the all-pay auction when a committee awards the prize

The All-Pay Auction When a Committee Awards the PrizeAuthor(s): J. Atsu AmegashieSource: Public Choice, Vol. 116, No. 1/2 (Jul., 2003), pp. 79-90Published by: SpringerStable URL: http://www.jstor.org/stable/30025869 .

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Public Choice 116: 79-90, 2003. © 2003 Kluwer Academic Publishers. Printed in the Netherlands.

The all-pay auction when a committee awards the prize *

J. ATSU AMEGASHIE Department of Economics, Simon Fraser University, Burnaby BC, Canada V5A 1S6, U.S.A.; e-mail: jamegash @ sfu. ca

Accepted 31 July 2002

Abstract. There is very little work on the equilibrium of the all-pay auction when the prize is awarded by two or more people. I consider an all-pay auction under committee administration with caps on the bids of the contestants. I show that for any number of committee members and contestants, there exists a pure-strategy equilibrium in which the contestants bid an amount equal to a suitably chosen cap. I argue that the cap is not an artificial restriction on the game, given that there are caps on political lobbying in the real world. I find that committee administration could result in higher aggregate expenditures, even if there is some probability that the committee will not award the prize. The intuition for this result is that the inclusion of additional administrators relaxes the effect of caps on lobbying. That is, caps on lobbying tend to be more effective the smaller is the size of the committee. Caps may also be a solution to the problem of majoritarian cycles in all-pay auctions under committee administration.

1. Introduction

The all-pay auction has been used to examine many allocation processes and interesting economic phenomena. Some examples include strategic trade policy (Konrad, 2000), and lobbying (Baye, Kovenock, and de Vries, 1993, 1996; Che and Gale, 1998; Ellingsen, 1991; Hillman and Riley, 1989).

Baye, Kovenock, and de Vries (1996) have characterized the equilibria for the all-pay auction with complete information when there is a single prize at stake. Clark and Riis (1998) consider the case of multiple prizes. Che and Gale (1997, 1998) examine the case of a single prize but with a cap on the bids of the lobbyists or contestants. In all these cases and in the papers cited above, it is assumed that the decision to award the prize is made by a single administrator or politician. Congleton (1984) considers the all-pay auction when a committee has to award the prize. However, he is unable to find the equilibrium because of the existence of majoritarian cycles.' Given that committees are widely used in many situations (for example, in the US con- gress) and that people do lobby committee members, it would be interesting to examine the nature of equilibria in the all-pay auction under committee

* My thanks are due to Doug Allen and Gordon Myers for comments. I also thank Kai

Konrad for his encouragement.



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administration. For example, Hall and Wayman (1990) find a significant relationship between campaign contributions and efforts of committee members in the US House of Representatives to influence legislation at the committee stage. Also, Romer and Synder (1994) find a significant relationship between committee assignments and political action committee (PAC) contributions.2 They find that more contributions are targeted towards more senior members.3

In this paper, I provide a solution to the all-pay auction under committee administration. I show that for any number of committee members and contestants, there exists a pure-strategy equilibrium for a suitably chosen lobbying cap. Indeed, one of the contributions of this paper is to show that there exists equilibria to the all-pay auction under committee administration if caps are placed on the bids of contestants. I find that committee administration could result in higher expenditures compared to expenditures under a single administrator. I also find that this result holds even if there is some probability that the committee will not award the prize as a result of its inability to reach a consensus. The intuition for this result is that the inclusion of additional administrators relaxes the effect of caps on lobbying. That is, caps on lobbying tend to be more effective, the fewer the number of administrators responsible for awarding the prize. I argue that caps may be a solution to the problem of majoritarian cycles in all-pay auctions under committee administration.

The paper is organized as follows: the next section examines all-pay auctions under committee administration. Some applications are considered in a sub-section. Section 3 concludes the paper.

2. All-pay auctions under committee administration

Consider two identical and risk-neutral agents contesting for a prize com- monly known to be valued at V > 0. Suppose that the prize will be awarded by a committee of two members (call them A and B).

Suppose the contest is an all-pay auction. In an all-pay auction, the contestant with the highest bid wins with certainty and ties are broken randomly; losers forfeit their bids. A contestant wins if both committee members vote for him. This occurs if he spends more on each of the two committee members than the other contestants. If the committee members do not vote for the same contestant, then a random mechanism is used to pick the winner.

