COMMON-VALUE ALL-PAY AUCTIONS WITH ASYMMETRIC INFORMATION AND BID CAPS. Ezra Einy, Ori Haimanko, Ram Orzach and Aner Sela. Discussion Paper No.

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1 COMMON-VALUE ALL-PAY AUCTIONS WITH ASYMMETRIC INFORMATION AND BID CAPS Ezra Einy, Ori Haimanko, Ram Orzach and Aner Sela Discussion Paper No. 4-0 September 04 Monaster Center for Economic Research Ben-Gurion University of the Negev P.O. Box 653 Beer Sheva, Israel Fax: Tel:

2 Common-Value All-Pay Auctions with Asymmetric Information and Bid Caps Ezra Einy, Ori Haimanko, Ram Orzach, Aner Sela September 6, 04 Abstract We study two-player common-value all-pay auctions (contests) with asymmetric information under the assumption that one of the players has an information advantage over his opponent and both players are budget-constrained. We generalize the results for all-pay auctions with complete information, and show that in all-pay auctions with asymmetric information, su ciently high (but still binding) bid caps do not change the players expected total e ort compared to the benchmark auction without any bid cap. Furthermore, we show that there are bid caps that increase the players expected total e ort compared to the benchmark. Finally, we demonstrate that there are bid caps which may have an unanticipated e ect on the players expected payo s one player s information advantage may turn into a disadvantage as far as his equilibrium payo is concerned. Keywords: Common-value all-pay auctions, asymmetric information, information advantage, bid caps. JEL Classi cation: C7, D44, D8. Ezra Einy, Ori Haimanko, Aner Sela: Department of Economics, Ben-Gurion University of the Negev, Beer-Sheva 8405, 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 48309, USA; orzach@oakland.edu

3 Introduction In an all-pay auction each player submits a bid (e ort) and the player with the highest bid wins the contest, but, independently of success, all players bear the cost of their bids. In the framework of all-pay auctions with complete information, where each player s type (value of winning the contest or ability) is common knowledge, and in the incomplete information framework, where each player s type is private and only the distribution from which the players types is drawn is common knowledge, it was shown that the behavior of the players may change drastically when constraints are placed on their budget or, alternatively, when bid caps are imposed by the contest designer. Che and Gale (998) calculated the bidding equilibrium of a complete information all-pay auction with two bidders who have di erent valuations for the prize and linear cost functions, and demonstrated that a bid cap can increase the players total e ort. Furthermore, they showed that a su ciently high (but still binding) bid cap changes the players strategies, but not their expected e orts nor their probabilities to win the contest. The e ect of bid caps on the players strategies may be di erent in other frameworks as Gavious, Moldovanu and Sela (003) demonstrated. They studied symmetric all-pay auctions with incomplete information and showed that, regardless of the number of bidders, if agents have linear or concave cost functions then setting a bid cap is not pro table for a designer who wishes to maximize the average bid. On the other hand, if agents have convex cost functions (i.e., increasing marginal costs), then e ectively capping the bids is pro table for a designer facing a su ciently large number of bidders. In this study we examine some e ects of bid caps in common-value all-pay auctions with asymmetric information. We consider a two-player common-value all-pay auction where the value of winning is the same for all the players in the same state of nature, but the information about which state of nature was realized can be di erent. 3 This model captures contests in which the value of winning is similar for di erent contestants, but is not precisely known at the time of making a bid. In our framework, the information of a player about the value of winning is described by a partition of the space of states of nature, which is assumed See, for example, Hillman and Samet (987), Hillman and Riley (989), Baye et al. (993, 996) and Siegel (009). See, for example, Amann and Leininger (996), Moldovanu and Sela (00, 006) and Moldovanu et al. (00). 3 Several researchers used the same framework as ours to analyze common-value second-price auctions and common-value rst price auctions (see Einy et al. 00, 00, Forges and Orzach 0, Malueg and Orzach 0 and Abraham et al. 0).

4 to be nite. Jackson (993) and Vohra (999) showed that this partition representation is equivalent to the more common Harsanyi-type formulation of Bayesian games. 4 In our model of asymmetric information we assume that information sets of each player are connected with respect to the value of winning the contest (see Einy et al. 00, 00 and Forges and Orzach 0). 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 these and all intermediate values. This assumption seems plausible in environments where the information of a player only allows him to put an upper and lower bound on the actual value of winning, without being able to rule out any outcome within the bounds. We also assume that one player has an information advantage over the other, which is re ected by having a ner information partition. Then, without loss of generality, it can be postulated that one player has the trivial information partition (he will be referred to as the uninformed player), and that the information partition of the other player is the nest one possible, enabling him to distinguish between all states of nature (he will be referred as the informed player). We nally assume that there is a cap constraining the players bids players can submit bids that are smaller than or equal to the bid cap. We generalize the result of Che and Gale (998) on bid caps in all-pay auction with complete information, and show that if the bid cap is su ciently high (but still binding) then its imposition changes the players strategies compared to the no-cap benchmark, but does not a ect the players expected e orts or their exante chances to win the contest. For su ciently low bid caps we show that when both players make a bid that is equal to the bid cap in every state of nature, then their expected total e ort decreases compared to the no-cap benchmark. However, in common with the all-pay auction under the complete information assumption, a bid cap may still increase the players expected e ort. This happens in an equilibrium in which the uninformed player submits a bid that is equal to the bid cap, but the informed player submits it only in some states of nature while in the other states he stays out. According to Siegel (04), in the common-value all-pay auction with asymmetric information without bid caps the expected payo of the informed player is always higher than that of the uninformed player. 5 Here, 4 Krishna and Morgan (997) analyzed the equilibrium strategies of the all-pay auction with interdependent types in the Harsanyi-type formulation of Bayesian games. They assumed that the players types are a liated and symmetrically distributed. 5 See Section of the online appendix to his work. 3

