Relative Di erence Contest Success Function

Transcription

1 Relative Di erence Contest Success Function Carmen Beviá Universitat Autònoma de Barcelona and Barcelona GSE Luis C. Corchón Universidad Carlos III de Madrid First version January 20 th, This version August 29 th, Abstract In this paper we present a Contest Sucess Function (CSF) which is homogeneous of degree zero and where the probabilities of winning the prize depend on the relative di erence of e orts. With two agents, we present a necessary and su cient condition for the existence of a Nash Equilibrium in pure strategies. This equilibrium is unique and interior. This condition does not depend on the size of the valuations as in absolute di erence CSF. We prove that several properties of Nash equilibrium and Leader-Follower equilibrium with the Tullock CSF still hold in our framework. Finally, we consider the case of n players, generalize the previous condition and show that it is su cient for the existence of a unique Nash equilibrium in pure strategies. This paper is dedicated to Gordon Tullock whose insightful work has being inspirational for both authors. We thank to Matthias Dahm, Joerg Franke, Magnus Ho mann, Guillaume Roger, Santiago Sanchez-Pages, Marco Serena, Anil Yildizparlak and the participants in a seminar in the SAET 2012 congress, Brisbane, for comments that improved substantially the quality of the paper. The rst author acknowledges nancial support from ECO and FEDER, SGR and Barcelona GSE research network. The second author acknowledges nancial support from ECO Both authors acknowledge the hospitality of the Australian School of Business during the revision of the paper. Electronic copy available at:

2 1. Introduction The concept of a Contest Success Function (CSF) plays a crucial role in the analysis of contests. This function encapsulates how the contest is played and what the consequences of the contest are. Thus the outcome of a contest is highly sensitive to the form of the CSF. There are two main families of CSF. The CSF proposed by Tullock (1980) (axiomatized by Skaperdas (1996)) in which the probabilities of winning the prize depend on relative e orts, and the di erence form proposed by Hirshleifer (1989) (also axiomatized by Skaperdas (1996), see, in addition, Baik (1998) and Che and Gale (2000)) in which the probabilities of winning the prize depend on the di erence between e orts. Both formulations have properties that one would like to have in an ideal CSF. The Tullock form is homogeneous of degree zero, so the units in which e orts are measured do not count. The Hirshleifer form captures "the tremendous advantage of being even just a little stronger than one s opponent" (Hirshleifer, 1991, p. 131). However this advantage cannot be independent of the size of the con ict. One more battalion is a great advantage in a battle of two battalions against one but of less help when two hundred battalions ght one hundred and ninety nine. Thus di erences must be scaled up to the size of the con ict. In this paper we present the Relative Di erence CSF (RDCSF) in which the probability of getting the prize depends on the di erence between e orts and is also homogeneous of degree zero. When there are two players, the RDCSF is a linear function of the di erence between a player e ort and the e ort of the other player weighted by a positive number, divided by the sum of e orts. When the weight of the e ort of the other player is zero we have the Tullock CSF, so the RDCSF generalizes the latter. The RDCSF is presented in Section 2 for the case of two players. In Section 3 we tackle the two players case. We present a necessary and su cient condition for the existence of a Nash equilibrium in pure strategies. This equilibrium is unique and all players make a positive e ort. This condition is satis ed when the sum of the intercept and the slope of the linear function de ning the RDCSF add up to less than one. We show that in equilibrium the player having a higher valuation of the prize regards e orts as strategic complements, whereas the player having a lower valuation regards e orts as strategic substitutes. This was shown by Dixit (1987) assuming either a di erence form or a logit form (a generalization of the Tullock CSF). Finally we consider the case in which players may move sequentially. Firstly, we compare the equilibrium 2 Electronic copy available at:

3 values of e orts and probabilities in the sequential model with those when moves are simultaneous. This allows us to see the impact of the change in the structure of moves on equilibrium. Secondly, we consider an endogenous order of moves, as in Baik and Shogren (1992) and Leininger (1993) again for logit or absolute di erence CSF. We show that the RDCSF replicates earlier ndings in these two issues, in particular the player with the smaller valuation has incentives to move rst. In Section 4 we consider the case of n players. The extension of the functional form presented in Section 2 to more than two players is not straightforward. First one has to choose a functional form that generalizes the one chosen before. But the main problem is to guarantee that this chosen functional form represents probabilities, i.e. yields non negative values and adds up to one. We solve this problem by introducing a rationing rule which gives priority to the players making the largest expenses. Probabilities are assigned until they add up to one. In some cases a player might be given zero probability of winning the contest despite the fact she is making a positive e ort but of course this cannot happen in equilibrium. With this procedure in hand we show that a Nash equilibrium exists and that this equilibrium is unique and all players make a positive e ort. Our results obtained in Sections 3 and 4 on the existence of a Nash equilibrium contrast with those obtained for Hirshleifer type CSF, where at most one contestant makes a positive e ort in any Nash equilibrium in pure strategies and a pure strategy Nash equilibrium does not exists when the value of the prize is su ciently large (see Hirshleifer (1989), Baik (1998) and Che and Gale (2000)). These two properties restrict severely the range of application of this kind of CSF. Finally Section 5 o ers some comments about possible extensions of our work. We are only aware of a CSF which depend on di erences and it is homogeneous of degree zero, the Serial Contest proposed by Alcalde and Dahm (2007). But this CSF does not generalize Tullock s CSF. A recent paper by Hwang (2012) present a CSF which depends on di erences and such that in the limit when a parameter tends to one, becomes the Tullock CSF. But this CSF is not of a di erence form and homogeneous of degree zero at the same time. 3

