All-Pay Auctions with Risk-Averse Players

Transcription

1 All-Pay Auctions with Risk-Aerse Players Gadi Fibich Arieh Gaious Aner Sela December 17th, 2005 Abstract We study independent priate-alue all-pay auctions with risk-aerse players. We show that: 1) Players with low alues bid lower and players with high alues bid higher than they would bid in the risk neutral case. 2) Players with low alues bid lower and players with high alues bid higher than they would bid in a first-price auction. 3) Players expected utilities in an all-pay auction are lower than in a firstprice auction. We also use perturbation analysis to calculate explicit approximations of the equilibrium strategies of risk-aerse players and the seller s expected reenue. In particular, we show that in all-pay auctions the seller s expected payoff in the risk-aerse case may be either higher or lower than in the risk neutral case. Keywords and Phrases: Priate-alue auctions, Risk aersion, Perturbation analysis. JEL Classification Numbers: D44, D72, D82. School of Mathematical Sciences, Tel Ai Uniersity, Tel Ai 69978, Israel, fibich@math.tau.ac.il Department of Industrial Engineering and Management, Faculty of Engineering Sciences, Ben-Gurion Uniersity, P.O. Box 653, Beer-Shea 84105, Israel, ariehg@bgumail.bgu.ac.il Department of Economics, Ben-Gurion Uniersity, P.O. Box 653, Beer-Shea 84105, Israel, anersela@bgu.ac.il 1

2 1 Introduction Consider n players who compete for a single item. Eery player submits a bid and the player with the highest bid receies the item. All players bear a cost of bidding which is an increasing function of their bids. This setup, which is called an all-pay auction, is commonly used to model applications such as job-promotion competitions, R&D competitions, political campaigns, political lobbying, sport competitions, etc. The literature on contests and particularly on all-pay auctions has dealt mostly with risk-neutral players. 1 In contrast to all-pay auctions, seeral studies on the classical auction mechanisms (first-price and second-price auctions) with risk-aerse players appear in the literature. In independent priate-alue second-price auctions, risk aersion has no effect on a player s optimal strategy which remains to bid her own aluation for the object. In independent priate-alue first-price auctions, on the other hand, risk aersion makes players bid more aggressiely (Maskin and Riley, 1984). Since the (risk-neutral) seller is indifferent to the first-price and second-price auctions when players are risk neutral, 2 she prefers the first-price auction to the second-price auction when players are risk aerse. Howeer, the seller s preference relations for auction mechanisms with risk-aerse players do not imply anything about the players preference relations for these auctions, since under risk 1 All-pay auctions with linear cost functions and incomplete information about the players alues include, among others: Weber (1985), Hillman and Riley (1989), Krishna and Morgan (1997), Kaplan et al. (2002). All-pay auctions with complete information about the players alues include, among others: Tullock (1980), Dasgupta (1986), Dixit (1987), Baye et al. (1993, 1996). 2 This follows from the Reenue Equialence Theorem (Vickrey (1961), Myerson (1981), and Riley and Samuelson (1981)). 2

3 aersion the combined reenue of the seller and the players is not a constant. Indeed, Matthews (1987) showed that risk aerse players with constant absolute risk aersion are indifferent to first and second-price auctions, and that players prefer the first-price auction if they hae increasing absolute risk aersion and the second price auction if they hae decreasing absolute risk aersion. 3 In this paper we analyze the role of risk aersion in all-pay auctions by comparing the situation where all players are risk neutral (henceforth referred to as the status quo), with the case where players are risk-aerse. In Section 2 we show that a risk-aerse player with a low aluation bids less aggressiely than in the status quo situation. On the other hand, a risk-aerse player with a high aluation bids more aggressiely than in the status quo. This behaior can be explained as follows. When a player s alue is small, she is most likely to lose. Therefore, as she becomes more risk aerse, she is willing to pay less, that is, she bids less aggressiely. On the other hand, when a player s alue is ery high, she is afraid of losing the object, therefore, she bids more aggressiely. These results are consistent with the experimental studies of Barut et al. (2002) and Noussair and Siler (2005), who obsered that players in single-unit and multiple-unit all pay auctions with low alues tend to bid below the risk-neutral equilibrium, and those with large alues tend to bid aboe the risk-neutral equilibrium. We can learn much about the all-pay auction with risk aerse players by comparing it to the first-price auction. Although the first-price auction is a classical auction whereas the all-pay auction is a contest, these models are similar since in both the highest player 3 This result was generalized by Monderer and Tennenholtz (2000) to all k-price auctions. 3