5 however, we will show that imposing a bid cap may result in the informed player having a lower expected payo. In other words, an information advantage may turn into a disadvantage as far as equilibrium payo s are concerned. Our work is related to the recent research of Siegel (04) and Einy et al. (04). Siegel (04) studies general asymmetric two-player all-pay auctions with interdependent valuations, where the private information of each player is represented by a nite set of possible types. It can be shown that his framework can accommodate our common value all-pay auctions where one player has an information advantage over his opponent. Einy et al. (04) study two-player common-value all pay auctions, but assume that no player has an information advantage over his opponent. Both works do not consider bid caps as we do here. The rest of the paper is organized as follows. In Section we present the model. In Section 3 we characterize the equilibrium in common-value all-pay auctions with high but binding bid caps. In Section 4 we show how an information advantage may turn into a payo disadvantage when there are bid caps. In Section 5 we demonstrate that bid caps may positively a ect the players total bid. Section 6 concludes. The model Consider the set N f; g of two players who compete in an all-pay auction, where the player with the highest bid (e ort) wins the contest but all the players bear the cost of their bid. We assume that each player can submit any non-negative bid that is lower than or equal to a given budget constraint d > 0. The uncertainty in our model is described by a nite set of states of nature, and a probability distribution p over that represents the common prior belief about the realized state of nature (w.l.o.g. p(!) > 0 for every! ): The uncertain common value of winning the contest is given by a function v :! R +, 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 by a move of nature that 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 [0; d]: Note that when the players have di erent information partitions they are ex-ante asymmetric. 4

6 The utility (payo ) of player i N is given by the function u i : R +! R de ned as follows: 8 >< u i (!; x) >: m(x) v(!) x i; if x i maxfx ; x g; x i ; if x i < maxfx ; x g; where m(x) denotes the number of players who exert the highest e ort, namely, m(x) ji N : x i maxfx ; x gj. A two-player common-value all-pay auction with incomplete information is fully described by and identi ed with the collection G ((; p); u ; u ; ; ; d): In all-pay auctions without a bid cap there is usually no equilibrium in pure strategies, and thus our attention will be given rst to mixed strategy equilibria. A (mixed) strategy of player i is a function F i : [0; d]! [0; ]; such that for every! ; F i (; ) is a cumulative distribution function (c.d.f.) of i s e ort on [0; d] ; and for all x R +, F i (; x) is a i -measurable function (that is, F i (; x) is constant on every element of i ): Slightly abusing notation, for any i i we will denote the constant value of F i (; x) on i by F i ( i ; x) ; whenever convenient. If player i s e ort choice given i is deterministic (pure); i.e., if the distribution represented by F i ( i ; ) is supported on some y [0; d] ; we will identify between F i ( i ; ) and y wherever appropriate. Furthermore, if i s strategy F i is pure, i.e., if i s e ort choice is deterministic in every! ; we will identify between F i and a function x i :! [0; d] that represents i s state-dependent (and i -measurable) e ort choice. Given a strategy pro le F (F ; F ), denote by E i (F ) the expected payo of player i when players use that strategy pro le, i.e., E i (F ) E( Z d Z d 0 0 u i (; (x ; x ))df (; x )df (; x )): For i i ; E i ( i ; F ) will denote the conditional expected payo of player i given his information set i ; i.e., E i ( i ; F ) E( " Z d Z d 0 0 u i (; (x ; x ))df (; x )df (; x ) # j i ): A pro le F (F ; F ) of mixed strategies constitutes a (Bayesian Nash) equilibrium in the commonvalue all-pay auction G if for every player i, and every mixed strategy F i of that player, the following inequality holds: E i (F ) E i (F i ; F i); 5

7 where i denotes i s rival. We will assume throughout that player has an information advantage over player. In our framework, where the players private information is represented by information partitions, the information advantage of player over player is tantamount to s partition being ner than s partition : Let us write as an indexed sequence, f! ; :::;! n g: () It can be easily shown (see Remark in the Appendix) that as far as equilibrium analysis is concerned, the information advantage assumption can be reduced to the postulate that fg and ff! g; f! g; :::; f! n gg; () i.e., that player has no information on the realized state of nature (other than the common prior distribution p, and thus he has the trivial information partition), while player knows the realized state precisely (and thus his information partition is the nest one possible, that consists of singletons). In addition to () and (), the following notation will be used throughout: v(! i ) and p i p(! i ) > 0 for every i ; :::; n: We will assume that the possible values of winning are distinct, and thus, w.l.o.g., strictly ranked as follows: 0 < v < v < ::: < : 3 The e ect of high (but binding) bid caps In this section we will consider the e ects on equilibrium e orts of a high but still binding bid cap, compared to the no-cap benchmark. Throughout the section we will assume that the bid cap d satis es the following: p j v j + p n < d p j v j : (3) Without the bid cap (d ) the contest has a unique equilibrium in mixed strategies 6, each of which h is supported on 0; P i n p jv j ; we will refer to this contest as the no-cap benchmark. Thus, the upper 6 See Section in the online appendix to Siegel (04). 6

