The Optimal Allocation of Prizes in Contests

Transcription

1 The Optial Allocation of Prizes in Contests Benny Moldovanu and Aner Sela July 4, 999 Abstract Westudyacontestwithultiple(notnecessarilyequal)prizes. Contestants have private inforation about an ability paraeter that a ects their costs of bidding. The contestant with the highest bid wins the rst prize, the contestant with the second-highest bid wins the second prize, and so on until all the prizes are allocated. All contestants incur their respective costs of bidding. The contest s designer axiizes the expected su of bids. Our ain results are: ) We display bidding equlibria for any nuber of contestants having linear, convex or concave cost functions, and for any distribution of abilities. ) If the cost functions are linear or concave, then, no atter what the distribution of abilities is, it is optial for the designer to allocate the entire prize su to a single rst prize. 3) We give a necessary and su cient conditions ensuring that several prizes are optial if contestants have a convex cost function.. Introduction In 90 Francis Galton posed the following proble: A certain su, say 00, is available for two prizes to be awarded at a forthcoing copetition; the larger one for the rst of the copetitors, the saller one for the second. How should the 00 be ost suitably divided between the two? What ratio should a rst prize bear to that of a second one? Does it depend on the nuber of copetitors, and if so, why? Galton (Bioetrika, Vol., 90) We wish to thank Karsten Fieseler, Martin Hellwig, Roan Inderst, Philippe Jehiel, Holger Müller, Georg Nöldeke and Jan Vleugels for helpful coents. Both authors are grateful for nancial support fro the SFB 504 at the University of Mannhei. Moldovanu: Econoics, University of Mannhei, Seinargebaeude A5, 683 Mannhei. old@pool.uni-annhei.de. Sela: Econoics, Ben-Gurion Universiy, P.O.B. 653, Beer-Sheva 8405, Israel.

2 In his article Galton proposes a ratio of 3 to to the above question. Since Galton does not explicitly state what is the contest designer s goal, his answer is soewhat arbitrary. Nevertheless, his work is iportant for it pioneered both the scienti c literature on contests and the use of order statistics (The ratio 3 to appears as the ratio of expected di erences involving the rst three order statistics for a large nuber of contestants whose abilities are norally distributed.) Many econoic, social and biological situations are contests where agents spend resources in order to increase to probability of winning a prize. Several well known exaples are: rent-seeking and lobbying inside organizations; R&D rivalry; sport copetitions; ars races and other econoic or biological ghts and wars of attrition ; job prootion in labor arkets; political capaigns; artistic copetitions such as piano or architectural contests. Given the wealth of exaples, it is not surprising that the econoic literature on contests is very large. Most of the literature has focused on the case of one prize. Models with coplete inforation about the value of the prize include: Tullock (980), Varian (980), Rosen (986), Hillan and Saet (987), Baye et al (996). The last paper o ers a coplete characterization of equilibriu behavior. Models with incoplete inforation about the prize s value to different contestants include Weber (985), Hillan and Riley (989), Aann and Leininger (996), and Krishna and Morgan (997). Clark and Riis (998) study contests with ultiple identical prizes under coplete inforation and focus on siultaneous versus sequential designs. Barut and Kovenock (998) and Glazer and Hassin (988) allow for non-identical prizes in a coplete inforation odel. The latter paper also includes a result for a odel with incoplete inforation (see below). The use of contests in order to extract e ort under oral hazard conditions has been ephasized by, aong others, Lazear and Rosen (98), Green and Stokey (98), and Nalebu and Stiglitz (983). A coon assuption in these papers is that (observed) output is a stochastic function of the unobservable e ort. All agents have the sae (known) ability. Lazear and Rosen derive the optial prize structure in a contest with two workers and two prizes (so that, in fact, only the di erence in prizes atters) and copare it to optial piece rates. They also brie y discuss a odel with heterogenous workers and the e ciency properties of having contests where such workers ix. In this paper we address Galton s proble in the following fraework: Several agents engage in a contest where ultiple prizes with known and coon values are awarded. Each contestant i subits a bid (or undertakes an observable e ort ): The contestant with the highest bid wins the rst prize, the contestant with the second-highest bid wins the second prize, and so on until all the prizes are allocated. All contestants (including those that did not win any prize) incur