4 wins for sure and pays her bid. Intuitiely, one can expect that as in the risk-neutral case, the equilibrium bids of risk aerse players in all-pay auctions should be lower than in firstprice auctions. We show that, indeed, in all-pay auctions, low types bid less aggressiely than they bid in first-price auctions. Howeer, high types bid more aggressiely in all-pay auctions than they bid in first-price auctions. In light of the aboe comparison of the players bids in first-price auctions and all-pay auctions, it is not clear in which auction the player s expected utility is larger. Neertheless, we show that, independent of the distribution of the players aluations and the number of players, the expected utility of a risk-aerse player in the first-price auction is always larger than in the all-pay auction. Consequently, a risk-aerse player will prefer the first-price auction to the all-pay auction. Rigorous analysis of all-pay auctions with risk-aerse players is limited since usually explicit expressions for the equilibrium strategies with risk-aerse players cannot be obtained. In order to oercome this difficulty, in Section 3 we consider the case of weakly risk-aerse players. The presence of the small risk-aersion parameter allows us to employ perturbation analysis, one of the most powerful tools in applied mathematics, to calculate explicit approximations of the equilibrium strategies of risk-aerse players and the seller s expected reenue. 4 The high accuracy of the explicit approximations of the equilibrium bids is illustrated by an example with two weakly risk-aerse players. We show that een when the risk-aersion parameter is not small, the agreement between the explicit 4 Fibich and Gaious (2003) and Fibich, Gaious and Sela (2004) employed perturbation analysis to study asymmetric auctions. 4

5 approximations obtained by the perturbation analysis and the exact alues obtained by numerical analysis is quite remarkable. 5 The approximate solutions in Section 3 show, for example, that risk aersion can lead to an increase, as well as a decrease, in the seller s expected reenue in all pay auctions. In addition, they shows that, roughly speaking, weak risk aersion leads to a larger departure from reenue equialence than weak asymmetry. Altogether, the combination of the quantitatie results of risk-aersion in all-pay auctions, together with the qualitatie results gien in Section 2 proide a clear picture of the behaior of risk-aerse players in all-pay auctions. Remark: For clarity, the proofs are delegated to the Appendices and presented in the order in which they are proed. The results in Section 2 are presented in a different order in which they are proed in the Appendices, in order to better present the economic results. 2 All-pay auctions with risk-aerse players Consider n players that compete to acquire a single object in an all-pay auction. The aluation of each player for the object is independently distributed according to a distribution function F () on the interal [, ], where 0. Each player submits a bid b and pays her bid regardless of whether she wins or not, but only the highest player wins the object. Each player s utility is gien by a function U( b) which is twice 5 This is, more often than not, the case in perturbation analysis (see, e.g., Bender and Orszag, 1978). 5

6 continuously differentiable, monotonically increasing, normalized such that U(0) = 0, and satisfies U < 0 (i.e., risk-aerse players). Gien that the equilibrium bid function b() is monotonically increasing, we can define the equilibrium inerse bid function = (b). The maximization problem of player i with aluation is gien by maxv (,b) =F n 1 ((b))u( b)+(1 F n 1 ((b)))u( b). b Differentiating with respect to b gies the first-order condition 0= V b = (n 1)F n 2 ((b))f((b)) (b)[u( b) U( b)] F n 1 ((b))[u ( b) U ( b)] U ( b). Therefore, the inerse bid function satisfies the ordinary differential equation (b) = (1) F ((b))[u ( b) U ( b)] (n 1)f((b))[U( b) U( b)] + U ( b) (n 1)F n 2 ((b))f((b))[u( b) U( b)], subject to the initial condition (0) =. Equation (1) is exact in the risk-neutral case U(x) =c x where c is a constant. In that case, its solution is gien by b all rn () =F n 1 () F n 1 (s) ds. (2) For comparison, the equilibrium bid in a first price auction with risk-neutral bidders is gien by b 1st rn () = 1 F n 1 () ball rn (). (3) 6

