Common-Value All-Pay Auctions with Asymmetric Information

Transcription

1 Common-Value All-Pay Auctions with Asymmetric Information Ezra Einy, Ori Haimanko, Ram Orzach, Aner Sela July 14, 014 Abstract We study two-player common-value all-pay auctions in which the players have ex-ante asymmetric information represented by nite partitions of the set of possible values of winning. We consider a family of such auctions in which no player has an information advantage over his opponent. We nd su cient conditions for the existence of equilibrium in monotonic strategies. We further show that the ex-ante distribution of equilibrium e ort is the same for every player (and hence the players expected e orts are equal), although their expected payo s are di erent and they do not have the same ex-ante probability of winning. Keywords: Common-value all-pay auctions, asymmetric information. JEL Classi cation: C7, D44, D. Ezra Einy, Ori Haimanko, Aner Sela: Department of Economics, Ben-Gurion University of the Negev, Beer-Sheva 4105, Israel; their respective addresses are: einy@bgu.ac.il, ori@bgu.ac.il, anersela@bgu.ac.il. Ram Orzach: Department of Economics, Oakland University, Rochester, MI 4309, USA; orzach@oakland.edu 1

2 1 Introduction All-pay auctions are used in diverse areas of economics, such as lobbying in organizations, R&D races, political contests, promotions in labor markets, trade wars, and biological wars of attrition. In an all-pay auction each player submits a "bid" (i.e., exerts e ort) and the player with the highest bid wins the contest. However, regardless of who the winner is, each player bears the cost of his bid. All-pay auctions have been studied when players have either private or common values of winning. 1 In this paper we focus on commonvalue all-pay auctions and consider contests with two ex-ante asymmetrically informed players where the value of winning is identical for both players in the same state of nature, but the information about which state of nature was realized is di erent. We assume that each player s information is represented by a nite partition of the set of states of nature that can be identi ed with the set of possible common values, but these partitions are di erent. When the state of nature is chosen, each player learns which element of partition contains the realized common-value, but the players do not necessarily know the exact value of winning the contest. 3 This model captures situations in which winning a contest is of similar bene t to each contestant, but the precise value of winning, which depends on several random parameters, may be unknown. In our model of asymmetric information we assume that the information sets of each player are connected with respect to the value of winning the contest (see Einy et al. (001, 00) and orges and Orzach (011)). This means that if a player s information partition does not enable him to distinguish between two possible values of winning, then he also cannot distinguish between all intermediate values. Connectness seems plausible in environments where the information of a player allows him to put upper and lower bounds on the actual value of winning, without ruling out any outcome within these bounds. In studying common-value all-pay auctions with asymmetric information, one particular case of interest 1 To mention just a few works, all-pay auctions have been considered by, e.g., Hillman and Riley (199), Baye et al. (1993, 1996), Amann and Leininger (1996), Che and Gale (199), Moldovanu and Sela (001, 006), Siegel (009) and Moldovanu et al. (010). This partition representation is equivalent to the more common Harsanyi-type formulation of Bayesian games (see Jackson (1993) and Vohra (1999)) 3 This framework has been used in several works to analyze common-value second-price auctions (see Einy et al. (001, 00), orges and Orzach (011), and Abraham et al. (01)), and common-value rst-price auctions (see Malueg and Orzach (009, 01)).