3 a cost that is a strictly increasing function of their bid. This function is coon knowledge. We di erentiate aong the cases where the cost function is, respectively, linear, concave or convex in e ort. The cost function of contestant i also depends on a paraeter (say ability ) that is private inforation to that player. The ain assuption we ake is one of separability between ability and bid in the cost function. Abilities are drawn independently of each other according to an arbitrary distribution function which is coon knowledge. We assue that each contestant chooses his bid in order to axiize expected utility (given the other copetitors bids and given the values of the di erent prizes). The goal of the contest designer is to axiize the total expected e ort (i.e, the expected su of the bids) at the contest. This ts well in the tournaent literature. Good exaples are sport copetitions, prootions in organizations, architectural contests and the grading of exas. The designer can deterine the nuber of prizes (i.e., the nuber of prizes having positive value) and the distribution of the xed total prize su aong the di erent prizes. It is worth to note here the theoretical distinction between having one prize (or several equal prizes) and having several unequal prizes. If there is only one prize, or if there are several equal prizes, each contestant perceives two payo -relevant alternatives: I win a prize, or I win nothing. Hence, bids will be deterined by one variable - the di erence in expected payo between those two alternatives. (Note that the sae logic applies if there are two unequal prizes but only two contestants). In contrast, if there are at least two unequal prizes and at least three contestants, each contestant perceives at least three payo relevant alternatives (I win the rst prize, I win the second prize,..., I win nothing). Bids will be deterined in a ore coplex way by several variables - the di erences in expected payo aong the various alternatives. We display bidding equilibria for any nuber of prizes and contestants, for any distribution of abilities, and for linear, concave or convex cost functions. In order to have a less technical exposition, we focus however on the designer s proble in the case where she can award two (potentially unequal) prizes, and where there are at least three contestants. As explained above, this case already displays the ain ingredient for coplexity in bidding. Moreover, it will becoe clear that none of our qualitative results changes if we allow for ore than two prizes. In soe odels, the designer has other goals. For exaple, in a lobying odel the contest designer ight not be the bene ciary of the wasteful lobying activities, and she ight wish to iniize the. Our analysis can be easily extended to other goal functions since we explicitly display bidding equilibria. The coplexity resebles the one appearing in ulti-object auctions. 3

4 Our answers to Galton s questions are as follows: ) We show that for any nuber of contestants having linear or concave cost functions, and for any distribution of abilities, it is optial for the designer to allocate the entire prize su to a single rst prize. ) We give a necessary and su cient conditions ensuring that (at least) two prizes are optial if the contestants have convex cost functions. Depending on the paraeters, the optial prize structure ay involve then several equal prizes or di erent prizes whose ratio can be easily coputed. Several related results appear in Barut and Kovenock (998) and Glazer and Hassin (988). Barut and Kovenock study a ulti-prize contest where players have linear cost functions and have the sae ability. In this syetric, coplete inforation environent, they show that the optial prize structure allows any cobination of K prizes, where K is the nuber of contestants. (In particular, allocating the entire prize su to a unique rst prize is optial.) Our result for linear cost functions (that allows for private inforation, asyetries aong contestants and is independent of the nuber of contestants) suggests that the only robust optial prize structure is the one involving a unique rst prize. Besides studying the syetric equilibria of a coplete inforation odel as above, Glazer and Hassin (988) also propose an incoplete inforation odel that is ore general than ours since it allows for both cost functions that are not necessarily separable in ability and bid and for a concave prize-valuation function. But they are not able to copute equilibria, and their indirect results (based on several iplicit and unproven assuptions) only deal with the case of a separable and linear cost function, a linear prize-valuation function and a unifor ability distribution such that the lowest ability type has an in nite cost of bidding. With these assuptions, they show that a unique rst prize is optial. We now want to describe the intuition behind our ain results. The equilibriu bid function for contestants with linear cost functions is a function of the values of the di erent prizes and of the ability of that contestant. If there are p prizes, this function involves the distribution of the rst p order-statistics. Moreover, the equilibriu bid is increasing in ability. A nice property of the di erential equation arising fro the rst-order condition of the contestants axiization proble allows us to easily copute the equilibriu bid for contestants with general (i.e., concave or convex) cost functions: this is siply equal to the inverse of the cost function applied to the equilibriu bid for linear cost functions. To siplify the exposition, we assue for the following descriptive details that the contestants have linear cost functions. The arginal e ect of the rst prize on the equilibriu bid function is positive for all possible abilities (or types ). Since a player with higher ability has a higher chance to win the rst prize, the arginal e ect of the rst price is an increasing function of ability. 4

