Tullock Contests with Asymmetric Information

Transcription

1 Tullock Contests with Asymmetric Information E. Einy y, O. Haimanko y, D. Moreno z, A. Sela y, and B. Shitovitz x September 203 Abstract We show that under standard assumptions a Tullock contest with asymmetric information has a pure strategy Bayesian Nash equilibrium. Next we study common-value Tullock contests. We show that in equilibrium the expected payo of a player is greater or equal to that of any other player with less information, i.e., an information advantage is rewarded. Moreover, if there are only two players and one of then has an information advantage, then in the unique equilibrium both players exert the same expected e ort, although the less informed player wins the prize more frequently. These latter properties do not extend to contests with more than two players. Interestingly, players may exert more e ort in a Tullock contest than in an all-pay auction. JEL Classi cation: C72, D44, D82. Moreno gratefully acknowledges nancial support from the Ministerio de Ciencia e Innovación, grant ECO y Department of Economics, Ben-Gurion University of the Negev. z Departamento de Economía, Universidad Carlos III de Madrid. x Department of Economics, University of Haifa.

2 Introduction In a Tullock contest see Tullock (980) each player s probability of winning the prize is the ratio of the e ort he exerts and the total e ort exerted by all players. Baye and Hoppe (2003) have identi ed a variety of economic settings (rent-seeking, innovation tournaments, patent races) which are strategically equivalent to a Tullock contest. Tullock contests also arise by design, e.g., in sport competition, internal labor markets. A number of studies have provided an axiomatic justi cation to such contests, see, e.g., Skaperdas (996) and Clark and Riis (998)). There is an extensive literature studying Tullock contests, and its variations, with complete information. Perez-Castrillo and Verdier (992), Baye Kovenock and de Vries (994), Szidarovszky and Okuguchi (997), Cornes and Hartley (2005), Yamazaki (2008) and Chowdhury and Sheremeta (2009) study existence and uniqueness of equilibrium. Skaperdas and Gan (995), Glazer and Konrad (999), Konrad (2002), Cohen and Sela (2005) and Franke et al. (20) look into the e ects of changes in the payo structure on the behavior of players, and Schweinzer and Segev (202) and Fu and Lu (203) study optimal prize structures. See Konrad (2008) for a general survey. Tullock contests with asymmetric information, the topic of the present paper, have been seldom investigated in the literature see Fey (2008) and Wasser (203) for a recent analysis of rent-seeking games under incomplete information. In our setting, each player s value for the prize as well as his cost of e ort depend on the state of nature. The set of states of nature is nite. Players have a common prior belief, but upon the realization of the state of nature, and prior to taking action, each player observes some event that contains the realized state of nature. The information of each player at the moment of taking action is described by a partition of the set of states of nature. A contest is therefore formally described by a set of players, a probability space describing players uncertainty and their prior belief, a collection of partitions of the state space describing the players information, a collection of state-dependent functions describing the players values and costs, and a success function specifying the probability distribution that is used to allocate the prize for each pro le of e orts. This representation is equivalent to Harsanyi s model of Bayesian games that uses players types see Jackson (993) and Vohra (999). (In a similar framework, Einy et al. (200, 2002), Forges and Orzach (20), and Malueg

3 and Orzach (2009, 202) study common-value rst- and second-price auctions.) We show that if players cost functions are continuously di erentiable, strictly increasing and convex with respect to e ort, then a Tullock contest has a pure strategy Bayesian Nash equilibrium. Our existence result applies regardless of whether players have private or common values, or whether their costs of e ort is the same or di erent, and makes no assumptions about the players private information. Moreover, our result extends to a general class of Tullock-like contests in which success function is given by the ratio between the score given to a player s e ort and the total scores given to all players, provided each player s score function is strictly increasing and concave. Our proof of existence of equilibrium involves constructing a sequence of equilibria of contests obtained from the original Tullock contest by truncating the action space so that it is a closed and bounded interval, whose lower bound approaches zero from above. We show that any limit point of a sequence of equilibria of these contests, which have an equilibrium by Nash s (existence) theorem, is an equilibrium of the original Tullock contest. A key step in the proof is to show that in any such limit point the total e ort exerted by players is positive in every state of nature. To our knowledge, there are no previous results in the literature on Tullock contests establishing existence of equilibrium by appealing to Nash s theorem. Next we study Tullock contests in which players have a common value for the prize and a common state independent linear cost function, to which we refer simply as common-value Tullock contests. We show that in any equilibrium of a commonvalue Tullock contests, the payo of a player is greater or equal to that of any other player with less information. Thus, a common-value Tullock contests rewards any information advantage. This result, which is a direct implication of a theorem of Einy, Moreno and Shitovitz (2002) showing that in any Bayesian Cournot equilibrium of an oligopolistic industry a rm s information advantage is rewarded, is established by observing the formal equivalence between a common-value Tullock contest and an oligopoly with asymmetric information. We then proceed to study other properties of the equilibria of common-value The payo functions in the truncated contests are continuous and concave in players own strategies, which allows the use of the Nash s theorem. However, it cannot be applied to the original, untruncated, contest, in which payo s have a discontinuity when all e orts are equal to zero. 2