8 boundary of the range of d in (3) is the minimal non-binding bid cap, under which the equilibrium strategies in the no-cap benchmark remain such when the e orts are capped. All lower caps, d < P n p jv j ; would lead to equilibrium strategies that are di erent from those of the no-cap benchmark. Accordingly, the right-hand inequality in (3) means that d is binding (unless it holds as equality), while the left-hand inequality means that d is nonetheless high. In what follows, we describe a mixed strategy pro le (F ; F ) of the all-pay auction with a bid cap d satisfying (3) that will turn out to be an equilibrium. Let x 0 0; and for each i ; :::; n; set Thus, x 0 < x < ::: < x n : In addition, let ix x i p j v j : (4) Condition (3) on the cap d implies that ex d p j v j : (5) x n < ex d x n (with all inequalities being strict if and only if d < P n p jv j ). Consider now a function F on [0; d] which is given by 8 x + P h i p j >< x F + P h n (x) p j ex + P h n p j i v j ; if x [x i ; x i ) for i ; :::; n ; i v j ; if x [x n ; ex); i v j ; if x [ex; d); >: ; if x d: It is easy to see that F is well de ned, strictly increasing on [0; ex), continuous on [0; d); and constant on [ex; d). Moreover, F (0) 0 and F (d). Thus, F is a c.d.f. of a probability distribution with full support on [0; ex] [ fdg (and a unique atom at d) if d < P n p jv j ; and a continuous distribution with full support on [0; ex] if d P n p jv j ( ex x n ). As the function F is state-independent, it can be viewed as a mixed strategy of the uninformed player. 7

9 Next, for each i ; :::; n ; consider a function F (! i ; ) on [0; d] which is given by 8 >< F (! i ; x) 0; if x < x i ; x P i pjvj p i ; if x [x i ; x i ); >: ; if x x i ; and for i n; a function F (! n ; ) given by 8 >< F (! n ; x) 0; if x < x n ; x P n pjvj p n ; if x [x n ; ex); ex P n pjvj p n ; if x [ex; d); >: ; if x d: Note that for i ; :::; n ; F (! i ; ) is well de ned, strictly increasing and continuous on [x i ; x i ], and F (! i ; x i ) 0 and F (! i ; x i ). Thus, F (! i ; ) is a c.d.f. of a continuous probability distribution with full support on [x i ; x i ]. In addition, the function F (! n ; ) is well de ned, strictly increasing on [x n ; ex), continuous on [0; d); and constant on [ex; d). Moreover, F (! n ; x n ) 0 and F (! n ; d). Thus, F (! n ; ) is a c.d.f. of a probability distribution with full support on [x n ; ex] [ fdg (and a unique atom at d) if d < P n p jv j ; and a continuous distribution with full support on [x n ; ex] if d P n p jv j ( ex x n ). In particular, F constitutes a mixed strategy of player. Proposition The strategy pro le (F ; F ) is a mixed strategy Bayesian Nash equilibrium in a common value all-pay auction with a bid cap satisfying (3). Proof. Note that E (f! i g; F ; x) F (x) x F (x i ) x i E (f! i g; F ; x i ) for every i ; :::; n and x [x i ; x i ); or i n and x [x n ; ex): Thus, given that! i was realized, the informed player is indi erent between all e orts in the interval [x i ; x i ) if i ; :::; n ; and is indi erent between all e orts in the interval [x n ; ex) if i n; provided that his rival acts according to F : Since the slopes of the function F (x) x E (f! i g; F ; x) are positive when 0 x < x i and negative when min(x i ; ex) < x < d; the set of player s pure best responses (given! i ) when the bids are constrained to be less then d contains the interval [x i ; x i ) if i ; :::; n ; and contains [x n ; ex) if i n: 8

10 Now note that the following holds for x d : E (f! i g; F ; d) (F (ex) + F (ex) ) d ( + F (ex) ) d (6) ( + d P n p jv j + P n p j [ v j ] ) d d( i ) X n ( p j v j v X n( p n + p j )) d ( ) Thus, if i < n; by using (6) and (3) we obtain + v i p j [ v j ] : On the other hand, E (f! i g; F ; d) ( p j v j + p n )( ( ) n p X n + p j ( v j ): ) + v i p j [ v j ] and hence E (f! i g; F ; x i ) F (x i ) x i (7) ( x Xi i ( v j ) + p j ) x i Xi p j ( v j ); E (f! i g; F ; d) E (f! i g; F ; x i ) ( ) p n + ji+ p j ( v j ) < 0: It follows that the set of player s pure best responses (given! i ) contains the interval [x i ; x i ) if i ; :::; n : Furthermore, if i n; then it follows from (6) and (7) (the latter does not require i < n) that E (f! n g; F ; d) p j [ v j ] E (f! n g; F ; x n ); hence the set of player s pure best responses (given! n ) contains the set [x n ; ex) [ fdg. Therefore, the mixed strategy F of player is his best response against player s strategy F ; since it has been shown that for each i ; :::; n; the distribution corresponding to F (! i ; ) is supported on a set of s pure best responses to F conditional on! i : 9

