Contests with Reimbursements Alexander Matros and Daniel Armanios University of Pittsburgh

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Text of 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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  • Plan Motivation Preliminary Results Model Results Examples Conclusion
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  • Motivation Contest literature has greatly expanded since Tullock (1980) Rosen (1986); Dixit (1987); Snyder (1989); Surveys: Nitzan (1994), Szymanski (2003), Konrad (2007)
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  • Motivation The contest literature is almost silent about the most realistic, real-life type, contests: contests with reimbursements.
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  • 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.
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  • 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.
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  • Motivation Kaplan, Luski, Sela, and Wettstein (JIE, 2002) The winner is reimbursed
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  • Motivation Politics Losers can also be reimbursed Security Dilemma. Yugoslavia: Serbia, Croatia, and Bosnia Herzegovina Multiple intrastate conflicts: the third party guarantor
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  • 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
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  • 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.
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  • Motivation Politics Cold War: the Soviet Union and the United States often opposed each other in their reimbursements Vietnam and Korea
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  • 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.
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  • 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.
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  • 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.
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  • 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
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  • 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.
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  • 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) x i and wins the prize with probability
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  • Player is problem
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  • Equilibrium In a symmetric equilibrium x 1 =... = x n = x * FOC becomes
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  • 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) x i and wins the prize with probability Matros (2007): r = 1, but V 1 V n > 0.
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  • Player is problem
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  • The Assumptions 0 < r 1 0 1 0 1 0 + < 2
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  • Open Question n = 2 risk-neutral contestants One prize Contestants' prize valuations are commonly known V 1 V 2 > 0. Player i exerts effort (buys lottery tickets) x i and wins the prize with probability
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  • Open Question: Player is problem
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  • Results FOC for the maximization problem
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  • Results In a symmetric equilibrium x 1 = = x n = x*. From FOC:
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  • Definitions Total spending in the symmetric equilibrium Z = nx*. Net total spending in the symmetric equilibrium T = nx* - x* - (n-1)x*.
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  • Designers objections Maximize or Minimize the Net total spending in the symmetric equilibrium. Choice of and ! Max/Min T = Max/Min (nx* - x* - (n-1)x*)
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  • Designers objections 1. Choice of ! Max/Min T = Max/Min (nx* - x* - (n-1)x*)
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  • Designers objections 1. Choice of ! Note that
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  • Designers objections 1. Choice of ! Maximize: = 1 Winner is reimbursed Minimize: = 0 Winner gets only the prize
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  • Designers objections 2. Choice of ! Maximize: = 1 Winner is reimbursed
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  • Designers objections 2. Choice of ! Maximize: = 1 Winner is reimbursed The Net Total Spending is independent from the Loser Premium!
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  • 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
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  • Designers objections 2. Choice of ! Minimize: = 0 Winner gets only the prize
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  • Designers objections: Minimize 2. Choice of ! Note that
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  • Designers objections: Minimize = 0 - Winner gets only the prize = 1 Losers are reimbursed
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  • 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
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  • 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
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  • 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
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  • 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.
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  • 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.
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  • Results: Properties of the equilibrium Corollary 2. The highest net total spending is achieved if r = 1 and T W = V.
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  • 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.
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  • Designers objections: Minimize = 0 - Winner gets only the prize = 1 Losers are reimbursed
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  • 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
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  • 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
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  • 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.
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  • 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.
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  • Results: Properties of the equilibrium Propositio