It is a standard result that the all-pay auction with complete information and exogenous valuations has no equilibrium in pure strategies (Hillman and Riley, 1989; Ellingsen, 1991; Baye, Kovenock, and de Vries, 1996).4 How- ever, it is known that there exists an equilibrium in mixed-strategies with or without a cap on bids, where contestants randomize their bids on some finite support. It is easy to show that if the prize is awarded by a committee, there



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is no equilibrium in pure strategies if there are no caps on bids. The proof is omitted because it is very similar to the proof for a single administrator.

In the case of a three-member committee, Congleton (1984) shows that not only is there no equilibrium in pure strategies but also there is a de-escalation of aggregate bids. Suppose, for the sake of argument, that player 1 knows that player 2 intends to allocate his resources among the three committee members as (100, 100, 100). Then player 1 can dominate player 2's strategy by allocating a smaller overall amount, say 202, in the proportion (101, 101, 0). However, player 2 can respond with a strategy, say (0, 102, 1). But player 1 can respond with a strategy, say (1, 0, 2). Using these adjustments, Congleton (1984) concludes that under committee administration, aggregate lobbying expenditures will tend to de-escalate. Note that although the aggregate bid de- escalates it will never get to zero since (0, 0, 0) cannot be a Nash equilibrium. But then when the aggregate bids de-escalate to some positive lower bound the bids will now have to escalate, since each player placing the minimum bid on any one, two or all three committee members cannot be a Nash equilibrium. Hence the players' strategy of targeting majoritarian coalitions results in cycling.5 The de-escalation of aggregate bids under committee administration led Congleton (1984) to conclude that aggregate expenditures will tend to be lower under committee administration than expenditures under a single administrator. I re-consider Congleton's conclusion in this paper.6

2.1. All-pay auctions under committee administration with lobbying caps

Suppose that both contestants face a common cap on their bids, b > 0. This cap is the maximum lobbying expenditure that can be targeted at a given committee member. For the cap to be binding we require b < V. Thus we require that each contestant's bid (per committee member) is not greater than b. Note that placing a cap on the bids is not an artificial restriction on the structure of game. For example, as noted by Che and Gale (1998: 643), in the USA "[p]olitical action committees can contribute at most $5000 per election to a candidate, while individuals can contribute at most $1000." I shall later argue that caps may also be a solution to the problem of majoritarian cycles in all-pay auctions under committee administration.

I state the following lemma:

Lemma 1: In an all-pay auction with a two-member committee responsible for awarding the prize and two contestants with common valuation, V, there exists an equilibrium in pure strategies, in which both contestants bid b on each of the committee members, if there is a cap b = V/4 on the contestants' bid (per committee member).



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Proof. see appendix.

Consider the all-pay auction above but with three committee members (A, B, and C) and three contestants (1, 2 and 3). The winner is determined by simple majority voting and if the members vote for three different contestants, a random mechanism picks the winner.

The following lemma holds.

Lemma 2: In an all-pay auction with a three-member committee responsible for awarding the prize and three contestants with common valuation, V, there exists an equilibrium in pure strategies, in which both contestants bid b on each of the committee members, if there is a cap b = V/9 on the contestants' bid (per committee member).

Proof: see appendix.

Given lemmas 1 and 2 one is inclined to conjecture that with an M-member committee and N contestants, there exists a pure strategy equilibrium in which the contestants bid V/N2 on each committee member if the cap is V/N2 and N = M. Thus I state the following proposition:

Proposition 1: In an all-pay auction with an M-member committee responsible for awarding the prize and N (= M) contestants with common valuation, V, there exists an equilibrium in pure strategies, in which all N contestants bid b on each of the committee members, if there is a cap b = V/N2 on the contestants' bid (per committee member), where N > 2.

Proof: The proof of this proposition is difficult since there are several possible deviations which have to be considered. In what follows, I offer an intuitive proof in the hope that future work will lead to a more rigorous proof. Indeed, I have confirmed the reasoning below for N = M E {2, 3, 4) (see appendix for N = M = 2, 3).

