contests with reimbursements alexander matros and daniel armanios university of pittsburgh
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Contests with Reimbursements
Alexander Matros and Daniel Armanios
University of Pittsburgh
Plan
Motivation Preliminary Results Model Results Examples Conclusion
Motivation
Contest literature has greatly expanded since
Tullock (1980)
…
Rosen (1986); Dixit (1987); Snyder (1989);…
Surveys:
Nitzan (1994), Szymanski (2003), Konrad (2007)
Motivation
The contest literature is almost silent about
the most realistic, real-life type, contests:
contests with reimbursements.
Motivation
Kaplan, Luski, Sela, and Wettstein (JIE, 2002) Politics: primary elections
Candidates raise and spend money to be the party's choice for the general election.
All losers pay the costs,
the winner advances and receives increased funding to compete.
Motivation
Kaplan, Luski, Sela, and Wettstein (JIE, 2002) Economics: JET contracts
Boening and Lockheed Martin were competing for a Joint Strike Fighter (JSF) contract.
Both companies built prototypes up-front to win this JSF government contract. This contract would enable the winning company to make more JSFs for the government purchase.
Motivation
Kaplan, Luski, Sela, and Wettstein (JIE, 2002) The winner is reimbursed
Motivation
Politics Losers can also be reimbursed
Security Dilemma. Yugoslavia: Serbia, Croatia, and Bosnia Herzegovina
Multiple intrastate conflicts: the third party guarantor
Motivation
Politics Losers can also be reimbursed
Kalyvas and Sambanis (2005): Bosnian Serbs performed massive atrocities towards Bosnian
Muslims, especially in Srebrenica UN and NATO intervene on behalf of the Bosnian Muslims
Motivation
In this paper we consider
contests with reimbursements.
Examples: conflict resolutions where not only
the winner but also loser(s) can be reimbursed
by third parties.
Motivation
Politics
Cold War: the Soviet Union and the United States
often opposed each other in their “reimbursements”
Vietnam and Korea
Preliminary Results
Classic Tullock's model with reimbursements:
There are continuum of reimbursement mechanisms which
maximize the net total effort spending in the contest.
In all these mechanisms, the winner has to be completely
reimbursed for her effort.
Preliminary Results
Classic Tullock's model with reimbursements:
There exists a unique reimbursement mechanism which
minimizes the total rent dissipation.
All losers have to be reimbursed in this case.
Applications
Casino and charity lotteries If the objective is to maximize the net total
spending, the winner has to receive the main prize and the value of her wager.
Related Literature: Auction literature
Riley and Samuelson (1981)
Sad Loser Auction: a two-player all-pay auction where the
winner gets her bid back and wins the prize.
Goeree and Offerman (2004)
Amsterdam auction
Related Literature: Auction literature
Sad Loser or Amsterdam auctions cannot produce more expected revenue than the optimal auction.
However, the contest when the winner gets her effort reimbursed provides the highest expected total effort.
It is strictly higher than the total effort in the Tullock's contest.
The Model
n ≥ 2 risk-neutral contestants One prize Contestants' prize valuations are the same and commonly known
V > 0.
Player i exerts effort (buys lottery tickets) xi and wins the prize with probability
n
jj
i
xf
xf
1
Player i’s problem
iiL
n
jj
ii
Wn
jj
in
jj
i
xxx
xf
xfx
xf
xfV
xf
xfi
111
1max
Equilibrium
In a symmetric equilibrium x1 = ... = xn = x*
FOC becomes
**
**
*
*
'1'
1
' xnxn
xxV
n
n
xf
xfLW
LW
The Model n ≥ 2 risk-neutral contestants One prize Contestants' prize valuations are the same and commonly known
V > 0.
Player i exerts effort (buys lottery tickets) xi and wins the prize with probability
Matros (2007): r = 1, but V1 ≥ … ≥ Vn > 0.
n
j
rj
ri
x
x
1
Player i’s problem
iin
j
rj
ri
in
j
rj
ri
n
j
rj
ri
xxx
x
xx
x
xV
x
xi
111
1max
The Assumptions
0 < r ≤ 1 0 ≤ ≤ 1 0 ≤ ≤ 1 0 ≤ + < 2
Open Question
n = 2 risk-neutral contestants One prize Contestants' prize valuations are commonly known
V1 ≥ V2 > 0.
Player i exerts effort (buys lottery tickets) xi and wins the prize with probability
n
j
rj
ri
x
x
1
Open Question: Player i’s problem
ii
j
rj
ri
i
j
rj
ri
i
j
rj
ri
xxx
x
xx
x
xV
x
xi
2
1
2
1
2
1
1max
Results
FOC for the maximization problem
iin
j
rj
ri
in
j
rj
ri
n
j
rj
ri
xxx
x
xx
x
xV
x
xi
111
1max
Results
In a symmetric equilibrium
x1 = … = xn = x*.
From FOC:
rV
rnnrnnn
nx
11
12
*
Definitions
Total spending in the symmetric equilibrium
Z = nx*.
Net total spending in the symmetric equilibrium
T = nx* - αx* - (n-1)x*.
rV
rnnrnnn
nx
11
12
*
Designer’s objections
Maximize or Minimize the Net total spending in
the symmetric equilibrium.
Choice of α and !