11 Next, observe that Xi E (x; F ) p j v j + p i F (! i ; x) x 0 (8) for every i ; :::; n and x [x i ; x i ); or i n and x [x n ; ex): Note also that E (d; F ) p i + p n (F (! n ; ex) + i p i + p n ( ex P n p jv j + p n ) d 0: i F (! n ; ex) ) d (9) It follows from (8) and (9) that player is (in expectation) indi erent between all e orts in [0; ex) [ fdg (and is obviously worse o when e orts are outside [0; ex) [ fdg) provided his rival acts according to F : Thus the mixed strategy F of player is his best response against player s strategy F ; since it has been shown that the distribution corresponding to F is supported on a set of s pure best responses to F : We conclude that (F ; F ) is indeed a mixed strategy equilibrium. Proposition can be strengthened. Call mixed strategy F of player monotone if, for every i ; :::; n; there is a set A i [0; d] such that the distribution corresponding to F (! i ; ) is supported on A i ; and the sets A ; :::; A n are "ordered" on [0; d] according to their index: if i < j n and x A i ; y A j ; then x y: This means that the bids of player are (weakly) increasing in the value of winning (which he knows by assumption). Note that the strategy F of player is monotone. Proposition The mixed strategy equilibrium (F ; F ) of Proposition is the unique Bayesian Nash equilibrium in a common value all-pay auction with a bid cap satisfying (3) in which the strategy of the informed player is monotone. Proof. See the Appendix. The inclusion of the no-cap benchmark contest (corresponding to the case of d P n p jv j ; that was studied in Siegel (04)) in our framework allows direct comparisons of expected equilibrium e orts and of ex-ante probabilities of winning in equilibrium in the scenario with and without caps. According to the next proposition, su ciently high but binding caps make no di erence as far as the expected e ort is concerned: Proposition 3 The expected e ort that each player exerts in the unique equilibrium of a common-value all-pay auctions with a bid cap d satisfying (3) is independent of d; and in particular is equal to the e ort exerted in the no-cap benchmark auction. 0

12 Proof. Player s expected e ort when he uses F is T E Z [x 0;x n ) n Z xi X i i i Z xdf (x) + xdf (x) + d ( [x n ;ex) F (ex)) (0) x i x dx + Z ex x i x i + (d x i x i x n x i p i n ex X dx + d ( p j P n p jv j ) x n d + d ( 0 p j v j + p i A ; v j ) P n p jv j p j v j )() and player s expected e ort when he uses F is T E Z p i i Z xi p i [x i ;x i) x i x i Z xi x i x i i p i Z xdf (! i ; x) + p n ( xdf (! n ; x) + d ( F (! n ; ex)) () [x n ;ex) p i dx + p n ( Z ex dx + x n x Z ex x n x p n dx + d ( dx + d (p n 0 p j v j + p i A d ex P n p jv j p n ) P n p jv j P n p jv j ) (3) (the last equality is obtained just as in the computation of T E above; since expression (3) is identical to expression ()). Thus, both T E and T E are independent of d. The proof of Proposition 3 reveals that the players ex-ante expected e orts are the same. The expected total e ort is therefore T E 0 p j v j + p i A ; (4) p i i and, just as the individual s expected e ort, it is independent of the bid cap d (provided (3) holds). This generalizes the result of Che and Gale (998) on the e ect of high but binding caps in all-pay auctions with complete information (which, in our framework, is captured by the information structure where the set consists of just one state of nature). The fact that the expected e orts of both players are identical (though not their equality across d) can also be obtained as a corollary of the following observation, which has already been noted by Siegel (04) for the no-cap benchmark auction. It turns out that the ex-ante distribution of equilibrium e ort is identical for both players:

13 Proposition 4 In the unique equilibrium (F ; F ) of a common-value all-pay auctions with a bid cap d satisfying (3), the ex ante distribution of equilibrium e ort of player is identical to F : In particular, each player wins with (ex-ante) probability. Proof. Denote by F (x) the ex-ante probability that player exerts an e ort that is smaller than or equal to x when acting according to his strategy F : For every i ; :::; n and x [x i ; x i ] ; or i n and x [x n ; ex); F (x) Xi Xi p j + p i F (! i ; x) p j + p i x P i p jv j p i x Xi + p j v j F (x) : When x [ex; d); F (x) p j + p n F (! i ; ex) p j + p n ex P n p jv j p n ex + p j v j F (x) ; and clearly F (d) F (d) : Thus, the ex-ante distribution of equilibrium e ort of player is identical to the distribution of equilibrium e ort of player. Since player s strategy F is state-independent, his chance to win only depends on the ex-ante distribution of his rival e orts, i.e., on F : But it has been shown that F F ; and hence the (ex-ante) probability to win is for each player. 4 Possible disadvantage of information advantage With su ciently high but still binding bid caps (described in the previous section) the expected payo of the uninformed player is zero in equilibrium, while his informed rival s expected payo is positive. The following example demonstrates the e ect of lower bid caps on the players equilibrium strategies, and, in particular, shows that for certain bid caps the expected payo to the uninformed player is higher than for the informed player.

14 Example Assume that n 3; and that in state! i the value of winning is v(! i ) i with probability of p i 3 ; i ; ; 3: Assume also that the players have the same bid cap d 5 6 : Then the following constitutes a pure strategy Bayesian equilibrium: player s bid is independent of the state of nature, x 5 6 ; and player s state-dependent bid is given by The expected payo of player is then 8 >< x (!) >: 0; if!! ; 5 6 ; if! 6! : and the expected payo of player is E ; E 3 ( 5 6 ) + 3 ( ) 5 8 : Thus, the expected payo of the uninformed player is higher than that of the informed player. In the next proposition we describe some su cient conditions (extending Example ) under which this unusual result of a higher expected payo to the uninformed player can be obtained. Proposition 5 Consider a common-value all-pay auction with a bid cap d. Suppose that there exists j n such that (i) (ii) mj+ p m v m d; (5) v j d v j+ ; (6) and (iii) jx p m (v m d) 0: (7) m Then there exists a pure strategy Bayesian Nash equilibrium in which the expected payo of the uninformed player (player ) is higher than that of the informed player (player ). 3