Recall that N = M. Suppose a contestant decides to deviate by bidding V/N2 on all the committee members except one member, given that all the other N - 1 contestants are bidding V/N2 on each committee member. If this contestant wants to deviate by bidding less than V/N2 on one committee member, the optimal deviation is to bid zero on this member. Hence he reduces his total bid by V/N2. This committee member will not vote for him. His expected value of the prize prior to deviating was V/N. With this deviation, his expected value of the prize falls to V(1/N - K) > 0, where



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K > 0. I shall now argue that K > 1/N2 which means that his expected value of the prize falls by more than V/N2. Consider M > 3.7 In the equilibrium in proposition 1, his probability of success is 1/N and he treats all the M members identically. Thus one could argue that the contribution of each member to his success probability is (1/M)(1/N) = 1/N2 , given M = N. However, if one committee member no longer votes for him, this committee member reduces his success probability by more than 1/N2. This is because when this member no longer votes for him, he essentially withdraws "his contribution" of 1/N2 but also exerts a negative effect on the contribution of all the other M-1 members because there are some instances where he votes with them to collectively contribute to this contestant's success probability.8 Hence if this committee member no longer votes for this deviating contestant, his success probability reduces by more than 1/N2.9 Thus when he deviates, his expected value of the prize reduces by more than V/N2 but his cost reduces by only V/N2. Therefore it is not profitable for him to deviate by bidding less than V/N2 on one committee member. A similar argument also shows that it is not profitable to deviate by bidding less than V/N2 on more than one committee member, given that the other N-1 contestants are bidding V/N2 on each of the M members.Q.E.D.

It is obvious that I could have directly stated the above proposition without first stating lemmas 1 and 2. I stated and proved both lemmas because they lend credence to the proof of the proposition.

Suppose there are M committee members and N > 2 contestants. Then we can state the following proposition:

Proposition la: In an all-pay auction with an M-member committee responsible for awarding the prize and N(< M) contestants with common valuation, V, there exists an equilibrium in pure strategies, in which all N contestants bid b on each of the committee members, if there is a cap b < V/MN on the contestants' bid (per committee member).

Proof: In this symmetric equilibrium each player has a success probability 1/N. As in the proof of proposition 1, we can view each committee member as contributing (1/M)(1/N) = 1/MN to this probability. A player who deviates reduces his bid by no more than V/MN but his expected value of the prize reduces by more than V/MN. Hence no player has a profitable deviation.

Q.E.D.

Given the proof of proposition 1, the following more general proposition follows.



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Proposition 2: In an all-pay auction with an M-member committee responsible for awarding the prize and N contestants with valuations, V1 > V2 > VN-1 > VN, there exists an equilibrium in pure strategies, in which all N contestants bid b on each of the committee members, if there is a cap b < VN/MN on the contestants' bid (per committee member).

Proof: Obvious from the proofs of propositions 1 and la.

In equilibrium the contestant with valuation VN has a zero expected payoff. All the other contestants have a positive expected payoff. Indeed, we can state the following corollary:

Corollary: In an all-pay auction with an M-member committee responsible for awarding the prize and N contestants with valuations, V1 > V2 > V3 VN1 > VN, the largest cap that can sustain a pure- strategy equilibrium is V2/2M. In equilibrium, only the top two contestants with valuations, V1 and V2, bid positive amounts equal to the cap on each member. The rest bid zero on each of the committee members.

Proof: Caps greater than V2/2M cannot sustain pure-strategy equilibria. If each contestant bids the cap on each of the committee members, then each contestant, with the possible exception of the contestant with the highest valuation, has a negative expected payoff. So only the contestant with the highest valuation will bid a positive amount. But then this contestant will not bid the cap; he is better off bidding a small but positive amount. However, if he did this the other contestants will now bid a positive amount. But since bidding below the cap cannot be a pure-strategy equilibrium and bidding the cap is also not an equilibrium, it follows that there is no pure-strategy equilibrium if the cap is greater than V2/2M. Q.E.D.

2.2. Discussion of results and some applications

For the sake of exposition, consider the case of non-identical contestants. Note that decreasing the number of committee members does not affect the equilibrium, holding the cap fixed. However, with fewer committee members, aggregate expenditures fall. Hence the smaller the number of committee members, the lower are aggregate lobbying expenditures. Also, if we are in an equilibrium where each contestant has a positive expected payoff, then small increases in the number of members increases aggregate expenditures. In- deed, aggregate expenditures are higher under committee administration than expenditures under a single administrator. I suggest the following intuition for



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this result. When a cap binds under a single administrator, then the contestants would like to spend more if their expected payoff is positive in equilibrium. The introduction of another administrator provides an additional avenue for lobbying expenditures. However, large increases in the number of members could result in a fall in aggregate expenditures because some contestants with very low valuations will drop out of the contest; given that the cap is fixed and the increase in committee members requires an increase in expenditures per contestant, their expected payoff may be negative. This means that the aggregate expenditures could be lower with more committee members. So while the possible de-escalation of aggregate bids observed by Congleton (1984) in the case of a three-member committee (or more generally in committees with M members) might give one cause to believe that aggregate expenditures will tend to be lower under committee administration, the fact that committee administration relaxes the effect of lobbying caps implies that the relative magnitude of expenditures under committee administration and one-person administration is a priori indeterminate. Since small increases in committee members increases aggregate expenditures but large increases reduce aggregate expenditures, it follows that there is a number of committee members which maximizes aggregate expenditures. If lobbying expenditures are bribes, then committee size will be chosen to maximize the total bribe per committee member for a given cap.

