Contests with Group-Specific Public-Good Prizes

Transcription

1 Contests wth Group-Specfc Publc-Good Przes Kyung Hwan ak * Department of Economcs Sungkyunkwan Unversty Seoul South Korea September 2005 Abstract I examne the equlbrum effort levels of ndvdual players and groups n contests n whch n groups compete wth one another to wn a group-specfc publc-good prze, the ndvdual players choose ther effort levels smultaneously and ndependently, and the probablty of wnnng for each group depends on the groups' effort levels. In the basc model, I frst show that, n each group, only the hghest-valuaton players expend postve effort and the rest expend zero effort I descrbe ths as the explotaton of the hghestvaluaton players by the rest. I also show that there s undernvestment n the contest for the group as a whole. Next, n the modfed model n whch the players are budgetconstraned, I show that low-valuaton players free rde on hgh-valuaton players' contrbutons, not vce versa, but the free-rder problem s allevated as compared wth the basc model. Keywords: Contest; Publc-good przes; udget-constraned players; the free-rder problem; Prvate provson of publc goods JEL classfcaton: D72; C72; H41 * I am grateful to Amy ak, Jon S. J. Charles, Sung Hyun Km, and Ka Konrad for ther helpful comments and suggestons. Earler versons of ths paper were presented at the 78th Annual Conference of the Western Economc Assocaton Internatonal, Denver, CO, July 2003, and at the 2004 Annual Conference of the Korean Econometrc Socety, Seoul, Korea, February Ths work was supported by the ran Korea 21 Project. Tel.: ; fax: E-mal address: khbak@skku.edu (K.H. ak).

2 1 1. Introducton Consder a stuaton n whch a local government has a budget for buldng a brdge and several communtes compete aganst one another to wn the budget. The local government selects the wnnng communty accordng to some rule whch s based on the voluntary contrbutons people n the communtes make. For example, the local government can use a lottery-lke wnner-selectng mechansm n whch people n the communtes purchase as many lottery tckets as they want from the government, and the wnnng communty s selected by drawng one tcket out of the tckets sold. The proceeds or contrbutons are nonrefundable and go nto general government revenues, where they may be used to fnance other local publc goods. The people n the wnnng communty are granted the budget and buld a brdge from whch they all get benefts n economc terms, the budget s a publc good wthn the communty. Contests nvolvng group-specfc publc-good przes, lke the motvatng or llustratve example above, are easly observed n the real world. Examples nclude class acton ltgaton, varous types of rent-seekng contests, budget-seekng contests, competton between domestc and foregn frms to obtan governmental trade polces favorable to them, R&D competton between consortums, electon campagns between poltcal partes, and competton between local governments to nvte busness frms nto ther dstrcts. 1 The purpose of ths paper s to examne how severe the free-rder problem s n such contests more precsely, to examne the equlbrum effort levels of ndvdual players and groups. To do so, I consder n-group contests wthout and wth budget constrants n whch the ndvdual players choose ther effort levels smultaneously and ndependently, and each group's contest success functon, whch s the rule that descrbes the relatonshp between effort levels of the groups and ts probablty of wnnng, s contnuous.

3 2 I frst consder the followng contest. There are n groups whch compete wth one another to wn a group-specfc publc-good prze. The ndvdual players choose ther effort levels smultaneously and ndependently. Each player's effort s rreversble. Unlke the selecton rule of all-pay auctons, each group's contest success functon s contnuous, and thus a group whch expends less effort than ts rvals may wn the prze. Each player's valuaton for the prze s publcly known, and the valuatons may dffer across the ndvdual players. In ths model, I show the followng. Frst, only the hghest-valuaton players n each group expend postve effort at a Nash equlbrum the "explotaton" of the hghest-valuaton players by the rest arses. Ths result bascally comes from the fact that, n each group, the hghest-valuaton players have the greatest gross margnal payoff at any effort level, whereas all the players (ncludng the hghest-valuaton players) have the same margnal cost that s constant at unty. Gven ths fact, at the effort level optmal for the hghest-valuaton players, the gross margnal payoffs for the other players n the group are less than the margnal cost. Ths, n turn, mples that unless they expend zero effort, the players whose valuatons for the prze are less than somebody else's n the group always have an ncentve to decrease ther effort levels. Second, the equlbrum effort level of each group and the equlbrum total effort level depend solely on n values, whch consst of each group's hghest valuaton for the prze. Ths mples that each group's equlbrum effort level and the equlbrum total effort level are ndependent of the number of players, the sum of valuatons, and the dstrbuton of valuatons n each group, as long as changes n these do not change the hghest valuaton for that group. Thrd, each group's equlbrum effort level depends on the hghest valuaton, not the sum of the valuatons of the players n the group, so that there s "undernvestment" n the contest for the group as a whole. Fnally, I pont out that, due to the free-rder problem, the equlbrum expected payoffs for the effort-expendng hghest-valuaton players n each group may be less than those for other players n the group.