4 Tullock contests. We show that a two-player contest in which one of the players has an information advantage over his opponent (i.e., the partition of one player is ner than that of his opponent) has a unique (pure strategy) Bayesian Nash equilibrium, which we characterize. 2 In equilibrium both players exert the same expected e ort, although the player with less information wins the prize more frequently. We also examine how players information a ects the e ort they exert and their payo s. Assuming that the distribution of the players value for the prize is not too disperse, we show that when one player is better informed than the other the total e ort exerted by the players is smaller, and thus the share of the total surplus they capture is larger, than when both players have the same information. However, these properties of equilibrium of two-player contests do not extend to contests with more than two players. Speci cally, we construct a three-player contest in which two of the players have symmetric information, which is superior to that of the third player, in which the expected e orts exerted by players di er. We also provide an example of a contest in which a player that has an information advantage over all other players, who have symmetric information, wins the prize with a greater ex-ante probability than that of any other player. Our results for two-player common-value Tullock contests are akin to those established by Warneryd (2003) in a setting where players common value is a continuous random variable, assuming that one player observes the value precisely while the other player does not observe anything. In our framework, our results apply whenever one player s information partition is ner than that of his opponent, i.e., whenever the players information endowments can be ranked. It is unclear whether this is also the case when players common value is a continuous random variable. Finally, we study the relative e ectiveness of Tullock contests and all-pay auctions in inducing the players to exert e ort. Einy et al. (203) characterize the unique equilibrium of a two-player common-value all-pay auction, which is in mixed strategies, and provide an explicit formula that allows to compute the players total e ort. Using the results in Einy et al. (203) and our results we show that the sign of the di erence in the total e ort exerted by players in a Tullock contest and an all-pay 2 Warneryd (202) establishes existence of equilibrium when there are two types of players, and investigates which players are active, i.e., make a positive e ort, in equilibrium. 3

5 auction is undetermined, and may be either positive or negative depending on the distribution of the players value for the prize. (Fang (2002) and Epstein, Mealem and Nitzan (20) study the outcomes of Tullock contests and all-pay auction under complete information.) The rest of the paper is organized as follows: in Section 2 we describe the general setting. In Section 3 we establish that every Tullock contest has a pure strategy Bayesian Nash equilibrium. In Section 4 we study common-value Tullock contests. Section 5 studies the relative e ectiveness of common-values Tullock contests and all pay auction in inducing players to exert e ort. Long proofs are given in the Appendix. 2 Tullock Contests A group of players N = f; :::; ng; with n 2; compete for a prize by choosing a level of e ort in R +. Players uncertainty about the state of nature is described by a probability space (; p); where is a nite set and p is a probability distribution over describing the players common prior belief about the realized state of nature. W.l.o.g. we assume that p(!) > 0 for every! 2. The private information about the state of nature of player i 2 N is described by a partition i of : The value for the prize of each player i is given by a random variable V i :! R ++, i.e., if! 2 is realized then player i s ( private ) value for the prize is V i (!). The cost of e ort of each player i 2 N is described by a function c i : R +! R +, which is continuously di erentiable, strictly increasing and convex in e ort x i, and such that c i (; 0) = 0 on : A contest starts by a move of nature that selects a state! from according to the distribution p: Every player i 2 N observes the element i (!) of i which contains! the set of states of nature between which i cannot distinguish given!. Then players simultaneously choose their e ort levels (x ; :::; x n ) 2 R n +. The prize is awarded in a probabilistic fashion, according to a success function ; which attributes to each pro le of e ort levels x 2 R n + a probability distribution (x) in the n-simplex according to which the prize recipient is chosen. Hence, the payo of player i 2 N; u i : R n +! R, is given for every! 2 and x 2 R n + by u i (!; x) = i (x) V i (!) c i (!; x i ) : () 4

6 Thus, a contest is described by a collection (N; (; p); f i g i2n ; fv i g i2n ; fc i g i2n ; ): In a contest, a pure strategy of player i 2 N is a i -measurable function X i :! R + (i.e., X i is constant on every element of i ); that represents i s choice of e ort in each state of nature following the observation of his private information. We denote by S i the set of strategies of player i, and by S = n i=s i the set of strategy pro les. For any strategy X i 2 S i and i 2 i ; X i ( i ) stands for the constant value that X i () takes on i. Also, given a strategy pro le X = (X ; :::; X n ) 2 S; we denote by X i the pro le obtained from X by suppressing the strategy of player i 2 N: Throughout the paper we restrict attention to pure strategies. Let X = (X ; :::; X n ) be a strategy pro le. We denote by U i (X) the expected payo of player i, which is given by U i (X) E[u i (; (X () ; :::; X n ())]: For i 2 i ; we denote by U i (X j i ) the expected payo of player i conditional on i ; i.e., U i (X j i ) E[u i (; (X () ; :::; X n ()) j i ]: An N-tuple of strategies X = (X; :::; XN ) is a Bayesian Nash equilibrium if for every player i 2 N, and every strategy X i 2 S i U i (X ) U i (X i; X i ); (2) or equivalently, for every i 2 i : U i (X j i ) U i (X i; X i j i ) (3) 3 Existence of Equilibrium in Tullock Contests Tullock contests are identi ed by a class of success functions T such that for x 2 R n +nf0g the probability that player i 2 N wins the prize is T i (x) = x i x ; (4) where x P N k= x k is the total e ort exerted by the players. Theorem establishes existence of equilibrium in Tullock contests. 5