7 Therefore, it immediately follows that b all rn () <b1st rn (), <<. (4) Although there are no explicit solutions of equation (1) for a general utility function U, we can derie some qualitatie results by comparing the equilibrium bids in the risk-aerse and the risk-neutral cases. These results are in the spirit of the ones obtained by Maskin and Riley (1984), who showed that in a first-price auction the equilibrium bid of a riskaerse player is higher than the equilibrium bid of a risk-neutral player with the same type, that is, b 1st rn () <b 1st (), <. (5) We also show how relation (4) is affected by risk aersion, by comparing the bids of risk-aerse bidders in all-pay auctions with the ones in first-price auctions, denoted by b 1st (). We first show that risk aersion affects low type players to bid less aggressiely: Proposition 1 In an all-pay auction the equilibrium bid of a risk-aerse player with low type is smaller than the equilibrium bid of a risk-neutral player with the same type, i.e., b all () <b all rn (), 0 < 1. (6) Proof. See Appendix B. The following result shows that risk-aersion affects high type players and low type players quite differently: 7

8 Proposition 2 In an all-pay auction, the equilibrium bid of a risk-aerse player with high type is higher than the equilibrium bid of a risk-neutral player with the same type, i.e., b all () >b all rn(), 0 1. (7) Proof. From equation (3) it follows that Similarly, from equation (5) it follows that b all rn( ) =b 1st rn ( ). (8) b 1st rn ( ) <b 1st ( ). (9) By Proposition 4, the equilibrium bid of a risk-aerse player with type in an all-pay auction is larger than in a first-price auction, that is, b 1st ( ) <b all ( ). (10) The combination of the three inequalities (8)+(9)+(10) completes the proof. Since in an all-pay auction a player pays her bid regardless of whether she wins, whereas in a first-price auction she pays only if she wins, it seems natural that players will bid more carefully (i.e., hae lower bids) in all-pay auctions than in first-price auctions. Indeed, the bid of a risk-neutral player in an all-pay auction is smaller than her bid in a first-price auction, see equation (5), and we can expect this relation to be een stronger for riskaerse players. Howeer, as Propositions 3 and 4 show, the relation of bids in first-price and all-pay auctions with risk-aerse players is more complex: Proposition 3 The equilibrium bid of a risk-aerse player with sufficiently low type in an all-pay auction is smaller than her bid in a first-price auction. 8

9 Proof. From Proposition 1, equation (3), and equation (9) we hae that b all () <b all rn () <b1st rn () <b1st (). (11) Proposition 4 The equilibrium bid of a risk-aerse player with sufficiently high type in an all-pay auction is larger than in a first-price auction. Proof. See Appendix C. ε = 0.25 ε = bids first price all pay bids first price all pay Figure 1: Bids of risk-aerse players (solid lines) and of risk-neutral players (dashed lines) in all-pay auctions and in first-price auctions. Example 1 Consider two players where each player s aluation is distributed on [0, 1] according to the uniform distribution function F () =. Assume that each player s utility function is U(x) =x εx 2. In Figure 1 we show the equilibrium bids of risk neutral and risk aerse bidders in first-price and all-pay auctions. The results illustrate our finding 9

10 that 1) Players in the all pay auction with low alues bid lower and players with high alues bid higher than they would bid in the risk neutral case. 2) Players in the all pay auction with low alues bid lower and players with high alues bid higher than they would bid in a first-price auction. Propositions 3 and 4 show that there is no dominance relation among the bids in firstprice and all-pay auctions. Neertheless, first-price auctions dominate all-pay auctions from the player s point of iew: 6 Proposition 5 The expected utility of a risk-aerse player with type < in the first-price auction is larger than her expected payoff in the all-pay auction. Proof. See Appendix A. 3 All-pay auctions with weakly risk-aerse players The results of the preious section leae many open questions. For example, because of the complex way that risk aersion affects the equilibrium bids, it is not clear whether, oerall, risk aersion leads to an increase or a decrease in the seller s expected reenue. In addition, the tools that we used in the preious section, which are standard in auction theory, typically proide qualitatie results (e.g., which of two possibilities is larger), but do not gie a quantitatie estimate (e.g., by how much). 6 Eso and White (2001, 2004) proed this property for DARA buyers. 10