5 The arginal e ect of the second-prize is abiguous. Consider, for exaple, a player with the highest possible ability. If such a player is present at the contest, he is sure to win the rst prize. Subtracting a penny fro the rst prize and adding it to the second prize lowers the expected utility of such a player (since he never gets the second prize), and he will consequently bid less. Siilarly, the arginal e ect of the second prize on the equilibriu bid function is negative for players with high enough abilities (and hence this e ect is lower that the arginal e ect of the rst prize for such types). The arginal e ect of the second price is, however, positive for iddle and low ability players. Moreover, for contestants with abilities below a certain threshold, the (positive) arginal e ect of the second prize is higher than the arginal e ect of the rst prize, since these types of player have a higher chance to get the second-prize. Consider now the contest designer: She has to allocate a xed prize su aong the two prizes, and she wants to axiize the average (i.e., expected) bid of each contestant. The relevant variable for the designer is then the average di erence between the arginal e ects of the second prize and the rst prize, respectively. This average di erence is precisely the arginal e ect of the second prize on the designer s revenue (i.e., the change in the designer s revenue if she subtracts a penny fro the value of the rst prize and adds it to the value of the secondprize). If the average arginal e ect of the rst prize is higher than the average arginal e ect of the second-prize then the average di erence is negative, the designer s goal function is a decreasing function of the value of the second-prize, and the designer should award only a unique ( rst) prize - this turns out to be the case for contestants with linear cost functions. To obtain the arginal e ect of the second prize on the designer s revenue for contestants with concave cost functions, one ultiplies the above di erence by a function that is increasing in ability. Hence the ters where the rst-prize is doinant (corresponding to high ability types) get agni ed in relation to the ters where the second prize is doinant (low ability types). Given the result for the linear cost functions, the designer s revenue is also a decreasing function of the value of the second-prize, and the designer should award only a unique ( rst) prize. The opposite happens for convex cost functions: the ters where the second prize is doinant get agni ed, and two prizes ay be optial (since, by de nition, the designer cannot award only a second prize). In such cases, the optial allocation of the prize su aong the two prizes depends on the nuber of contestants, the distribution of abilities, and the e ort cost function. The paper is organized as follows: In Section we present the contest odel with ultiple prizes and private inforation about a paraeter (e.g. ability) entering cost functions. In Section 3 we focus on linear cost functions. We derive the equilibriu bid functions and we forulate the contest designer s proble. 5

6 We then prove several properties that characterize the arginal e ects of the rstand the second-prize on each type of contestant. Finally, we use these properties to prove that it is always optial to award a single prize. In Section 4 we use the result obtained above in order to study the optial prize structure for contestants with concave and convex cost functions. Finally, we illustrate the (non-trivial) optial prize structure in an exaple with convex cost functions. In Section 5 we gather several concluding coents.. The Model Consider a contest where p prizes are awarded. The value of the j th prize is V j,wherev V ::: V p. The values of the prizes are coon knowledge. We assue that P p i= V i =- this is just a noralization. The set of contestants is K = f; ; :::; kg. Without loss of generality we can assue that k p (i.e., there are at least as any contestants as there are prizes). At the contest each player i akes a bid x i : Bids are subitted siultaneously. Abidx i causes a disutility (or cost) denoted by c i (x i ); where : R +! R + is a strictly increasing function with (0) = 0; and where c i > 0 is an ability paraeter 3.Notethatalow c i eans that i has a high ability(i.e.,lowercost) and vice-versa. The ability (or type) of contestant i is private inforation to i: Abilities are drawn independently of each other fro an interval [; ] accordingtothedis- tribution function F ( ); which is coon knowledge. We assue that F ( ) has a continuous density F 0 ( ) > 0: In order to avoid in nite bids caused by zero costs, we assue that ; the type with highest possible ability, is strictly positive 4. The contestant with the highest bid wins the rst prize V. The contestant with the second highest bid wins the second prize V, and so on until all the prizes are allocated 5. That is, the payo of contestant i who has ability c i,and subits a bid x i is either V j c i (x i ) if i wins prize j; or c i (x i ) if i does not win a prize. Each contestant i chooses his bid in order to axiize expected utility (given the other copetitors bids and the values of the di erent prizes.) The contest designer deterines the nuber of prizes having positive value and the distribution of the total prize su aong the di erent prizes in order to 3 The treatent of the case in which i 0 s cost function is given by ±(c i ) (x i ),where±( ) is strictly onotone increasing, is copletely analogous. The ain assuption here is the separability of ability and bid. 4 The case where =0can be treated as well, but requires slightly di erent ethods. Thechoiceoftheinterval[; ] is a noralization. 5 If h> bids tie for a prize, each respective bidder gets the prize with probability h : 6