7 Theorem. Every Tullock contest has a (pure strategy) Bayesian Nash equilibrium. Theorem makes no assumptions about players private information, and applies regardless of whether players have private or common values, or whether their costs of e ort are the same or di erent. Theorem also implies existence of a Bayesian Nash equilibrium in a Tullock contest in the Harsanyi types model, where each player s uncertain type represents his private information, and players have a common prior distribution over all possible realizations of types. (In the discrete case, these two models of incomplete information games are equivalent see Jackson (993) and Vohra (999).) The proof of Theorem, given in the Appendix, relies on Nash s theorem. Nash s theorem, however, cannot be applied directly to a Tullock contest because the expected payo functions have a discontinuity when all e orts are equal to zero in some state of nature. Truncated contests, where players are forced to choose e orts from a compact interval with a positive lower bound, need to be considered instead then the expected payo functions are continuous and concave in players own strategies, which allows the use of the Nash s theorem to deduce the existence of equilibrium. However, in order for a limit point of the sequence of equilibria of truncated contests with a lower bound on players e orts approaching zero to be an equilibrium in the original contest the upper hemi-continuity property the expected payo functions must be continuous at the limit strategy pro le, and this holds provided the total e ort in the limit pro le is positive in all states of nature. Establishing the latter property is the crux of the proof of Theorem. A direct implication of Theorem is Corollary below, which establishes existence of equilibrium for a general class of success functions. The proof of Corollary proceeds by rst showing the existence of equilibrium in scores, and then in e orts, which is an argument familiar in the literature studying equilibrium under complete information see Corchon (2007). Existence of equilibrium in contests with success functions belonging to this class was established for the complete information case by Szidarovszky and Okuguchi (997). Theorem and Corollary extend their result into the incomplete information setting, employing a noticeably di erent method of proof. 6

8 Corollary. Every contest in which the success function is given for x 2 R n +nf0g and i 2 N by i (x) = g i (x i ) P n j= g j (x j ) ; where, for every j 2 N; his score function g j : R +! R + is a strictly increasing and concave bijection 3, has a Bayesian Nash equilibrium. Proof. Let C = (N; (; p); f i g i2n ; fv i g i2n ; fc i g i2n ; ) be a contest satisfying the assumptions of Corollary for (g ; :::; g n ). The Tullock contest (N; (; p); f i g i2n ; fv i g i2n ; fc i g i2n ; T ) where c i (; ) = c i ; g i () for every i 2 N and T (0) = (0) has an equilibrium X = (X ; :::; X n) by Theorem. It is easy to see that Y = (g X; :::; gn Xn) is an equilibrium of C: 4 Common-Value Tullock Contests Henceforth we study contests in which players have a common value for the prize and a common state-independent linear cost function, i.e., for all i 2 N; V i = V; and c i (; x) x on. We refer to these contests as common-value contests, and they are described by a collection (N; (; p); ( i ) i2n ; V; ). We say that player i 2 N has an information advantage over player j 2 N if partition i is ner than partition j : Thus, if i has an information advantage over j; then i (!) j (!) for every! 2 ; i.e., player i knows the realized state of nature with at least the same precision as player j. We begin by establishing in Theorem 2 a general property of common-value Tullock contests: these contests reward information advantage. This is a direct implication of the theorem of Einy, Moreno and Shitovitz (2002), that shows that information advantage is rewarded in any Bayesian Cournot equilibrium of a symmetric oligopolistic industry in which the rms cost function is linear. 3 Functions g j do not, in fact, need to be bijections, for our claim to hold. This can be shown using the same argument as in the proof below, provided Theorem is extended to hold for contests where the levels of e ort are restricted to be in a set [a; b) for 0 a b and b 2 R + [ fg (under the additional assumption that lim x!b c i (; x) = ). This extension of Theorem can be obtained by essentially the same proof as the one given in the Appendix. 7

9 Theorem 2. Let X = (X ; :::; X n) be any equilibrium of an n-player commonvalue Tullock contest. If player i has an information advantage over player j, then U i (X ) U j (X ). Proof. An n-player common-value Tullock contest (N; (; p); ( i ) i2n ; V ) is formally identical to an oligopolist industry (N; (; p); P; c; ( i ) i2n ); where the market demand P and the cost function c are de ned for (!; x) 2 R ++ as and P (!; x) = V (!) x ; c(!; x) = x; respectively. With this convention, the state-dependent pro t of rm i 2 N in the industry coincides with the payo of player i 2 N in the contest, i.e., for! 2 and X 2 S; u i (!; X) = V (!) P n s= X s X i (!) X i (!) = P (!; nx X s (!))X i (!) c(!; X i (!)): s= Theorem 2 then follows from the theorem of Einy, Moreno and Shitovitz (2002). 4 Next we study other properties of common-value Tullock contests. Let us index the set of states of nature as = f! ; :::;! m g: For k = ; :::; m, write and w.l.o.g. assume that p(! k ) = p k and V (! k ) = v k ; 0 < v v 2 ::: v m : We begin by studying two-player contests in which player 2 has an information advantage over player. Thus, we may assume w.l.o.g. that the only information player 4 The demand function P (!; x) is not di erentiable at x = 0 it is not even de ned and therefore does not formally satisfy the assumptions of Einy, Moreno and Shitovitz (2002). However, it is easy to see that in any equilibrium X of a common-value Tullock contest the total e ort is positive in all states of nature, i.e., X() > 0: Thus the non-di erentiability at 0 is irrelevant, and the proof of the theorem in Einy, Moreno and Shitovitz (2002) applies in this case with no change. 8

10 has about the state is the common prior belief, i.e., = fg, whereas player 2 has perfect information about the state of nature, i.e., 2 = ff! g; :::; f! m gg. In such contests a strategy pro le is a pair (X; Y ); where X can be identi ed with a number x 2 R + specifying player s unconditional e ort, and Y can be identi ed with a vector (y ; :::; y m ) 2 R m + specifying the e ort of player 2 in each of the m states of nature. Thus, abusing notation, we shall write X = x and Y = (y ; :::; y m ) whenever appropriate. The following notation will be useful in characterizing the pure strategy Bayesian Nash equilibria of a contest. For k 2 f; :::; mg write Note that A k =! p vs s=k + A = E(p V ) : 2 Lemma establishes a key property of the sequence fa k g m k= :! : (5) s=k Lemma. If k > k: p vk > A k for some k < m; then p v k > A k and A k > A k for all Proposition shows that a two-player common-value Tullock contest in which player 2 has an information advantage has a unique pure strategy equilibrium, which is calculated explicitly. Let k 2 f; :::; mg be the smallest index such that p v k > A k : Since k is well de ned. p vm > p m p vm = A m ; ( + p m ) Proposition. A two-player common-value Tullock contest in which player 2 has an information advantage has a unique Bayesian Nash equilibrium (X ; Y ) given by x = A 2 k ; and 8 < 0 if k < k yk = : p A k vk A k otherwise. 9