Committee administration may lead to higher aggregate expenditures even if there is some probability that the committee will not award the prize if it unable to reach a majority decision. Without loss of generality consider a two- member committee, with two identical contestants, and a cap of V/8. With a single administrator, the probability that the prize will not be awarded is zero. There exists a pure-strategy equilibrium in which the contestants each bid V/8, with an expected payoff of (1/2)V - V/8 > 0 and aggregate expenditure of V/4. Under committee administration and a unanimity rule, there exists a pure-strategy equilibrium in which the contestants bid V/8 on each of the committee members resulting in an expected payoff of (1/2)(1/2)V - V/8 - V/8 = 0 and aggregate expenditure of V/2.10 Hence aggregate lobbying expenditures under committee administration could be higher than expenditures under one-person administration, even if there is some probability that the committee will not award the prize." Again the intuition for this result is due to the observation that committees relax the effect of lobbying caps. If this "relaxation" effect is sufficiently strong, then it will outweigh the effect of contestants reducing their bids stemming from a lower probability of the rent

being awarded. Consider proposition 2. Suppose VN-1 > VN. Let the cap be VN-1/MN.

Then aggregate expenditure is [(N - 1)/N]VN-1, since all the contestants bid



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the cap on each of the committee members but the contestant with valuation VN bids zero on each member. Now suppose there is a tighter cap equal to VN/MN. Then aggregate expenditure is VN. Now VN > [(N - 1)/N]VN-1, if VN-1 - VN is sufficiently small. Hence tighter lobbying caps could paradoxically result in higher aggregate expenditures. Che and Gale (1998) obtained this result in the case of a single administrator. The above analysis shows that this result also holds for committees. The intuition for this result is the same as in Che and Gale (1998): tighter lobbying caps encourage the participation of low-valuation contestants. If the effect of the increase in participation is strong enough to outweigh the effect of the fall in the cap, then aggregate expenditures will increase.

Consider M > 3. This case results in majoritarian cycles as noted in section 1. Suppose lobbying expenditures take the form of bribes. Then the members of the committee have the incentive to solve this problem instead exposing themselves to the situation of not knowing what the equilibrium of the game will be (i.e., cycles). One solution will be to collude, give one member the sole right of determining the winner and hence collecting the bribes. They will then agree to share the income from bribes equally. What this does is reduce the committee all-pay auction to a single-administrator all- pay auction (whose solution is knownl2). However, collusion will be harder to sustain given that the designated member might ex post renege on the agreed- upon distribution of the surplus. Lobbying caps may be used to get round this problem by solving the problem of majoritarian cycles in all-pay auctions under committee administration. The committee members may use caps to implement a given equilibrium instead of facing the situation of not knowing what the actual equilibrium will be (i.e., cycling) if there are no caps.

There is also the issue of what the optimal cap should be. In the case of identical contestants (i.e., proposition 1), a cap of V/N2 exhausts the entire surplus and therefore maximizes aggregate expenditures. If lobbying expenditures take the form of bribes or cash, it would seem that caps will be chosen to extract surpluses from contestants.13 In the case of non-identical contestants, the optimal cap might be such that all contestants will be included in the game or it might exclude some contestants with very low valuations.

3. Concluding remarks

The paper provides a solution to the all-pay auction under committee administration with caps on the bids of contestants. For any number of contestants and administrators, I show that there exists a pure-strategy equilibrium for a suitably chosen cap.

Another result is that committee administration could result in higher aggregate lobbying expenditures than expenditures under one-person admin-



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istration, if there is a cap on the bid of contestants. This is because the inclusion of additional administrators relaxes the effect of lobbying caps. Indeed, committee administration may result in higher aggregate expenditures compared to expenditures under one-person administration, even if there is a probability that the committee will not award the prize as result of its inability to reach a consensus. The paper also shows that tighter lobbying caps may paradoxically increase aggregate lobbying expenditures. Finally, I argue that caps may be a solution to the problem of majoritarian cycles in all-pay auctions under committee administration.