7 axiize the expected value of the su of the bids P k i= x i (given the contestants equilibriu bid functions). 3. Linear Cost Functions In this Section we assue that the cost functions are linear, i.e., (x) =x: As we shall see below, both the equilibriu bid functions and the optial prize structure in this case will be very iportant for the derivations in the other cases. The next Proposition displays the equilibriu bid when there are two prizes. As entioned in the introduction, if there are only two contestants, the situation is isoorphic to the one where there is a unique prize whose value is equal to the di erence between the two prizes. Hence, it is trivially true that awarding a unique prize is optial for the contest s designer. Therefore, we assue below that the nuber of contestants is at least three, i.e., k 3. In the Appendix we also provide the general forula for the bid functions with p> prizes. Proposition 3.. In a syetric equilibriu, the bid function of each contestant is given by b(c) =A(c)V + B(c)V where: A(c) =(k ) c a ( F (a))k F 0 (a)da (3.) B(c) =(k ) c a ( F (a))k 3 [((k )F (a) ]F 0 (a)da (3.) Proof. See Appendix 3.. The Designer s Proble Let V = and V =, where0 (since the second prize ust be saller than the rst). By Proposition 3., each contestant s equilibriu bid function is given by b(c) =( )A(c) + B(c) =A(c) + (B(c) A(c)): The average bid of each contestant is given by (A(c)+ (B(c) A(c)))F 0 (c)dc: Since there are k contestants, the seller s proble is: ax k (A(c)+ (B(c) A(c)))F 0 (c)dc 0 7

8 The above proble is equivalent to: ax (B(c) A(c)))F 0 (c)dc (3.3) 0 The solution to Proble 3.3 is extreely siple: if the integral is positive, then the optial is (i.e., award two equal prizes). Otherwise, the optial is zero (i.e., award a unique prize). The next Proposition shows that the integral appearing in Proble 3.3 is negative. We also list several other properties of the arginal e ects, A(c) and B(c); thatareusedintheproofandinotherpartsofthepaper. Proposition 3.. Let b(c) =A(c)V + B(c)V be the syetric equilibriu bid function for contestants having linear cost functions. Then the following properties hold:. A() = B() = 0. 8c [; ); A(c) > 0; and A 0 (c) < 0 3. Let c be such that F (c ) = k : Then B0 (c ) = 0; B 0 (c) > 0 for all c [; c ),andb 0 (c) < 0 for all c (c ; ] 4. j B 0 (c) j>j A 0 (c) j for c in a neighborhood of : 5. B() < 0 6. For any k>;there exists a unique point c 6=such that A(c )=B(c ): 7. R (B(c) A(c))F 0 (c)dc < 0 Proof. See Appendix. Proposition 3.-7 shows that the solution to Proble 3.3 ust be =0 6 : Hence we have obtained: Proposition 3.3. For any nuber of contestants with linear cost functions, and for any distribution of abilities in the population, it is optial to allocate the entire prize su to a single rst prize. 6 As noted in the introduction, we can easily deal with the case where the designer wants,say, to iniize the the expected su of bids. In that case it is obvious by the above analysis that two equal prizes ( = ) are optial. 8

9 The result above holds even if, a-priori, the seller is allowed to award ore than two prizes. Indeed, assue that the optial prize structure is (V ;V ; :::; V p ): Let = P p j=3 V j : The arginal e ects of the rst and second prizes on the equilibriu bids do not depend on (this is siilar to the fact that the arginal e ect of the rst prize does not depend on the value of the second-prize and vice versa - see the general equilibriu bid forula in the Appendix). For any xed, Proposition 3.3 shows that V =0: Since V j V for all j>; we obtain that V =and that V j =0for all j : The following exaple illustrates the above results. In particular, we give explicit forulas for the equilibriu bid functions for the case where the distribution of abilities is unifor. Exaple 3.4. Assue that F (c) = c ; i.e., abilities are uniforly distributed on the interval [;]: We obtain that: A(c) =( )k ( k)( k X s= ( c) s s +lnc) B(c) = ( )k (k )[ k X s= ( c) s +lnc s +( c) k k 3 X ( c) s + (k )( +lnc)] s= s Assue now that k =3;= on the interval [ ; ]). The forulas above yield: and F (a) =a (i.e., unifor distribution A(c) = 8+8c 8lnc: B(c) =6 6c +lnc (B(c) A(c))F 0 (c)dc = (4 4c +0lnc)dc = ln = 0: 37 : 9

10 c A(c) - thick line ; B(c) -thinline; (A(c)+B(c)) - dotted line Note that the curve A(c) also describes the equilibriu bid when there is a unique price V =: For coparison, we have also plotted the resulting equilibriu bid function when there are two equal prizes, V = V = : 4. Concave and Convex Cost Functions Assue now that bidder i with ability c has a cost function given by c ( ) such that (0) = 0; 0 ( ) > 0: Let g( ) = ( );and observe that g 0 ( ) > 0: Proposition 4.. In a syetric equilibriu, the bid function of each contestant is given by b(c) =g[a(c)v + B(c)V ] where A(c) and B(c) are de ned by equations 3. and 3., respectively. Proof. See Appendix. 4.. The Designer s Proble Let V = ; and V =,where0 : Analogous to the case of linear cost functions, the designer s proble is given by ax k g[a(c)+ (B(c) A(c))]F 0 (c)dc 0 The designer s revenue as a function of the value of the second prize is: R( ) =k g[a(c)+ (B(c) A(c))]F 0 (c)dc (4.) 0