4 3 Then, I consder a modfed model n whch the players are budget-constraned. In ths model, knowng the valuatons and budgets of the other players n hs group, each player fgures out the effort levels to be expended by the other players n hs group who have hgher valuatons than hmself. If he expects ther effort levels to be enough from hs perspectve, he expends zero effort. Otherwse, he expends postve effort. ased on these (and others), I clam that low-valuaton players free rde on hgh-valuaton players' contrbutons, not vce versa, and descrbe t as the explotaton of the hgh-valuaton players by the low-valuaton players. Next, I show that, overall, n the case where the players are budget-constraned, more players exert postve effort and thus the free-rder problem s allevated, compared wth the basc model n whch the players have no budget constrants. Fnally, I pont out that a group's equlbrum effort level obtaned n the modfed model may be greater than that obtaned n the basc model. Katz et al. (1990), Ursprung (1990), ak (1993), Raz et al. (1995), Djkstra (1998), and ak et al. (2001) study contests wth group-specfc publc-good przes. Among them, ak (1993) and ak et al. (2001) are closely related to ths paper. These papers employ models n whch players have no budget constrants, and establsh that the free-rder problem s the severest form that s, n each group, all the players except hghest-valuaton players free rde. Specfcally, ak et al. (2001) consder contests wth two groups, and model them as frst-prce all-pay auctons: A group whch expends more effort than ts rval wns the prze wth certanty, and the wnnng group pays the hgher "bd,".e., ts own bd. Adoptng the model n ak (1993), ths paper analyzes t completely, and then extends t to the model n whch the players are budget-constraned. In ths paper, the ndvdual players n each group choose ther effort levels noncooperatvely to wn ther publc-good prze. In other words, they play a game of the prvate (also called voluntary) provson of a publc good. Accordngly, ths paper s related to the lterature on the prvate provson of publc goods the lterature whch deals wth stuatons n whch publc goods are fnanced by voluntary contrbutons of

5 4 ndvduals (see, for example, Olson, 1965; ergstrom et al., 1986; Konrad, 1994; Varan, 1994; Vcary, 1997; Robledo, 1999; Marx and Matthews, 2000). The paper proceeds as follows. Secton 2 develops the basc model. Secton 3 obtans the Nash equlbra of the game. Secton 4 presents and analyzes the modfed model n whch the players are budget-constraned. Fnally, Secton 5 presents conclusons. 2. The basc model Consder a contest n whch n groups (1 through n) compete wth one another to wn a prze, where n 1. Group conssts of m rsk-neutral players who expend effort to wn the prze, where m 1. The prze s a publc good wthn each group thus, t s called a group-specfc publc-good prze. The ndvdual players' valuatons for the prze may dffer. Let v k represent the valuaton for the prze of player k n group. Each player's valuaton for the prze s postve and publcly known. Assumpton 1. Wthout loss of generalty, I assume that vs 1 v s 0 for s œ 2,..., m. Let x represent the effort level expended by player k n group, and let X k represent the effort level expended by all the players n group, so that X x 1 m k= k. Each player's effort s rreversble each player cannot recover hs effort expended whether or not hs group wns the prze. Effort levels are nonnegatve, and are measured n unts commensurate wth the prze. Let p denote the probablty that group wns the prze. I assume that each group's probablty of wnnng depends on the other groups' effort levels as well as ts own. Specfcally, I assume that the contest success functon for group s

6 5 p p( X,..., X ), œ 1 n where 0 Ÿ p Ÿ 1, the functon p has the propertes specfed n Assumpton 2 below, and n j j= 1 p œ ` j Assumpton 2. I assume that p Î` X 0, ` p Î`X Ÿ 0, `p Î`X Ÿ 0, and j j ` p Î` X 0. I further assume that `p Î`X 0 and ` p Î`X 0 when X 0 for j j some j, and `p Î`X 0 and ` p Î`X 0 when X 0. Assumpton 2 says that, gven the rval groups' effort levels, each group's probablty of wnnng s ncreasng n ts own effort level at a decreasng rate. It also says that each group's probablty of wnnng s decreasng n a rval group's effort level at a decreasng rate, gven that the effort levels of the rest reman constant. Under Assumpton 2, the group expendng the largest effort level does not wn the prze wth certanty that s, t may lose the prze when there are at least two groups whch expend postve effort levels. Let 1 k represent the expected payoff for player k n group. Then the payoff functon for player k n group s 1 k œ v k p ( X1,..., X n ) x k. I assume that all the players n the contest choose ther effort levels ndependently and smultaneously that s, when a player chooses hs effort level, he does not know the other players' effort levels. Fnally, I assume that all of the above s common knowledge among the players, and employ Nash equlbrum as the soluton concept. 4

7 6 3. The explotaton of the hghest-valuaton players by the rest Ths secton obtans the pure-strategy Nash equlbra of the game, and shows that although the players n each group have the common goal of wnnng the group-specfc publc-good prze, only the hghest-valuaton players n that group are "actve" n equlbrum; the rest expend zero effort and free rde. k Let x denote the best response of player k n group, gven effort levels of all the other players n the contest. y defnton, t s the effort level whch maxmzes hs expected payoff 1 k œ v k p ( X1,..., X n ) x k subject to the nonnegatvty constrant, x 0. Thus x satsfes the frst-order condton: k k or v ( `pî`x) 1 œ 0 for x 0 (1) k k v ( `pî`x) 1 Ÿ 0 for x œ 0. (2) k k k k In the case where x 0, the margnal gross payoff, v ( `pî`x ), for player k n group s equal to hs margnal cost, 1, at that postve effort level. In the case where xk œ 0, hs margnal gross payoff does not exceed hs margnal cost at that zero effort. Note that under Assumpton 2, hs margnal gross payoff, v ( `pî`x), decreases n hs effort level, k x. Ths mples that hs payoff functon, 1, s strctly concave n hs effort level, x, k k k whch n turn mples that the second-order condton for maxmzng 1 k s satsfed and x k s unque. As a prelmnary step to obtan the Nash equlbra of the game, I obtan ts group-- specfc equlbra. Gven effort levels of the other groups, a group--specfc equlbrum s defned as an m-tuple vector of effort levels, one for each player n group, at whch