3 is when one player has an information advantage (henceforth, IA) over another, which means that his information partition is ner than that of his opponent. This assumption simpli es the analysis when there are only two players, since it can usually be reduced to the postulate that one player is completely informed about the state of nature, while the other player has no information about it. Signi cant results have been obtained in the IA case. Siegel (014) has shown 4 the existence of a unique equilibrium in mixed strategies in a two-player common value auction with IA, in which the ex-ante distribution of e ort is the same for both players, as is the ex-ante probability of winning the contest. In this paper we study two-player common-value all-pay auctions in which, except in the extreme states of nature (corresponding to the lowest and the highest possible values of winning), neither player has an IA over his opponent. 5 We construct a "candidate" for an equilibrium with monotonic strategies (i.e., strategies where more favorable signals do not lead to lower bids), and then nd su cient conditions for the "candidate" to be a true equilibrium. Our results show that, although the players have asymmetric strategies that yield di erent expected payo s as well as di erent chances to win the contest unlike in the IA case, the ex ante distributions of both players e orts are the same in common with the IA case. Thus the asymmetry of information between the players manifests itself not in unequal expected e orts, but in their chances to win the contest and the allocation of payo s between the players (since in our model no player has an inherent advantage over his rival, the expected payo of a player may be higher or lower than that of his opponent 6 ). We also show that a player s expected payo, conditional on an element of his partition, monotonically increases with the values of winning. Thus, higher/lower values in the information set of a player lead to a higher/lower conditional expected payo. As was already mentioned, our work is related to the recent research of Siegel (014), who studies general asymmetric two-player all-pay auctions with interdependent valuations, where private information of each player is represented by a nite set of possible types. He shows that a unique equilibrium exists in his set-up, and provides an algorithm to calculate the equilibrium strategies. Siegel makes strong assumption 4 See Section of the online appendix to his work. 5 Malueg and Orzach (009) studied the rst-price auction with information partitions of this type. 6 As Siegel (014) has shown, in the unique equilibrium of the IA case, the expected payo of the uninformed player is zero, while the expected payo of the informed payer is positive. 3

4 ("Condition M") of joint monotonicity of conditional densities and valuations, that presupposes the common prior distribution of type pro les having full support. Thus, given his information, a player cannot rule out any speci c type of his rival. In contrast, in our model, given one player s information there is always some information set of the other player which is ruled out. The paper is organized as follows. In Section we present the model. In Section 3 we give a numerical example that demonstrates how to nd an equilibrium in our model. In Section 4 we nd su cient conditions for the existence of equilibrium and explicitly describe such an equilibrium. In Section 5 we analyze the players expected e orts and payo s. Section 6 concludes. Some of the proofs are in the Appendix. The model Consider the set N = f1; g of two players who compete in an all-pay auction where the player with the highest e ort (output) wins the contest, but all the players bear the cost of their e ort. The uncertainty in our model is described by a nite set of states of nature and a probability distribution p over which can be interpreted as the common prior belief about the realized state of nature (w.l.o.g. p(!) > 0 for every! ): A function v :! R + represents the common value of winning the contest, i.e., if! is realized then the value of winning is v(!) for every player. The private information of each player i N is described by a partition i of : A common-value all-pay auction starts when nature chooses a state! form according to the distribution p: Each player i is informed of the element i (!) of i which contains! (thus, i (!) constitutes the information set of player i at!), and then he chooses an e ort x i R + : The players will typically have di erent information partitions, and thus are ex-ante asymmetric. The utility (payo ) of player i N is given by the function u i : R +! R as follows: >< 1 m(x) u i (!; x) = v(!) x i; if x i = maxfx 1 ; x g; >: x i ; if x i < maxfx 1 ; x g; where m(x) denotes the number of players who exert the highest e ort, namely, m(x) = ji N : x i = maxfx 1 ; x gj. A two-player common-value all-pay auction with di erential information is fully described by and identi ed with the collection G = ((; p); u 1 ; u ; 1 ; ): 4

5 In all-pay auctions, there is usually no equilibrium in pure strategies. Thus our attention will be given to mixed strategy equilibria. A mixed strategy of player i is a function i : R +! [0; 1]; such that for every! ; i (; ) is a cumulative distribution function (c.d.f.) on R + ; and for all x R +, i (; x) is a i -measurable function (that is, i (; x) is constant on every element of i ): Slightly abusing notation, for any i i we will denote the constant value of i (; x) on i by i ( i ; x) ; whenever convenient. If player i plays a pure strategy given i ; i.e., if the distribution represented by i ( i ; ) is supported on some y R + ; we will identify between i ( i ; ) and y wherever appropriate. Given a mixed strategy pro le = ( 1 ; ), denote by E i ( ) the expected payo of player i when players use that strategy pro le, i.e., E i ( ) E( Z 1 Z u i (; (x 1 ; x ))d 1 (; x 1 )d (; x )): or i i ; E i ( i ; ) will denote the conditional expected payo of player i given his information set i ; i.e., Z 1 E i ( i ; ) E( 0 Z 1 0 u i (; (x 1 ; x ))d 1 (; x 1 )d (; x ) j i ): A pro le = (1 ; ) of mixed strategies constitutes a (Bayesian Nash) equilibrium in the commonvalue all-pay auction G if for every player i, and every mixed strategy i of that player, the following inequality holds: E i ( ) E i ( i ; i); where i denotes i s rival. 3 Example We begin with a simple example of the behavior of players in our model. Consider a two-player commonvalue all-pay auction with three states of nature. or i = 1; ; 3; state! i occurs with probability p i = 1 3 and the value for the prize in! i is v(! i ) = i: Player 1 s information partition is 1 = ff! 1 g; f! ;! 3 gg ; while player s information partition is = ff! 1 ;! g; f! 3 gg: It can be easily veri ed that the corresponding common-value all-pay auction does not have an equilibrium in pure strategies. However, there does exist a mixed strategy equilibrium. In this equilibrium, player 1 s 5