15 Proof. Consider a pure strategy pro le (x ; x ) in which x d; (8) and 8 >< 0, for k ; :::; j; x (! k ) >: d, for k j + ; :::; n : (9) The expected payo of player when (x ; x ) is played is E (x ; x ) p m ( v m d); (0) mj+ and the expected payo of player is jx E (x ; x ) p m (v m d) + p m ( v m d): () m mj+ We will now check that (x ; x ) is an equilibrium. If player submits a bid of 0 x < d; then by (5) jx E (x ; x ) p m v m x E (x ; x ) : m Thus x is player s best response to x : If player unilaterally deviates from x to a strategy x with 0 < x (! k ) " d for some k j; then E (f! k g ; x ; x ) max "; v k d 0 E (f! k g ; x ; x ) ; where the second inequality is implied by (6). But, also by (6), if player unilaterally deviates from x to a strategy x with 0 x (! k ) " < d for some j + k n; then E (f! k g ; x ; x ) " 0 v k d E (f! k g ; x ; x ) : By taking expectation over! k ; it follows that E (x ; x ) E (x ; x ) 4

16 for any pure strategy x of player that obeys his bid cap, and thus x is player s best response to x : We conclude that (x ; x ) is an equilibrium. By comparing the players expected payo s given by (0) and (), we obtain by (7) that the expected payo of player is higher than that of player. Q:E:D: 5 The advantage of bid caps Che and Gale (998) have shown that low bid caps may lead to higher e orts by players. This occurs in an equilibrium where both players make bids that are equal to the bid cap. We will rst examine whether this can also happen in our common-value all-pay auctions with incomplete information. Assume that both players make a bid that is equal to the bid cap in all the states of nature: x d; x (! j ) d, j ; :::; n: The players expected payo s when (x ; x ) is played are: and, for j ; ; :::; n; E (x ; x ) p j v j d; E (f! j g ; x ; x ) v j d: The necessary and su cient condition for (x ; x ) to be an equilibrium is d v ; since then both players have non-negative expected payo s in all states of nature, and therefore none of them has an incentive to deviate from the above strategies. The total e ort in this equilibrium is equal to d ( v ): On the other hand, in the common-value all-pay auction without a bid cap of Siegel (04) (which is equivalent to the assumption that d P n p jv j ; considered among other cases in Section 3), the players expected total e ort is given by (4), and hence T E i p i 0 p j v j + p i A > i p i 0 p j v + p iv A v : 5

17 Thus, a simple adaptation of the bid that is equal to the bid cap from the complete information case does not increase the expected total e ort. The following result, however, demonstrates that a bid cap may increase the players expected e ort compared to the no-cap benchmark in some equilibria. Proposition 6 The players expected total e ort in a common-value all-pay auction with a bid cap may be higher than in the same auction without a bid cap: Proof. Consider a common-value all-pay auction with a bid cap d. Suppose that there are two states of nature (i.e., n ) that occur with probabilities (p ; p ); and that v d p v : As in Proposition 5 there is a pure strategy Bayesian Nash equilibrium (x ; x ) ; in which x d; and 8 >< 0, for i ; x (! i ) >: d, for i : The players expected total e ort is then dt E ( + p )d: On the other hand, by (4), in the common-value all-pay auction without a bid cap the player expected total e ort is given by T E v (p + p p ) + p v : Since (+p)p > p for all p < ; we obtain that for such p ; d that is close to pv, and a su ciently low value of v, d T E > T E: Q:E:D: 6 Concluding remarks In this paper we generalize some of the results of Che and Gale (998), who studied all-pay auctions with bid caps under the complete information assumption. We consider common-value all-pay auctions with asymmetric information and show that high (but still binding) levels of bid caps do not change the players 6

18 expected e orts, but other levels of bid caps may drastically a ect them, and in particular increase the total expected e ort compared to the no-cap benchmark. We also show that if players face bid caps, the information advantage might become a disadvantage as far as equilibrium payo s are concerned in that the player with the information advantage may have a lower expected payo than his opponent. This unusual result implies that by imposing bid caps, the contest designer can control the relation between the players expected payo s. We provide a partial characterization of the equilibrium strategies for high (but binding) levels of bid caps. It is not clear, however, whether equilibrium exists, or is unique, for other levels of bid caps. The study of all-pay auctions under asymmetric information with and without bid caps remains an important endeavor, and much more work is needed for a complete analysis. 7 Appendix Remark As far as equilibrium analysis is concerned, the assumption that player has an information advantage over player can be reduced to (). Indeed, in the general case of being ner than note the following. Given ; the event is common knowledge at any!. Thus, the equilibrium analysis can be carried out separately for each ; as conditional on the occurrence of the auction G can be viewed as a distinct common-value all-pay auction G 0, where the set of states of nature is 0 and the conditional distribution p( j ) serves as the common prior distribution p 0. In G 0, player has the trivial information partition, 0 f 0 g : Thus, we may w.l.o.g. assume that fg in the original auction G. Now consider another commonvalue all-pay auction G 00 in which the set of states of nature 00 consists of information sets of player in G; the common value function v 00 is given by v 00 ( ) E(v j ) for every 00 ; player has the full information partition, consisting of all singleton subsets of 00 ; i.e., 00 ff g j 00 g ; and player has the trivial information partition 00 f 00 g : Naturally, every mixed strategy F (; ) of player in G corresponds to a unique mixed strategy F 00 (; ) of in G 00 that is given by F 00 ( ; ) F (!; ) 7

