SIGNALING IN CONTESTS Tomer Ifergane and Aner Sela Discussion Paer No. 17-08 November 017 Monaster Center for Economic Research Ben-Gurion University of the Negev P.O. Box 653 Beer Sheva, Israel Fax: 97-8-647941 Tel: 97-8-64786
Signaling in Contests Tomer Ifergane and Aner Sela y May 15, 017 Abstract We analyze whether re-contest communications would occur in contest models with asymmetric information. We nd that in Tullock contests signals can be e ectively used in equilibrium. We then study all-ay contests and show that such signals are not e ective, and therefore re-contest communications will not occur in equilibrium. Keywords: Contests, signaling, asymmetric information, incomlete information. JEL classi cation: C70, D7, D8, D44 1 Introduction In signaling games some layers are uninformed about the tyes of their oonents in which case their oonents may either send or not send signals to reveal their tyes. The ower and the comlexity of signaling games has already been demonstrated in the seminal works of Nobel rize-winners Akerlof (1970) and Sence (1973) as well as in many other research works. Our goal in this aer is to examine the ower of signaling in the two main contest models: the Tullock contest 1 in which the contest success function is stochastic such that the robability of a layer to win is equal to the ratio of his e ort and the total e ort Deartment of Economics, Ben-Gurion University of the Negev, Beer Sheva 84105, Israel. y Deartment of Economics, Ben-Gurion University of the Negev, Beer Sheva 84105, Israel. Email: anersela@bgu.ac.il 1 Numerous researchers have analyzed the Tullock contests with either comlete and incomlete information. See, among others, Tullock (1980), Skaerdas (1996), Szidarovszky and Okuguchi (1997), Clark and Riis (1998), Fey (008) and Wasser (013). 1
exerted by all the layers, and the all-ay contest in which the contest success function is deterministic such that the layer with the highest e ort wins. 3 In this aer we ask whether or not re-contest communication between contestants with asymmetric information is ossible. In other words, is there a lace for signaling in these two contest models under incomlete information. Alternatively, we ask whether the layers who have rivate information about their tyes have any incentive to send costly signals in order to reveal their tyes and by doing so to change the contest with incomlete information into a contest with comlete information. The incentive to send a costly signal could be whether the layer is strong (has a high value of winning) or weak (has a low value of winning). In both cases, the role of the signal is to reduce the layers costs of e ort. We assume that one of the layers (referred to as the informed layer) has only two ossible tyes (values of winning) and his tye is rivate information. The tye of the other layer (referred to as the uninformed layer) is common knowledge. In the rst stage, the informed layer decides whether or not to send a costly signal in order to reveal his tye. In the second stage, both layers comete in an asymmetric contest with either comlete or incomlete information. We consider rst the Tullock two-stage contest, analyze its erfect Bayesian equilibrium, and show that, deending on the arameters of the model, there may exist a searating erfect Bayesian equilibrium as well as a ooling erfect Bayesian equilibrium. We rst nd that the informed layer (when he is relatively strong) might have an incentive to send a costly signal in order to reveal his tye. In that case, the weak layer who wishes to retend to be strong may not nd it ro table to send the same signal of the strong layer. Seci cally, we show that the informed layer has an incentive to send such a signal when he is strong but the robability of his tye is relatively low. Otherwise, when the robability of his tye is relatively high, he does not have an incentive to send a costly signal since he does not need to convince his oonent that he is indeed strong. We also nd that the weak layer might have an incentive to send a costly signal, but the signal he needs to send in order to reveal his tye is not Numerous researchers have analyzed the all-ay contest with either comlete or incomlete information. See, among others, Hlilman and Riley (1989), Baye, Kovenock and de Vries (1996), Amann and Leininger (1996), Krishna and Morgan (1997), Che and Gale (1998), Moldovanu and Sela (001, 006) and Siegel (009). 3 The Tullock contest as well as the all-ay contest have several alications including rent-seeking and lobbying in organizations, R&D races, olitical contests, romotions in labor markets, trade wars, military uroses and biological wars of attrition.