11 In order to address such questions, we consider the case of weak risk aersion, 7 i.e., U x. This is the case, for example, for players with a constant absolute risk aersion (CARA) utility function U(x) =[1 exp( ɛx)]/ɛ, or for players with constant relatie risk aersion (CRRA) utility function U(x) =x 1 ɛ,if0<ɛ 1. Therefore, in general, the utility function of weakly risk-aerse players can be written as U(x) =x + εu(x)+o(ε 2 ), ε 1. (12) Thus, ε is the risk aersion parameter and ε 1 implies weak risk aersion. Note that u(0) = 0 and u < 0. On the other hand, u can be either positie or negatie (gien that u (x) > 1 ) since in either case U = x + ɛu is monotonically increasing. ε The existence of a small risk aersion parameter enables us to use perturbation methods to calculate explicit approximations to the bidding strategies: Proposition 6 The symmetric equilibrium bid function in an all-pay auction with weakly risk-aerse players is gien by b all () =b all rn ()+εball 1 ()+O(ε2 ), where b all rn() is the equilibrium bid in the risk-neutral case (2), and b all 1 () = u( ball [ ] rn ()) + F n 1 () u( b all rn ()) u( ball rn ()) F n 1 (s)u (s b all rn(s)) ds. (13) 7 The assumption of weak risk aersion is quite reasonable. Indeed, while most people would prefer to receie $500 dollar with probability 1 rather than $1000 with probability 1/2, much fewer would prefer receiing $300 dollar with probability 1 rather than $1000 with probability 1/2. 11

12 Proof. See Appendix D. We thus found an explicit expression for εb all 1 (), i.e., the leading-order effect of riskaersion on the equilibrium strategy. Roughly speaking, for a 10% leel risk aersion, we calculated the corresponding 10% change in the equilibrium strategy with 1% accuracy. ε = 0.25 ε = 0.5 bids bids Figure 2: Bids of risk-aerse buyers (solid lines) and their explicit approximation (equation (14), dotted lines) in all-pay auctions. Example 2 The results of our perturbation analysis can be illustrated by the following example. Consider two players where each player s aluation is distributed on [0, 1] according to the distribution function F () = α. Assume that each player s utility function is U(x) =x εx 2. From Proposition 6 the equilibrium bid function in the all-pay auction is gien by b all () = α ( 1+α 1+α + ε α 2+α 2+α + α 1+α 2+2α ) + O(ε 2 ). (14) 12

13 In Figure 2 we compare the approximation (14) with the exact bid functions (i.e., the numerical solutions of equation (1)), for the case α =1. Atɛ =0.25, the approximations are almost indistinguishable from the exact bids. Although when ɛ = 0.5 the risk-aersion parameter is not small, 8 the agreement between the explicit approximations and the exact alues is quite remarkable. 9 In addition for proiding quantitatie predictions for the equilibrium bids, the explicit approximations obtained in Proposition 6 can be used to approximate the seller s expected reenue under risk aersion: Proposition 7 In an all-pay auction with weakly risk-aerse players, the seller s expected reenue is gien by R all = R rn + (15) { [ εn F n 1 ()u( b all rn()) + ( 1 F n 1 () ) ] u( b all rn()) f() d } F n 1 ()(1 F ())u ( b all rn()) d where R rn is the expected reenue in the risk-neutral case. + O(ε 2 ), Proof. See Appendix E. As we hae said, unlike first price auctions, the effect of risk aersion of the seller s reenue in all pay auctions is not obious, since it lowers the bids for low alues but increases the 8 In fact, ɛ =0.5 is the largest possible alue of ɛ for which U = x ɛx 2 is monotonically increasing. 9 Such good agreement was also obsered in numerous other comparisons that we made with different distribution functions and utility functions. 13