8 7 each player's effort level s the best response to the other players' effort levels (ncludng those of the players n the other groups). I begn by defnng group 's player-k-best response to the other groups' effort levels. Let X denote an ( n 1)-tuple vector of effort levels, one for each group except group : X ( X 1,..., X 1, X 1,..., X n ). Defnton 1. Gven the other groups' effort levels,, group 's player-k-best response, X X ( X ; vk), s defned as the effort level whch maxmzes v p( X, ) X k X subject to the nonnegatvty constrant, X 0. whole to Group 's player-k-best response to X represents the best response of group as a X when the valuaton for the prze of player k n group s taken nto account. To put t dfferently, t s group 's best response to the other groups' effort levels whch s computed wth player k's valuaton. Naturally, group 's player-k-best response, X ( ; v k ), satsfes the frst-order condton: X or k X k v ( `pî`x) 1 œ 0 for X ( ; v ) 0 (3) vk( `pî`x) 1 Ÿ 0 for X ( X ; vk) œ 0. (4) The term, v ( `pî`x), n expressons (3) and (4) decreases n X due to Assumpton 2. k Therefore, the second-order condton s satsfed and group 's player-k-best response, X X ( ; v ), s unque. k Lemma 1 compares group 's best responses computed wth dfferent players' valuatons to the other groups' effort levels.

9 8 Lemma 1. Gven the other groups' effort levels,, X group 's best response computed wth a hgher valuaton s greater than or equal to that computed wth a lower valuaton. s 1 s œ Thus X ( X ; v ) X ( X ; v ) holds for s 2,..., m. Proof. It s trval to see that for any h and z, X ( X ; v ) œ X ( X ; v ) holds f v œ v. Next, consder the case where v v holds. Snce the term, v ( `pî`x), n h z h z k expressons (3) and (4) decreases n X, t s straghtforward to obtan that for any h and z, h z h h X ( X ; v ) œ X ( X ; v ) œ 0 holds f X ( X ; v ) œ 0, and X ( X ; v ) X ( X ; v ) holds f X ( X ; v ) 0. The above together wth Assumpton 1 yelds Lemma 1. z h h z Lemma 1 comes from two facts. One s that gven a postve effort level from the other groups, group 's gross margnal payoff computed wth a hgher valuaton s greater than that computed wth a lower valuaton at any effort level. The other s that group 's margnal cost of ncreasng effort s constant. Lemma 1 says that gven the other groups' effort levels, group 's player-1-best response s the largest, ts player-m-best response s the smallest, and the rest are nbetween. Now I am prepared to fnd group--specfc equlbra. Let an m -tuple vector of * * m effort levels, ( x,..., x ), represent a group--specfc equlbrum, gven the other groups' effort levels, 1 X. At the group--specfc equlbrum, the effort level of each player n group s the best response to the other players' effort levels t satsfes ether expressons (1) or (2), gven the other players' effort levels. Lemma 2 shows that group 's * effort level, X, at a group--specfc equlbrum gven must be equal to group 's player-1-best response to X : X X ( X ; v ). Usng ths result, n Lemma 3, I * œ 1 construct group--specfc equlbra gven X. X Lemma 2. Group 's effort level at a group--specfc equlbrum gven greater nor less than group 's player-1 -best response to X. X s nether

10 9 Proof. Let X be group 's effort level such that X X ( X ; v 1 ). Then, player 1 n group has an ncentve to ncrease hs effort level because X level from hs perspectve. falls short of the optmal effort Next, let X be group 's effort level such that X X ( X ; v 1 ). Then, due to 1 h Ÿ Lemma 1, X X ( X ; v ) X ( X ; v ) holds for any h, where 1 h Ÿ m. It follows from ths and Assumpton 2 that v ( `pî`x) 1 0 holds at X for any h (see h expressons (3) and (4)). Ths means that any player expendng a postve effort level has an ncentve to decrease hs effort level because hs margnal gross payoff s less than hs margnal cost (see expressons (1) and (2)). Lemma 2 says that gven the other groups' effort levels, group 's effort level at a group--specfc equlbrum s exactly the same as the optmal effort level for the hghest- valuaton player n group ; consequently, t depends only on the hghest valuaton of the group. Lemma 3. ( a) Gven the other groups' effort levels, X, f X ( X ; v ) 0, then there s 1 œ a unque group--specfc equlbrum at whch all the players n group expend zero X effort. ( b) If X ( ; v ) 0 and v v, then there s a unque group--specfc * * 1 1 s equlbrum at whch x œ X ( X ; v ) and x œ 0 for s œ 2,..., m. ( c ) If X ( X ; v 1) 0 and v1 œ vt v t 1 for some t, then there are multple group--specfc equlbra t * * k X 1 s at whch x œ X ( ; v ) and x œ 0 for s œ t 1,..., m, where 2 Ÿ t Ÿ m. Proof. ( a) If group 's player-1-best response to X s zero, then by Lemma 1, X ( X ; v ) œ 0 holds for any k, where 1 Ÿ k Ÿ m. Then t follows from expresson (4) that f all k the players n group expend zero effort and thus group 's effort level s zero, then