6 mixed strategy 1 is a state-dependent c.d.f. given by and 1 (f! 1 g; x) = 1 (f! ;! 3 g; x) = >< >: >< >: 0; if x < 0; x; if 0 x 1 ; 1; if x > 1 0; if x < 1 ; x 1, if 1 x 1; x ; if 1 < x 5 ; 1; if x > 5 : Player s mixed strategy is a state-dependent c.d.f. given by (f! 1 ;! g; x) = >< >: 0; if x < 0; x; if 0 x 1; 1; if x > 1 and (f! 3 g; x) = >< >: 0; if x < 1; 3 x 3 ; if 1 x 5 ; 1; if x > 5 : In order to see that the above strategies are an equilibrium, note that when player uses mixed strategy, player 1 s expected payo conditional on the event f! 1 g is E 1 (f! 1 g; x; ) = 1 x x = 0 for any e ort x [0; 1 ]. It is easy to see that e orts above 1 would result in a non-positive conditional expected payo to player 1. Thus, any e ort in [0; 1 ] is a best response of player 1 to conditional on f! 1 g. urthermore, player 1 s expected payo conditional on the event f! ;! 3 g is as follows: when 1 exerts e ort x (1; 5 ]; E 1 (f! ;! 3 g; x; ) = ( 3 x 3 ) x = 0; 6

7 and when 1 exerts e ort x [ 1 ; 1], E 1 (f! ;! 3 g; x; ) = 1 x x = 0: It is easy to see that exerting e orts below 1 or above 5 would lead to non-positive conditional expected payo s, and thus any e ort in [ 1 ; 5 ] is a best response of player 1 to conditional on f! ;! 3 g. Thus the mixed strategy 1 (supported on [0; 1 ] given f! 1g and on [ 1 ; 5 ] given f! ;! 3 g) is an (unconditional) best response of player 1 to. Similarly, when payer 1 uses mixed strategy 1, player s expected payo conditional on the event f! 1 ;! g payo is as follows: when exerts e ort x ( 1 ; 1]; and when 1 exerts e ort x [0; 1 ]; E (f! 1 ;! g ; 1 ; x) = (x 1 ) x = 0; E (f! 1 ;! g ; 1 ; x) = 1 1 x x = 0: It is easy to see that any e ort above 1 would result in a negative expected payo. Also, conditional on the event f! 3 g ; the expected payo of player when he exerts e ort x [1; 5 ] is E (f! 3 g ; x); 1 ; x) = 3 ( x ) x = 1 : Any e ort above 5 or below 1 would lead to a conditional expected payo smaller than 1. Thus the mixed strategy (supported on [0; 1] given f! 1 ;! g and on [1; 5 ] given f! 3g) is an (unconditional) best response of player to 1. Hence, the pair = ( 1 ; ) is a mixed strategy equilibrium. The ex-ante distributions of each player s equilibrium e ort are identical indeed, the ex-ante probability to exert an e ort smaller than or equal to x is the same for both players and is given by 0; if x < 0; >< (x) = >: 3x, if 0 x 1; x ; if 1 < x 5 ; 1; if x > 5 : 7