14 bids for large alues. Indeed, the result of Proposition 7 shows that risk-aersion can lead to an increase, as well as to a decrease, of the seller s expected reenue in all-pay auctions: Example 3 Consider n =2risk aerse players with distribution functions F ()= α in [0, 1], such that U(x) =x ɛx 2. Substituting u(x) = x 2 in (15) and integrating gies R all = R rn + ɛ R + O(ɛ 2 ), R = (2 α) α 2 (2 + 5 α +3α 2 )(α +2). We thus see that depending on the alue of α, R can be either positie or negatie. Hence, we conclude that risk-aersion can lead to an increase, as well as to a decrease, of the seller s expected reenue in all-pay auctions. An immediate, yet important consequence from Proposition 7 is as follows: Proposition 8 An O(ɛ) risk aersion leads to an O(ɛ) difference in the seller s reenue among different auction mechanisms. Proof. Since risk-aersion does not affect the reenue in a second price auction, the result follows from Proposition 7. In Fibich et al. (2004) we showed that if ɛ is the leel of asymmetry among the distribution functions of the players aluations, then weak asymmetry only leads to an O(ɛ 2 ) difference in the seller s reenue among different auction mechanisms. Hence, Proposition 8 shows that, roughly speaking, weak risk aersion leads to a larger reenue differences among different auction mechanisms than weak asymmetry. We can also use the explicit expression obtained in Proposition 6 to analyze the effect of weak risk aersion on the players expected utility. 14

15 Proposition 9 The expected utility of a weakly risk aerse player with type in an all-pay auction is gien by V all () =V rn ()+ε F n 1 (s)u (s b all rn (s)) ds + O(ε2 ), where V rn () = F n 1 () d is the expected utility in the risk-neutral case. Proof: See Appendix F. Note that the difference between the expected payoffs of a weakly risk-aerse player and a risk-neutral player does not depend on the alue of u, but depends on the alue of u. That is, if the utility function of a risk-aerse player U(x) always larger or equal than the utility function of a risk-neutral player U rn (x) =x, it does not necessarily imply that the expected utility of the risk-aerse player is larger than the expected utility of the risk-neutral player. A natural question that arises is whether in the case of weak-risk aersion one cannot simply approximate the bidding functions using the risk-neutral expressions. In other words, when ɛ is small, is there an adantage for the approximation b all (; ɛ) b all rn()+ɛb all 1 () oer the continuous approximation b all (; ɛ) b all (; ɛ =0)=b all rn ()? The answer is that the accuracy of the first approximation is O(ɛ 2 ), whereas that of the second approximation is only O(ɛ). Therefore, the first approximation is significantly more accurate when ɛ is moderately small (but not negligible). Indeed, comparison of Figures 2 and 1 shows that the (exact) bids in the risk-aerse case are well-approximated with the explicit approximation that we deried, but are not well-approximated with the bids in the risk-neutral case. 15

16 A Proof of proposition 5 When all players follow their equilibrium bidding strategies, a player s expected utility gien that his type is and that he plays as if his type is t is V all (t ) = F n 1 (t)u( b all (t)) + (1 F n 1 (t))u( b all (t)), (16) V 1st (t ) = F n 1 (t)u( b 1st (t)), for all-pay and first-price auctions, respectiely. Let V all () =V all ( ) and V 1st () = V 1st ( ). By a standard argument, in equilibrium Therefore, V j (t ) t =0, j = all, 1st. (17) t= ( V j () ) = F n 1 ()U ( b j ()), j = all, 1st. (18) In addition, V all () =V 1st (), since in both auctions the lowest type expects a zero utility. We proe by negation. Assume that for some type, <<, we hae V all () V 1st (). Then, by (16) it follows that b 1st () >b all (). From the concaity of U it follows that U ( b all ()) <U ( b 1st ()). Thus ( V all () ) < (V 1st ()). Let y = V all V 1st. Then, y() = 0, and for <<, y() 0 implies that y () < 0. Therefore, it follows that y<0 for <<. To complete the proof, we now proe that y( ) =V all ( ) V 1st ( ) < 0. Since y() < 0 for <<, we only need to proe that it is not possible to hae y( ) = 0. Assume, therefore, by negation that y( ) = 0. We will show that this implies that y ( ) = 0 and 16