4 2. The Relative Di erence Contest Success Function We start with the case of two players to gain intuition on the relative contest success function. In Section 4 we present the generalization for the n players case. We motivate our CSF by means of the following properties. Our CSF should be based on a function, denoted by f i (), which depend on the di erence between weighted e orts G 1 and sg 2 where s is a positive number. If s = 1 the function depends on the di erence G 1 is to write f i ((G 1 G 2. We also want our CSF to be homogeneous of degree zero. A possibility sg 2 )=G j ) as in Alcalde and Dahm (2007) where j could be 1 or 2 but in this case we have to rede ne the function each time G j is zero. Instead, we divide the di erence by the aggregate e ort G 1 + G 2 so we have to rede ne our CSF only when both e orts are zero. Thus, f i () is written as f i ((G 1 sg 2 )=(G 1 + G 2 )). Next we take f i () to be the simplest possible form, namely linear. Last, we assume symmetry, so the intercept and the slopes of f 1 () and f 2 () are identical. A function with all these properties must have the following form: f i (G i ; G j ) = + G i sg j P 2 k=1 G ; if k 2X G k 6= 0; i; j 2 f1; 2g; i 6= j; (2.1) k=1 f i (G i ; G j ) = 1 2X 2 if G k = 0; i; j 2 f1; 2g; i 6= j: (2.2) k=1 with > 0 to guarantee that f i is increasing in G i and 2 [0; 1] to have f i (sg j ; G j ) 2 [0; 1]: When = s = 0 and = 1 we get the Tullock CSF. 1 The part (2.2) takes care of the case in which all e orts are zero and it is the usual expedient when de ning the Tullock CSF. In the sequel we will call the function de ned in (2.1) and (2.2) the notional CSF. For f 1 () and f 2 () to be a real CSF the two properties below must hold. a) f i (G) 2 [0; 1] for i 2 f1; 2g; and b) P 2 i=1 f i(g) = 1: where G = (G 1 ; G 2 ). Conditions a) and b) are satis ed in some special cases, i.e. when = s = 0 and = 1, i.e. when the notional CSF is the Tullock CSF. But in general, we need extra conditions to convert (2.1)-(2.2) into a CSF. To take care of part a) we de ne our CSF as min(max(f i (G); 0); 1): This minmax operator is used by Che and Gale (2000) with the same purpose we do, namely, to (5.1). 1 Actually it generalizes the Tullock CSF in ratio form but the generalization to general Tullock CSF is simple, see 4

5 guarantee that winning probabilities are between zero and one. To guarantee part b) above, we add up (2.1) over all players and nd that 2 + (1 s) = 1: (2.3) Condition (2.3) also guarantees that G 1 = G 2 implies that p 1 = p 2, i.e. that our notional CSF is symmetric. Note that this condition plus 2 [0; 1] implies 1 (1 s) 1. Now we are ready to o er our main de nition De nitio. The Relative Di erence CSF (RDCSF) for two players is de ned as follows: p i = min(max(f i (G i ; G j ); 0); 1); i; j 2 f1; 2g; i 6= j: (2.4) where 2 + (1 s) = 1. We note two things: 1. A more exact description of this CSF would be "Relative Weighted Di erence" but we chose the shorter name in the hope that this does not yield to any confusion. 2. When s = 0 (when the RDCSF becomes the Tullock CSF) the CSF in De nitio is no longer a "Relative Di erence" CSF. But we may assume that s > 0 and in this case the Tullock CSF is the limit case of the RDCSF when s! 0. We decided to retain the case s = 0 by simplicity. With the RDCSF, a player making zero e ort might win the contest (as it happens in the absolute di erence CSF of Che and Gale (2001)). A historical example of this is when the Mongol navy was wiped out by the divine wind (kamikaze) i281 without any intervention of the Japanese eet. 2 When it is reasonable to impose that a player making zero e ort has zero probability of winning, then by (2.1), + = 1 (which given condition (2.3), is equivalent to s = 0). As we will see later this equality implies the existence of a Nash equilibrium. In Figure 1 we represent the winning probability for player 1 as a function of the relative di erences for the case s = 1: In this case condition (2.3) does not impose any restriction on : When = 0; the outcome of the contest does not depend on relative di erences. When tends to in nity the contest coincides with the all pay auction. 2 Actually, before the typhoon destroyed the mongolian ships, an inconclusive battle took place between two small forces of japanese and mongol warriors. outcome of the war. But it could be argued that this encounter did not play any role in the 5

