Abstract
We study a sequential Tullock contest with two stages and two identical prizes. The players compete for one prize in each stage and each player may win either one or two prizes. The players have either decreasing or increasing marginal values for the prizes, which are commonly known, and there is a constraint on the total effort that each player can exert in both stages. We analyze the players’ allocations of efforts along both stages when the budget constraints (effort constraints) are either restrictive, nonrestrictive or partially restrictive. In particular, we show that when the players are either symmetric or asymmetric and the budget constraints are restrictive, independent of the players’ values for the prizes, each player allocates his effort equally along both stages of the contest.
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Notes
Several papers deal with sequential auctions. These include, Pitchik and Schotter (1988) who analyzed sequential first and second price auctions with a budget constraint and two different prizes; Pitchik (2009) who analyzed a sequential auction with a budget constraint under incomplete information, and Brusco and Lopomo (2008, 2009) who considered sequential auctions with a budget constraint and with and without a synergy between the values of the prizes.
We assume tat the players’ values are in the interval \([0,1].\) This is only a normalization. Considering higher values wold not qualitatively affect the results.
The uniqueness of the subgame perfect equilibrium is obtained by the uniqueness of the equilibrium in the one-stage Tullock contest with two players.
The uniqueness of the subgame perfect equilibrium is obtained by the uniqueness of the equilibrium in the one-stage Tullock contest with two players.
References
Amegashie J, Cadsby C, Song Y (2007) Competitive burnout: theory and experimental evidence. Games Econ Behav 59:213–239
Benoit JP, Krishna V (2001) Multiple-object auctions with budget constrained bidders. Rev Econ Stud 68:155–180
Brusco S, Lopomo G (2008) Budget constraints and demand reduction in simultaneous ascending-bid auctions. J Ind Econ 56(1):113–142
Brusco S, Lopomo G (2009) Simultaneous ascending auctions with complementarities and known budget constraints. Econ Theory 38(1):105–125
Che Y-K, Gale I (1997) Rent dissipation when rent seekers are budget constrained. Public Choice 92(1): 109–126
Che Y-K, Gale I (1998) Caps on political lobbying. Am Econ Rev 88:643–651
Gavious A, Moldovanu B, Sela A (2003) Bid costs and endogenous bid caps. Rand J Econ 33(4):709–722
Harbaugh R, Klumpp T (2005) Early round upsets and championship blowouts. Econ Inq 43:316–332
Klumpp T, Polborn M (2006) Primaries and the New Hampshire effect. J Public Econ 90:1073–1114
Konrad KA (2004) Bidding in hierarchies. Eur Econ Rev 48:1301–1308
Leininger W (1993) More efficient rent-seeking—a Munchhausen solution. Public Choice 75:43–62
Matros A (2006) Elimination tournaments where players have fixed resources. Working paper, Pittsburgh University
Morgan J (2003) Sequential contests. Public Choice 116:1–18
Pitchik C (2009) Budget-constrained sequential auctions with incomplete information. Games Econ Behav 66(2):928–949
Pitchik C, Schotter A (1988) Perfect equilibria in budget-constrained sequential auctions: an experimental study. Rand J Econ 19:363–388
Robson ARW (2005) Multi-item contests. Working paper, The Australian National University
Sela A (2009) Sequential two-prize contests. Econ Theory 51(2):383–395
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Appendix
Appendix
1.1 The proof of Proposition 1
If the budget constraint is nonrestrictive both of the restrictions in the maximization problem (3) are nonrestrictive such that
Then the first-order conditions of the maximization problems in the second stage (1) and (2) are
Because of the symmetry we denote
The solution of the above two first-order conditions is:
The first-order condition of the maximization problem in the first stage (Eq. (3)) is
where \({\widetilde{x}}_{i}^{a},{\widetilde{x}}_{j}^{a},{\widetilde{x}}_{i}^{b}, {\widetilde{x}}_{j}^{b}\) are given by (21). Because of the symmetry we denote
Then the solution of the above first-order condition is
By normalizing (\(v^{a}=1\)) we obtain
Since the expression \((v^{b})^{2}+v^{b}-2\) is negative for all \(0<v^{b}<1,\) the difference \({\widetilde{x}}^{a}-x^{a}\) is always positive. Furthermore,
Since the expression \((v^{b})^{2}-3v^{b}+2\) is positive for all \(0<v^{b}<1,\) the difference \(x^{a}-{\widetilde{x}}^{b}\) is always positive.