ro table. Therefore there is only a searating equilibrium in which the strong layer is the only one who might send a costly signal. Then we consider the two-stage all-ay contest and analyze its erfect Bayesian equilibrium where the uninformed layer s tye is either higher or lower than the two ossible tyes of the informed layer. We demonstrate that in the all-ay contest, a searating erfect Bayesian equilibrium does not exist, which means that the informed layer has no incentive to send a costly signal in order to reveal his tye. The reason is that for both tyes of the informed layer a searating equilibrium is not ro table comared to the ooling equilibrium and therefore each of the tyes does not have any incentive to send a costly signal. Furthermore, even when the uninformed layer s tye is higher than the informed layer s high tye and lower than his low tye (for which we do not have an exlicit characterization of the equilibrium strategies) it is clear that a searating equilibrium is not ossible. These results emhasize the di erent e ects of the deterministic and stochastic contest success functions on signalling in contests under incomlete information. The issue of signaling in contests has already been studied (see, for examle, Amegashie 005 and Zhang and Wang 009). However, there is a key di erence between our model and the other ones. In the other aers, the signaling occurs through early eriod e orts that a ect these layers robabilities of winning. On the other hand, in our aer the signaling occurs through a costly signal that has no e ect on the layers robabilities of winning, but only on their exected ayo s. The rest of the aer is organized as follows. In Section we analyze the two-stage Tullock contest, in Section 3 we analyze the two-stage all-ay contest, and Section 4 concludes. The roofs aear in an Aendix. The two-stage Tullock contest We rst consider a two-stage Tullock contest with two layers. The value of winning (tye) for layer j is v j while the value of winning for layer i is v il with a robability of L or v ih with a robability of H. The tye of layer i is rivate information while the tye of layer j is commonly known. In the second stage, if layers i and j exert e orts of x i ; x j ; then layer i wins with a robability of with a robability of x i x i+x j ; layer j wins x j x i+x j ; and the layers costs in that stage are x i and x j resectively. In the rst stage, 3
however, layer i can send a signal s i given his tye in order to reveal it, and then his cost in that stage is equal to s i : The layers wish to maximize their utility functions which are given by u i (x i ; x j ; s i ) = v i x i x i + x j x i s i u j (x i ; x j ) = v j x j x i + x j x j We analyze the Perfect Bayesian Nash Equilibrium (PBNE) of the above two-stage Tullock contest that consists of strategy ro les of both layers and the belief of layer j (the uninformed layer) after he observes the signal s i of layer i (the informed layer). The layers strategies are sequentially rational given the beliefs and the strategies of their oonent, and the beliefs of the uninformed layer that based on the signal received from the informed layer satisfy the Bayes rule whenever ossible..1 The second stage In order to analyze the erfect Bayesian Nash Equilibrium of the two-stage Tullock contest we begin with the second stage and go backwards to the rst one..1.1 Pooling equilibrium Consider rst that both tyes of layer i send the same signal in the rst stage which can be either ositive or zero. Then, the maximization roblems of layer i with tyes H and L in the second stage are max x ih v ih x ih x ih + x j x ih (1) max x il v il x il x il + x j x il In that case, layer j does not know the tye of layer i and therefore his maximization roblem is The F.O.C. of (1) and () are x j x j max v j ( H + P L ) x j () x j x j + x ih x j + x il v ih v il x j (x ih + x j ) = 1 x j (x il + x j ) = 1 4