6 Winning Probaility for 1 1 1/(2β) 1/(2β) Relative Differences Winning Probability for Player 1 as a function of the Relative Differences when α=1/2 and s=1. Figure 1 A microfoundation to the RDCSF is given using the framework in Corchón and Dahm (2010). There, CSF is derived as the choice of a utility maximizing planner whose type is unknown to the contestants. Let be the type of the planner, assumed to be a random variable uniformly distributed in [0; 1]. Let U i (; G i ) be the planner s utility if i wins the contest. U 1 (; G 1 ) (resp. U 2 (; G 2 )) is increasing (decreasing) in ; so is her degree of favoritism towards contestant 2. U i (; ) is increasing in G i since expenses are correlated with the quality of the project which in turn is positively correlated with the planner s utility. Assume that the U i () 0 s are separable in and G i so U i (; G i ) = V (F i1 (); F i2 (G i )). V () is a symmetric Cobb-Douglas function so, U i (; G i ) = F i1 ()F i2 (G i ). Take F 0 s to be the simplest possible form, namely linear and assume that the comparison between the contestants is independent of the units in which expenses are measured so it only depends on the ratio G i =G j. Assume also symmetry such as F 12 () = F 22 () and df 11 () d = df 21() d (recall that F i1 () describes how the planner is partial to contestant 2 and thus F 11 () and F 21 () cannot be identical). Thus U 1 (; G 1 ) = (a 1 b)dg 1 and U 2 (; G 2 ) = (a 2 b)dg 2. Without lost of generality take d = 1. Write U 1 (; G 1 ) = (1 + )G 1 and U 2 (; G 2 ) = (s + )G 2: (2.5) Let (G) be such that U 1 (; G 1 ) = U 2 (; G 2 ). If U 1 (; G 1 ) > U 2 (; G 2 ) (resp <) for all take (G) = 1 (= 0). Assuming that when G1 = G 2, (G) = 1=2, we obtain (2.3). Since U 1 (; G 1 ) (resp. U 2 (; G 2 )) is decreasing (increasing) in, the probability that 1 wins the contest is (G) and this yields the RDCSF. Note that when! 1, U 1 (; G 1 ) > U 2 (; G 2 ) i G 1 > sg 2 which is the 6

7 (biased when s 6= 1 and unbiased when s = 1) all pay auction. To end this section, note that the RDCSF, denoted by p 1 (G); p 2 (G), yields payo functions i (G) = p i (G)V i G i, i = 1; 2 where V i is i s valuation of the prize. A Nash equilibrium is a pair G = (G 1 ; G 2 ) such that i(g ) i (G i ; G j ) j 6= i for all G i, i = 1; Equilibrium with Two Players. In this section we study the existence and uniqueness of Nash equilibrium and the properties of equilibrium strategies and payo s in the case of two players. Later, we consider the case in which players may act in di erent moments of time and we will endogenize the order of moves. We see that the results obtained with the Tullock CSF still hold with the RDCSF Existence and Uniqueness of a Nash Equilibrium Let us de ne V i P 2 k=1 1 V k ; i 2 f1; 2g: Note that for all players > 1 and that for the player with the smallest valuation, say j, Y j 2. If valuations of the two players are identical, = 2. To prove the existence of a Nash equilibrium we need the following assumption: ( + ) (1 + s)(2 1 ), i = 1; 2: (3.1) If players are identical and given (2.3) (2 + (1 s) = 1) condition (3.1) is 2 (1 + s) or + 3=2. If = 1 and s = = 0 (the Tullock CSF), condition (3.1) is + 1= 2 which always holds. In the Appendix we show that Assumption (3.1) is implied by + 1. All these conditions say that the probability of getting the prize cannot be very sensitive to relative di erences in e orts. This is because, as we saw before, when is large, the RDCSF approaches the all pay auction in which a Nash equilibrium in pure strategies does not exist (Hillman and Riley, 1989, Baye, Kovenock and de Vries, 1996). Note that, the ratio of valuations is limited by (3.1) but this assumption says nothing about the absolute value of valuations like in the Hirshleifer type CSF studied in Baik (1998) and Che and Gale (2000). Next proposition shows the existence of a Nash equilibrium in pure strategies. Propositio. Under (3.1) there is a unique Nash equilibrium in pure strategies in which both players are active. 7

8 Proof. Let us see rst that (0; 0) cannot be an equilibrium. In (0; 0) the payo for each player is V i =2: Let " > 0 su ciently close to zero. Consider that, say, player 1 deviates and increases his e ort to ": Let G = ("; 0): In this case, f 1 (G) = + ; and f 2 (G) = s: If + 1, given that 2 + (1 s) = 1; then s 0: Thus, p 1 = 1. So, player 1 will win the contest for sure and his payo will be V 1 " which is greater than V 1 =2: If f 1 (G) = + < 1; p 1 = + : Since 1 = 2 + (1 ( + )V i " > V i =2: s) 2 + < 2 + 2; then + > 1=2; for " su ciently close to zero, To prove that there is an equilibrium with positive e ort, consider an auxiliary game with payo s FOC in an interior equilibrium are ^ i = ( + G i sg j P 2 k=1 G )V i G i ; for all i 2 f1; 2g: (3.2) k (1 + s)g j ( P 2 k=1 G k) 2 V i 1 = 0, i 2 f1; 2g. (3.3) Adding up over players all equations in (3.3) and setting X P 2 k=1 G k we obtain that Plugging (3.4) in (3.3) we nd that X = (1 + s) P 2 k=1 1 V k : (3.4) G i = (1 + s)v i (1 1 ); i 2 f1; 2g. (3.5) Since > 1; G i > 0: From the de nition of probabilities we have that f i (G 1; G 2) = + (1 1 + s ); i 2 f1; 2g. (3.6) Note that f i (G 1 ; G 2 ) 0 if and only if ( + ) (1 + s); i 2 f1; 2g. (3.7) Under Assumption (3.1) condition (3.7) always holds. Furthermore, since P 2 k=1 f k(g 1 ; G 2 ) = 1; f i (G 1 ; G 2 ) 1 for all i 2 f1; 2g. Pro ts in equilibrium amount to i = V i [( + ) (1 + s)(2 1 )]; i 2 f1; 2g: (3.8) Assumption (3.1) implies that pro ts are non negative so (G 1 ; G 2 ) is an equilibrium of the auxiliary game. Next, we see that G = (G 1 ; G 2 ) is also an equilibrium of the original game. Clearly, no 8