Now we examine the conditions under which the budget constraint is nonrestrictive. If the restrictions are nonrestrictive we have
Since \(x^{a}+{\widetilde{x}}^{a}>x^{a}+{\widetilde{x}}^{b}\) we obtain that the constraints are nonrestrictive iff \(w>x^{a}+{\widetilde{x}}^{a}.\) Thus, the condition that implies nonrestrictive budget constraints is
\(\square \)
1.2 The proof of Proposition 2
We proved in Proposition 1 that if \(v^{a}\) is normalized to be 1, the budget constraint is nonrestrictive if
In this case the total effort in the first stage of the contest is
and the total effort in the second stage of the contest is
The difference between the total efforts in both stages when the budget constraint is nonrestrictive is
Since \(v^{b}<v^{a}=1\) (decreasing marginal values) this difference is negative and therefore \(TE_{1}<TE_{2}.\) \(\square \)
1.3 The proof of Proposition 3
If the budget constraint is nonrestrictive both of the restrictions in the maximization problem (6) are nonrestrictive such that
The first-order conditions of the maximization problems in the second stage (4) and (5) are
Because of the symmetry we denote
The solution of the above two first-order conditions is
The first-order condition of the maximization problem in the first stage (Eq. (6)) is
where \({\widetilde{x}}_{i}^{a},{\widetilde{x}}_{j}^{a},{\widetilde{x}}_{i}^{b},{\widetilde{x}}_{j}^{b}\) are given by (22). Because of the symmetry we denote
Then the solution of the above first-order condition is
By using the normalization (\(v^{a}=1\)) we obtain
Since \(0<v^{b}<1,\) the difference \({\widetilde{x}}^{a}-{\widetilde{x}}^{b}\) is always positive. Furthermore,
Since the expression \(-2(v^{b})^{2}+v^{b}+1\) is positive for all \(0<v^{b}<1,\) the difference \(x^{b}-{\widetilde{x}}^{b}\) is always positive. We also have
Since the expression \(2(v^{b})^{2}-3v^{b}+1\) is positive for all \(0<v^{b}<0.5 \) and negative for all \(0.5<v^{b}<1\) we obtain that the difference \(x^{b}-{\widetilde{x}}^{a}\) is positive for all \(0<v^{b}<0.5\) and is negative for all \(0.5<v^{b}<1.\) The relations between a player’s allocations of effort is therefore
Now we examine the conditions under which the budget constraint is nonrestrictive. If the restrictions are nonrestrictive we have
Since \(x^{b}+{\widetilde{x}}^{a}>x^{b}+{\widetilde{x}}^{b}\) we obtain that the constraints are nonrestrictive iff \(w>x^{b}+{\widetilde{x}}^{a}.\) Thus, the condition that implies nonrestrictive budget constraints is
\(\square \)
1.4 The proof of Proposition 4
We proved in Proposition 3 that if \(v^{a}\) is normalized to be 1, the budget constraint is nonrestrictive if
In this case the total effort in the first stage of the contest is
and the total effort in the second stage of the contest is
The difference between the total efforts in both stages when the budget constraint is nonrestrictive is
Since \(v^{b}<v^{a}=1\) (increasing marginal values) this difference is positive and therefore \(TE_{1}>TE_{2}.\) \(\square \)
1.5 The proof of Proposition 5
(1) Assume first that the players have decreasing marginal values. If the budget constraint is restrictive both of the restrictions in the maximization problem (3) are restrictive such that
Player \(i\)’s maximization problem in the first stage is then
Therefore the first-order condition is
Because of the symmetry we denote
Then, the solution of the above first-order condition is:
and then by our assumption
In the second stage, player \(i\)’s maximization problems are given by (1) and (2). The first-order conditions of these maximization problems are
In order that both constraints will be restrictive these first-order conditions of the maximization problems in the second stage should be positive. Thus, both constraints are restrictive iff
In this case the total effort in the first stage of the contest is
and the total effort in the second stage is
Therefore
(2) Assume now that the players have increasing marginal values. When the budget constraint is restrictive both of the restrictions in the maximization problem (6) are restrictive such that
Player \(i\)’s maximization problem in the first stage is then
The first-order condition is
Because of the symmetry we denote
The solution of the above first-order condition is
and then by our assumption
In the second stage, player \(i\)’s maximization problems are given by (4) and (5). The first-order conditions of these maximization problems are
In order that both constraints will be restrictive these first order conditions of the maximization problems in the second stage should be positive. Thus, both constraints are restrictive iff
In this case the total effort in the first stage of the contest is
and the total effort in the second stage is
Therefore
\(\square \)
1.6 The proof of Proposition 6
If the budget constraint is partially restrictive only the second restriction in the maximization problem (3) is restrictive such that
Thus, if player \(i\) does not win in the first stage his effort in the second stage is \({\widetilde{x}}_{i}^{a}=w-x_{i}^{a}.\) If, on the other hand, he wins in the first stage his maximization problem in the second stage is
The first-order condition of this maximization problem is
Player \(i\)’s maximization problem in the first stage is then
Therefore the first-order condition is
Because of the symmetry we denote
The solution of the first-order conditions (when \(v^{a}=1\)) from both stages (24) and (23) implies that the equilibrium effort in the first stage \(x^{a}\) is determined by the following equation
where
According to Propositions 1 and 5, the budget constraint is partially restrictive iff
\(\square \)
1.7 The proof of Proposition 7
If the budget constraint is partially restrictive only the first restriction in the maximization problem (6) is restrictive such that
Thus if player \(i\) wins in the first stage his effort in the second stage is \({\widetilde{x}}_{i}^{a}=w-x_{i}^{b}.\) If, on the other hand, he does not win in the first stage his maximization problem in the second stage is
The first order of this maximization problem is
Player \(i\)’s maximization problem in the first stage is then
The first-order-condition is
Because of the symmetry we denote
The solution of the first-order conditions (when \(v^{a}=1\)) from both stages (26) and (25) implies that the equilibrium effort in the first stage \(x^{b}\) is determined by the following equation
where
According to Propositions 3 and 5 the budget constraint is partially restrictive iff
\(\square \)
1.8 The proof of Theorem 1
If the budget constraint is restrictive all of the four restrictions in the maximization problems (13) and (14) are restrictive such that
Thus we denote
Then the first-order condition of player 1’s maximization problem (15) is
Similarly, the first-order condition of player 2’s maximization problem (16) is
Thus, it can be verified that the solution of the above first-order conditions (27) and (28) is
The budget constraints are restrictive if the first-order conditions of the maximization problems in the second stage (9), (10), (11) and (12) are positive. Thus,
This happens when
In this case the total effort in the first stage of the contest is
The total effort in the second stage of the contest is
Therefore
\(\square \)
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Megidish, R., Sela, A. Sequential contests with synergy and budget constraints. Soc Choice Welf 42, 215–243 (2014). https://doi.org/10.1007/s00355-013-0723-5
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DOI: https://doi.org/10.1007/s00355-013-0723-5