11 Note that and that: R 0 ( ) =k (B(c) A(c))g 0 [A(c)+ (B(c) A(c))]F 0 (c)dc (4.) R 00 ( ) =k (B(c) A(c)) g 00 [A(c)+ (B(c) A(c))]F 0 (c)dc (4.3) Observe also that and that dg 0 [A(c)+ (B(c) A(c))] dc = g 00 [A(c)+ (B(c) A(c))] (4.4) [( )A 0 (c)+ B 0 (c)] ( )A 0 (c)+ B 0 (c) < 0 (4.5) since this last ter is the derivative of the equilibriu bid function for contestants having linear cost functions. We can now prove the following: Proposition 4.. For any nuber of contestants with concave cost functions, and for any distribution of abilities in the population, it is optial to allocate the entire prize su to a single rst prize. Proof. The cost function of contestant i with ability c, c ( ); has the additional feature that 00 ( ) 0: Hence g 00 ( ) =( ( )) 00 0: By equations 4.4 and 4.5 we obtain that the positive function g 0 [A(c)+ (B(c) A(c))] is decreasing in c. This eans that in the integral de ning R 0 ( ); all negative ters B(c) A(c) (corresponding to c [; c )) are ultiplied by relatively high values of g 0 ( ), while all positive ters B(c) A(c) (corresponding to c (c ; )) are ultiplied by relatively lower values. By Proposition 3.-7, we obtain: R 0 ( ) =k g 0 [A(c)+ (B(c) A(c))](B(c) A(c))F 0 (c)dc < 0 (4.6) Hence, the designer s payo function has a axiu at =0; and a single prize is optial. Proposition 4.3. Consider a contest where contestants have convex cost functions. A necessary and su cient condition for the optiality of two prizes is given by (B(c) A(c))g 0 (A(c))F 0 (c)dc > 0 (4.7)

12 If condition 4.7 is satis ed 7 then it is either optial to award two prizes V = and V =,where > 0 is deterined by the equation R 0 ( )=0,ortoaward two equal prizes, V = V = : Proof. The cost function of contestant i with ability c, c ( ); has the additional feature that 00 ( ) 0: Hence g 00 ( ) =( ( )) 00 0: By equations 4.4 and 4.5 we obtain that the positive function g 0 [A(c) + (B(c) A(c))] is increasing in c. This eans that in the integral de ning R 0 ( ) all negative ters of the for B(c) A(c) (corresponding to c [; c )) are ultiplied by relatively low values of g 0 ( ), while all positive ters B(c) A(c) (corresponding to c (c ; )) are ultiplied by higher values. Moreover, for all [0; ] we have R 00 ( ) =k (B(c) A(c)) g 00 [A(c)+ (B(c) A(c))]F 0 (c)dc 0 If condition 4.7 is satis ed then we have: R 0 (0) = k ((B(c) A(c))g 0 (A(c))F 0 (c)dc > 0 (4.8) Hence, the revenue function R( ) cannot have a axiu at =0: It either has a axiu at such that R 0 ( )=0or at = : For the converse, assue that two prizes are optial. This eans that =0 is not a axiu of R( ). If condition 4.7 is not satis ed we obtain R 0 (0) 0. Together with R 00 ( ) 0 for all [0; ] we obtain a contradiction. Exaple 4.4. Let k =3;= and F (a) =a (i.e., unifor distribution on the interval [ ; ]). Let the cost function be c (x) =cx : We have (x) = g(x) =x and g 0 (x) = x : By the results in Exaple 3.4, we obtain: A(c) = 8+8c 8lnc B(c) = 6 6c +lnc B(c) A(c) = 4 4c +0lnc (B(c) A(c))g 0 (A(c))F 0 (c) = p g 0 (A(c)) = ( 8+8c 8lnc) 6 6c +5lnc dc =0: 9 q( +c ln c) 7 Note that the condition involves only priitives of the odel: the distribution function, the cost function, and the nuber of contestants.