9 player has a pro table deviation G i such that 0 f i (G i ; G i ) 1 because G is an equilibrium of the auxiliary game. In particular, Gi ~ such that f i ( G ~ i ; G i ) = 1 is not a pro table deviation. Thus, a pro table deviation G 0 i can only occur when f i(g 0 i ; G i ) > 1; or f i(g 0 i ; G i ) < 0: In the rst case, since f i is increasing in G i ; G 0 i > G ~ i ; and p i = 1: Payo at (G 0 i ; G i ) are less than payo s at ( ~ G i ; G i ): Therefore, G0 i can not be a pro table deviation. If f i(g 0 i ; G i ) < 0; p i = 0 and payo for player i in this deviation is not larger than zero. Thus, this is not a pro table deviation. Finally we will show that there cannot be an equilibrium with (G i ; 0), G i case, i = ( + )V i G i so G i > 0 cannot maximize the payo s of i. > 0. If this were the Note that, contrarily to the case in which the CSF is based on absolute di erences, the RDCSF yields a Nash Equilibrium in pure strategies in which all contestants make a positive e ort, even if they value the prize highly. Furthermore, condition (3.1) is not only su cient for the existence of equilibrium but also necessary. 3 Proposition 2. If Assumption (3.1) does not hold, there is no Nash equilibrium in pure strategies. Proof. We can not have an equilibrium with both players doing positive e ort because in this case, and given that condition (3.1) does not hold, payo s would be negative. A player by doing no e ort will be better o. Thus, if an equilibrium exists, it has to be with some player making zero e ort. Suppose G = (G i ; 0): Since condition (3.1) does not hold, it should be the case that + > 1 (otherwise condition (3.1) will hold as we prove in the Appendix). Therefore, f i (G) > 1; which implies that f j (G) < 0 for player j: Thus, p i (G i ; 0) = 1: But for any G i > 0; player i by decreasing an " his e ort will be better o. Thus, a Nash equilibrium in pure strategies does not exist. To build intuition on Propositions 1 and 2, we study the best reply functions in the case of a common valuation V. From (3.3) we obtain that G i (G j ) = q (1 + s)g j V G j : (3.9) Equation (3.9) is the best reply for player i whenever i gets non negative pro ts and f(g i (G j ); G j ) 1: If pro ts are positive at (G i (G j ); G j ); but f(g i (G j ); G j ) > 1; then the best reply for player i is to chose G i such that f(g i ; G j ) = 1: If pro ts are negative at (G i (G j ); G j ) then the best reply for 3 See Malueg and Yates (2005, 2006) for su cient conditions for the existence of a Nash equilibrium when the CSF is homogeneous of degree zero. 9

10 player i is G i = 0: We draw the best reply of each player for two cases. First when a Nash equilibrium in pure strategies exists and second when it does not. To simplify calculations we take V = 1; = 1=2; and s = 1. Given the condition (3.1), under these parameters a Nash equilibrium in pure strategies exists i 1: In the rst case we take = 1 and in the second case we take = 1:5: In Figures 2 and 3, the solid line represents the best reply for player 1 and the dash line the best reply for player 2. The curvilinear line in both gures corresponds to the part of the best reply characterized by the rst order condition as in (3.9). The straight lines with positive and nite slope correspond to the case where the best reply is determined by the probability of winning being equal to 1: And nally, the dashed and the solid lines on the axes corresponds to the case where pro ts are negative at (G i (G j ); G j ) with G i (G j ) determined by (3.9) and therefore the best reply is zero e ort. Note that the best reply are neither convex value nor continuous, even though they are upper-hemi-continuous. G Figure 2. Beta equals 1 G1 10

11 G G1 Figure 3. Beta equals 1.5 To close this section we analyze some properties of the Nash equilibrium. First, from (3.5) we obtain that the player with the highest valuation exerts the highest e ort: G i > G j, V i > V j, i; j = 1; 2: (3.10) Second, the player having a higher valuation of the prize regards e orts as strategic complements, whereas the player having a lower valuation regards e orts as strategic substitutes. Dixit (1987) proved an identical result assuming either a di erence form or a logit form (a generalization of the Tullock CSF). This point is easy to show by computing the slope of best reply. Totally di erentiating the rst order condition (3.3) we obtain 2 i 2 dg i i (G i ) dg j = 0: (3.11) j From (3.11) it follows that the sign of the slope of the best reply is the sign 2 i (G)=@G j. From (3.3) we obtain 2 i (G) = (1 + s)(g i G j j (G i + G j ) 3 which evaluated in equilibrium yields the desired 2 i (G j > 0, G i > G j, i; j = 1; 2: (3.12) Finally, the player with the highest valuation gets the highest winning probability. From (3.6) p i > p j, V i > V j, i; j = 1; 2: (3.13) By considering (3.10), (3.12) and (3.13) we obtain the following result: 11