11 10 v ( `pî`x) 1 Ÿ 0 holds for any k. Ths mples that f all the players n group expend k zero effort, expresson (2) s satsfed for each player n group. Therefore, gven zero effort levels of the other players n group, zero effort s the best response of each player n group. Next, the unqueness of the group--specfc equlbrum follows mmedately from Lemma 2 and the fact that effort levels are nonnegatve. ( b) Usng expresson (3) and Assumpton 1, I obtan: If group 's effort level s s equal to X ( X ; v 1 ), then v 1 ( `pî`x) 1 œ 0 and v ( `pî`x) 1 0 for s œ 2,..., m hold. Hence, f player 1 expends X ( ; v ) and the rest of the players n group X 1 choose zero effort levels, expresson (1) s satsfed for player 1, and expresson (2) s satsfed for each of the rest. Therefore, the proposed effort level of each player n group s hs best response to X and the proposed effort levels of the other players n group. Next, I show that no other m-tuple vector of effort levels (of the players n group ) consttutes a group--specfc equlbrum. Frst, t follows from Lemma 2 that a vector of effort levels, ( x,..., x ), does not consttute a group--specfc equlbrum f 1 m m xk X X v1 x1 m xm xk X œ X X v 1 Á ( ; ). Second, suppose on the contrary that a vector of effort levels, (,..., ), consttutes a group- -specfc equlbrum, where ( ; ) and x 0 for some t, where 2 Ÿ t Ÿ m. Then because expresson (1) must be satsfed for t player t, at the vector of effort levels v ( `pî`x) 1 œ 0 must hold. On the other hand, t t follows from expresson (3) that v1( `pî`x) 1 œ 0 must hold at the vector of effort t levels because X œ X ( X ; v 1 ) 0. Ths mples that v ( `pî`x) 1 0 must hold because v v v due to the condtonal statement of part ( b) and Assumpton 1. Ths 1 2 t leads to a contradcton. ( c) The proof of part ( c) s smlar to that of part ( b), and therefore omtted. X and Part ( b) says that f just one player has the hghest valuaton n group, there s a unque group--specfc equlbrum at whch only the hghest-valuaton player expends

12 11 postve effort and the rest expend zero effort. Part ( c) says that f more than one player has the hghest valuaton n group, there are multple group--specfc equlbra. At these equlbra, f a player expends postve effort, then he must have the hghest valuaton for the prze n hs group; however, there may be hghest-valuaton players who expend zero effort. Now I obtan the pure-strategy Nash equlbra of the game. Let a ( n j= 1 m )-tuple j N N N N 11 1m n1 nm vector of effort levels, ( x,..., x,..., x,..., x ), represent a Nash equlbrum. 1 n At the Nash equlbrum, each player's effort level s the best response to the other players' effort levels, whch occurs f and only f the n groups each are n group-specfc equlbrum. Thus, usng Lemma 3, I obtan Proposton 1. Proposton 1. The followng strategy profles consttute the Nash equlbra of the game. ( a) For group wth v1 v 2, ts players play the strateges : x œ X ( X N ; v 1) and N s j1 jt jt 1 t N N jk j X N j j1 js x œ 0 for s œ 2,..., m. ( b ) For group j wth v œ v v for some t, ts players use strateges such that x œ X ( ; v ) and x œ 0 for s œ t 1,..., m, where 2 Ÿ t Ÿ. m j N 1 j Proposton 1 mples the followng. Frst, each group's equlbrum effort level s equal to ts player-1-best response to the other groups' equlbrum effort levels. Second, n the case where a group's player-1-best response to the other groups' equlbrum effort levels s postve, at least one of the hghest-valuaton players n the group expends 5 postve effort. Thrd, a player whose valuaton for the prze s less than somebody else's n hs group expends zero effort namely, he s a free rder. Followng Olson (1965), I descrbe ths result that only the hghest-valuaton players exert effort, and the rest free rde 6 as the explotaton of the hghest-valuaton players by the rest. Fourth, each group's equlbrum effort level and the equlbrum total effort level are equal to those

13 12 obtaned n a reduced contest n whch only n players one for each group, who s one of the hghest-valuaton players n that group 7 compete to wn the prze. Consequently, to obtan the groups' equlbrum effort levels and/or the equlbrum total effort level, one only needs to solve ths reduced n-player game. Ffth, the number of players, the sum of valuatons, and the dstrbuton of valuatons n each group only affect the groups' equlbrum effort levels and the equlbrum total effort level f changes n them come wth a change n the hghest valuaton for the group. Sxth, each group's equlbrum effort level depends on the hghest valuaton, not the sum of the valuatons of the players n the group, so that there s "undernvestment" n the contest for the group as a whole. Fnally, t s very lkely that, due to the free-rder problem, the equlbrum expected payoffs for the effort-expendng hghest-valuaton players n each group s less than those for other players n the group. Furthermore, t s lkely that the actual payoffs for the effortexpendng hghest-valuaton players n the wnnng group s less than those for other players n the group. Why do only the hghest-valuaton players n each group expend postve effort at the Nash equlbra? Why do the other players expend zero effort? The answer to these questons comes from the fact that, n each group, the hghest-valuaton players have the greatest gross margnal payoff at any effort level, whereas all the players (ncludng the hghest-valuaton players) have the same margnal cost that s constant at unty. Gven ths fact, at the effort level at whch the gross margnal payoff for the hghest-valuaton players equals the margnal cost, the gross margnal payoffs for the other players n the group are 8 less than the margnal cost. Ths, n turn, mples that, at the effort level optmal for the hghest-valuaton players, the players expendng postve effort except the hghest- valuaton players have an ncentve to decrease ther effort levels. Therefore, unless effort s exerted only by hghest-valuaton players, an equlbrum cannot occur.

14 13 4. udget-constraned players I have shown n Secton 3 that, n each group, only the hghest-valuaton players expend postve effort and the rest expend zero effort. In ths secton, I present a modfed model n whch the players are budget-constraned. Consder a model whch s the same as the basc model wth the excepton that the players are budget constraned. Let b k represent the budget of player k n group. Wthout loss of generalty, I assume that f v œ v for some s, then b b holds, s 1 s s 1 s where 2 Ÿ s Ÿ m. Each player's budget s postve and publcly known. In ths game, then, player k n group faces the followng maxmzaton problem: Maxmze œ v p ( X,..., X ) x 1 k k 1 n k subject to hs budget constrant 0 Ÿ x Ÿ b. k k b k Let x denote the best response of player k n group, gven effort levels of all the other players n the contest. y defnton, then, t satsfes the frst-order condton: or or b k k k v ( `p/ ` X) 1 0 for x œ b, (5) b k k k v ( `p/ `X) 1 œ 0 for 0 x b, (6) v ( `p/ `X) 1 Ÿ 0 for x œ 0. (7) k b k b k k k In the case where x œ b, the margnal gross payoff for player k n group, v ( `p/ `X), s ether greater than or equal to hs margnal cost, 1, at that effort level. Ths s the case where hs budget constrant s bndng. When hs margnal gross payoff s greater than hs margnal cost, t s benefcal to the player to ncrease hs effort level, but he cannot do so