and v j ( ih x ih (x ih + x j ) + il x il (x il + x j ) ) = 1 The solution of the F.O.C. yields that the layers e orts in the second stage are x il = v il( H v j vil v ih + L v j v ih ) (v il v ih + P H v j (v il vil v ih )) (v il v ih + H v il v j + L v ih v j ) (3) x ih = v il v ih ( H v j vil v ih + L v j v ih ) ( v il v ih L v j v il (v il vil v ih )) (v il v ih + H v il v j + L v ih v j ) and x j = v il( H v j vil v ih + L v j v ih ) (v il v ih + H v il v j + L v ih v j ) (4) We focus here on interior erfect Bayesian equilibrium and therefore we need the following su cient condition that layer i with tye L exerts a ositive e ort (the e orts of layer i with tye H and layer j are always ositive): v il v ih + P H v j (v il vil v ih ) > 0 Then, layer i s exected ayo s in the second stage are u ih = v vil v ih + L v j (v ih vil v ih ) ih (5) v il v ih + H v il v j + L v ih v j u il = v vil v ih + H v j (v il vil v ih ) il v il v ih + H v il v j + L v ih v j where u ik is layer i s exected ayo if his value is v ik; k fh; Lg:.1. Searating equilibrium Consider now that both tyes of layer i send di erent signals in the rst stage. Then, the maximization roblems of layer i with tyes H and L are max x ih v ih x ih x ih + x j max x il v il x il x il + x j x ih x il In that case, layer j is able to distinguish between the tyes of layer i and therefore his maximization roblem is max x j v j x j x j + x i x j (6) 5
where x i = x ih or x i = x il according to the tye of layer i: Then, the equilibrium e orts are given by the solution of the standard two-layer Tullock contest as follows: x i = x j = v i v j (v i + v j ) (7) v j v i (v i + v j ) where x i = x ih and v i = v ih or x i = x il and v i = v il according to the tye of layer i: Then, layer i s exected ayo s in the second stage are u s ih = u s il = v 3 ih (v ih + v j ) (8) v 3 il (v il + v j ) where u s ik is layer i s exected ayo if his value is v ik; k fh; Lg: Denote now by eu s il the exected ayo of layer i with tye L when layer j believes that he has tye H: By (7), the strategy of layer j will be x j = v j v ih (v ih +v j) : Then, the maximization roblem of layer i with tye L is Similarly, denote by eu s ih x il max v il x x il x il + v j v il (9) ih (v ih +v j) the exected ayo of layer i with tye H when layer j believes that he has tye L: By (7), the strategy of layer j will be x j = with tye H is The solution of the maximization roblems (9) and (10) yields v j v il (vl+v j) : Then, the maximization roblem of layer i x ih max v ih x x ih x ih + v j v ih (10) il (v il +v j) Proosition 1 In the two-stage Tullock contest 1. If layer j believes that layer i with tye L has tye H, then layer i has an exected ayo of eu s vil (v j + v ih ) il = v j + v ih v j vih (11). If layer j believes that layer i with tye H has tye L, then layer i has an exected ayo of eu s vih (v j + v il ) ih = v j + v il v j vil (1) 6
Proof. See Aendix. The following result rovides the conditions under which layer i has an incentive to retend that he has a di erent tye. Proosition In the two-stage Tullock contest 1. eu s il > us il i v j < v ih v il ; i.e., for such layers values of winning, layer i with tye L has a higher exected ayo if layer j believes that he has tye H:. eu s ih > us ih i v j > v ih v il ; i.e., for such layers values of winning, layer i with tye H has a higher exected ayo if layer j believes that he has tye L: Proof. See Aendix..1.3 Searating equilibrium vs. ooling equilibrium In the following we comare layer i s exected ayo s under the ooling and the searating equilibrium. We rst denote the di erences of both tyes of layer i s exected ayo s when layer j has the true beliefs about the tyes of layer i; namely, he believes that tye k is indeed tye k; k fh; Lg. Formally, H s = u s ih u ih (13) L s = u s il u il We then show that the tyes of layer i necessarily have di erent references about these two tyes of equilibrium (ooling or searating). Proosition 3 In the two-stage Tullock contest it is not ossible that both tyes of layer i refer the searating equilibrium over the ooling equilibrium and vice versa. In articular, 1. If v j < v ih v il then H s > 0 and L s < 0; i.e., layer i with tye H (L) has a higher (lower) exected ayo under the searating equilibrium than under the ooling equilibrium.. If v j > v ih v il then H s < 0 and L s > 0; i.e., layer i with tye L (H) has a higher (lower) exected ayo under the searating equilibrium than under the ooling equilibrium. 3. If v j = v ih v il then H s = L s = 0; i.e., both tyes of layer i; H and L, have the same exected ayo under the searating and the ooling equilibrium. 7