12 Proposition 3. In equilibrium, G i > G j i (G j > 0 i p i > p j i V i > V j, i; j = 1; 2: 3.2. Leader-Follower Games We assume now that e orts are done sequentially. Firstly, we compare the equilibrium values of e orts and probabilities with those obtained in the case of simultaneous moves (Nash equilibrium) to see the impact of the change in the structure of moves on equilibrium. This was done by Dixit (1987) assuming a logit or a absolute di erence CSF. Secondly, we will consider an endogenous order of moves, as in Baik and Shogren (1992), Leininger (1993). They show that in the subgame perfect Nash equilibrium where in the rst stage players decide if they spend e ort "early" or "late" and in the second stage they exert e ort according to the order decided in the previous stage, the player with the lower valuation plays "early". We will see that the RDCSF replicates the earlier ndings in these two issues. A leader-follower equilibrium is a pair of e orts such that given the e ort of the leader, the e ort of the follower maximizes her payo s and the e ort of the leader maximizes her payo s taking into account the best reply of the follower. In a sequential game in which the leader moves in the rst stage, a leader-follower equilibrium is a subgame perfect Nash equilibrium. We denote by j the leader and by i the follower. To prove the existence of a subgame perfect Nash equilibrium we need the following assumptions: 3=2; (or equivalently, 2V i > V j ); (3.14) ( + )( 1) (1 + s)(1 1 ); (3.15) 4( 1) (1 + s) ( + ) 1: (3.16) 4( 1) If both players have the same valuation, condition (3.14) always holds because = 2: Given that 2+(1 s) = 1; condition (3.16) also holds and nally condition (3.15) is equivalent in this case to 2 (1 + s); which it coincides with condition (3.1) of existence of equilibrium in the simultaneous game of the previous section. Proposition 4. Under conditions (3.14), (3.15), and (3.16) a leader-follower equilibrium in pure strategies exists in which both players make a positive e ort. 12

13 Proof. Let us rst nd the best reply function of i, the follower. From FOC of payo maximization we obtain that G i = q (1 + s)v i G j G j. (3.17) Plugging this function into the payo s of the leader we obtain that j = ( + G j(1 + s) s p (1 + s)v i G j p )V j (1 + s)vi G j G j : (3.18) This expression can be simpli ed as follows FOC of payo maximization are j = ( s)v j + s G j (1 + s) V i V j G j : (3.19) s 1 (1 + s) V j 2 G j V i 1 = 0: (3.20) Clearly, SOC of payo maximization hold. Thus, in a leader-follower equilibrium G S j = (1 + s)v 2 j 4V i = (1 + s)v j 4( 1) : (3.21) Plugging the value of G S j into the best reply function of i we obtain the equilibrium value of G i, namely G S i = (1 + s)v j (2V i V j ) = (1 + s)v j (2 4V i 4 1 ): (3.22) ( 1) The follower s e ort is positive from 3=2. Total e ort in equilibrium, denoted by X S is The probability that in equilibrium j and i win is X S = (1 + s)v j : (3.23) 2 p S j = + 2V i (V j (1 + s) 2sV i ) = (3.24) = ( s) + (1 + s) 2( 1) : (3.25) p S i = + (2V i V j (1 + s)) = (3.26) 2V i (1 + s) = + 2( 1) : (3.27) Condition (3.16) guarantees that p S j 0; which implies that ps i that p S i 0 which implies that p S j 1: 1; and condition (3.15) guarantees 13

14 Finally, equilibrium payo s are: (1 + s) j = V j (( s) + ): (3.28) 4( 1) (1 + s) i = ( + 2( 1) )V (1 + s)v j 1 i (2 4 ( 1) ) = (3.29) (1 + s) = V j ( + 2( 1) )(Y (1 + s) 1 i 1) (2 4 ( 1) ) = (3.30) 1 = V j ( + )( 1) (1 + s)(1 4( 1) ) : (3.31) Condition (3.16) guarantees that j 0; and condition (3.15) guarantees that i 0: Remark 1. Note that if both players have the same valuation, the leader-follower equilibrium coincides with the Nash equilibrium of the simultaneous game. Aggregate e ort is smaller in the leader follower game than in the simultaneous game if and only if the leader is the player with the smallest valuation. Next we consider endogenous moves. The game is as follows. In the rst period players decide if they spend e ort early or late. In the second period they exert e ort. If both choose the same period the e orts correspond to the (simultaneous) Nash equilibrium. If, say, player 1 chooses early and player 2 chooses late the e orts are those corresponding to leader-follower equilibrium with 1 as a leader, etc. We will say that the player with the smaller (resp. larger) prize value will be called "the underdog" (resp. "the favorite"). We have the following: Proposition 5. In equilibrium, the favorite moves after the underdog. than it is in the Nash Equilibrium of the simultaneous-move game. Aggregate e ort is less Proof. Virtually identical to Baik and Shogren (1992). First note that, after lengthy calculations it can be shown that G S j > G j, V j > V i so as in Dixit (1987) the favorite "has incentive to overcommit e ort" (p. 891). Next, the expected payo of any player is decreasing when we move along her best reply e orts with respect to the e ort of the other player. From these two results we see that to play early for the underdog is a dominant strategy. Given this the favorite plays late. Finally from (3.23) and (3.4) we have that X S > X, V j > V i 14