15 14 due to the lmted budget. As mentoned n Secton 3, 1 k s strctly concave n x k, and b therefore the second-order condton for maxmzng 1 k s satsfed and x k s unque. In order to characterze the Nash equlbra of ths modfed game, I frst obtan ts, a group- group--specfc equlbra. Recall that, gven the other groups' effort levels, X -specfc equlbrum s defned as an m-tuple vector of effort levels, one for each player n group, at whch each player's effort level s the best response to the other players' effort levels. At a group--specfc equlbrum, thus, the effort level of each player n group satsfes ether expressons (5), (6) or (7), gven the other players' effort levels. Let an m - ** ** m tuple vector of effort levels, ( x,..., x ), represent a group--specfc equlbrum, gven 1 the other groups' effort levels, X. Recall that group 's player-k-best response to X, denoted by X ( X ; v ), s the best response of group as a whole to X when the k valuaton for the prze of player k n group s taken nto account. I begn by obtanng ** group 's effort level, X, at the group--specfc equlbra gven. Lemmas 4 and 5 show the results. X Lemma 4. ( a) Gven the other groups' effort levels, X, f X ( X ; v ) Ÿ b holds, then 1 1 group 's effort level at a group--specfc equlbrum s equal to group 's player-1-best ** response to X : X œ X ( X ; v ). ( b) If X ( X ; v ) b holds, then group 's 1 m k effort level at the group--specfc equlbrum s equal to the sum of the budgets of the ** players n group : X œ bk. m m The proof of part ( a) s smlar to that of Lemma 2, and the proof of part ( b) s mmedate. Thus ther proofs are omtted. Part ( a) says that, gven the other groups' effort levels, f the budget of player 1 n group s greater than or equal to group 's player-1-best response, then group 's effort level at a group--specfc equlbrum s equal to group 's player-1-best response. Ths mples that, as long as player 1's budget s enough to cover

16 15 group 's player-1-best response, ntroducng the players' budget constrants does not affect group 's effort level at a group--specfc equlbrum (see Lemma 2). Part ( b) shows the case where all the budget constrants of the players n group are bndng at the group-- specfc equlbrum. Due to Lemma 1, the condtonal statement of part ( b) mples that m h k X ( X ; v ) b holds for any h, where 1 Ÿ h Ÿ m. That s, the total budget of group does not exceed any of m group 's best responses recall that group 's player-k-best response represents group 's optmal effort level from player k's perspectve. Next, consder the case where X ( X ; v ) b and X ( X ; v ) Ÿ b hold. 1 1 m k q 1 X q 1 k Let q be an nteger between 2 and m, nclusve, such that X ( ; v ) b and X ( ; v ) Ÿ b hold. That s, q s an nteger such that the total budget of top q 1 X q k q players (accordng to valuaton for the prze) n group cannot cover group 's player- ( q 1)-best response, but the total budget of top q players s enough to cover group 's player-q-best response. Usng Lemma 1, t s easy to see that qs unque. m Lemma 5. ( a) Gven the other groups' effort levels, X, f X ( X ; vq) Ÿ 1 bk holds, then group 's effort level at the group--specfc equlbrum s equal to the sum of the ** budgets of players 1 through q 1: X œ b. ( b) If b X ( ; v ) Ÿ b q 1 q 1 q k k X q k holds, then group 's effort level at a group--specfc equlbrum s equal to group 's player-q-best response to X : X œ X ( X ; v ). ** q q Proof. ( a) Let X be group 's effort level such that X 1 bk. Then, there s a player among players 1 through q 1 who does not exhaust hs budget. He has an ncentve to ncrease hs effort level because X ( X ; vh) 1 bk holds for any h, where 1 Ÿ h Ÿ q 1, and thus X falls short of group 's optmal effort level from hs perspectve. q q

17 16 Next, let X be group 's effort level such that X 1 bk. Then, there s a player say, player d among players q through m who expends postve effort. Due to the condtonal statement of part ( a) and Lemma 1, I have X b X ( ; q d d q q 1 k X v ) X ( X ; v ). It follows from ths and Assumpton 2 that v ( `p/ `X) 1 0 holds at X (see expressons (3) and (4)). Ths means that player d has an ncentve to decrease hs effort level because hs margnal gross payoff s less than hs margnal cost (see expressons (5), (6), and (7)). ( b) The proof of part ( b) s smlar to that of part ( a), and therefore omtted. Part ( a) says that f group 's player-q-best response to X s less than or equal to the total budget of the top q 1 players n group, then group 's effort level at the group- -specfc equlbrum s equal to the total budget of the top q 1 players. Ths means that group 's effort level at the group--specfc equlbrum s greater than or equal to group 's 9 player-q-best response. On the other hand, n part ( b), f group 's player-q-best response s greater than the total budget of the top q 1 players, then group 's effort level s equal to group 's player-q-best response. Usng Lemmas 4 and 5, I obtan group--specfc equlbra gven X. Lemma 6. ( a) Gven the other groups' effort levels, X, f X ( X ; v ) 0, then there s 1 œ a unque group--specfc equlbrum at whch all the players n group expend zero X effort. ( b) Consder the case where 0 X ( ; v ) Ÿ b holds. If v v, then there ** ** 1 X 1 s s a unque group--specfc equlbrum at whch x œ X ( ; v ) and x œ 0 for s œ 2,..., m. If v œ v v for some t, then there are multple group--specfc 1 t t 1 t ** ** k X 1 s m X m k equlbra at whch x œ X ( ; v ) and x œ 0 for s œ t 1,..., m, where 2 Ÿ t Ÿ m. ( c) If X ( ; v ) b holds, then there s a unque group--specfc