Max/Min T = Max/Min (nx* - αx* - (n-1)x*)
Designer’s objections
1. Choice of α!
Max/Min T = Max/Min (nx* - αx* - (n-1)x*)
rV
rnnrnnn
nnnT
11
112
Designer’s objections
1. Choice of α!
Note that
rVn
rnnrnnn
nnrnnrnnrnnnT1
11
111122
2
011
11112
nnr
nnrnnrnnrnnn
Designer’s objections
1. Choice of α! Maximize: α = 1 – Winner is reimbursed Minimize: α = 0 – Winner gets only the prize
Designer’s objections
2. Choice of ! Maximize: α = 1 – Winner is reimbursed
Vrn
rx
1*
rVrn
nT
1
Vrn
nrnxZ
1*
Designer’s objections
2. Choice of ! Maximize: α = 1 – Winner is reimbursed
The Net Total Spending is independent from the Loser
Premium!
rVrn
nT
1
Results: Maximize
Proposition 1. The contest designer should always
return the winner's spending. Moreover, there is
continuum optimal premie. They can be described by
The highest Net Total Spending is
.101 and
rVrn
nT
1
Designer’s objections
2. Choice of ! Minimize: α = 0 – Winner gets only the prize
rV
rnnn
nx
1
12
*
rV
rnnn
nnnT
1
112
Designer’s objections: Minimize
2. Choice of !
Note that
rVn
rnnn
nnrnrnnnT 2
22
2
11
11
.0112 rnnnrnrnnn
Designer’s objections: Minimize
α = 0 - Winner gets only the prize
= 1 – Losers are reimbursed
rV
rnn
nxT LL
1
1
Results: Maximize
Proposition 1. The contest designer should always
return the winner's spending. Moreover, there is
continuum optimal premie. They can be described by
The highest Net Total Spending is
.101 and
rVrn
nT
1
Winner gets her effort reimbursed
Proposition 2. Suppose that 0 ≤ r ≤ 1 and n ≥ 2, then the
contest when the winner gets her effort reimbursed has a unique
symmetric equilibrium. In this equilibrium
.01 and
rVrn
nT
1
Vrn
rxW
Vrn
nrnxZ
*
Vrn
rW
1
Winner gets her effort reimbursed
Corollary 1. Suppose that r = 1 and n ≥ 2, then the contest
when the winner gets her effort reimbursed has a unique
symmetric equilibrium. In this equilibrium
VT
VnxW
1
1
VVn
nnxZ
1*
0W
Results: Properties of the equilibrium
Proposition 3. Suppose that 0 ≤ r ≤ 1, then in the symmetric equilibriumthe individual effort and the expected individual payoff are decreasingfunctions of the number of players and the (net) total spending is anincreasing function of the number of players.
0
n
xW
0
n
Z W
0
n
W
0
n
T W
Results: Properties of the equilibrium
Proposition 4. Suppose that n ≥ 2, then in the symmetricequilibrium the individual effort and the (net) total spendingare increasing functions of the parameter r and the expectedindividual payoff is a decreasing function of the parameter r.
0
r
xW
0
r
Z W
0
r
W
0
r
T W
Results: Properties of the equilibrium
Corollary 2. The highest net total spending is achieved if
r = 1 and TW = V.
Results: Properties of the equilibrium
Proposition 5. Suppose that 0 ≤ r ≤ 1 and n ≥ 2, then in thesymmetric equilibrium the individual effort, the expectedindividual payoff, and the (net) total spending are increasingfunctions of the prize value V.
0
V
xW
0
V
Z W
0
V
W
0
V
T W
Designer’s objections: Minimize
α = 0 - Winner gets only the prize
= 1 – Losers are reimbursed
rV
rnn
nxT LL
1
1
Losers get their effort reimbursed
Proposition 6. Suppose that 0 ≤ r ≤ 1 and n ≥ 2, then the
contest when losers get their effort reimbursed has a unique
symmetric equilibrium. In this equilibrium
.10 and
rV
rnn
nnnxZ LL
1
1
Vrnn
L
1
1
rV
rnn
nxT LL
1
1
Losers get their effort reimbursed
Corollary 3. Suppose that r = 1 and n ≥ 2, then the contest
when losers get their effort reimbursed has a unique
symmetric equilibrium. In this equilibrium
V
n
nxT LL
12
1
Vn
L
12
1
Results: Properties of the equilibrium
Proposition 7. Suppose that 0 ≤ r ≤ 1, then in the symmetric equilibriumthe individual effort and the (net) total spending are increasing functions of the number of players and the expected individual payoff is a decreasing function of the number of players.
0
n
xL
0
n
Z W
0
n
W
0
n
T W
Results: Properties of the equilibrium
Proposition 8. Suppose that n ≥ 2, then in the symmetricequilibrium the individual effort and the (net) total spendingare increasing functions of the parameter r and the expectedindividual payoff is a decreasing function of the parameter r.
0
r
xL
0
r
Z L
0
r
L
0
r
T L
Results: Properties of the equilibrium
Proposition 9. Suppose that 0 ≤ r ≤ 1 and n ≥ 2, then in thesymmetric equilibrium the individual effort, the expectedindividual payoff, and the (net) total spending are increasingfunctions of the prize value V.
0
V
xL
0
V
Z L
0
V
L
0
V
T L
Comparison with Tullock (1980)
Example 1.
Suppose that r = 0.5 and V = 100Then
Example 1.
Suppose that r = 0.5 and V = 100Then
Example 1.
Suppose that r = 0.5 and V = 100Then
Example 2.
Suppose that n = 2 and V = 100Then
Example 2.
Suppose that n = 2 and V = 100Then
Example 2.
Suppose that r = 2 and V = 100Then
Conclusion1. Symmetric equilibria in contests with transfers
2. Maximize/Minimize net total spending
3. Winner gets her effort reimbursed
4. Losers get their effort reimbursed Individual spending is increasing in the number of players
5. Properties are discussed
6. Applications: Lotteries, Charities
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