8 In particular, the expected equilibrium e orts of both players are identical and equal to :47. The expected payo of the players in are E 1 ( 1 ; ) = 0; E ( 1 ; ) = = 1 6 : The ex-ante probability of player to win is given by P = 1 3 [ Z = Z [ Z 5 1 Z x 0 Z x 1 Z x 1 ds 1dx + 1ds! 1dx Z ds 3 dx + 1dx] Thus, the ex-ante probabilities of players 1 and to win are distinct, 11 4 Z 1 1 1ds] and 13 4 respectively. In the next section we will present su cient conditions for the existence of a mixed-strategy equilibrium, and describe the monotonic equilibrium strategies in a general two-player common-value all-pay auction without information advantage. 4 Equilibrium analysis Assume that = f! 1 ;! ; :::;! n g, where n > 1 is without loss of generality an odd number. or each state of nature! i ; denote v i = v(! i ) and p i = p(! i ) > 0; (1) and assume that that the possible values are positive and ranked as follows: 0 < v 1 v ::: v n : () Also consider: Assumption1: Each partition i ; i = 1; is connected with respect to the common value function v; i.e., for every element j j ; if! i ;! i+k j and k > 1; then also! i+1 ; :::;! i+k 1 j.

9 Assumption 1 means that each information set is an "interval" in that it contains only consecutive elements of (there are no "holes"). Assumption : The partitions 1 and are overlapping, i.e., for any 1 1 and the following holds:! 1 ;! n = 1 =) 1 *! 1 ;! n = =) * 1 Assumption means that except in the extreme states of nature (containing the lowest and the highest values of winning), after observing their own signals neither player has an information advantage over his opponent. Malueg and Orzach (009) showed that the environment where the players partitions are connected and overlapping can be viewed in the following simpler but equivalent form: Proposition 1 Every common-value all-pay auction with information partitions i ; i = 1; which are connected and overlapping is strategically equivalent to a common-value all-pay auction with information partitions given by 1 = ff! 1 g; f! ;! 3 g; f! 4 ;! 5 g; :::; f! n 1 ;! n gg (3) = ff! 1 ;! g; f! 3 ;! 4 g; :::; f! n ;! n 1 g; f! n gg In accordance with Proposition 1, we will assume from now on that information partitions 1 and are given by (3), unless stated otherwise. It can be shown that some environments not satisfying Assumptions 1 and can nevertheless be reduced to strategically equivalent environments that do satisfy Assumptions 1 and. or this purpose consider the following example given by Malueg and Orzach (009). In that example = f1; ; 3; 4; 5g; 1 = ff1g; f; 3g; f4; 5gg; = ff1; g; f3g; f4g; f5gg; and v i = i for every i = 1; :::; 5: It obviously does not satisfy Assumption. However, denote e = f1; ; 3g and b = f4; 5g: It is clear that regardless of the realized state, it is common knowledge whether the state is in e or b : Note that each of e and b together with the partitions induced by 1 and satis es Assumption and therefore this environment is strategically 9

10 equivalent (conditional on a realization of either e or ) b to the common-value all-pay auctions with information partitions given by (3). Malueg and Orzach (009) provided other examples depicting environments that can be transformed into the setting analyzed in this paper. urthermore, it is worth noting that our analysis remains valid if is an in nite set of states of nature provided the partitions are nite. To see this, simply replace with a nite 0, which is the coarsest partition of that re nes all f n g nn ; and, for each 0 ; let the value of winning at, v(); be equal to the conditional expectation E(v () j ): Hence, all-pay auctions with the information structure satisfying Assumptions 1 and are suitable to analyze a larger family of all-pay auctions, with more general information partitions. In order to describe now equilibrium strategies we introduce the following notations. Denote f! i ;! i+1 g; if i = 1; ; :::; n 1; >< i f! 1 g; if i = 0; >: f! n g; if i = n: Thus, 1 consists of the sets i for every even integer 0 i n 1; and consists of the sets i for every odd integer 1 i n: Also, for every i = 1; :::; n 1; let p i;i+1 p(! i j i ) (= p i p i + p i+1 ) and p i+1;i 1 p i;i+1 be the conditional probabilities of the states! i and! i+1 given the event i : Additionally, set p n;n+1 = p 1;0 1; p n+1;n = p 0;1 0: In what follows, we describe a mixed strategy equilibrium ( 1 ; ) of the all-pay auction. Let x 0 = 0; and for every i = 1; :::; n set x i Given 0 = f! 1 g; player 1 s mixed strategy is ix p j;j 1 p j;j+1 v j : j=1 10