13 c B(c) A(c) - thick line; g 0 (A(c)) -thinline Nuerical calculations reveal that = is (an approxiate) solution to the e equation R 0 ( ) =0: Hence the optial prize structure is V = ¼ 0: 63, and e V = e ¼ 0: 37. The ratio of prizes is V V = e ¼ : 7; and the di erence V V is about one quarter of the prize su. 5. Concluding Coents We have studied the optial prize structure in ulti-prize contests where players have private inforation about their abilities. In order to axiize the expected su of bids, the designer should award a single prize if contestants have linear or concave cost functions. If the contestants have convex cost functions, then two prizes (or ore) ay be optial. The optial proportion between the prizes values depends then on the nuber of contestants, the distribution of abilities in the population, and on the exact for of the cost function. What can we say about Galton s proposal to have a rst prize which is three ties higher than the second prize? Assue that contestants have linear cost functions. It can be easily shown that, when the nuber of contestants goes to in nity (this was the actual case envisaged by Galton), the average di erence between the arginal e ects of the two prizes goes to zero. Hence, the designer becoes indi erent between awarding one prize or two prizes (in any proportion). In particular, Galton s 3:proportion becoes approxiately optial. We see several avenues for future research: ) Cobine the odel of private inforation about abilities with the earlier odels where output depend stochastically on e ort, and only output can be observed. ) Perfor coparative static exercises about the ne e ects of changes in the distribution of abilities or in the 3

14 for of the cost functions when these are convex. 3) Copare siultaneous contests with sequential contests. 4) Conduct epirical studies about the optial prize structure in observed contests, such as architectural copetitions. Francis Galton concluded his article with the following reark: I now coend the subject to atheaticians in the belief that those who are capable, which I a not, of treating it ore thoroughly, ay nd that further investigations will repay trouble in unexpected directions (Galton, 90) The challenge was iediately picked by the faous statistician Karl Pearson, at that tie editor of Bioetrika. His notes at the end of Galton s article contain a coplete solution of the statistical proble posed (calculate the ratio of expected di erences aong order statistics), but does not refer to the original question of prize allocation in contests. It is now up to the reader to decide whether we, huble econoists, have ade a contribution. 6. References Aann, E., and Leininger, W. (996). Asyetric all-pay Auctions with Incoplete Inforation: The Two-Player Case. Gaes and Econoic Behavior, 4, -8. Baye, M., Kovenock, D., and de Vries, C. (996). The All-Pay Auction with Coplete Inforation, Econoic Theory, 8, Barut, Y., and Kovenock, D. (998). The Syetric Multiple Prize All-Pay Auction with Coplete Inforation, European Journal of Political Econoy, 4, Clark, D., and Riis, C. (998). Copetition over More than One Prize., Aerican Econoic Review, 88, Galton, F. (90). The Most Suitable Proportion Between The Values Of First And Second Prizes, Bioetrika, (4), Glazer, A., and Hassin, R. (988). Optial Contests, Econoic Inquiry 6, Green, J., and Stokey, N. (983). A Coparison of Tournaents and Contracts. Journal of Political Econoy, 9(3), Hilan, A., and Riley, J. (989). Politically Contestable Rents and Transfers. Econoics and Politics,, Hilan, A., and Saet, D. (987). Dissipation of Contestable Rents by Sall Nubers of Contenders. ; Public Choice, 54,

15 Krishna, V., and Morgan, J. (997). An Analysis of the War of Attrition and the All-Pay Auction., Journal of Econoic Theory, 7, Lazear, E., and Rosen, S.(98). Rank Order Tournaents as Optiu Labor Contracts ; Journal of Political Econoy, 89, Nalebu, B., and Stiglitz, J. (983). Prizes and Incentives: Towards a General Theory of Copensation and copetition., Bell Journal of Econoics, 4, Pearson, K. (90). Note On Francis Galton s Proble, Bioetrika, (4), Rosen, S. (986). Prizes and Incentive in Eliination Tournaents, Aerican Econoic Review, 76, Tullock, G. (980), E cient rent-seeking, in J.Buchanan et.al., eds, Towards a Theory of the Rent-Seeking Society, Texas A&M University Press, College Station. Varian, H. (980), A Model of Sales, Aerican Econoic Review, Weber, R. (985), Auctions and copetitive bidding, in H.P. Young, ed., Fair Allocation, Aerican Matheatical Society, Appendix ProofofProposition3.: Assue that all contestants in the set Knfig bid according to b( ), and assue that the bid function is strictly onotonic and di erentiable. Player i s axiization proble reads: ax x [V ( F (b (x))) k +(k )V F (b (x))( F (b (x))) k cx] Let y( ) denote the inverse function of b( ): Using strict onotonicity and syetry, the rst order condition is: = (k )(V V )y 0 y ( F (y))k F 0 (y) (k )(k )V y 0 y F (y)( F (y))k 3 F 0 (y) Note that the right hand side of the FOC is a function of y only (i.e., this is a di erential equation with separated variables). A contestant with the lowest possible ability c =can either never win a prize (if k>) or wins for sure the second prize (if k =): Hence the optial bid of this type is always zero, and this yields the boundary condition y(0) = ; 5

