Contests with Reimbursements

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

2. Plan • Motivation • Preliminary Results • Model • Results • Examples • Conclusion

3. Motivation Contest literature has greatly expanded since Tullock (1980) … Rosen (1986); Dixit (1987); Snyder (1989);… Surveys: Nitzan (1994), Szymanski (2003), Konrad (2007)

4. Motivation The contest literature is almost silent about the most realistic, real-life type, contests: contests with reimbursements.

5. 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.

6. 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.

7. Motivation Kaplan, Luski, Sela, and Wettstein (JIE, 2002) • The winner is reimbursed

8. Motivation Politics • Losers can also be reimbursed Security Dilemma.Yugoslavia: Serbia, Croatia, and Bosnia Herzegovina Multiple intrastate conflicts: the third party guarantor

9. 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

10. 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.

11. Motivation Politics Cold War: the Soviet Union and the United States often opposed each other in their “reimbursements” Vietnam and Korea

12. 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.

13. 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.

14. 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.

15. 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

16. 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.

17. 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

18. Player i’s problem

19. Equilibrium In a symmetric equilibrium x1 = ... = xn = x* FOC becomes

20. 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.

21. Player i’s problem

22. The Assumptions • 0 < r ≤ 1 • 0 ≤  ≤ 1 • 0 ≤  ≤ 1 • 0 ≤  +  < 2

23. 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

24. Open Question: Player i’s problem

25. Results FOC for the maximization problem

26. Results In a symmetric equilibrium x1 = … = xn = x*. From FOC:

27. Definitions Total spending in the symmetric equilibrium Z = nx*. Net total spending in the symmetric equilibrium T = nx* - αx* - (n-1)x*.

28. 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*)

29. Designer’s objections 1. Choice of α! Max/Min T = Max/Min (nx* - αx* - (n-1)x*)

30. Designer’s objections 1. Choice of α! Note that

31. Designer’s objections 1. Choice of α! • Maximize: α = 1 – Winner is reimbursed • Minimize: α = 0 – Winner gets only the prize

32. Designer’s objections 2. Choice of ! • Maximize: α = 1 – Winner is reimbursed

33. Designer’s objections 2. Choice of ! • Maximize: α = 1 – Winner is reimbursed The Net Total Spending is independent from the Loser Premium!

34. 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

35. Designer’s objections 2. Choice of ! • Minimize: α = 0 – Winner gets only the prize

36. Designer’s objections: Minimize 2. Choice of ! Note that

37. Designer’s objections: Minimize α = 0 - Winner gets only the prize  = 1 – Losers are reimbursed

38. 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

39. 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

40. 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

41. Results:Properties of the equilibrium Proposition 3. Suppose that 0 ≤ r ≤ 1, then in the symmetric equilibrium the individual effort and the expected individual payoff are decreasing functions of the number of players and the (net) total spending is an increasing function of the number of players.

42. Results:Properties of the equilibrium Proposition 4. Suppose that n ≥ 2, then in the symmetric equilibrium the individual effort and the (net) total spending are increasing functions of the parameter r and the expected individual payoff is a decreasing function of the parameter r.

43. Results:Properties of the equilibrium Corollary 2. The highest net total spending is achieved if r = 1 and TW = V.

44. Results:Properties of the equilibrium Proposition 5. Suppose that 0 ≤ r ≤ 1 and n ≥ 2, then in the symmetric equilibrium the individual effort, the expected individual payoff, and the (net) total spending are increasing functions of the prize value V.

45. Designer’s objections: Minimize α = 0 - Winner gets only the prize  = 1 – Losers are reimbursed

46. 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

47. 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

48. Results:Properties of the equilibrium Proposition 7. Suppose that 0 ≤ r ≤ 1, then in the symmetric equilibrium the 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.

49. Results:Properties of the equilibrium Proposition 8. Suppose that n ≥ 2, then in the symmetric equilibrium the individual effort and the (net) total spending are increasing functions of the parameter r and the expected individual payoff is a decreasing function of the parameter r.

50. Results:Properties of the equilibrium Proposition 9. Suppose that 0 ≤ r ≤ 1 and n ≥ 2, then in the symmetric equilibrium the individual effort, the expected individual payoff, and the (net) total spending are increasing functions of the prize value V.