11 1 0 ; x = >< >: 0; if x < 0; x p 1; v 1 ; if 0 x x 1 ; 1; if x > x 1 : (4) Note that the function 1 0 ; is well de ned, strictly increasing on [x 0 ; x 1 ], continuous, 1 0 ; x 0 = 0 and 1 0 ; x 1 = 1. Thus, 1 0 ; is a c.d.f. of a continuous probability distribution supported on [x 0 ; x 1 ]. or i = 1; ; :::; n 1; given i = f! i ;! i+1 g and assuming that i j for player j, the mixed strategy of player j is j i ; x = >< >: 0; if x < x i 1 ; x x i 1 p i;i 1 v i ; if x i 1 x x i ; x+p i+1;i+ v i+1 x i+1 p i+1;i+ v i+1 ; if x i < x x i+1 ; 1; if x > x i+1 : (5) The function j i ; is well de ned, strictly increasing on [x i 1 ; x i+1 ], continuous, j i ; x i 1 = 0, j i ; x i+1 = 1 and j i ; x i = p i;i+1. Thus, j i ; is a c.d.f. of a continuous probability distribution supported on [x i 1 ; x i+1 ]. inally, given n = f! n g; player s mixed strategy is ( n ; x) = >< >: 0; if x < x n 1 ; x x n 1 p n;n 1 v n ; if x n 1 x x n ; 1; if x > x n. (6) The function ( n ; ) is well de ned, strictly increasing on [x n 1 ; x n ], continuous, ( n ; x n 1 ) = 0 and ( n ; x n ) = 1. Thus, ( n ; ) is a c.d.f. of a probability distribution supported on [x n 1 ; x n ]. Proposition Suppose that p i+1;i+ v i+1 v i 0 for every i = 1; ; :::; n 1: (7) Then, the strategy pro le = ( 1 ; ) described in (4), (5), and (6) is a mixed strategy equilibrium of 11

12 the common-value all-pay auction G: 7 Proof. See Appendix. Condition (7) is satis ed whenever each value v i+1 is su ciently above its closest predecessor v i : It is easy to see that the pair need not be an equilibrium if (7) does not hold (in particular, when v 1 = v = ::: = v n ). 5 Results In this section we derive some comparative results about the players expected payo s and e orts in equilibrium (described in (4), (5), and (6) in the previous section). The proof of Proposition (given in the Appendix) provides an explicit formula for expected equilibrium payo s of the players conditional on each event i ; for i = 0; :::; n: If j is the player for whom i j ; it follows from (11), (1), and (13) that Xi 1 E j ( i ; ) = p k;k+1 (p k+1;k+ v k+1 v k ); () k=1 when the sum in () is de ned as 0 if i 1: This yields the following immediate result that compares the players conditional expected payo s. Proposition 3 Under condition (7), 1) for every i = 1; :::; n; and player j such that i j, 9 E j ( i ; ) E j ( i 1 ; ) = p i 1;i (p i;i+1 v i v i 1 ) 0: It follows, in particular, that ) the expected payo of each player j, conditional on i j ; is increasing in i: E 1 ( 0 ; ) E 1 ( ; ) ::: E 1 ( n 1 ; ) 7 Note that (7) can be assumed to hold only for i = 1; ; :::; n : As for i = n 1; the inequality p n;n+1 v n v n 1 = v n v n 1 0 holds trivially by our assumption that v i v i 1 ; 1 < i n: We additionally use the fact that, by de nition, j i ; is supported on [x i 1 ; x i+1 ] if 1 i n 1 and i j ; 1 0 ; is supported on [x 0 ; x 1 ]; and (n ; ) is supported on [x n 1 ; x n]: 9 As in the proof of Proposition, we use the convention that when i = 1; v i 1 = v 0 is de ned as 0. 1