18 17 ** k m q 1 1) 1 ( X ; m ) Ÿ k. ( X ; q) Ÿ k equlbrum at whch x œ b k for k œ 1,..., m. ( d) Consder the case where X ( X ; v b and X v b hold If X v b holds, then there s a ** ** k k s q 1 q k X q k unque group--specfc equlbrum at whch x œ b for k œ 1,..., q 1, and x œ 0 10 for s œ q,..., m. If b X ( ; v ) Ÿ b holds, then I have the followng two subcases. () If v v v, then there s a unque group--specfc equlbrum at q 1 q q 1 q ** ** ** whch xk œ b k for k œ 1,..., q 1, xq œ X ( X ; vq) 1 b k, and x s œ 0 for s œ q 1,..., m. ( ) If v v œ v œ v v for some h and t, then there are multple group-h 1 h q t t 1 ** ** specfc equlbra at whch x œ b for k œ 1,..., h 1, x œ X ( ; v ), and k k k ** s x œ 0 for s œ t 1,..., m, where 1 Ÿ h Ÿ q 1 and q 1 Ÿ t Ÿ m. t X q The proof of Lemma 6 s smlar to that of Lemma 3, and therefore omtted for concse exposton. Parts ( a) and ( b) are stated the same as Lemma 3. However, f there s more than one hghest-valuaton player and at least one of the top players has a budget less than group 's player-1-best response to X, X ( X ; v ), then group--specfc equlbra obtaned here are not exactly the same as those obtaned n the basc model (or Lemma 3). Part ( b) says that n the case where player 1 n group has a budget enough to cover group 's optmal effort level from hs perspectve (.e., group 's player-1-best response), only the hghest-valuaton players n the group expend postve effort. Part ( c) says that f the total budget of the players n group does not exceed even group 's player-m-best response the smallest of m group 's best responses then all the players n group exhaust ther budgets. In part ( c), the gross margnal payoff for each player s greater than the margnal cost at any effort level up to hs budget because hs budget s far less than group 's optmal effort level from hs perspectve so that t s benefcal to the player to expend all hs budget. 1

19 18 In part ( d), the total budget of top q 1 players n group cannot cover group 's player-( q 1)-best response, but that of top q players s enough to cover group 's player- X q-best response. The frst case of part ( d) s then the one where X ( ; q q 1 k X q 1 q 1 q v ) Ÿ b X ( ; v ) holds. In ths case, v v holds due to Lemma 1, and group 's effort level at the group--specfc equlbrum s equal to the total budget of the top q 1 players (see Lemma 5). The top q 1 players exhaust ther budgets at the group--specfc equlbrum because ther budgets fall short of group 's optmal effort levels from ther perspectves, and because they cannot get any help from players q through m due to ther smaller valuatons. On the other hand, players q through m expend zero effort because group 's optmal effort levels from ther perspectves do not exceed the total effort level of the top q 1 players. They smply wat for the top q 1 players to expend all ther budgets. Next, consder the second case of part ( d) where group 's player-q-best response s greater than the total budget of the top q 1 players n group. In ths case, f player q's valuaton dffers from any other player's n group, then there s a unque group--specfc equlbrum at whch the top q 1 players exhaust ther budgets, player q expends postve effort to attan group 's optmal effort level from hs perspectve, and players q 1 through m expend zero effort. Smlarly to the precedng paragraph, the top q 1 players exhaust ther budgets because ther budgets fall short of group 's optmal effort levels from ther perspectves, and because the valuatons of players q through m are smaller than thers. Even though the top q 1 players exhaust ther budgets, ther total effort level s less than group 's player-q-best response, and thus player q makes up the dfference. Players q 1 through m have no ncentve to expend postve effort because group 's optmal effort levels from ther perspectves are less than the total effort level of the top q players. Fnally, n the second subcase where some players n group have the valuaton equal to player q's, there are multple group--specfc equlbra. At these equlbra, a

20 19 player whose valuaton s greater than player q's exhausts hs budget, a player whose valuaton s less than player q's expends zero effort, and a player whose valuaton s equal to player q's expends ether postve or zero effort. Now, usng Lemma 6, t s straghtforward to obtan the Nash equlbra of the modfed game. Note that a Nash equlbrum occurs f and only f the n groups each are n group-specfc equlbrum. I do not repeat the complcated statements n Lemma 6. Instead, I hghlght the followng nterestng results obtaned at the Nash equlbra. Proposton 2. ( a ) If the total budget of the players n a group does not exceed any of the group's best responses to the other groups' equlbrum effort levels, then all the players n the group exhaust ther budgets. ( b ) If the total budget of top α players n a group s less than the group's player-α -best response to the other groups' equlbrum effort levels, and the valuaton for the prze of player α s greater than that of player α 1, then the top α players n the group exhaust ther budgets. ( c ) If a player expends postve effort, then the players n hs group who have hgher valuatons than hmself exhaust ther budgets. ( d ) If (" 1) -best response to the other groups' equlbrum effort levels, and the valuaton for the prze of player " s greater than that of player " 1, then the players n the group whose valuatons for the prze are less than player "'s expend zero effort. ( e ) Each the total budget of top " players n a group s greater than or equal to the group's player- group's equlbrum effort level never exceeds ts player-1-best response to the other groups' equlbrum effort levels. 11 Interestngly, parts ( a) and ( b) mply that f some player, say player h, exhausts hs budget, then players 1 through h 1 all exhaust ther budgets. Part ( d) mples that f some player, say player t, expends zero effort, then all the players whose valuatons for the prze are less than player t's expend zero effort. Therefore, based on parts ( a) through ( d), I clam that low-valuaton players free rde on hgh-valuaton players' contrbutons, not