19 for every 00 and every! (which is well-de ned because of -measurability of F in its rst coordinate), and the mixed strategies of player in G and G 00 are identical as they are state-independent. The correspondence (F ; F ) $ (F ; F 00 ) between mixed strategy-pro les in G and G 00 obviously preserves both players expected payo s. Thus, G and G 00 are payo -equivalent, and hence identical in terms of equilibrium analysis. However, G 00 has the property that player has no information on the realized state of nature while player is completely informed of it. Thus, in studying equilibria in two-player common-value all-pay auctions, attention can be con ned w.l.o.g. to auctions with the latter property. Proof of Proposition Fix an equilibrium (F ; F ) in the auction G, in which F is monotone. We will prove that (F ; F ) (F ; F ): In what follows, for k ; and! ; F k (!; ) will be treated either as a probability measure on [0; d] ; or as the corresponding c.d.f., sometimes in the same context. Accordingly, for A [0; d], F k (!; A) will stand for the probability that the equilibrium e ort of player k; conditional on!; belongs of the set A; and for x [0; d]; F k (!; x) will stand for the probability that the equilibrium e ort of player k; conditional on!; is less or equal to x. Also, as F is state-independent, F (!; ) will be shortened to F () ; whenever convenient. Notice that F k (; fcg) 0 for any e ort c (0; d) and k ; : Indeed, if F k (!; fcg) > 0 for some c (0; d); k and!; then F k (! 0 ; (c "; c]) 0 for the other player k and every! 0 ; and some su ciently small " > 0: But then k would be strictly better o by shifting the probability from c to c ", a contradiction to F k being an equilibrium strategy. Thus, F () ; F (!; ) are non-atomic on (0; d) for every! : Notice also that there is no interval (a; b) (0; d) on which in some state of nature only one player places a positive probability according to his equilibrium strategy. Indeed, otherwise there would exist a 0 > 0 such that only one player places a positive probability on (a 0 ; b), and it would then be pro table for that player to deviate (in at least one state of nature, if this is the informed player ) by shifting a positive probability from (a 0 ; b) to a 0. Suppose now that there is a (non-degenerate) interval (a; b) (0; d) such that F ((a; b)) 0 (and thus F (!; (a; b)) 0 for every! ; by the previous paragraph), but F ([b; d)) > 0. By increasing 8

20 b < d if necessary, it can also be assumed that (a; b) is maximal with respect to this property, i.e., that F ([b; b + ")) > 0 for every small enough " > 0: However, the expected payo of player at a+b is strictly bigger than his payo for any e ort in [b; b + "), if " > 0 is small enough: This contradicts the assumption that F is an equilibrium strategy, and shows that there exists no interval (a; b) as above. Non-existence of such (a; b) together with non-atomicity of F () on (0; d) imply that there exist 0 d such that F () is supported on [0; ] [ fdg (where [0; ] denotes the interval [0; ) if > 0 and [0; 0] f0g); and either the restriction of F () to [0; ] has full support on it, or F () is concentrated on d (i.e., F (fdg) ). Note also that for every! ; F (!; ) must also be supported on [0; ] [ fdg (though there need not be full support on [0; ] ) if F (fdg) <, and on f0g [ fdg if F (fdg) ; since otherwise there would be an open subinterval of (0; d) where only player places positive probability, and this was ruled out. Observe next that F () cannot be concentrated on d: For otherwise each F (! i ; ) would be supported on f0g [ fdg: It cannot be that all F (! i ; ) are concentrated 0; or all F (! i ; ) are concentrated on d (in the rst case, player would have a pro table deviation to a positive e ort near 0; and in the second case player would have a pro table deviation to 0 since his expected payo under F would be negative, as follow from assumption (3)). Thus, since the strategy F is, by assumption, monotonic, there exists i 0 such that F (! i 0; ) is supported on f0g [ fdg; whereas F (! i ; ) is concentrated on 0 for i < i 0 and F (! i ; ) is concentrated on d for i > i 0 : If i 0 6 n; however, F (which is concentrated on d) gives a negative expected payo by assumption (3) on d; and thus cannot be a best response. We conclude that i 0 n: But then, by lowering his e ort from d to a positive e ort near zero; player will save almost the entire cost of e ort d, while his loss of utility from winning will not exceed p n : By assumption (3), this will be a pro table deviation, in contradiction to F being an equilibrium strategy. Thus, F cannot be concentrated on d; and consequently the restriction of F () to [0; ] has full support on it. Note further that the interval [0; ] is non-degenerate, i.e., that 0 < : Indeed, if 0 then both equilibrium strategies prescribe mixtures of e ort 0 and e ort d: It was shown that F () is not concentrated on d; and thus it is supported on f0g [ fdg with F (f0g) > 0. Then the only strategy F of that may constitute a best response to F would prescribe d with probability at every! ; and then F must be concentrated on 0 (since choosing d will give a negative expected payo by assumption (3) on d). However, 9