13 and E ( 1 ; ) E ( 3 ; ) ::: E ( n ; ): urthermore, 3) no player is dominant with respect to the expected payo, i.e., for every i = ; 4; :::; n 1 10 E 1 ( 1 (! i ); ) E ( (! i ); ) and for every i = 1; 3; :::; n E 1 ( 1 (! i ); ) E ( (! i ); ): The next result shows that although the players are asymmetrically informed (i.e., have di erent information partitions), their ex-ante distributions of equilibrium e ort are identical. Proposition 4 In equilibrium, both players have the same ex-ante distribution of e ort. In particular, the expected e orts of both players are the same: Proof. Let x i 1 x x i ; for i = ; :::; n 1; and let j be the player for whom i j. Then the ex-ante probability that player j exerts an e ort smaller than or equal to x is j (x) = = = Xi 1 p k + (p i + p i+1 )j i ; x (9) k=1 Xi 1 p k + (p i + p i+1 ) x x i 1 p i;i 1 v i k=1 Xi 1 p k + (p i + p i+1 )(p i + p i 1 )(x x i 1 ) : p i v i k=1 The ex-ante probability that j s rival, player j, exerts an e ort smaller than or equal to x is then j(x) = = = Xi p k + (p i 1 + p i ) j i 1 ; x (10) k=1 Xi p k + (p i 1 + p i ) x + pi;i+1 v i x i p i;i+1 v i k=1 Xi 1 p k + (p i + p i+1 )(p i + p i 1 )(x x i 1 ) p i v i k=1 = j (x): 10 Recall that j (! i ) j denotes the element of j that contains! i : 13

14 With the convention that P i k=1 p k = P i 1 k=1 p k = 0 when i < and that p 0 = p n+1 = 0; (9) and (10) also hold for i = 1 and i = n: Since we showed that 1 (x) = (x) for every x [0; x n ] ; and since obviously 1 (x) = (x) = 1 for x > x n ; the ex-ante distributions of e ort in equilibrium are identical for both players. It is worth noting that although the players have the same ex-ante distribution of e ort, we showed that their ex-ante probabilities to win the contest are not the same. 6 Concluding remarks In models with asymmetric information, di erences in players information usually result in di erent equilibrium strategies, probabilities of winning, and expected payo s. In our model where no player has information advantage over his opponent, we show that equilibrium e orts have the same distribution for all players, and thus the expected e orts of all players are equal. We also show that the conditional expected payo s of the players increase in the expected value of winning (as conveyed by the revealed information). We present su cient conditions for the existence of the equilibrium in our model. One of them is that the information set of each player is connected with respect to the value of winning the contest. This condition is also su cient for the existence of equilibrium in the framework when the di erent information endowments can be ranked (see Einy et al (013a)). Our results yield the conclusion that in common-value all-pay auctions the players information does not a ect the ratio of the players expected e orts. It is interesting to note that this property does not necessarily hold for other forms of contests such as Tullock contests with or without the assumption of information advantage (see Einy et al. (013b) and Warneryd (003, 01)). 7 Appendix Proof of Proposition or any i = 0; :::; n consider the player j for whom i j ; and assume that j s rival (denoted j) uses the strategy j : The expected payo of player j conditional on the event i is given as follows. If 1 14

15 i n; and j exerts e ort x [x i 1 ; x i ]; then E j ( i ; x; j) = (11) = p i;i+1 v i j i 1 ; x x = p i;i+1 v i x + p i;i+1 v i x i p i;i+1 v i x = p i;i+1 v i x i : If 1 i n 1 and j exerts e ort x [x i ; x i+1 ]; then E j ( i ; x; j) = (1) = p i;i+1 v i + p i+1;i v i+1 j i+1 ; x x = p i;i+1 v i + p i+1;i v i+1 x x i p i+1;i v i+1 x = p i;i+1 v i x i : Now set v 0 = 0: Then (1) applies also when i = 0 (in which case j = 1), i.e. (1) holds for every 0 i n 1: Equalities (11) and (1) establish the following fact: act 1. When player j s opponent uses j ; player j is: (i) indi erent between all e orts in [x i 1; x i+1 ] given the event i j for 1 i n 1; (ii) indi erent between all e orts in [x 0 ; x 1 ] given 0 (if j is player 1); (iii) indi erent between all e orts in [x n 1 ; x n ] given n (if j is player ): It can be shown by induction on i that, for i = ; 3; :::; n; Xi 1 p i;i+1 v i x i = p k;k+1 (p k+1;k+ v k+1 v k ) 0: (13) k=1 The expression in (13) is non-negative as every summand in P i 1 k=1 pk;k+1 (p k+1;k+ v k+1 v k ) is non-negative by assumption (7). When i = 0 or i = 1; equality (13) remains meaningful if the sum is de ned as 0. It then follows from (13) and (11), (1) that: act. The conditional expected payo s of player j considered in (11) and (1) are non-negative for the corresponding e orts. Next consider i j ; for some 0 i n and player j: Notice that, given the event i ; if 15