21 20 vce versa, and descrbe t as the explotaton of the hgh-valuaton players by the lowvaluaton players. The followng elaborate upon part ( e). If player 1 n a group has a budget enough to cover the group's player-1-best response to the other groups' equlbrum effort levels, or f all the players n a group have the same valuaton and ther total budget s greater than or equal to the group's unque best response, then the group's equlbrum effort level s equal to ts player-1-best response. Otherwse, t s less than the group's player-1-best response. Now, an nterestng queston s: How does the equlbrum effort level of a group obtaned here compare to that obtaned n the basc model (or Proposton 1)? At a frst glance of part ( e) and Proposton 1, t may appear that the former does not exceed the latter. ut ths appearance may be wrong. Notce that "the other groups' equlbrum effort levels" obtaned here may dffer from those obtaned n the basc model. In fact, the answer to the queston depends on whch specfc contest success functons are used as well as how stff budget constrants the players face. Proposton 2 together wth Proposton 1 establshes that, overall, n the case where the players are budget-constraned, more players exert postve effort and thus the free-rder problem s allevated, compared wth the basc model n whch the players have no budget constrants. Especally, f the total budget of the players n a group cannot cover any of the group's best responses to the other groups' equlbrum effort levels, then all the players n the group exhaust ther budgets. These suggest that when publc goods are fnanced by voluntary contrbutons of ndvduals, ntroducng caps on ndvdual contrbutons may allevate the free-rder problem. 5. Conclusons I have examned the equlbrum effort levels of ndvdual players and groups n contests n whch n groups compete wth one another to wn group-specfc publc-good przes, and the ndvdual players choose ther effort levels smultaneously and

22 21 ndependently. I frst consdered the basc model n whch the players have no budget constrants, and then consdered the modfed model n whch the players are budgetconstraned. In the basc model, I showed that only the hghest-valuaton players n each group expend postve effort at a Nash equlbrum. I also showed that the equlbrum effort level of each group and the equlbrum total effort level depend solely on n values, whch consst of each group's hghest valuaton for the prze. Ths mples that the equlbrum effort level of each group and the equlbrum total effort level are ndependent of the number of players, the sum of valuatons, and the dstrbuton of valuatons n each group, as long as changes n these do not change the hghest valuaton for that group. Fnally, I showed that because each group's equlbrum effort level depends on the hghest valuaton, not the sum of the valuatons of the players n the group, there s "undernvestment" n the contest for the group as a whole. In the modfed model, I showed that f some player, say player h, exhausts hs budget, then players 1 through h 1 all exhaust ther budgets; f a player expends postve effort, then the players n hs group who have hgher valuatons than hmself exhaust ther budgets. I also showed that f some player, say player t, expends zero effort, then all the players whose valuatons for the prze are less than player t's expend zero effort. ased on these, I clamed that low-valuaton players free rde on hgh-valuaton players' contrbutons. Fnally, I showed that overall, n the case where the players are budgetconstraned, more players exert postve effort and thus the free-rder problem s allevated, compared wth the basc model n whch the players have no budget constrants. In Secton 4, I ponted out that a group's equlbrum effort level obtaned n the modfed model may be greater than that obtaned n the basc model. Ths leads to an nterestng queston: How does the equlbrum total effort level obtaned n the modfed model compare to that obtaned n the basc model? The answer depends on specfc

23 22 contest success functons used and the players' budget constrants mposed. I leave t for future research.

24 23 Footnotes 1. A contest s a stuaton n whch ndvdual players or groups compete wth one another by expendng rreversble effort or resources to wn a prze. Due to ther prevalence and mportance n economes, contests have been studed by many economsts. Examples nclude Tullock (1980), Rosen (1986), Dxt (1987), Hllman and Rley (1989), Ntzan (1991), Che and Gale (1998, 2003), Clark and Rs (1998), Hurley and Shogren (1998), Moldovanu and Sela (2001), aye and Hoppe (2003), Szymansk (2003), and Konrad (2004). 2. Throughout the paper, when I use and j at the same tme, I mean that Á j. 3. A specfc form of the functon p s p( X1,..., X n) œ XÎS f S 0 1În f S œ 0, where S n X. Ths specfes that group 's probablty of wnnng the prze s equal to jœ1 j ts effort level dvded by the groups' total effort level f the total effort level s postve, and t s equal to 1/ n f all the groups expend zero. Ths smplest logt-form contest success functon s extensvely used n the lterature on the theory of contests. Examples nclude Tullock (1980), Hllman and Rley (1989), Ursprung (1990), Ntzan (1991), Che and Gale (1997), ak and Lee (2001), Szymansk (2003), and Sten and Rapoport (2004). 4. I assume that the strategy profle at whch all the players expend zero effort does not consttute a Nash equlbrum, whch mples that some player has an ncentve to expend postve effort when the other players expend zero effort.