Proof. See Aendix. We now denote the di erences of both tyes of layer i s exected ayo s when layer j has wrong beliefs about layer i s tye (namely, he believes that tye L is tye H and vice versa) as follows: e H s = eu s ih u ih e L s = eu s il u il In the following we show that both tyes of layer i may refer the searating equilibrium over the ooling equilibrium if layer j has the wrong beliefs about their tyes. Proosition 4 In the two-stage Tullock contest 1. If v j < v ih v il, then e L s > 0; i.e., layer i with tye L has a lower exected ayo under the ooling equilibrium than under the searating equilibrium when layer j believes that he has tye H:. If v j > v ih v il, then e H s > 0; i.e., layer i with tye H has a lower exected ayo under the ooling equilibrium than under the searating equilibrium when layer j believes that he has tye L: Proof. See Aendix. We also denote the di erences of both tyes of layer i s exected ayo s when layer j has the wrong and the right beliefs by H s s = eu s ih u s ih (14) L s s = eu s il u s il Then, we can conclude from Proositions (3) and (4) that Conclusion 1 In the two stage Tullock contest 1. If v j < v ih v il, layer i with tye H refers the searating equilibrium over the ooling equilibrium, i.e., H s > 0: In that case, layer i with tye L refers a searating equilibrium to the ooling equilibrium i layer j believes that he has tye H, i.e., L s s > 0:. If v j > v ih v il, layer i with tye L refers the searating equilibrium over the ooling equilibrium, i.e., L s > 0: In that case, layer i with tye H refers a searating equilibrium to the ooling equilibrium i layer j believes that he has tye L, i.e., H s s > 0: 8
. The rst stage In order to characterize the erfect Bayesian equilibrium of the two-stage Tullock contest we de ne the following beliefs of layer j about the tye of layer i according to the signal sent by layer i in the rst stage. De nition In the two-stage Tullock contest, if v j < v ih v il, layer j s beliefs are as follows: If H s > L s, then layer j believes that each signal s i L s s is sent by layer i with tye H, and each signal s i < L s s is sent by tye L. And, if H s L s, then layer j believes that each signal s i is sent by both tyes of layer i according to their riors: The rationale behind De nition is that when v j < v ih v il, by Proosition 3, tye H of layer i refers the searating equilibrium to the ooling equilibrium, while tye L of layer i has the oosite reference. If H s > L s, the di erence of tye H s ayo from the searating equilibrium comared to the ooling equilibrium is higher than the di erence of tye L s ayo from the ooling equilibrium comared to the searating equilibrium. Then, if tye H sends a signal higher than L s s it is clear that this signal was sent by him since tye L will have a negative ayo if he would send the same signal. Any signal lower than L s s can be sent by both tyes of layer i and therefore layer j does not distinguish between layer i s tyes for such signals. If, on the other hand, H s L s, the di erence between tye H s ayo from the searating equilibrium comared to the ooling equilibrium is lower than the di erence of tye L s ayo from the ooling equilibrium comared to the searating equilibrium. Then, every signal that tye H will send can be sent by tye L as well. Thus, layer j believes that any signal could be sent by each of the tyes of layer i according to their riors. De nition 3 In the two-stage Tullock contest, if v j > v ih v il, layer j s beliefs are as follows: If L s > H s then layer j believes that each signal s i H s s is sent by layer i with tye L, and that each signal s i < H s s is sent by tye H. And, if L s H s, then layer j believes that each signal s i is sent by both tyes of layer i according to their riors. The rationale behind De nition 3 is that when v j > v ih v il ; by Proosition 3, tye L of layer i refers the searating equilibrium to the ooling equilibrium while tye H of layer i has the oosite 9
reference. If L s > H s, the di erence of tye L s ayo from the searating equilibrium comared to the ooling equilibrium is higher than the di erence of tye L s ayo from the ooling equilibrium and the searating equilibrium. If tye L sends a signal higher than H s s it is clear that this signal was sent by him since tye H will have a negative ayo if he would send the same signal. Any signal lower than H s s can be sent by both tyes of layer i and therefore layer j does not distinguish between layer i s tyes for such signals. If, on the other, L s H s, the di erence of tye H s ayo from the searating equilibrium comared to the ooling equilibrium is lower than the di erence of tye H s ayo from the ooling equilibrium comared to the searating equilibrium and then every