17 y ( ) > 0, which is in contradiction with the fact that y<0 for <<. Indeed, y( ) =0 = U( b all ( )) = U( b 1st ( )) = b 1st ( ) =b all ( ) = ( V all) ( ) = ( V 1st ) ( ) = y ( ) =0. In addition, substituting t = = in (17) gies that Therefore, by (18), ( ) b all ( ( ) b 1st ) (n 1)f( ) ( ) = U ( b) U( ball ( )) > 0. y ( ) = ( V all) ( ) [ ( ) V 1st (b ( ) = U ( b all all ( )) ) ( ) ] ( ) b 1st ( ) > 0. B Proof of Proposition 1 Since b all () =b all rn() = 0, we can proe the result by showing that (b all ) () < (b all rn ) (), 0 < 1. (19) Let us first note that (17) implies that ( b all () ) =(n 1)F n 2 U( b all ()) U( b all ()) ()f() F n 1 ()U ( b all ()) + (1 F n 1 ())U ( b all ()). In particular, in the case of risk neutrality (b all rn ) () =(n 1)F n 1 ()f(). Therefore, (b all ) () (b all rn ) () (n 1)F n 1 ()f() = U( b all ()) U( b all ()). (20) F n 1 ()U ( b all ()) + (1 F n 1 ())U ( b all ()) 17

18 Let us begin with the case when > 0. Since b all () = 0, then [ ] (b all ) () (b all rn ) () U() U(0) U (x) (n 1)F n 1 ()f() = = = U (0) U (0) 1, where 0 <x<. By the concaity of U, U (x) U (0) To proe (19) when = 0, we first expand, < 1. Therefore, we proed (19) for > 0. U( b all ()) = U( b all ()) + U ( b all ()) U ( b all ()) + O( 3 ), U ( b all ()) = U ( b all ()) + U ( b all ()) + O( 2 ). Therefore, = = = U( b all ()) U( b all ()) F n 1 ()U ( b all ()) + (1 F n 1 ())U ( b all ()) = U ( b all ()) U ( b all ()) + O( 3 ) U ( b all ()) + F n 1 ()[U ( b all ()) + O( 2 )] = U ( b all ()) + U ( b all ()) O(3 ) + U ( b all ()) O(2 ) = 1+F n 1 () U ( b all ()) [ 1+ U ( b all ()) 2 = + 2U ( b all ()) U ( b all ()) ][ U ( b all ()) + O(2 ) 1 F n 1 () U ( b all ()) [ ] 1 2 F n 1 () + O( 3 ). ] U ( b all ()) + O(2 ) Therefore, by (20), ( ) (b all ) () (b all rn) () (n 1)F n 1 ()f() = 2U ( b all ()) 1 U ( b all ()) 2 F n 1 () + O( 3 ) < 0. C Proof of Proposition 4 By Proposition 5, the expected utility of a risk-aerse player with type in the firstprice auction is larger than her expected payoff in the all-pay auction (V all ( ) >V 1st ( )). 18

19 Since V j ( ) =U( b j ( )) for j = all, 1st, see equation (16), and since U is monotonically increasing, the result follows. D Proof of Proposition 6 We can write the equilibrium bid as (b) = rn (b)+ε 1 (b)+o(ε 2 ), where rn (b) is the inerse function of the risk-neural equilibrium strategy in all-pay auctions (2). For clarity, we drop the superscript all. We first note that when ε 1, F ((b)) = F ( rn )+ε 1 F ( rn )+O(ε 2 ), f((b)) = f( rn )+ε 1 f ( rn )+O(ε 2 ), U((b) b) U( b) = (b)+ε[u((b) b) u( b)] = rn (b)+ε[ 1 (b)+u( rn (b) b) u( b)] + O(ε 2 ), U ((b) b) U ( b) = ε[u ((b) b) u ( b)] = ε[u ( rn (b) b) u ( b)] + O(ε 2 ). Substitution in (1) and expanding in a power series in ε, the equation for the O(1) term is identical to the one in the risk-neutral case and therefore is automatically satisfied. The equation for the O(ε) terms is 1(b) = F ( rn(b))[u ( rn (b) b) u ( b)] u ( b) + (n 1)f( rn (b)) rn (b) (n 1)F n 2 ( rn (b))f( rn (b)) rn (b) (n 2) 1 (b) (n 1)F n 1 ( rn (b)) rn (b) 1 f ( rn (b)) (n 1)F n 2 ( rn (b))f 2 ( rn (b)) rn (b) [ 1(b)+u( rn (b) b) u( b)] (n 1)F n 2 ( rn (b))f( rn (b))rn 2 (b), 19