25 24 5. If the group has more than one hghest-valuaton player, then multple Nash equlbra exst (see part ( b) of Proposton 1). In ths case, there are equlbra at whch some, but not all, hghest-valuaton players n the group expend zero effort. 6. Olson (1965, pp ) consders a stuaton n whch a collectve good s provded for a small unequal group by ts members. y unequal group he means "group composed of members of greatly dfferent sze or nterest n the collectve good." Olson argues that "there s a systematc tendency for 'explotaton' of the great by the small," whch means that "the largest member, the member who would on hs own provde the largest amount of the collectve good, bears a dsproportonate share of the burden of provdng the collectve good." 7. If a group has just one hghest-valuaton player, then the equlbrum effort level of the hghest-valuaton player, too, s equal to hs equlbrum effort level obtaned n the reduced n-player contest. 8. Proposton 1 establshes that each group's equlbrum effort level s equal to ts player-1-best response to the other groups' equlbrum effort levels. Hence, f a group's equlbrum effort level s postve, then the gross margnal payoff for the hghest-valuaton players n the group must equal the margnal cost at the group's equlbrum effort level. q 1 9. ecause b X ( ; v ) holds, group 's effort level at the group--specfc k X q 1 equlbrum s less than group 's player-( q 1)-best response. 10. Recall from Lemma 5 that q s an nteger between 2 and, nclusve, such that X ( X ; v ) b and X ( X ; v ) Ÿ b hold. q 1 q 1 k q k q m

26 Part ( d) mples that f player 1 n a group has a budget greater than or equal to the group's player-2-best response to the other groups' equlbrum effort levels, then the players n the group whose valuatons for the prze are less than the hghest one expend zero effort.

27 26 References ak, Kyung Hwan. "Effort Levels n Contests: The Publc-Good Prze Case." Economcs Letters, 1993, 41(4), pp ak, Kyung Hwan; Km, In-Gyu and Na, Sunghyun. "ddng for a Group-Specfc Publc-Good Prze." Journal of Publc Economcs, December 2001, 82(3), pp ak, Kyung Hwan and Lee, Sanghack. "Strategc Groups and Rent Dsspaton." Economc Inqury, October 2001, 39(4), pp aye, Mchael R. and Hoppe, Hedrun C. "The Strategc Equvalence of Rent-Seekng, Innovaton, and Patent-Race Games." Games and Economc ehavor, August 2003, 44(2), pp ergstrom, Theodore; lume, Lawrence and Varan, Hal. "On the Prvate Provson of Publc Goods." Journal of Publc Economcs, February 1986, 29(1), pp Che, Yeon-Koo and Gale, Ian. "Rent Dsspaton When Rent Seekers Are udget Constraned." Publc Choce, July 1997, 92(1-2), pp Che, Yeon-Koo and Gale, Ian L. "Caps on Poltcal Lobbyng." Amercan Economc Revew, June 1998, 88(3), pp Che, Yeon-Koo and Gale, Ian. "Optmal Desgn of Research Contests." Amercan Economc Revew, June 2003, 93(3), pp Clark, Derek J. and Rs, Chrstan. "Competton over More Than One Prze." Amercan Economc Revew, March 1998, 88(1), pp Djkstra, ouwe R. "Cooperaton by Way of Support n a Rent Seekng Contest for a Publc Good." European Journal of Poltcal Economy, November 1998, 14(4), pp Dxt, Avnash. "Strategc ehavor n Contests." 1987, 77(5), pp Amercan Economc Revew, December

28 27 Hllman, Arye L. and Rley, John G. "Poltcally Contestable Rents and Transfers." Economcs and Poltcs, Sprng 1989, 1(1), pp Hurley, Terrance M. and Shogren, Jason F. "Effort Levels n a Cournot Nash Contest wth Asymmetrc Informaton." Journal of Publc Economcs, August 1998, 69(2), pp Katz, Elakm; Ntzan, Shmuel and Rosenberg, Jacob. "Rent-Seekng for Pure Publc Goods." Publc Choce, Aprl 1990, 65(1), pp Konrad, Ka A. "The Strategc Advantage of eng Poor: Prvate and Publc Provson of Publc Goods." Economca, February 1994, 61, pp Konrad, Ka A. "ddng n Herarches." European Economc Revew, December 2004, 48(6), pp Marx, Lesle M. and Matthews, Steven A. "Dynamc Voluntary Contrbuton to a Publc Project." Revew of Economc Studes, Aprl 2000, 67(2), pp Moldovanu, enny and Sela, Aner. "The Optmal Allocaton of Przes n Contests." Amercan Economc Revew, June 2001, 91(3), pp Ntzan, Shmuel. "Collectve Rent Dsspaton." Economc Journal, November 1991, 101(409), pp Olson, Mancur. The logc of collectve acton: Publc goods and the theory of groups. Cambrdge, MA: Harvard Unversty Press, Raz, Khald; Shogren, Jason F. and Johnson, Stanley R. "A General Model of Rent Seekng for Publc Goods." Publc Choce, March 1995, 82(3-4), pp Robledo, Julo R. "Strategc Rsk Takng When There Is a Publc Good To e Provded Prvately." Journal of Publc Economcs, March 1999, 71(3), pp Rosen, Sherwn. "Przes and Incentves n Elmnaton Tournaments." Amercan Economc Revew, September 1986, 76(4), pp

29 28 Sten, Wllam E. and Rapoport, Amnon. "Asymmetrc Two-Stage Group Rent-Seekng: Comparson of Two Contest Structures." Publc Choce, January 2004, 118(1-2), pp Szymansk, Stefan. "The Economc Desgn of Sportng Contests." Journal of Economc Lterature, December 2003, 41(4), pp Tullock, Gordon. "Effcent Rent Seekng," n James M. uchanan, Robert D. Tollson, and Gordon Tullock, eds., Toward a theory of the rent-seekng socety. College Staton, TX: Texas A&M Unversty Press, 1980, pp Ursprung, Henrch W. "Publc Goods, Rent Dsspaton, and Canddate Competton." Economcs and Poltcs, July 1990, 2(2), pp Varan, Hal R. "Sequental Contrbutons to Publc Goods." Journal of Publc Economcs, February 1994, 53(2), pp Vcary, Smon. "Jont Producton and the Prvate Provson of Publc Goods." Journal of Publc Economcs, February 1997, 63(3), pp