11 Proposition in particular implies uniqueness and symmetry of equilibrium in the complete information case, i.e., when m =. (Note that in this case k = ; and therefore y = A ( p v A ) = v =2 v =4 = A 2 = x. This result is well known in the literature.) When m >, we have p v > A = E( p V )=2 (and hence k = ) whenever the distribution of values is not too disperse; e.g., this inequality holds when v m < 4v : When this is the case, the unique equilibrium is interior. For future references we state this observation in Remark 2. Remark 2. Consider a two-player common-value Tullock contest in which player 2 has an information advantage. The unique Bayesian Nash equilibrium is interior if and only if p v > E( p V )=2, i.e., the distribution of values is not too disperse. Interestingly, when one player has superior information the expected e ort exerted by players in the equilibrium of the contest is the same. Proposition 2. In a two-player common-value Tullock contest in which player 2 has an information advantage both players exert the same (expected) e ort, i.e., E(Y ) = A 2 k = x = X : (6) Hence the expected total e ort is T E = X + E(Y ) = 2A 2 k : Proof. By Proposition, E(Y ) = ys s= = A k ( p v k s=k A k ) = p A k vk A 2 k s=k = A 2 k + m X = A 2 k : s=k! s=k X m A 2 k p s s=k 0

12 In a two-player common-value Tullock contest in which player 2 has an information advantage the equilibrium probability that player wins the prize when the state is! k is k := T (x ; y k) = A 2 k + A k A 2 k p vk = A k p A k vk when k k ; whereas the probability that player 2 wins the prize is 2k = k : Thus, the larger is the realized value of the prize, the smaller (larger) is the probability that player (player 2) wins the prize, i.e., k 0 k and 2k 0 2k for k0 > k k ; with a strict inequality if v k 0 prize, the larger is the e ort of player 2, i.e., > v k : Of course, the larger is the realized value of the y k 0 = A k (p v k 0 A k ) A k ( p v k A k ) = y k: for k 0 > k k (with a strict inequality if v k 0 > v k ). Additionally, for k 0 > k k ; k 0v k 0 = A k p vk 0 A k p vk = kv k (with a strict inequality if v k 0 > v k ), i.e., the larger is the realized value of the prize, the larger is the conditional expected payo of player ; also, 2k 0v k 0 2kv k 0 2kv k (with a strict inequality if v k 0 > v k ), i.e., the larger is the realized value of the prize, the larger is the conditional expected payo of player 2. Write i = E( i ) for the ex-ante probability that player i wins the prize. Proposition 3 establishes another interesting property of equilibrium. Proposition 3. Consider a two-player common-value Tullock contest in which player 2 has an information advantage. If v < v 2 < ::: < v m, then the ex-ante probability that player wins the prize is greater than that of player 2, i.e., > 2: The next Remark 3 is a corollary of propositions and 2, and its straightforward proof if omitted. Remark 3. A two-player common-value Tullock contest in which players have symmetric information has a unique pure strategy equilibrium, in which each player exerts the same expected e ort, equal to E(V )=4:

13 The surplus captured by the players in a contest is the di erence between the expected (total) surplus E(V ) and the expected total e ort they exert. In Proposition 4 below we show that asymmetry of information leads, in an interior equilibrium, to exertion of less e ort, and therefore to the capture a greater share of surplus, compared to the symmetric information case. Proposition 4. Consider a two-player common-value Tullock contest in which player 2 has an information advantage. If v < v m and the distribution of values is not too disperse, i.e., p v > E( p V )=2, then the players exert less e ort and hence capture a greater share of the surplus than when both players have symmetric information. Proof. When player 2 has an information advantage, then p v > E( p V )=2 implies that the equilibrium is interior by Remark 2, and therefore the expected total e ort is T E = 2A 2 = E( p 2 V ) =2 by Proposition 2. When players have symmetric information the expected total e ort T E is T E = E(V )=2 by Remark 3. v < v m together with Jensen s inequality imply T E T E = E(V ) 2 E( p V ) 2 2 > 0: The following example illustrates our ndings for two-player contests. Then Example. Let m = 2, p = p; v = ; and v 2 = v; where p 2 (0; ) and v 2 (; ): Then E(V ) = p( v), E( p p V ) = p( v); A = E( p V )=2, and A 2 = p p v=( + p): If v < ( + p) 2 =p 2 ; then p v = > A and k = ; otherwise k = 2: In a Tullock contest in which player 2 observes the value but player does not, the unique equilibrium is X = A 2 ; Y = (A ( A ) ; A p v A ); and the total e ort is T E = 2A 2 = [ p( Otherwise, the unique equilibrium is p v)] 2 =2 when v < ( + p) 2 =p 2. X = A 2 2; Y = (0; A 2 p v A2 ); and the total e ort is T E = 2A 2 2 = 2p 2 v=(+p) 2 : If v < ( + p) 2 =p 2 ; then the ex-ante probability that player wins the prize is = ( p) A + p A p v = 2 p + ( p)p v p + p p v p v + p > 2 : 2

