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

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Page 1: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

Contests with Reimbursements  

Alexander Matros and Daniel Armanios

University of Pittsburgh

Page 2: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

Plan

Motivation Preliminary Results Model Results Examples Conclusion

Page 3: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

Motivation

Contest literature has greatly expanded since

Tullock (1980)

Rosen (1986); Dixit (1987); Snyder (1989);…

Surveys:

Nitzan (1994), Szymanski (2003), Konrad (2007)

Page 4: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

Motivation

The contest literature is almost silent about

the most realistic, real-life type, contests:

contests with reimbursements.

Page 5: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

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.

Page 6: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

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.

Page 7: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

Motivation

Kaplan, Luski, Sela, and Wettstein (JIE, 2002) The winner is reimbursed

Page 8: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

Motivation

Politics Losers can also be reimbursed

Security Dilemma. Yugoslavia: Serbia, Croatia, and Bosnia Herzegovina

Multiple intrastate conflicts: the third party guarantor

Page 9: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

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

Page 10: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

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.

Page 11: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

Motivation

Politics

Cold War: the Soviet Union and the United States

often opposed each other in their “reimbursements”

Vietnam and Korea

Page 12: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

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.

Page 13: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

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.

Page 14: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

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.

Page 15: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

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

Page 16: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

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.

Page 17: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

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

Page 18: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

Player i’s problem

iiL

n

jj

ii

Wn

jj

in

jj

i

xxx

xf

xfx

xf

xfV

xf

xfi

111

1max

Page 19: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

Equilibrium

In a symmetric equilibrium x1 = ... = xn = x*

FOC becomes

**

**

*

*

'1'

1

' xnxn

xxV

n

n

xf

xfLW

LW

Page 20: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

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

Page 21: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

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

Page 22: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

The Assumptions

0 < r ≤ 1 0 ≤ ≤ 1 0 ≤ ≤ 1 0 ≤ + < 2

Page 23: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

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

Page 24: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

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

Page 25: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

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

Page 26: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

Results

In a symmetric equilibrium

x1 = … = xn = x*.

From FOC:

rV

rnnrnnn

nx

11

12

*

Page 27: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

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

*

Page 28: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

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*)

Page 29: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

Designer’s objections

1. Choice of α!

Max/Min T = Max/Min (nx* - αx* - (n-1)x*)

rV

rnnrnnn

nnnT

11

112

Page 30: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

Designer’s objections

1. Choice of α!

Note that

rVn

rnnrnnn

nnrnnrnnrnnnT1

11

111122

2

011

11112

nnr

nnrnnrnnrnnn

Page 31: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

Designer’s objections

1. Choice of α! Maximize: α = 1 – Winner is reimbursed Minimize: α = 0 – Winner gets only the prize

Page 32: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

Designer’s objections

2. Choice of ! Maximize: α = 1 – Winner is reimbursed

Vrn

rx

1*

rVrn

nT

1

Vrn

nrnxZ

1*

Page 33: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

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

Page 34: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

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

Page 35: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

Designer’s objections

2. Choice of ! Minimize: α = 0 – Winner gets only the prize

rV

rnnn

nx

1

12

*

rV

rnnn

nnnT

1

112

Page 36: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

Designer’s objections: Minimize

2. Choice of !

Note that

rVn

rnnn

nnrnrnnnT 2

22

2

11

11

.0112 rnnnrnrnnn

Page 37: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

Designer’s objections: Minimize

α = 0 - Winner gets only the prize

= 1 – Losers are reimbursed

rV

rnn

nxT LL

1

1

Page 38: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

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

Page 39: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

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

Page 40: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

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

Page 41: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

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

Page 42: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

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

Page 43: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

Results: Properties of the equilibrium

Corollary 2. The highest net total spending is achieved if

r = 1 and TW = V.

Page 44: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

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

Page 45: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

Designer’s objections: Minimize

α = 0 - Winner gets only the prize

= 1 – Losers are reimbursed

rV

rnn

nxT LL

1

1

Page 46: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

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

Page 47: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

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

Page 48: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

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

Page 49: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

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

Page 50: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

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

Page 51: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

Comparison with Tullock (1980)

Page 52: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

Example 1.

Suppose that r = 0.5 and V = 100Then

Page 53: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

Example 1.

Suppose that r = 0.5 and V = 100Then

Page 54: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

Example 1.

Suppose that r = 0.5 and V = 100Then

Page 55: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

Example 2.

Suppose that n = 2 and V = 100Then

Page 56: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

Example 2.

Suppose that n = 2 and V = 100Then

Page 57: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

Example 2.

Suppose that r = 2 and V = 100Then

Page 58: Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

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