾱj ].
⇒ Maximum is invariant to such changes:
max n f (α) = max n f (α̂) with
α∈(0,∞)
A = {α̂ ∈ (0, ∞)n |
α̂∈(0,1)
n
P
n
P
α̂i = 1.
i=1
α̂i = 1} is bounded, but not closed.
i=1
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Introduction
The Contest Game
The Maximization Problem
Conclusion
Addressing 2. Problem: Theorem 3.3, 3.4
i) objective function f (α̂) is in fact continuous on set A.
ii) objective function f (α̂) can be continously extended onto
Ā
iii) global maximizer of f (α̂) in Ā implies that f (α) obtains a
global maximum in (0, ∞)n even if the maximizer of f (α̂)
belongs to Ā\A
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Introduction
The Contest Game
The Maximization Problem
Conclusion
Result 1 (Existence): Corollary 3.5
I There exist optimal weights α∗ that maximize total
equilibrium effort.
I The optimal weights can be chosen such that f (α∗ ) is
locally differentiable.
Characterization of Global Maximum:
f (α) differentiable ⇒ optimal weights α∗ satisfy ∇f (α∗ ) = 0.
This system can be solved explicitly for the optimal weights α∗ .
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Introduction
The Contest Game
The Maximization Problem
Conclusion
Result 2: Theorem 4.3
Setting optimal weights α∗ implies:
I Set of active contestants K (α∗ ) is indirectly characterized
by following (necessary) condition:
P
βj for all i ∈ K (α∗ ).
(k(α∗ ) − 2)βi <
j∈K (α∗ )
∗
I
Optimal
( weights α are:
αi (k(α∗ ), β), for all i ∈ K (α∗ ),
αi∗ =
αi ∈ [0, ᾱi ] , for all i ∈
/ K.
I
Closed form expression for total equilibrium effort:
f (α∗ ) = f (k(α∗ ), β).
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Introduction
The Contest Game
The Maximization Problem
Conclusion
Necessary condition: (k − 2)βi <
P
j∈K
βj for all i ∈ K
Example 1: n = 5 and β = (1, 1, 1, 10, 10)
Then the index sets
{1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}, {1, 2, 4},
{1, 2, 5}, {1, 3, 4}, {1, 3, 5}, {2, 3, 4}, {2, 3, 5}, {2, 4, 5},
{3, 4, 5}
trivially satisfy the necessary condition.
So do {2, 3, 4, 5}, {1, 3, 4, 5} and {1, 2, 4, 5}.
But {1, 2, 3, 4}, {1, 2, 3, 5} and {1, 2, 3, 4, 5} do not.
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Introduction
The Contest Game
The Maximization Problem
Conclusion
Question: How to find the global maximum?
Definition: An index set K ⊆ N is called
I feasible, if it satisfies the necessary condition
I maximal, if it is feasible and not contained in another
feasible index set
I optimal, if it maximizes f (K , β) over all feasible sets
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Introduction
The Contest Game
The Maximization Problem
Conclusion
Characterization of unique optimal set
Exclusion Result: Any βi contained in a feasible index set is
strictly smaller than the sum of the smallest three elements.
Inclusion Result: Only maximal sets can be optimal.
Implication: An optimal set contains at least three elements.
(Theorem 4.7)
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Introduction
The Contest Game
The Maximization Problem
Conclusion
Representation
Result: One maximal
feasible set is given by
n
o
Pi
∗
K = i ∈ N|(i − 2)βi < j=1 βj if β1 , ≤ . . . , βn
Implication: K ∗ = {1, 2, . . . , k ∗ }; i.e. only the most efficient
players are active.
Finally: This K ∗ is the unique optimal index set. (Theorem
4.12)
Corollary: The ’exclusion principle’ of Baye et al. (1993)
never holds.
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Introduction
The Contest Game
The Maximization Problem
Conclusion
How do the closed-form solutions determined by K ∗
look like?
Should the organizer ”level” the field?
I.e. set
β1
α∗1
=
β2
α∗2
= ... =
βk
?