14 Otherwise, this probability is = ( p) + p A 2 p v = ( p) + p2 + p = + p > 2 : Hence, consistent with Proposition 3 the uninformed player wins the prize more frequently than the informed player. Further, if v < ( + p) 2 =p 2, then 2 [U 2 (X ; Y ) U (X ; Y )] = ( p) A ( A ) A 2 A 2 + A ( A ) + pv A p ( v A ) A 2 A 2 + A ( p v A ) p 2 = ( p) p v And if v ( + p) 2 =p 2, then > 0: 2 [U 2 (X ; Y ) U (X ; Y )] = ( p) + pv A 2 ( p v A 2 ) A 2 2 A A 2 ( p v A 2 ) = p (p(v p + ) ) > p p > 0: That is, consistent with Theorem 2, the payo of the informed player is greater or equal to that of the uninformed player. Under symmetric information the equilibrium total e ort in a Tullock contest is E(V )=2 > maxf2a 2 ; 2A 2 2g; i.e., the total e ort when player 2 has an information advantage is less than when both players have the same information. Warneryd (2003) establishes counterparts to propositions to 4 when the players common-value V is a continuous random variable, and also shows that the fully informed player obtains a greater payo than the uninformed player. This latter result also holds when V is a discrete random variable, according to Theorem 2, and in fact extends to contests with three or more players in a very general way when a player has an information advantage (not necessarily an extreme one) over some other player, then in any equilibrium the expected payo of the former is greater or equal to that of the latter. The following examples show that the properties established in propositions 2 and 3 do not extend to common-value Tullock contests with more than two players. 3

15 In Example 2 player has only the prior information whereas players 2 and 3 have complete information. In equilibrium the expected e ort of the uninformed player is below that of each of the informed players. Example 2. Consider a 3-player common-value Tullock contest in which m = 2; p = p 2 = =2, v = and v 2 = 2: Player has no information, i.e., his information partition is = f! ;! 2 g; and players 2 and 3 have complete information, i.e., their information partitions are 2 = 3 = ff! g; f! 2 gg: In the interior equilibrium of this contest, which is readily calculated by solving the system of equations formed by the players reaction functions, the e ort of player is X = 0:30899 while the e orts of players 2 and 3 are X 2 = X 3 = (0:20342; 0:46933). Note that X = 0:30899 < 2 (0: :46933) = E(X 2) = E(X 3); i.e., the e ort of player is less than the expected e ort of players 2 and 3. In Example 3 there is an informed player and a number of uninformed players. In equilibrium, the ex-ante probability that the informed player wins the prize is above that of the uninformed players. Thus, the natural extension of Proposition 3 to contests with more than two players does not hold. Example 3. Consider an eight player common-value Tullock contest in which m = 2; p = p 2 = =2, v = and v 2 = 2: Players to 7 have no information, i.e., their information partition is i = f! ;! 2 g for i 2 f; :::; 7g; and player 8 is completely informed, i.e., his information partitions is 8 = ff! g; f! 2 gg: This contest has a (corner) equilibrium given by X = ::: = X 7 = 0:555; X 8 = (0; 0:38694) : In equilibrium, the ex-ante probability that player i 2 f; 2; :::; 7g wins the prize is i = 2 ( 7 + 0:55 5 7(0:55 5) + 0: ) = 0:243; whereas the ex-ante probability that player 8 win the prize is 8 = 7(0:243) = 0:309: Thus, the informed player wins the prize more frequently than an uninformed player. 4

16 5 Common-Value Tullock and All-Pay Auction Contests Contests that arise in many economic and political applications are e ectively all-pay auctions either by design (e.g., sports or political competition) or by the nature of the problem (e.g., patent races). Here we study whether the players expected total e ort in all pay auctions and Tullock contests can be ranked. A common-value all-pay auction is a common-value contest in which the success function is given for x 2 R n + by AP A (x) = =m(x) if x i = maxfx j g j2n ; and AP A (x) = 0 otherwise, where m(x) = jk 2 N : x k = maxfx j g j2n j. Einy et. al. (203) show that in the unique equilibrium of a two-player common-value all-pay auction in which v < ::: < v m and player 2 observes the value while player does not, the players total expected e ort is! Xs T E AP A = 2 p k v k + 2 v s = 2 s= k= s= X p k v k + s k= p 2 sv s : Hence the di erence between total e orts in an all-pay auction and a Tullock contest is := T E AP A T E = 2 s= X p k v k + s k= s= p 2 sv s 2A 2 k : For simplicity, consider the case where there are only two states of nature, i.e., m = 2: If the equilibrium of the Tullock contest is interior, then = 2p p 2 v + p 2 v + p 2 2v 2 2A 2 = 2p p 2 v + p 2 v + p 2 2v 2 2 p p p 2 v + p 2 v2 4 = 2p p 2 v + 2 (p p p v p 2 v2 ) 2 > 0: Hence an all-pay auction generates more e ort that a Tullock contest. However, if the Tullock contest has a corner equilibrium, then = 2p p 2 v + p 2 v + p 2 2v 2 2A 2 2 = 2p p 2 v + p 2 v + p 2 2v 2 2 p p 2 2 v2 ( + p 2 ) 2 = p v ( + p 2 ) p 2 2 2v 2 ( + p 2 ) 2 : 5 s=