21 no strategy of can be a best response to such F, a contradiction. We conclude that, indeed, > 0; and the interval [0; ] [0; ) is non-degenerate. In keeping with our earlier notation, for 0 a b d denote by [a; b] the interval [a; b) if b > 0 and the set f0g if a b 0: Given i ; :::; n, we will now show that there is a (possibly empty or degenerate) subinterval [a i ; b i ] of [0; ) such that F (! i ; ) is supported on [a i ; b i ] [ fdg; and, if F (! i ; fdg) < ; the restriction of F (! i ; ) to [0; ) has full support on [a i ; b i ]. If F (! i ; ) is supported on f0g [ fdg; the claim obviously holds for 0 < a i b i (an empty [a i ; b i ] ), or a i b i 0: Assume now that F (! i ; ) is not supported on f0g [ fdg, but the claim does not hold. Then there are 0 < a < b < such that F (! i ; (a; b)) 0, but F (! i ; [0; a]) > 0 and F (! i ; [b; )) > 0: Since F ((a; b)) > 0, there must be j 6 i such that F (! j ; (a; b)) > 0: Assume that i < j (the opposite case is treated similarly). Then there are x [b; ) and y (a; b) such that F (x) x E f! i g ; F ; x () E f! i g ; F ; y F (y) y (3) and v j F (x) x E f! j g ; F ; x (4) E f! j g ; F ; y v j F (y) y: (5) But x > y; and therefore (v j )F (x) > (v j )F (y) (6) since < v j and the c.d.f. F is strictly increasing on [0; ) (as the restriction of this distribution to [0; ) has full support). Adding (6) to the inequality in ()-(3) contradicts the inequality obtained in (4)-(5), and therefore no such (a; b) exists. Consequently, the restriction of each F (! i ; ) to [0; ) has full support on some [a i ; b i ] : If there is i n such that F (! i ; fdg) > 0, denote by i 0 the smallest index with this property. Since F is a monotone strategy, F (! i ; ) is concentrated on d for every i > i 0 ; and hence, if i 0 < n; F (! n ; ) is concentrated on d: Thus E (d 0 ; F ) p j v j + p n d 0 < 0 0

22 by assumption (3) for all e orts d 0 d that are su ciently close to d. Consequently, the equilibrium strategy F of player will not have support in some left-hand neighborhood of d. Thus player will not exert e ort d at! n with positive probability, a contradiction to F (! n ; ) being concentrated on d. Thus, even if i 0 is de ned, it must be equal to n; implying that F (! i ; fdg) 0 for every i ; :::; n : Thus, the distribution F (! i ; ) has full support on [a i ; b i ] [0; ) for i ; :::; n (and in particular each [a i ; b i ] is non-empty). As F (! n ; ) cannot be concentrated on d; [a n ; b n ] is also non-empty. It furthermore follows from monotonicity of F that, if i < j; then [a i ; b i ] _ lies below [a j ; b j ] _ (or coincides with [a j ; b j ] _ ; if [a j ; b j ] _ f0g). Thus, the intervals [a i ; b i ] _ n are disjoint (barring the set f0g), and "ordered" according to the index i i on the interval [0; ): Moreover, [ n i [a i; b i ] [0; ); since otherwise there would be a "gap" (a; b) on which only player places positive probability, which is impossible as we have seen earlier. It follows that there are points 0 y 0 y ::: < y n such that [a i ; b i ] _ [y i ; y i ] _ for every i ; ; :::; n; i.e., for i ; :::; n ; F (! i ; ) has full support on [y i ; y i ] _ ; and the restrictions of F () and F (! n ; ) to [0; ) have full support on [0; y n ) and [y n ; y n ), respectively. We denote by i 0 the smallest integer with y i0 > 0: 7 Since F () has full support on [0; y n ) (when restricted to [0; y n )) and F (!; ) has no atoms in (0; y n ); player is indi erent between any two e orts in (0; y n ). Thus, the following equality must hold for every i i 0 ; :::; n and every positive x [y i ; y i ) : Xi p j v j + p i F (! i ; x) x E (x; F ) lim y&0 E (y; F ) e 0: In particular, F (! i ; x) x P i p jv j + e p i (7) for every i i 0 ; :::; n and every x [y i ; y i ): Since, for i i 0 + ; :::; n; F (! i ; ) has full support (after restriction to [0; ) for i n) on [y i ; y i ) (0; ); we have F (! i ; y i ) 0; and thus y i for every i i 0 ; :::; n : ix p j v j e (8) 7 Since each interval [y i ; y i ] _ is either non-degenerate or f0g, 0 y 0 ::: y i0 < y i0 < ::: < y n:

23 Since, for i i 0 ; :::; n; F (! i ; ) has full support (after restriction to [0; ) for i n) on [y i ; y i ) and F () has no atoms (except, possibly, at 0) on [0; ); player is indi erent between all positive e orts in [y i ; y i ). Thus, the following equality must hold for every positive x [y i ; y i ) : and in particular F (x) x E (f! i g; F ; x) lim y&y i E (f! i g; F ; y) F (y i ) y i ; F (x) x + F (y i ) y i : (9) The rest of the proof will separately consider the following cases. Case : Assume that e > 0 or i 0 > : The assumption of Case implies that either F (! i0 ; ) is concentrated on 0 (if i 0 > ) or F (! ; f0g) > 0 by (7) (if i 0 but e > 0). Thus F () has no atom at 0 (otherwise shifting mass from 0 to a sightly higher e ort would constitute a pro table deviation), and hence F (0) 0: Using this, (8) and (9); we obtain F (x) x + e P i p jv j + for every i i 0 ; :::; n; and every positive x [y i ; y i ): ix ji 0+ p j + P i0 p jv j e 0 (30) Case.: Assume that F (fdg) > 0: It is a corollary of this assumption that F (! n ; fdg) > 0 (since otherwise for some small " > 0 and every i we would have F (! i ; (d "; d]) 0; and then player could pro tably shift probability from d to a lower e ort). Note further that having an atom at d by both F () and F (! n ; ) implies that these distributions place no probability on an open interval (d "; d) for some small " > 0; and hence < d; (3) and F (! n ; fdg) F (! n ; ): 8 As F (! n ; ) has no atom at ; F (! n ; ) is given by (7) for x and i n: Since F (fdg) > 0; the expected payo of player from choosing d must be equal to the expected 8 In the LHS F is viewed as a probability measure, and in the RHS as a c.d.f.