20 subject to 1 (0) = 0. Since, by (1), rn(b) = the equation for 1(b) can be rewritten as 1 (n 1)F n 2 ( rn (b))f( rn (b)) rn (b), (21) 1 (b)+ 1(b)B(b) =G(b) (22) where B(b) = [ rn (b) rn (b) + f ( rn (b)) f( rn (b)) rn (b)+(n 2) f( ] rn(b)) F ( rn (b)) rn (b), and G(b) = rn(b) { [ ] u( rn (b) b) u( b) (n 1)F n 2 ( rn (b))f( rn (b)) rn(b) (23) ( ) } + F n 1 ( rn (b)) u ( rn (b) b) u ( b) + u ( b). The solution of (22) is gien by 1 (b) =e brn b B ( C 1 brn where b rn = b rn ( ). It is easy to erify that (see (21)) e brn b B = rn(b) rn( b rn ). b ) G(x)e brn x B dx, Thus, as b 0, rn (b) and e brn b B. Therefore it follows that C 1 = b rn G(x)e brn x 0 B dx and that b 1 (b) = rn(b) G(x)/ rn(x) dx. 0 In addition, we note that if we differentiate the identity = (b(; ε); ε) with respect to ε and set ε = 0, we get that 1 (b rn ()) + rn(b rn ())b 1 () =0orb 1 () = 1 / rn(b). Thus, 20

21 we get that b 1 () = brn() 0 G(x)/ rn(x) dx. Substitution of G from (23) gies b 1 () = brn() 0 { [ u( rn (b) b) u( b) ] (F n 1 ( rn (b)) ( ) } F n 1 ( rn (b)) u ( rn (b) b) u ( b) u ( b) db. A few more technical calculations completes the proof. E Proof of Proposition 7 The seller s reenue is gien by R all = n b(s)f(s) ds. Substituting b = b rn + εb 1 + O(ε 2 ), we hae R all = n (b rn + εb 1 ) f(s)ds + O(ε 2 )=n b rn f(s)ds + εn b 1 f(s)ds + O(ε 2 ) = R rn + εn Substituting b 1 from (13) yields b 1 f(s) ds + O(ε 2 ). b 1 f(s) ds = (1 F n 1 ())u( b rn ())f() d + [ ] F n 1 (s)u (s b rn (s)) ds f() d. F n 1 ()u( b rn ())f() d Integrating by parts the double integral gies [ ] F n 1 (s)u (s b rn (s)) ds f() d = F n 1 ()(1 F ())u ( b rn ()) d. Therefore, the result follows. 21

22 F Proof of Proposition 9 The expected utility for a player with type in all-pay auctions in equilibrium is gien by V all () =F n 1 ()U( b()) + [1 F n 1 ()]U( b()). In the case of weak risk aersion (12), [ ( ) ] V all () =F n 1 () b()+ε F n 1 () u( b()) u( b()) + u( b()) + O(ε 2 ). Using the relation b() =b rn ()+εb 1 ()+O(ε 2 ), we hae [ ( ) ]} V all () =Vrn {b all () ε 1 () F n 1 () u( b rn ()) u( b rn ()) + u( b rn ()) +O(ε 2 ). By the reenue equialence theorem, V all rn () =V rn () = F n 1 (s) ds is independent of the auction mechanism. Substituting (13) in the last equation yields the result. References [1] Barut, Y., Koenock, D., Noussair, C.: A Comparison of Multiple-Unit All-Pay and Winner-Pay Auctions Under Incomplete Information. International Economic Reiew 43(3), (2002) [2] Baye, M., Koenock, D., de Vries, C.: Rigging the lobbying process. American Economic Reiew 83, (1993) [3] Baye, M., Koenock, D., de Vries, C.: The all-pay auction with complete information. Economic Theory 8, (1996) 22

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