15 as desired. 4. The Case of n Players. In this section we tackle the case of n players. First we generalize the CSF de ned in Section 2. The construction of a CSF for more than two players based on the idea of relative di erences requires some extra work with respect to the two person case because the minmax operator introduced there (see (2.4)) does not guarantee that the sum of the intended probabilities add up to one. 4 We introduce an extra device which guarantees that and show, that under a condition which is a generalization of assumption (3.1), a Nash equilibrium exists and is unique The Relative Di erence CSF in the case of n Players. Let G = (G 1 ; :::; G n ): The notional CSF (2.1) can be generalized in several ways. In order to keep as close as possible to the case of n = 2 we propose the following function: P f i (G) = + G j6=i i s G j P n j=1 G ; i 2 f1; ::; ng if j nx G j 6= 0; (4.1) f i (G) = 1 nx n if G j = 0; i 2 f1; ::; ng: (4.2) j=1 We require that for all G; P n i=1 f i(g) = 1: Adding up (4.1) over players the following condition must hold j=1 n + (1 s) = 1: (4.3) This condition is a generalization to n players of (2.3). The Tullock CSF arises when = 0, = 1 and s = 0. Note that the notional CSF is anonymous, whenever G i = G j ; f i (G) = f j (G): Based on (4.1), we de ne a contest success function using the following auxiliary function: i(g) = min(max(f i (G); 0); 1): (4.4) Unfortunately, when n > 2, the minmax operator does not guarantee that P i(g) = 1 so an extra construction is needed. In particular let the players with the largest expenses be given the probabilities corresponding to (4.4). Once probabilities add up to one the remaining players are 4 We note that the di culties of extending CSF for more than two players are shared by other CSF like the di erence of Baik (1998) and Che and Gale (2000) or the serial contest of Alcalde and Dahm (2007). 15

16 given zero probability. In order to formally introduce the CSF we need some extra notation. Given G; a type i player is a player that exercise e ort G i : Let k i be the number of players of type i; and let l i be the last player of type i: De nition 2. The Relative Di erence CSF for any number of players is de ned as follows: Given G, order the type of players by decreasing order of G i. Let m be the type of player such that P lm j=1 j(g) 1 and P l m+1 j=1 j(g) > 1: Then, p j (G) = j(g); for all j 2 f1; ::; l m g; (4.5) p m+1 (G) = (1 where n + (1 s) = 1. l m X j=1 j(g))=(k m+1 ); for all players of type m + 1; and (4.6) p h (G) = 0 for all h 2 fl m+1 + 1; ::; ng: (4.7) We may regard conditions (4.5), (4.6) and (4.7) as a rationing rule: players are rationed depending on their expenditures. Agents with the largest expenses are given priority and their probabilities of winning the contest are given by (4.4). Agent with less expenses might be given zero probability of winning despite the fact that (4.4) would recommend a positive probability. This rationing rule re ects the competitive aspect of the contest by rewarding players who make more e ort. Note that if G 6= (0; ::; 0) and 0 f i (G) 1 for all i 2 f1; ::; ng; then p i = f i (G); and if G = (0; ::; 0); p i = 1=n for all i 2 f1; ::; ng: For the case of n = 2; the CSF (4.5) can be written as: p i = i(g); and p j = 1 i(g); (4.8) because whenever f i (G) 0; then f j (G) 1 (given that condition (4.3) is satis ed). So in this case the extra conditions (4.5), (4.6) and (4.7) are not needed Nash Equilibrium with n Heterogeneous Players As in Section 3, let V i P n j=1 1 V j : To prove the existence of a Nash equilibrium we need the following assumption which is a generalization of (3.1): ( + ) (n 1 + s)(2 ), i 2 f1; ::; ng: (4.9) When all players have identical valuations, condition (4.9) collapses in n (n 1 + s) which when either n is su ciently large or s = 1, is just 1. In the Appendix we show that condition (4.9) is 16

17 implied by + 1. The latter implies that when all players but i make zero e ort, the notional CSF yields a number for player i less or equal than one. In order to simplify the presentation, we focus on Nash equilibria where all players exert a positive e ort. In order to guarantee this we make the following assumption: > (); i 2 f1; ::; ng (4.10) which, of course, holds in the symmetric case where = n for all i and when n = 2. An identical assumption guarantees that when the CSF is of the Tullock type, all players are active in equilibrium, see Franke et al. (2012), Theorem 2.2. Now we have the following. Proposition 6. Under Assumption (4.9) and Assumption (4.10) there is a Nash equilibrium in pure strategies with all players active. Proof. Note rst that (0; 0; ::; 0) can not be an equilibrium. In (0; :::; 0) the payo for each player is V i =n: Let " > 0 su ciently close to zero. And consider that player i deviates and increases his e ort, G i = ": Let G = (0; ::; "; ::; 0): In this case, f i (G) = + : If + 1, given that n + (1 s) = 1; then (s)=() 0: Thus, p i = 1. So, player i wins the contest for sure and her payo will be V i " which is greater than V i =n: If f i (G) = + < 1; p i = + : Since 1 = n + (1 ( + )V i " > V i =n: s) n + < n + n; then + > 1=n; for " su ciently close to zero, Consider now the auxiliary game where the payo s of each player are de ned by FOC in an interior equilibrium are P j6=i G j ^ i = ( + G i s P n j=1 G j )V i G i ; i 2 f1; ::; ng, (4.11) Pj6=i G j(1 + s ) ( P n j=1 G j) 2 V i 1 = 0: (4.12) We readily see that the second order conditions hold so payo s of i are concave in G i. Write (4.12) as follows: Adding up (4.13) over all players and simplifying we obtain that P j6=i G j ( P n j=1 G j) 2 = 1 V i (1 + s (4.13) ): nx G j = j=1 ()(1 + s ) P n j=1 1 V j : (4.14) 17