24 payo from choosing e ort levels in (0; ); i.e., Therefore e lim E (y; F ) E (d; F ) y&0 p j v j + p n F (! n ; ) + ( F (! n ; )) p j v j + p n [F (! n ; ) + ] " p j v j + P n p nv p # jv j + e n + p n p j v j + + e d: Since F (! n ; ) has full support on [y n d d d p j v j + e : (3) ; ) [ fdg (as we have seen, d is chosen with positive probability), the conditional expected payo of player at! n from choosing d must be equal to his conditional expected payo from choosing e ort levels close to (but smaller than) ; i.e., d F () lim E (f! n g; F ; x) x% E (f! n g; F ; d) F () + ( F ()) d [ + F ()] d: It follows, by using (30) and (3) for x and i n, 9 that 0 [ F ()] + d d i 0 X P n p jv j + e + e P n p jv j p j v j + e A ji 0+ p j p j v j 0 P i0 d p jv j e 5 + e : 3 ji 0+ p j P i0 p jv j e Distribution F () does not have an atom at ; as follows from (3), and hence formula (30) can be used for x although it is stated for x < y n : 3

25 But the expression in brackets is positive under the assumptions of Case, a contradiction. Thus, the properties assumed in Case and Case. cannot hold jointly in equilibrium. Case.: Assume that F (fdg) 0: This assumptions implies that F (! n ; fdg) 0 as well (since otherwise for some small " > 0 we would have F ((d "; d]) 0; and then player could pro tably shift mass from d to a lower e ort at! n ). Thus, F () and F (! i ; ) for each i i 0 ; :::; n are supported on [0; ), and non-atomic on (0; ): In particular, formulas (7) and (30) hold for x y n and i n; and F (! n ; ) F () : (33) Thus It follows from (7) and (33) that y n p j v j e : F () + e Pn ji 0+ ji 0+ P n p jv j + ji 0+ p jv j e + e P n p jv j + p j + p j + P i0 p jv j e 0 i 0 X p j v j 0 e 0 : p j + ji 0+ P i0 p jv j e 0 p j + P i0 p jv j e 0 The last expression is smaller than, however, under the assumptions of Case, a contradiction. We conclude that the assumptions of Case cannot hold in equilibrium, and consider next the complimentary Case : Case : Assume that i 0 and e 0: Given the assumptions, it follows that (7), (8), and (9) can be rewritten in the following way: ix y i p j v j ( x i ) (34) 4

26 for every i ; :::; n (where x i was de ned in (4)), and F (! i ; x) x P i p jv j p i ; (35) F (x) x + F (y i ) y i (36) for every i ; :::; n and every x [y i ; y i ): Denote e F (y 0 ) 0: From (36) and (34) we obtain F (x) x Xi + p j v j + e (37) for every i ; :::; n and every x [y i ; y i ): Case.: Assume that F (fdg) > 0: Arguing as in Case. (but substituting e 0), we obtain where ex was de ned in (5), and y n d p j v j ( ex); (38) F () [ + F ()] d: Using these equalities and (37), it follows that 0 [ F ()] + d v P n n d 4 p n jv j X p j 0 p j v j A d e ; and we conclude that e 0: This turns (37) into v j e 3 5 for every i ; :::; n and every x [y i ; y i ): F (x) x Xi + p j v j (39) Due to (34), (35), (38) and (39), the equilibrium strategy pro le (F ; F ) is identical to (F ; F ) of Proposition. 5

27 Case.: Assume that F (fdg) 0: Arguing as in Case. (but substituting e 0 and i 0 ), we obtain F (! n ; ) F () (40) and y n p j v j : (4) (Since d P n p jv j ; (4) shows that Case. can only occur when d P n p jv j :) It follows from (37), (40), and (4) that P n p n jv j X + p j v j + e + e ; and hence e 0: Thus, (39) also holds in Case., for every i ; :::; n and every x [y i ; y i ): It follows from this, together with (34), (35), and (38), that the equilibrium strategy pro le (F ; F ) is identical to (F ; F ) of Proposition in the case of d P n p jv j. References [] Abraham, I., Athey, S., Babaiof, M., Grubb, M.: Peaches, Lemons, and Cookies: Designing auction markets with dispersed information. Working paper, Harvard University (04) [] Amman, E., Leininger,W.: Asymmetric all-pay auctions with incomplete information: the two-player case. Games and Economic Behavior 4, -8 (996) [3] Baye, M. R., Kovenock, D., de Vries, C. G.: Rigging the lobbying process: an application of the all-pay auction. American Economic Review 83, (993) [4] Baye, M., Kovenock, D., de Vries, C.: The all-pay auction with complete information. Economic Theory 8, (996) [5] Che, Y-K., Gale, I.: Caps on political lobbying. American Economic Review 88(3), (998) 6

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