17 We show that may be either positive or negative depending on the distribution of the players common value. Consider the environment described in Example. In an all pay auction in which player 2 observes the value but player does not, the equilibrium total e ort is T E AP A = 2 ( p) p + ( p) 2 + p 2 v = ( p) ( + p) + p 2 v: As we have shown above, if v < ( + p) 2 =p 2 ; then the expected total e ort in the unique equilibrium of the Tullock contest, which is interior, is T E = 2A 2 2 = [ p( p v)] 2 =2: Hence T E AP A T E = ( p) ( + p) + p 2 v ( p( 2 p v)) 2 > 2p > 0: However, if v ( + p) 2 =p 2 ; then the expected total e ort in the unique equilibrium of the Tullock contest, which is a corner equilibrium, is T E = 2p 2 v= ( + p) 2 : Assume that p = =4. Then T E AP A T E = v: Hence T E AP A < T E for v > 375=7. Therefore the level of e ort generated by these two contests cannot be ranked in general. 6 Appendix Proof of Theorem. Let C = (N; (; p); f i g i2n ; fv i g i2n ; fc i g i2n ; T ) be a Tullock contest. Since the cost function of each player is strictly increasing and convex in the player s e ort, it follows from () that there exists Q > 0 such that u i (; x) < 0 for every i 2 N and every x 2 R n +; provided x i Q: For any 0 < " < Q consider a variant of the contest, denoted by C " ; in which the e ort set of each player i is restricted to be the bounded interval ["; Q] : In C " ; the set of strategies of player i, S i;", is identi able with the compact set ["; Q] i via the the bijection x i! (x i ( i )) i 2 i. Player i s expected payo function U i is continuous on S " = n i=s i;" (since the success function in (4) is continuous if e orts are restricted to ["; Q]), and it is concave in i s own strategy (as the state-dependent payo function u i (; x) is concave in the variable x i if e orts are restricted to ["; Q]). Nash s theorem thus guarantees 6

18 existence of a Bayesian Nash equilibrium in C " ; pick one such equilibrium and denote it by X " = (X ;"; :::; X n;"). We show that lim inf "!0+ X " () > 0: Indeed, suppose to the contrary that there is a vanishing positive sequence f" k g k= such that and x! 2 such that lim min k!!2 X " k (! ) = min!2 X " k (!) = 0; (7) X " k (!) (8) for in nitely many k (and thus, w.l.o.g., for every k). Since the expected payo of player i is negative in every state of nature when x i = Q; for any su ciently small " k the equilibrium strategy X i;" k satis es X i;" k () < Q: Thus, for a given i 2 i ; X i ( i ) 2 [" k ; Q): Additionally, X i and X i ( i ) can both be viewed as the argument of the function U i (X i;" k ; X i j i ); since X i ( i ) is the only numerical input needed to determine the conditional expected payo of player i given i ; when the equilibrium strategies of players other than i are X i;" k. Since the equilibrium strategy X i;" k (local) maximizer of U i (X i;" k ; X i j i ) by (3), du i (X i;" k ; X i ; j i ) dx i ( i ) 0: Xi ( i )=Xi;"k ( i) is a That is, de[u i (; X i;" k () ; X i ( i ) j i ] dx i ( i ) 0; Xi ( i )=Xi;"k ( i) or, equivalently, dui (; X i;" E k () ; Xi;" k ( i )) j i 0: dx i Using (4) and () we calculate the derivative explicitly, " V i () Xi;" E k ( i ) V i () d X " k () X " k () 2 c i ; Xi;" dx k ( i ) # j i 0: i Thus E Vi () X " k () d dx i c i ; X i;" k ( i ) j i X i;" k ( i ) E " # V i () X " k () 2 j i 0; 7

19 which leads to X i;" k ( i ) Vi () d E c X " i ; Xi;" k () dx k ( i ) j i i " # : (9) V i () E X " k () 2 j i Inequality (9) holds, in particular, for i = i (! ) : Since X i;" k (! ) = X i;" k ( i (! )) (as, by de nition,! 2 i (! )); (9) yields Vi () d E c X Xi;" k (! " i ; Xi;" ) k () dx k (! ) j i (! ) i " # : (0) V i () E X " k () 2 j i (! ) Summing over i 2 N we obtain Vi () d E c nx X X " k (! i ; Xi;" " ) k () dx k (! ) j i (! ) i " # ; i= V i () E X " k () 2 j i (! ) or (since X " k (! ) n" > 0) X "k (! ) E nx X " k () V i() X "k (! ) d dx i c i ; Xi;" k (! ) j i (! ) " # : () X "k (! ) 2 E X " k () 2 V i() j i (! ) i= By the de nition of X " k (! ) (see (8)), 0 X " k (! ) X " k (!) for every! 2 : Hence we assume w.l.o.g. (by moving to a subsequence if necessary) that the limit a (!) = lim k! X "k (! ) X " k (!) exists for every! 2 : Note also that a (!) = for! =!, which occurs with positive probability by our assumption on p; and thus " # X lim E "k (! ) 2 k! X " k () 2 V i() j i (! ) = E a () 2 V i () j i (! ) > 0: (2) 8

20 Also, (7) and (8) imply lim E k! " X " k (! ) dc i ; Xi;" k (! ) # j i (! ) = 0: (3) dx i Taking limit of the right-hand side of (), which exists by (2) and (3), we get nx i= E[a () V i () j i (! )] E[a () 2 V i () j i (! )] : Furthermore, as 0 a () 2 a () ; we obtain nx i= E[a () V i () j i (! )] E[a () 2 V i () j i (! )] n: Since by assumption n 2; we have reached a contradiction. indeed, lim inf "!0+ This proves that, X " () > 0: (4) Now let f" k g k= be a vanishing positive sequence such that the limit X i (!) lim k! X i;" k (!) exists for every i 2 N and! 2 : (Such a sequence exists since all X i;" (!) belong to the compact interval [0; Q]:) Obviously, X = (X ; :::; X n) constitutes a strategy pro le in the contest C; and it follows from (4) that X () > 0: (5) We show that X is a Bayesian Nash equilibrium of C. Since the state-dependent payo function u i (; x) is continuous at any point x with x > 0, for every i 2 N, every i 2 i ; and every sequence fy k g k=0 of strategy pro les such that Y 0 () > 0 and Y i;0 (!) = lim k! Y i;k (!) for every i and!; we have lim U i(y ;k ; :::; Y n;k j i ) = U i (Y ;0 ; :::; Y n;0 j i ): (6) k! Since every X " is a Bayesian Nash equilibrium in C " ; for every su ciently large k and every strategy X i of player i satisfying 0 < X i () Q we have U i (X " k j i) U i (X i;" k ; X i j i ): (7) 9