signal that tye L will send can be sent by tye H as well. Thus, layer j believes that any signal could be sent by each of the tyes of layer i according to their riors. De nition 4 In the two-stage Tullock contest, if v j = v ih v il, layer j believes that each signal s i is sent by both tyes of layer i according to their riors. The rationale behind De nition 4 is that when v j = v ih v il, by Proosition 3, tyes H and L of layer i are indi erent between the searating equilibrium and the ooling equilibrium. Thus they both do not have an incentive to send a costly signal, and therefore layer j believes that each signal s i is sent by both tyes of layer i according to their riors. Given layer j s beliefs, we can characterize the erfect Bayesian equilibrium in the two-stage Tullock model. The equilibrium characterization is divided into three arts (Theorems 5, 6 and 7) according to the relation between the layers values of winning. 1. If v j < v ih v il we have a searating equilibrium as well as a ooling equilibrium. Theorem 5 In the two-stage Tullock contest, if v j < v ih v il and if the layers beliefs are given by De nition, then (i). If H s L s, there is a ooling erfect Bayesian equilibrium in which both tyes of layer i, L and H, do not send any signal in the rst stage. Then, in the second stage, the layers strategies are given by (3) and (4). (ii). If H s > L s ; there is a searating erfect Bayesian equilibrium in which layer i with tye H sends a signal in the rst stage s ih = L s s and layer i with tye L does not send any signal. Then, in 10
the second stage, the layers strategies are given by (7). In that case, a su cient condition for a searating erfect Bayesian equilibrium is a su ciently small value of H : Proof. See Aendix. According to Theorem 5, we can see that there is a searating erfect Bayesian equilibrium in the twostage Tullock contest if the robability of tye H of layer i is relatively small and his value is signi cantly larger than the value of his oonent. Note that only if the robability of tye H is small, layer i with tye H has an incentive to send a signal. Otherwise, if the robability of tye H is high, there is no need to send a signal since layer j already believes that his oonent robably has tye H:. If v j > v ih v il we have only the ooling equilibrium. Theorem 6 In the two-stage Tullock contest, if v j > v ih v il and if the layers beliefs are given by De nition 3, there is only a ooling erfect Bayesian equilibrium in which both tyes of layer i, L and H, do not send any signal in the rst stage. Then, in the second stage, the layers strategies are given by (3) and (4). Proof. See Aendix. 3. If v j = v ih v il, as in the revious case, we have only the ooling equilibrium. Theorem 7 In the two-stage Tullock contest, if v j = v ih v il and if the layers beliefs are given by De nition 4, there is a ooling erfect Bayesian equilibrium in which both tyes of layer i, L and H, do not send any signal in the rst stage. Then, in the second stage, the layers strategies are given by (3) and (4). Proof. See Aendix. 3 The two-stage all-ay contest We now consider a two-stage all-ay contest with two layers. The value of winning (tye) for layer j is v j while the value of winning for layer i is v L with robability L, or v H with robability H. The tye of layer i is rivate information and the tye of layer j is commonly known. In the second stage, each layer i submits a bid (e ort) x [0; 1) and the layer with the highest bid wins the rst rize and all the layers 11
ay their bids. In the rst stage, however, layer i can send a signal s i in order to reveal his tye and then his cost in that stage is equal to s i : The layers wish to maximize their utility functions which are given by 8 u i (x i ; x j ; s i ) = u j (x i ; x j ) = >< >: 8 >< v i x i s i if x i > x j 1 v i x i s i if x i = x j x i s i if x i < x j v j x i if x j > x i 1 v j x i if x j = x i >: x i if x j < x i We analyze the Perfect Bayesian Nash Equilibrium (PBNE) of the above two-stage all-ay contest that consists of strategy ro les of both layers and the belief of layer j (the uninformed layer) after he observes the signal s i of layer i (the informed layer). The layers strategies are sequentially rational given the beliefs and the strategies of their oonent, and the beliefs of the uninformed layer that based on the signal received from the informed layer satisfy the Bayes rule whenever ossible. 