α∗k
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Introduction
The Contest Game
The Maximization Problem
Conclusion
Solution (in closed form as a function of k ∗ ):
αi∗ =
αi∗
2(k ∗ −1)βi
i = 1, . . . , k ∗
(k−2)βi
1+ Pk ∗
∗
β
j=1 j
i = k ∗ + 1, . . . , n
< (k − 1) · βi
Implication:
"
xi∗ (α∗ ) =
f (α∗ ) =
1
4βi
1
4
1−
(k ∗ −2)β
Pk ∗
j=1
Pk ∗
1
j=1 βj
−
2 #
i
βj
(k ∗ −2)2
Pk ∗
j=1 βj
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Introduction
The Contest Game
The Maximization Problem
Conclusion
Value of aggregator in optimal solution:
y=
n
P
yi =
i=1
xi∗ =
1
(1
4βi
k∗
P
αi∗ xi∗ with αi∗ =
i=1
2(k ∗ −1)βi
1+
(k ∗ −2)βi
∗
kP
βj
j=1
and
∗
−2)βi 2
− ( (k P
))
k∗
βj
j=1
Result: y =
(k ∗ −1)k ∗
2
− (k ∗ − 2) > 0 as k ∗ ≥ 3
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Introduction
The Contest Game
The Maximization Problem
Conclusion
Value of aggregator in equilibrium given
α = (α1 , · · · , αn ):
equilibrium effort: xi∗ =
⇒
P
i∈K
αi xi∗ =
1
(1
αi
βi (k−1)
βj
αi P
j∈K αj
) P(k−1)βj
j∈K αj
k−1
P
βj
j∈K αj
=
−
k−1
HM( αβ11 , · · ·
k
, αβkk )
⇒ free-entry equilibrium, not ZPHEE
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Introduction
The Contest Game
The Maximization Problem
Conclusion
Role of number of active players k ∗ in optimal solution:
βi
1
≤ ∗
∗
αi
k −1
βi
1
> ∗
∗
αi
k −1
player is active (n = 2 : true)
player is inactive
Heterogeneity Lemma
K ∗ ⊆ {i | βi < β1 + β2 + β3 }
Example 1: n = 5
β = (1, 1, 1, 10, 10)
k∗ = 3
i.e.
K ∗ = {1, 2, 3}
α1∗ = α2∗ = α3∗ = α4∗ = α5∗ = 3
→ Lottery
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Introduction
The Contest Game
The Maximization Problem
Conclusion
If N ≥ 3 it follows from our formula:
αi∗
βi
R
⇐⇒ βi Q βj
∗
αj
βj
resp.
βj
βi
Q ∗ ⇐⇒ βi Q βj
∗
αi
αj
Corollary 4.13: Under the optimal weighting scheme α∗
active players with higher βi obtain higher αi∗ ; but α∗ does not
level the field except for n = 2.
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Introduction
The Contest Game
The Maximization Problem
Conclusion
Tullock Lottery: αi = α
i = 1, . . . , n ≥ 3
A Tullock lottery is maximizing total effort only if the set of
active players is homogeneous.
Fang’s inclusion principle, which refers to the Tullock lottery
with heterogenous agents, is extended by our solution:
Lemma: Let β = (β1 , . . . , βn ) be given and denote by K (β)
the set of active players in the lottery.
Then K (β) ⊆ K (α∗ )
and the inclusion may be strict.
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Introduction
The Contest Game
The Maximization Problem
Conclusion
Recall the Example from Fang’s lottery:
V1 = 50
V2 = 40
we know:
K=
in our language : β1 =
∗
⇒ K (α ) =
V3 = 38
V4 = V
{1, 2, 3},
V < 28
{1, 2, 3, 4}, V ≥ 28
1
50
β2 =
1
40
β3 =
1
30
β4 =
1
V
{1, 2, 3},
V < 15 (β4 < 0.0713)
{1, 2, 3, 4}, V ≥ 15 (β4 > 0.0713)
i.e. V4 ∈ [15, 28) is inactive in lottery, but is active in
optimally biased lottery.
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Introduction
The Contest Game
The Maximization Problem
Conclusion
Concluding Remarks
Analysis of optimal (total effort maximizing) design of
asymmetric (weighted) CSF:
I Existence and characterization of optimal CSF.
I Leveling the playing field is profitable to some extent.
I Exclusion of strong contestants is never optimal.
I At least 3 (the most strongest) contestants must be
active.
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Introduction
The Contest Game
The Maximization Problem
Conclusion
Epstein, Mealem and Nitzan, 2010 JPubE forthcomming:
2 Player Model
ui = pi · (x1 , x2 ) · Vi − xi
i = 1, 2
2
x
pi = 2 i
i = 1, 2
x1 + α · x22
Designer’s objective function
G = γ(E (u1 ) + E (u2 )) + (1 − γ)(x1 + x2 )
γ = “political culture”
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Introduction
The Contest Game
The Maximization Problem
Conclusion
Morgan, 2000 ReStud, Financing Public Goods by Means of
Lotteries
n- player model
ui
n
X
xi
= wi − xi + P
R + hi (
xj − R)
n
j=1
xj
(1)
j=1
Theorem 1: Fixed- prize Raffles yield higher provision of
public good than voluntary contributions.
Research question: Could optimally biased lotteries achieve
the first- best provision? (→ ”Lindahl-Prices”’)
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