16 The solution to the di erential equation with the boundary condition is given by : Z 0 x Denote dt = V ((k )) V (k ) y y t ( F (t))k F 0 (t)dt + t ( F (t))k 3 [ (k )F (t)]f 0 (t)dt (7.) G(y) = V ((k )) V (k ) y y t ( F (t))k F 0 (t)dt + t ( F (t))k 3 [ (k )F (t)]f 0 (t)dt (7.) We obtain that x = G(y) =G(b (x)); and therefore that b( ) =G( ): Thus, the bid function of every player is given by b(c) =A(c)V + B(c)V ; where: A(c) =(k ) c a ( F (a))k F 0 (a)da B(c) =(k ) c a ( F (a))k 3 [(k )F (a) ]F 0 (a)da We now check that the candidate equilibriu function b( ) is strictly onotonic decreasing (it is clearly di erentiable). Note rst that for all c [; ):We have also A 0 (c) = (k ) c ( F (c))k F 0 (c) < 0 B 0 (c) =(k ) c ( F (c))k 3 F 0 (c)[( (k )F (c)]f 0 (c) Because V V we obtain for all c [; ) : b 0 (c) = A 0 (c)v + B 0 (c)v V (A 0 (c)+b 0 (c)) = V (k )(k ) c F (c)( F (c))k 3 F 0 (c)) < 0 6

17 Assuing that all contestants other than i bid according to b( ); we nally need to show that, for any type c of player i; the bid b(c) axiizes the expected utility of that type. The necessary rst-order condition is clearly satis ed (since this is how we guessed b(c) to start with). We now show that a su cient second-order condition (called pseudoconcavity ) is satis ed. Let ¼(x; c) =V ( F (b (x)) k +(k )V F (b (x))( F (b (x)) k cx be the expected utility of player i with type c that akes a bid x: We will show that the derivative ¼ x (c; x) is nonnegative if x is saller than b(c) and nonpositive if x is larger than b(c): As ¼(x; c) is continuous in x; this iplies that ¼(x; c) is axiized at x = b(c). Note that ¼ x (x; c) = (k )(V V ) db (x) ( F (b (x))) k F 0 (b (x)) dx db (k )(k )V (x) F (b (x))( F (b (x))) k 3 F 0 (b (x)) c: dx Let x<b(c); and let bc be the type who is supposed to bid x; that is b(bc) =x: Note that bc >csince b( ) is strictly decreasing. Di erentiating ¼ x (x; c) with respect to c yields ¼ xc (x; c) = < 0: That is, the function ¼ x (x; ) is decreasing in c. Since bc >c;we obtain ¼ x (x; c) ¼ x (x; bc) Since x = b(bc) we obtain by the rst order condition that ¼ x (x; bc) =0;and therefore that ¼ x (x; c) 0 for every x<b(c): A siilar arguent shows that ¼ x (x; c) 0 for every x>b(c): The syetric equilibriu with p prizes: Fix agent i; and let F s (a); s p; denote the probability that agent i with type a eets k copetitors such that s of the have lower types, and k s have higher types. Recall that in equilibriu we expect i to bid ore than copetitors with higher types (lower ability). Hence F s ( ) is exactly the probability of winning the s 0 th prize. We have then F s (a) = The corresponding derivatives are given by and by (k )! (s )!(k s)! ( F (a))k s (F (a)) s F 0 (a) = (k )( F (a))k (7.3) F 0 s(a) = (k )! (s )!(k s)! ( F (a))k s (F (a)) s F 0 (a) (7.4) [( k)f (a)+(s )] for s>. 7

18 Note that A(c) = R c F 0 a (a)da and that B(c) =R c F 0 a (a)da: Analogously to the case of two prizes, the equilibriu bid for any nuber of prizes p,any nuber of contestants k p with linear cost functions is given by: b(c) = px V s s= c a F 0 s (a)da (7.5) ProofofProposition3.: Recall that : A(c) =(k ) R c ( F a (a))k F 0 (a)da and B(c) =(k ) R c ( a F (a))k 3 [(k )F (a) ]F 0 (a)da. This is obvious by de nition.. A(c) > 0 for c [; ) is obvious by de nition. Further we have A 0 (c) = (k ) c ( F (c))k F 0 (c) < 0 for all c [; ) and A 0 () = 0: 3. B 0 (c) =(k ) ( c F (c))k 3 F 0 (c)[( (k )F (c)] For c such that F (c )= we obtain k B0 (c )=0: Moreover, B 0 (c) > 0 for all c [; c ),andb 0 (c) < 0 for all c (c ; ): Finally, B 0 () = 0: 4. For for all c [c ; ) we obtain that j B 0 (c) j ja 0 (c) j = B 0 (c)+a 0 (c) = (k ) c ( F (c))k 3 F 0 (c)(kf(c) ) For k> we obtain that j B 0 (c) j ja 0 (c) j is positive for c close enough to(sincekf(c) > for such types.) 5. We have Z c B() = (k )( a ( F (a))k 3 [(k )F (a) ]F 0 (a)da +(k )( c a ( F (a))k 3 [(k )F (a) ]F 0 (a)da < (k ) ( F (a)) k 3 [(k )F (a) ]F 0 (a)da The last inequality follows by noting that the integrand in the rst integral is negative and that the integrand of the second integral is positive. If we ultiply both integrands by the increasing function h(a) =a we strictly increase the value of the su of the two integrals. In order to prove that 8