18 Now realize that (4.13) can be written as P n j=1 G j G i ( P n j=1 G j) 2 = and taking into account (4.14) we obtain that 1 V i (1 + s (4.15) ); G i = s ()(1 + )V i (1 ): (4.16) Since Assumption (4.10) requires > ; for all i 2 f1; ::; ng; G i 0: From the de nition of the notional CSF f i we have that in the Nash equilibrium P n f i (G ) = + G i (1 + s ) s j=1 G j P n j=1 G j = (4.17) = + Taking into account (4.15) we have that G P i n j=1 G j (1 + s ) s : (4.18) G i P n j=1 G j = 1 () ; (4.19) and we obtain f i (G ) = + (1 + (1 () )(1 + s ) s = (4.20) ()(1 + s ) ): (4.21) Note that f i (G ) 0 if and only if (+) ()(1+ s ). But, given assumptions (4.9) and (4.10), this condition always hold. Furthermore, since P n i=1 f i(g) = 1, we have that f i (G ) 1. Finally, pro ts in equilibrium amount to i = ( + (1 which can be rewritten as: ()(1 + s ) ()(1 + s ))V )V i i (1 ); (4.22) i = ( + )V i ()(1 + s n 1 )V i ()(1 + s )V i (1 ); (4.23) i = V i (( + ) ()(1 + s s ) ()(1 + )(1 ); (4.24) i = V i (( + ) ()(1 + s )(2 )): (4.25) 18

19 Assumption (4.9) implies that pro ts are non negative so G = (G 1 ; ::; G n) is an equilibrium of the auxiliary game. Let us nally see that G is also an equilibrium of the original game. Clearly, no player have a pro table deviation G i such that 0 f i (G i ; G i ) 1 because G is an equilibrium of the auxiliary game. In particular, Gi ~ such that f i ( G ~ i ; G i ) = 1 is not a pro table deviation. Thus, a pro table deviation G 0 i can only occur when f i(g 0 i ; G i ) > 1; or f i(g 0 i ; G i ) < 0: In the rst case, since f i is increasing in G i ; G 0 i > G i : Furthermore since player i has incresed hihs e ort, the f j of the other players is reduced, thus f j (G 0 i ; G i ) f j(g ) 1: Therefore, by the rationing rule p i = 1: Thus, payo at (G 0 i ; G i ) are less than payo s at ( G ~ i ; G i ): Therefore, G0 i can not be a pro table deviation. If f i (G 0 i ; G i ) < 0; p i = 0 and then payo for player i in this deviation can not be greater than zero. Thus, this is not a pro table deviation. Note that when all players have identical valuations, V; aggregate equilibrium e orts are ( + s)v n : (4.26) Assumption (4.9) implies that aggregate e ort is less or equal than V. Thus, unless n = (n 1+s) (which happens if s = = 1) rents are not fully dissipated, as it happened in the two person case. Again, aggregate e ort is minimized when s = 0 which includes the Tullock CSF (where in addition = 0) in which total e ort is V=2. Comparing Proposition 6 with the results obtained in the two person case, we note that assumption (4.9) is only a su cient condition for the existence of equilibrium. Nash equilibrium is unique. Next we show that Proposition 7. If + > 1 there is at most a Nash equilibrium which is interior. Proof. Suppose that there is a Nash equilibrium with G i = 0 some i. Then, p i = s = + 1 < 0: (4.27) Since probabilities add up to one, this implies that some active agent, say j, is rationed. Assume that G j 6= G r for any r. But then j can decrease in nitesimally G j, obtain the prize with the same probability and increase payo s. Assume that there are r and j with G r = G j. But player j by increasing in nitesimally G j takes all the probability allocated to r so we have a contradiction. 19

20 We now tackle the case + 1. A coalition C is a subset of the set of players. Let c be the number of agents in coalition C. Consider the following condition: For any coalition C and player i =2 C, V i X j2c 1 V j > c 1: (4.28) Lemma 1 shows that condition (4.28) is always satis ed whenever assumption (4.10) holds. Lemma 1. If Assumption (4.10) holds for any coalition C and player i =2 C, V i Pj2C 1 V j > c 1: Proof. Suppose that condition (4.28) does not hold. Then, there is a coalition C and a player i =2 C such that V i Pj2C 1 V j c 1: In particular, for any player k =2 C such that V k V i ; V k Pj2C 1 V j c 1: Let k min =2 C be such that V kmin V k for all k =2 C: Let us see that Y kmin in contradiction with the assumption of the Proposition. Given that Y kmin = V kmin P n j=1 1 V j ; we can rewrite Y kmin as: Y kmin = V kmin X j2c 1 X 1 + V kmin : (4.29) V j V k Given that V kmin P j2c 1 V j c 1; and V kmin P k =2C 1 V k n c; Y kmin : We are now ready to prove the counterpart of Proposition 7 for the case + 1: k =2C Proposition 8. If Assumption (4.10) holds and + 1 there is at most a Nash equilibrium which is interior. Proof. Suppose we have a Nash equilibrium in which only agents 1; 2; :::; k are active. First we show that player i = k + 1; ::::n is not rationed. Indeed p i = s = 1 0: (4.30) Thus player i maximizes, at least in a neighborhood of the Nash equilibrium, over the notional CSF. The FOC of payo maximization for i is Now computing the expenses made by active players we see that i = (1 + P k i j=1 G V i 1: (4.31) j kx s (k 1)(1 + G j = P ) k (4.32) j=1 1 V j j=1 20