At this point let me indicate how my paper is similar to and differs from the work of Van Cayseele, Deneckere, and de Vries (1993). They consider an all-pay auction with a single administrator but with identical contestants com- peting over multiple dimensions. The number of dimensions in their model is equivalent to the number of committee members in my model. Van Cayseele et al. (1993) do not focus on any pure-strategy equilibria. Furthermore, they are not interested in how the number of dimensions (the size of the committee) or the size of lobbying caps affect aggregate expenditures. Finally, Van Cayseele et al. (1993) do not provide a solution for the case of M > 2 dimensions (i.e., M committee members).14

It is obvious that this paper has not provided a complete characterization of equilibria in all-pay auctions under committee administration. It is hoped that future research effort will be directed towards this issue given that it may have potentially useful applications.

Notes

1. Congleton (1984) does not use the phrase "all-pay auction". 2. See note 2 in Cite and Gale (1998). 3. This is consistent with Amegashie (2000). He finds that the stronger the voting power

of a committee member (i.e., the more influential), the higher is the rent-seeking effort targeted at this member.

4. However, see Amegashie (2001) for an all-pay auction with endogenous valuations and a pure-strategy equilibrium. Che and Gale (1998) also find that there exists a pure-strategy equilibrium if there are caps on bids.

5. See also Myerson (1993). 6. In the case of the all-pay auction, Congleton (1984) reaches this conclusion based on

the argument that aggregate expenditures, in the limit, is equal to 2V under a single administrator. It is not clear why this limiting aggregate expenditure is relevant since it is not a Nash equilibrium. In such an "equilibrium", the players' expected payoff is negative.

7. I shall later come to the case of M = 2. 8. In this case, voting is like team production where there is some positive synergy among

the members of the team such that their aggregate output, if they work separately is smaller than their aggregate output if they work as a team. In team production with positive synergy, if M identical workers produce an aggregate output of y, then when one worker



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is no longer a member of the team, output will fall by a disproportionate amount; that is, the fall in output will be greater than y/M.

9. This can easily be checked for the case of M = 3 above. I have also confirmed it for M = 4. In the case of M = 2, his success probability reduces by exactly, 1/N2. Hence his expected value of the prize falls by exactly V/N2. Thus there is also no profitable deviation for M = 2.

10. The proof is straight forward. Obviously bidding less than V/8 on each committee member, given that the other contestant bids V/8 on each member is not an equilibrium. Also bidding less than V/8 on one committee member and V/8 on the other member, given that the other contestants bids V/8 on each member is not an equilibrium, since one member will certainly not vote for the contestant bidding less than V/8 on him. Given that unanimity is required the probability of winning the contest is zero for the contestant who bids less than V/8 on one member. Also, this contestant has no incentive to deviate to a bid of zero on each committee member.

11. A similar result is obtained in Amegashie (2000) in a lottery model with three committee members.

12. My conjecture is that there is no equilibrium, even in mixed strategies, for M > 3, if there are no caps. In the case of M = 2, there is a mixed-strategy equilibrium with no caps (see Van Cayseele et al., 1993).

13. If caps are chosen optimally, then it may not be necessary to look for equilibria with different levels of the cap.

14. Myerson (1993) considers a model with multiple candidates contesting for political office. Each voter votes for the candidate who offers him/her a higher payoff based on the candidate's campaign promise. However, Myerson's model is not an all-pay auction because the candidates do not incur the cost of lobbying; the campaign promises are not costly to the contestants.

15. Note that since he bids less than b on one of the committee members, this member will not vote for him given that the other player bids b. However, there is a probability of 1/2 that the other committee member will vote for him. If this is the case, then the committee members have voted for different contestants. So with probability 1/2 a random process will break the tie in his favor. Thus his probability of winning the prize is (1/2)(1/2).

16. See Che and Gale (1998) for a similar result when one person awards the prize. 17. Without loss of generality, suppose this player bids V/9 each on A and B. He bids less than

V/9 on C, so C will not vote for him. Suppose this player is player 1. With probability (1/3) A will vote for him and with the same probability B will vote for him. So he has a probability of (1/3)(1/3) of winning in this case. But there are other ways that he could win the prize. With probability (1/3) A will vote for him; with probability (1/3) B votes for player 2 and with probability (1/2) C votes for player 3. Since the members have voted for three different contestants, there is a probability of (1/3) that he will win. Or A votes for him with probability (1/3), B votes for player 3 with probability (1/3) and C votes for player 2 with probability (1/2), and then he wins the random draw with probability (1/3). Hence player 1 has a probability, 2(1/3)(1/3)(1/3)(1/2) of winning the prize in this case where A votes for him but B and C vote for different contestants. Similarly, he has a probability of 2(1/3)(1/3)(1/3)(1/2) of winning, when B votes for him but A and C vote for different contestants. Hence he has a probability, [(1/3)(1/3) + 2(1/3)(1/3)(1/3)(1/2) + 2(1/3)(1/3)(1/3)(1/2)] of winning the prize.