16 y [x i+1 ; x i+ ] then E j ( i ; y; j) = p i;i+1 v i + p i+1;i v i+1 j i+1 ; y y (14) = p i;i+1 v i + p i+1;i v i+1 y + p i+;i+3 v i+ x i+ p i+;i+3 v i+ y p i;i+1 v i + p i+1;i v i+1 x i+1 + p i+;i+3 v i+ x i+ p i+;i+3 v i+ x i+1 = p i;i+1 v i + p i+1;i v i+1 x i+1 x i p i+1;i v i+1 x i+1 = p i;i+1 v i x i = E j ( i ; x i ; j): The inequality in (14) holds since, by (7), p i+;i+3 v i+ > p i+1;i v i+1 ; and the last equality in (14) holds by (1). Since obviously, if y > x i+ and i n ; E j ( i ; y; j) E j ( i ; x i+ ; j) (15) and, if y > x n ; E 1 ( n 1 ; y; ) E 1 ( n 1 ; x n ; ) = E 1 ( n 1 ; x n 1 ; ); (16) E ( n ; y; 1 ) E ( n ; x n ; 1 ) Then (15), (16) and (14) establish the following: act 3. When player j s rival uses j ; player j (weakly) prefers e ort x i to any e ort above x min(i+1;n) ; given the event i j for 0 i n: Now consider i j ; for some i n and player j. Given the event i ; if y [x i ; x i 1 ] then E j ( i ; y; j) = (17) = p i;i+1 v i j i 1 ; y y = p i;i+1 v i y x i p i 1;i v i 1 y p i;i+1 v i x i 1 x i p i 1;i v i 1 x i 1 = p i;i+1 v i x i = E j ( i ; x i ; j): The inequality in (17) holds since by (7) p i;i+1 v i p i 1;i v i 1 ; and the last equality in (17) holds by (11). Note also that when i and 0 y x i, E j ( i ; y; j) 0: (1) 16

17 Then (17), (1), and act lead to the following: act 4. When player j s rival uses j ; player j (weakly) prefers e ort x i to any e ort below x i 1 ; given the event i j for i n: acts 1, 3, and 4 show that for any 0 i n; conditional on the event i j ; any e ort in the support of j i ; is a best response of player j against the mixed strategy j of his rival. Thus j is also an unconditional best response of player j, which means that is indeed an equilibrium of G. Q.E.D. References [1] Abraham, I., Athey, S., Babaiof, M., Grubb, M.: Peaches, Lemons, and Cookies: Designing auction markets with dispersed information. Working paper, Harvard University (01) [] Amman, E., Leininger,W.: Asymmetric all-pay auctions with incomplete information: the two-player case. Games and Economic Behavior 14, 1-1 (1996) [3] Baye, M. R., Kovenock, D., de Vries, C. G.: Rigging the lobbying process: an application of the all-pay auction. American Economic Review 3, 9-94 (1993) [4] Baye, M., Kovenock, D., de Vries, C.: The all-pay auction with complete information. Economic Theory, (1996) [5] Che, Y-K., Gale, I.: Caps on political lobbying. American Economic Review (3), (199) [6] 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, (001) [7] Einy, E., Haimanko, O., Orzach, R., Sela, A.: Dominance solvability of second-price auctions with di erential information. Journal of Mathematical Economics 37, 47-5 (00) [] Einy, E., Haimanko, O., Orzach, R., Sela, A.: Common-value all-pay auctions with asymmetric information. CEPR Discussion Paper No. DP9315 (013a) 17

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19 [] Siegel, R.: All-pay contests. Econometrica 77 (1), 71-9 (009) [3] Siegel, R.: Asymmetric contests with interdependent valuations. orthcoming in Journal of Economic Theory (014) [4] Vohra, R.: Incomplete information, incentive compatibility and the core. Journal of Economic Theory 54, (1999) [5] Warneryd, K.: Information in con icts. Journal of Economic Theory 110, (003) [6] Warneryd, K.: Multi-player contests with asymmetric information. Economic Theory 51, 77-7 (01) 19