19 B() < 0 it is then enough to prove that R ( F (a))k 3 [(k )F (a) ]F 0 (a)da =0: By the change of variable z = F (a);we obtain = 0 ( F (a)) k 3 [(k )F (a) ]F 0 (a)da ( z) k 3 [(k )z ]dz =0 6. This follows by cobining all properties above. 7. We know that B(c) A(c) > 0 for all c [; c ) and that B(c) A(c) < 0 for all c (c ; ): This yields:. Z c (B(c) A(c))F 0 (c)dc = (B(c) A(c))F 0 (c)dc + (B(c) A(c))F 0 (c)dc c Z c ( F (a)) k 3 = (k ) [ (kf(a) )F 0 (a)da]f 0 (c)dc + c a ( F (a)) k 3 (k ) [ (kf(a) )F 0 (a)da]f 0 (c)dc c c a < (k ) Z c [ ( F (a)) k 3 (kf(a) )F 0 (a)da]f 0 (c)dc + c c (k ) c = (k ) = [ c 0 c Z v [ c c [ c ( F (a)) k 3 (kf(a) )F 0 (a)da]f 0 (c)dc ( F (a)) k 3 (kf(a) )F 0 (a)da]f 0 (c)dc (k )( z) k 3 (kz )dz]dv =0 The last equality follows by the changes of variables F (a) =z and F (c) =v Proof of Proposition 4.: Assue that all contestants in the set Knfig bid according to b( ), and assue that the bid function is strictly onotonic and di erentiable. Let y( ) denote the inverse function of b( ): Player i s axiization proble reads: ax x [V ( F (b (x))) k +(k )V F (b (x))( F (b (x))) k c (x)] Using strict onotonicity and syetry, the rst order condition is: 9

20 0 (x) = (k )(V V )y 0 y ( F (y))k F 0 (y) (k )(k )V y 0 y F (y)( F (y))k 3 F 0 (y) Note that this is also an ordinary di erential equation with separated variables (i.e. the left hand side of the rst equation is a function of x only; while the right hand side is a function of y only: Integration and the use of the boundary condition y() = 0 yield (x) =G(y), where G(y) is de ned exactly as in the proof of Proposition 3. (see equation 7.) Hence, we obtain that x = (G(y)) = g(g(b (x))) and that b( ) =g(g( )): Note that the candidate equilibriu bid function b(c) = (A(c)V +B(c)V )= g(a(c)v + B(c)V ) is strictly decreasing since, for all c [; ); it holds: dg dc (A(c)V + B(c)V ) = g 0 (A(c)V + B(c)V ) (A 0 (c)v + B 0 (c)v ) < 0 The last inequality follows because g 0 ( ) > 0 by assuption, while A 0 (c)v + B 0 (c)v < 0 since this is the derivative of the bid function with linear cost functions (seeproofofproposition3.). For the su cient second-order condition we proceed exactly as in the proof of Proposition 3.. Using the notation eployed in that proof, we have ¼(x; c) =V ( F (b (x)) k +(k )V F (b (x))( F (b (x)) k c (x) and ¼ x (x; c) = (k )(V V ) db (x) ( F (b (x))) k F 0 (b (x)) dx db (k )(k )V (x) F (b (x))( F (b (x))) k 3 F 0 (b (x)) c 0 (x): dx Di erentiating ¼ x (x; c) with respect to c yields ¼ xc (x; c) = 0 (x) < 0: That is, the function ¼ x (x; ) is decreasing in c, exactly as in the linear case. The proof of pseudoconcavity is concluded as in that case. Analogously to the case of two prizes, the equilibriu bid for any nuber of prizes p, and any for nuber of contestants k p with cost functions of the for c (x) is given by: b(c) = ( px V s s= where F 0 s (a) is given in forulas 7.3 and 7.4. c a F 0 s (a)da) (7.6) 0