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References

Amegashie, J.A. (2002). Committees and rent-seeking effort under probabilistic voting. Public Choice 112: 345-350.

Amegashie, J.A. (2001). An all-pay auction with a pure-strategy equilibrium. Economics Letters 70: 79-82.

Baye, M.R., Kovenock, D. and de Vries, C.G. (1993). Rigging the lobbying process: An application of the all-pay auction. American Economic Review 83: 289-294.

Baye, M.R., Kovenock, D. and de Vries, C.G. (1996). The all-pay auction with complete information. Economic Theory 8: 291-305.

Che, Y.K. and Gale, I. (1997). Rent dissipation when rent seekers are budget constrained. Public Choice 92: 109-126.

Che, Y.K. and Gale, I. (1998). Caps on political lobbying. American Economic Review 88: 643-651.

Clark, D.J. and Riis, C. (1998). Competition over more than one prize. American Economic Review 88: 276-289.

Congleton, R.D. (1984). Committees and rent-seeking effort. Journal of Public Economics 25: 197-209.

Ellingsen, T. (1991). Strategic buyers and the social cost of monopoly. American Economic Review 81: 648-657.

Hall, R.L. and Wayman, EW (1990). Buying time: Moneyed interests and mobilization of bias in congressional committees. American Political Science Review 84: 797-820.

Hillman, A.L. and Riley, J. (1989). Politically contestable rents and transfers. Economics and Politics 1: 17-39.

Konrad, K.A. (2000). Trade contests. Journal of International Economics 51: 317-334. Myerson, R.B. (1993). Incentives to cultivate favored minorities under alternative electoral

systems. American Political Science Review 87: 856-869. Romer, T. and Synder, J.M., Jr. (1994). An empirical investigation of the dynamics of PAC

contributions. American Political Science Review 38: 745-769. Van Cayseele, P.J., Deneckere, R.J. and de Vries, C.G. (1993). A nontrivial multidimensional

all-pay tournament. Unpublished manuscript.



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Appendix

Proof of Lemma 1

Suppose player 1 bids b each on A and B. Then if player 2 also bids b each on A and B, both players get an expected payoff of V/2 - 2b = 0. None of the

players has the incentive to deviate from this equilibrium by bidding less than b on each of the committee members (given that the other player is bidding b on each committee member) since that guarantees that they will lose with certainty resulting in a nonpositive payoff. Also, no one has the incentive to deviate by bidding b on one committee member and less than b (i.e., b - e' > 0) on the other, given that the other player is bidding b on each committee member. If he did this, his expected payoff, given that the other player is bidding b each on both members, will be

(1/2)(1/2)V - (b + b - e') < 0, given b = V/4 and 0 < b - e' < b.15 Hence if b = V/4, then there is a pure-strategy Nash equilibrium in which both players bid b on each of the committee members.16 Q.E.D.

Proof of Lemma 2

If each of the three contestants bids V/9 on each committee member, then each contestants gets an expected payoff of zero. None of the contestants has an incentive to deviate from this equilibrium. Suppose two contestants bid V/9 on each of the members, then the third contestant has no incentive to deviate and bid less than V/9 on each of the members since he will get a non-positive payoff from doing so. It is not optimal for him to bid V/9 on one member and less than V/9 on each of the other two members, since he will lose the contest with certainty by doing so. The other strategy is to bid V/9 on each of two members and less than V/9 on one member. If he did this, his payoff will be [(1/3)(1/3) + 2(1/3)(1/3)(1/3)(1/2) +

2(1/3)(1/3)(1/3)(1/2)] V - [V/9 + V/9 + (V/9 - e)]<0,17 given that the other two contestants are bidding V/9 on each of the three members (where 0 < e < V/9). It is also easy to show that a joint deviation by any two contestants where they both bid V/9 each on two members but less than V/9 on one member is not profitable.

Q.E.D.



the all-pay auction when a committee awards the prize

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