21 Applying the limit as k! to both sides of inequality (7), it follows from (6) (and the fact (5)) that U i (X j i ) U i (X i; X i j i ) (8) for every strategy X i of player i satisfying 0 < X i () Q and every i 2 i : It is easy to see that lim inf x i!0+ U i(x i; x i j i ) U i (X i; 0 j i ); where x i > 0 (respectively, x i = 0) is identi ed with a strategy of i for which X i ( i ) = x i (respectively, X i ( i ) = 0): Thus (8) in fact holds for every strategy X i satisfying 0 X i () Q (i.e., the deviations of i may be zero at some states of nature). Finally, note that player i can improve upon any strategy X i for which X i (!) > Q at some! by lowering the e ort on i (!) to zero and thus receiving non-negative expected payo conditional on i (!): Thus, in contemplating a unilateral deviation from X i ; player i is never worse o by limiting himself to strategies X i satisfying 0 X i () Q: But this implies that (8) holds for every strategy X i 2 S i : Since this is the case for every i 2 N, we have shown that X is a Bayesian Nash equilibrium of C. Proof of Lemma. Assume that p v k > A k for some k < m: We show that p v k > A k for all k > k: Suppose not; let ^k > k be the rst index k > k such that for p v k A k : Note that v^k v^k and p v^k > A^k imply + s=^k A p v^k > = = + + s=^k + s=^k s=^k A p v^k p^k p v^k A A^k p^k p v^k p vs p^k p v^k s=^k which contradicts the assumption that p v^k A^k: 20 A A^k;

22 Now we show that A k > A k for all k > k: Suppose not; let ~ k > k be the rst index k > k such that A k A k : Since p v ~k > A ~k (as we have just shown), + Hence + s= ~ k s= ~ k i.e., 0 Thus, A k A + Proof of Proposition. A A ~k = s= ~ k p vs = p ~k p v~k + 0 p vs s= ~ k > p ~k A ~k + 0 A A ~k p ~k A ~k + s= ~ k 0 A A ~k + s= ~ k s= ~ k s= ~ k A A ~k : > A ~k ; which contradicts the choice of ~ k. A A ~k : A A ~k ; Let (X; Y ); where X = x and Y = (y ; :::; y m ), be a Bayesian Nash equilibrium, whose existence is guaranteed by Theorem. We show that x > 0: If x = 0; then T 2 (0) = ; since otherwise player 2 does not have a best response against x = 0: But then y = y 2 = ::: = y m = 0; and therefore player can pro tably deviate by exerting an arbitrarily small e ort " > 0: Hence x > 0: Moreover, y k > 0 for some k 2 f; :::; mg since otherwise x > 0 is not a best response of player : Since x > 0 maximizes player s payo s= v s x x + y s! x = y s v s (x + y s ) 2 = 0: (9) And since y s maximizes player 2 s payo in state! s given y s x v s y s = v s x + y s (x + y s ) 2 0; (20) (with equality if y s > 0) for each s = ; :::; m. s= 2

23 Notice next that if y k > 0 for some k < m; then y k 0 > 0 for all k 0 > k: Since x > 0; if y k > 0 then y k = p x p p v k x by (20), and since vk 0 v k for all k 0 > k; p p p x vk 0 x > 0, i.e., x v k 0 > 0; x 2 for all k 0 > k: Then y k 0 = 0 would violate inequality (20) for s = k 0 : Hence y k 0 > 0: Let k be the smallest index such that y k > 0: Thus, (9) implies y s v s (x + y s= s ) 2 = y s v s (x + y s=k s ) 2 = ; and (20) implies y k 0 = p x p p v k 0 x > 0 for all k 0 k : Hence x = A 2 k ; y k = p A k vk A k for all k k ; and y k = 0 for all k < k : We now show that k = k, which establishes that the pro le (x ; y ; :::; y m) identi ed in Proposition is the unique equilibrium. Assume rst that k < k : Then p v k A k since k is the smallest index such that p v k > A k ; and hence y k = p x p p p v k x = Ak vk A k 0; a contradiction as yk > 0 by the de nition of k : Assume next that k > k : In this case, y k = 0: Since p v k > A k ; by Lemma and therefore x v k x 2 This stands in contradiction to (20), as y k conclude that indeed k = k. A 2 k > A2 k = x; (2) = A2 k A 4 k v k A 2 k > 0: = 0 by the de nition of k (> k ): We Proof of Proposition 3. Let us be given a two-player common-value Tullock contest in which player 2 has an information advantage over player. Given (y k ; ::::; y m ) 2 R m k + + de ne the function p 2 (y k ; :::; y m ) := k=k p k y k y k + P m s=k y s : Hence, recalling (6), 2 = p 2 (y k ; :::; y m). We show that a maximum point y of p 2 on K = f(y k ; ::::; y m ) 2 R m k + + j y k y k+ ::: y m g must satisfy y k = ::: = y m : Hence max K p 2 = P m s=k + P m s=k 2 : (22) 22