21 and thus and applying condition (4.28) we obtain i = V P k i j=1 1 V i k 1 1 i > k i k 1 1 = 0 (4.34) which contradicts that the best reply of i is zero. 5. Conclusions In this paper we have presented a new CSF, the Relative Di erence CSF (RDCSF), which combines properties of the Tullock and the Hirshleifer CSF s. The RDCSF is analytically tractable yielding closed form solutions that generalize those of the Tullock CSF. When the valuations of both players are not very di erent and the impact of relative di erences on the winning probability is not large, all contestants make a positive e ort in a Nash equilibrium. Recall that in the Hirshleifer CSF this only happen when the value of the prize times the impact of absolute di erences on the winning probability is less than one. The RDCSF looks like a mild extension of the Tullock CSF. However new issues surface. 1. To convert the notional RDCSF into a RDCSF. In the case of two players this is accomplished with the minmax operator like in Gale and Che (2000). But with more than two players the minmax operator is not enough and we have to introduce a full- edged rationing rule that distributes the available probabilities among players. 2. Nash equilibrium not always exists. In the case of two players we present a necessary and su cient condition for the existence of equilibrium. In the case of more than two players we present a su cient condition for the existence of equilibrium. We show that equilibrium is always unique. A drawback of the RDCSF is that it is not continuous when aggregate e ort tends to zero, but this is a general property of all homogeneous of degree zero CSF (see Corchón (2000)). We hope that the RDCSF can be useful in the analysis of con icts in which there is an advantage of being stronger but this advantage must be weighted by a measure of how large the con ict is. 21

22 Since many papers in this area used the Tullock CSF, the use of the RDCSF would provide a generalization of this work. Moreover, from the point of view of revenue maximization, the RDCSF provides a simple CSF for which, for some parameters, full rent dissipation occurs for any number of players. The RDCSF is amenable to certain extensions. A simple one would be based on the following form: f i (G) = + G i sg j P 2 j=1 G j ; i = 1; 2, 2 [0; 1]: (5.1) It is easy to show that the methods used in this paper su ce to show the existence of a Nash equilibrium when the CSF is of the (5.1) form. An alternative to the RDCSF with n players presented in this paper would be the following form: f i (G) = + G i s max j6=i G P j n j=1 G ; i 2 f1; ::; ng. (5.2) j in which we substract from G i not the weighted average of the actions of all other competitors but only that of the best competitor. When players are homogeneous both formulations yield identical results but not when players are heterogeneous. References [1] Alcalde J. and M. Dahm (2007) Tullock and Hirshleifer: a meeting of the minds. Rev. Econ. Design (2007) 11, 2, [2] Baik KH (1998) Di erence-form contest success functions and e ort level in contests. Eur J Polit Eco4, [3] Baik, Kyung Hwan and Jason F. Shogren, Strategic Behavior in Contests: Comment," American Economic Review, 1992, 82 (1), [4] Baye, M., D. Kovenock and C. G. de Vries (1996). "The all-pay auction with complete information". Economic Theory, 8, [5] Corchón L. (2000) On the allocative e ects of rent-seeking. J Public Econ Theory 2, 4, [6] Corchón, L. and M. Dahm (2010). "Foundations for contest success functions," Economic Theory, 43, 1, 81-98, 22

23 [7] CheY-K and I. Gale (2000) Di erence-form contests and the robustness of all-pay auctions. Games Econ Behav 30, [8] Dixit, Avinash, Strategic Behaviour in Contests, American Economic Review, 1987, 77 (5), [9] Hillman, A. L. and Riley, J. G. (1989). Politically contestable rents and transfers. Economics and Politics 1, [10] Hirshleifer J (1989) Con ict and rent-seeking success functions: ratio vs. di erence models of relative success. Public Choice 63, [11] Hirshleifer J (1991) The Technology of Con ict as an Economic Activity. The American Economic Review, 81, 2, Papers and Proceedings, [12] Hwang, S.-H. (2012). "Technology of military con ict, military spending, and war". Journal of Public Economics, 96, [13] J. Franke, C. Kanzow, W. Leininger and A. Schwartz (2012). "E ort Maximization in Asymmetric Contests Games with Heterogeneous Contestants" Economic Theory, forthcoming. [14] Leininger, Wolfgang, More E cient Rent-Seeking - A Munchhausen Solution, Public Choice, 1993, 75, [15] Malueg D and A Yates (2005). "Equilibria and comparative statics in two-player contests" European Journal of Political Economy, 21, 3, [16] Malueg D and A Yates (2006). "Equilibria in rent-seeking contests with homogeneous success functions". Econ Theory, 27, [17] Nitzan, Shmuel, More on E cient Rent Seeking and Strategic Behavior in Contests: Comment, Public Choice, 1994, 79, [18] Skaperdas S (1996) Contest success functions. Econ Theory 7: [19] Tullock G (1980). "E cient rent-seeking". In: Buchanan JM, Tollison RD, Tullock G (eds) Towards a theory of a rent-seeking society. Texas A&M University Press, College Station, pp

24 6. Appendix We show that the su cient condition for the existence of a Nash equilibrium (and necessary in case of two players) is implied by the inequality + 1. Recall that the su cient condition is ( + ) (n 1 + s)(2 ), i 2 f1; ::; ng: (6.1) The minimum of the left hand side of (6.1) is achieved when = ( + s) + and the corresponding value of the left hand side of (6.1) is 2 ( + s) (( + s)() Thus the the left hand side of (6.1) is always positive if (n (4.3) (i.e. n + (1 s) = 1) is ) s which taking into account 24