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									Cohen and Sela (2005)
Cohen and Sela (2005)
Cohen and Sela (2005)
Cohen and Sela (2005)
Cohen and Sela (2005)
      Cohen and Sela (2005): Review

The authors study the classical Tullock’s model and show
that by a simple non-discriminating rule the contest
designer is able to manipulate the outcome of the contest
such that the probabilities to win are not ordered according
to the contestant’s abilities.

2 players only?
N players?
The contest designer maybe able to manipulate?
   Sad Loser Contests


   Alexander Matros
Department of Economics
 University of Pittsburgh
Plan
   Motivation
   Preliminary Results
   Model
   Results
   Examples
   Conclusion
         Motivation
                  How to Raise Money?
                How to reduce Rent-Seeking?

1.    Contest design
2.    Prize structure

     What if the design and the prize structure are fixed?
Motivation: Example


Charity Lottery with a given prize.

        Rent-Seeking activities
Motivation
               This paper:

Consider transfers of players’ contributions
Preliminary Results

  Winner-get-her-money-back contests
        answer both questions!
      Example

The Amsterdam auction:
a loser gets premium which depends only on her own bid.
    The Model: different values
   n ≥ 2 risk-neutral contestants
   One prize
   Contestants' prize valuations are commonly known

                             V1 ≥ … ≥ V n ≥ 0

   Player i exerts effort (buys lottery tickets of the amount) xi and wins the
    prize and her contribution with probability          xi
                                                      n

                                                     x
                                                     j 1
                                                            j


   Cohen and Sela (EL, 2005): n = 2 and a particular case for n = 3.
Player i’s problem

          xi
max   n
                   Vi  xi   xi
      x
 xi
               j
      j 1
      Results
Based on the approach in (Hillman and Riley, 1989)
                                      n
Define the total spending    s ( n)   xi
                                     i 1




                   s n   xi
FOC
         Vi  xi  2          
                                  xi
                                        1
                      s n      s n 
      Results
FOC

x  Vi  2sn xi  sn sn   Vi   0,
 2
 i

             Vi  Vi
xi  sn           .
                2
     Results
Proposition 1. ( 0,...,0,xi,0,...,0) is i-equilibrium, if

                 V1 , if i  1,
            xi  
                 V2 , if i  1.
Cohen and Sela (2005)
Results
Results
Results
Cohen and Sela (2005)
    Example: V1 ≥ V2 ≥ V3.
                    i-type equilibria

1-type: (x1, 0,0), where x1 ≥ V2;
2-type: (0, x2, 0), where x2 ≥ V1;
3-type: (0,0, x3 ), where x3 ≥ V1.
     Example: V1 ≥ V2 ≥ V3.
                    Internal equilibria
(V2, V1, 0) and (V3, 0, V1).

If V1 ≤ V2 + V3, then

(0, V3, V2) and

   1
     V2  V3  V1 , V1  V3  V2 , V1  V2  V3 .
   2
      Results: Properties of equilibria

Proposition 4. The expected payoff of the active player i
in the equilibrium of i-type is Vi.


Proposition 5. The expected payoff of each player in any
internal equilibrium is zero.
    Results: Properties of equilibria
Proposition 6. In each internal equilibrium with k active
players,
                xi1  ...  xik
                and
                pi1  ...  pik
                if and onlyif
                Vi1  ...  Vik .
     Results: Properties of equilibria

Corollary 2. A stronger player always has a smaller
chance to win the contest than a weaker player in any
internal equilibrium.
Cohen and Sela (2005)
Results: Total spending
Results: Rent Dissipation
Results: Rent Dissipation
Results: Rent Dissipation
Results: Rent Dissipation
Example: with or without reimbursement
Example: with or without reimbursement
Results: Rent Dissipation
      Conclusion
1. All equilibria in asymmetric contests

2. Properties are discussed

3. A higher value (stronger) player always exerts less
effort than a lower value (weak) player

4. The expected net total spending might be higher

5. Applications: Lotteries, Charities

								
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