24 Since yk < ::: < y m (the inequalities are strict, which follows from our assumption that v < v 2 < ::: < v m and the expressions for (y k )m k=k given in Proposition ), (22) implies which establishes Proposition 3. 2 = p 2 (y k ; :::; y m) < max K p 2 =2; Di erentiating p 2 with respect to y k for k 2 fk ; :::; mg we p k = p k t=k ;t6=k p t y t (y k + P m s=k y s ) 2 t=k ;t6=k p t y t (y t + P m s=k y s ) 2! : (23) For every (y k ; :::; y m ) 2 K such that y k < y k + ::: y m p 2 =@y k (y) > 0; and therefore necessarily y k = y k +. Suppose now that it has already been shown that y k = y k + = ::: = y k ; m k > : We show that y k+ = y k as well. Indeed, if y k = y k + = ::: = y k < y k+ ::: y m, then by (23) we obtain p 2 =@y k (y) > 0; a contradiction. Thus y k = ::: = y m : 23

25 References [] Baye, M., Hoppe, H.: The strategic equivalence of rent-seeking, innovation, and patent-race games. Game and Economic Behavior 44(2), (2003). [2] Baye, M., Kovenock, D., de Vries, C.: The solution to the Tullock rent-seeking game when r > 2: mixed-strategy equilibrium and mean dissipation rates. Public Choice 8, (994). [3] Chowdhury, S., Sheremeta, R.: Multiple equilibria in Tullock contests. Economics Letters, 2, (20). [4] Clark, D., Riis, C.: Contest success functions: an extension. Economic Theory, (998). [5] Cohen, C., Sela, A.: Manipulations in contests. Economics Letters 86, (2005). [6] Corchon, L.: The Theory of Contests: A Survey. Review of Economic Design, (2007). [7] Cornes, R., Hartley, R.: Asymmetric contests with general technologies. Economic Theory 26, (2005). [8] Einy, E., Haimanko, O., Orzach, R., Sela, A.: Dominant strategies, superior information and winner s curse in second-price auctions. International Journal of Game Theory 30, (200). [9] Einy, E., Haimanko, O., Orzach, R., Sela, A.: Dominance solvability of secondprice auctions with di erential information. Journal of Mathematical Economics 37, (2002). [0] Einy, E., Haimanko, O., Orzach, R., Sela, A.: Common-value all-pay auctions with asymmetric information. Working paper, Ben-Gurion University of the Negev (203). [] Einy, E., Moreno, D., Shitovitz, B.: Information advantage in Cournot oligopoly, Journal of Economic Theory 06, 5-60 (2002). 24

26 [2] Epstein, G., Mealem Y., Nitzan S.: Political culture and discrimination in contests. Journal of Public Economics 95(-2), (20) [3] Fang, H.: Lottery versus all-pay auction models of lobbying. Public Choice 2, (2002). [4] Fey, M.: Rent-seeking contests with incomplete information. Public Choice 35(3), (2008). [5] Forges, F., Orzach, R.: Core-stable rings in second price auctions with common values. Journal of Mathematical Economics 47, (20). [6] Franke, J., Kanzow, C., Leininger, W., Schwartz, A.: E ort maximization in asymmetric contest games with heterogeneous contestants. Economic Theory, forthcoming (20). [7] Fu, Q., Lu, J.: The optimal multi-stage contest. Economic Theory 5(2), (202). [8] Glazer, A., Konrad, K.: Taxation of rent-seeking activities. Journal of Public Economics 72, 6-72 (999). [9] Jackson, M.: Bayesian implementation. Econometrica 59, (993) [20] Konrad, K.: Investment in the absence of property rights: the role of incumbency advantages. European Economic Review 46(8), (2002). [2] Konrad, K.: Strategy and dynamics in contests. Oxford University Press, Oxford (2008). [22] Malueg, D., Orzach, R.: Revenue comparison in common-value auctions: two examples. Economics Letters 05, (2009). [23] Malueg, D., Orzach, R.: Equilibrium and revenue in a family of common-value rst-price auctions with di erential information. International Journal of Game Theory 4(2), (202). [24] Nitzan, S.: Modelling rent-seeking contests. European Journal of Political Economy 0(), 4-60 (994). 25

27 [25] Perez-Castrillo, D., Verdier, T.: A general analysis of rent-seeking games. Public Choice 73, (992). [26] Schweinzer, P., Segev, E.: The optimal prize structure of symmetric Tullock contests. Public Choice 53(), (202). [27] Skaperdas, S.: Contest success functions. Economic Theory 7, (996). [28] Skaperdas, S., Gan, L.: Risk aversion in contests. Economic Journal 05, (995). [29] Szidarovszky, F., Okuguchi, K.: On the existence and uniqueness of pure Nash equilibrium in rent-seeking games. Games and Economic Behavior 8, (997). [30] Tullock, G.: E cient rent-seeking, in J.M. Buchanan, R.D. Tollison and G. Tullock (Eds.), Toward a theory of rent-seeking society. College Station: Texas A.&M. University Press (980). [3] Vohra, R.: Incomplete information, incentive compatibility and the core. Journal of Economic Theory 54, (999). [32] Yamazaki, T.: On the existence and uniqueness of pure-strategy Nash equilibrium in asymmetric rent seeking contests. Journal of Public Economic Theory 0, (2008). [33] Warneryd, K.: Information in con icts, Journal of Economic Theory 0, 2 36 (2003). [34] Warneryd, K.: Multi-player contests with asymmetric information, Economic Theory 5: (202). [35] Wasser, C.: Incomplete information in rent-seeking contests. Economic Theory, 53: (203). 26