3.1 The second stage 3.1.1 Searating equilibrium Consider rst that both tyes of layer i send di erent signals in the rst stage. Then, assume that layer i s value in the second stage is v i where v i = v il or v i = v ih, and, without loss of generality, assume also that the layers values satisfy v i > v j : According to Baye, Kovenock and de Vries (1996), there is always a unique mixed-strategy equilibrium in which layers i and j randomize on the interval [0; v j ] according to their e ort cumulative distribution functions F i ; F j, which are given by v i F j (x) x = v i v j v j F i (x) x = 0 Thus, layer i s equilibrium e ort in the second stage is uniformly distributed as follows: F i (x) = x v j 1
while layer j s equilibrium e ort in the second stage is distributed according to the following cumulative distribution function F j (x) = v i v j + x v i The resective exected ayo s in the second stage are u i = v i v j (15) u j = 0 3.1. Pooling equilibrium Consider now that both tyes of layer i send the same signal in the rst stage where this signal could be either ositive or zero. Then, we consider two ossible scenarios as follows: 1. Assume rst that v j > v ih > v il : In that case, there is a mixed-strategy equilibrium in which layer i with tyes H and layer j randomize on the interval [0; v ih ] according to their e ort cumulative distribution functions F ih ; F j, which are given by v j ( L + H F ih (x)) x = v j v ih v ih F j (x) x = 0 However, layer i with tye L chooses to stay out of the contest. Thus, Proosition 5 In the two-stage all-ay contest, if v j > v ih > v il, layer i with tye L chooses x il = 0 with robability one in the second stage, while the equilibrium e ort of layer i with tye H in the second stage is distributed according to the cumulative distribution function 8 >< F ih (x) = 0 if x maxf0; v ih H v j g H v j v ih +x H v j if maxf0; v ih H v j g < x v ih (16) >: 1 if x > v ih Player j s equilibrium e ort in the second stage is uniformly distributed as follows: 13
8 >< F j (x) = >: 0 if x maxf0; v ih H v j g x v ih if maxf0; v ih H v j g < x v ih 1 if x > v ih (17) The resective exected ayo s in the second stage are then u il = u ih = 0 (18) u j = v j v ih Proof. See Aendix.. Assume now that v j < v il < v ih : Then, there is a mixed-strategy equilibrium in which layer i with tye L and layer j randomize on the interval [0; L v j ] according to their e ort cumulative distribution functions F il ; F j, which are given by v j L F il (x) x = 0 v il F j (x) x = v il v j ( H v il v ih + L ) And layer i with tyes H and layer j randomize on the interval [ L v j ; v j ] according to their e ort cumulative distribution functions F ih ; F j, which are given by v j ( L + H F ih (x)) x = 0 v ih F j (x) x = v ih v j Thus, we obtain Proosition 6 In the two-stage all-ay contest, if v j < v il < v ih, the equilibrium e ort of layer i with tye L in the second stage is distributed according to the cumulative distribution function 8 >< x L v j if 0 x L v j F il (x) = (19) >: 1 if x > L v j and the equilibrium e ort of layer i with tye H in the second stage is distributed according to the cumulative 14
distribution function 8 >< F ih (x) = >: 0 if 0 x L v j x L v j H v j if L v j < x v j 1 if x > v j (0) Player j s equilibrium e ort in the second stage is distributed according to the cumulative distribution function 8 >< F j (x) = v il v j( H v il v + L )+x ih v il v ih v j+x v ih if 0 x L v j if L v j < x v j (1) >: 1 if x > v j The resective exected ayo s in the second stage are then u il = v il v j ( H v il v ih + L ) () u ih = v ih v j u j = 0 Proof. See Aendix. 3. The rst stage Based on the analysis of the second stage we show that in contrast to the two-stage Tullock contest we obtain: Proosition 7 In the two-stage all-ay contest there is no searating erfect Bayesian equilibrium. Proof. See Aendix. The result of Proosition 7, according to which there is no searating equilibrium in the two-stage allay contest, is roved only for the case when v j > v ih > v il and v j < v il < v ih since there we exlicitly calculate the equilibrium strategies. For the other case when v il < v j < v ih it is quite comlex to exlicitly calculate the equilibrium strategies. However, by similar arguments used in the roof of Proosition 7, even without such a calculation, it can be shown that a searating erfect Bayesian equilibrium does not exist. 15
4 Concluding remarks We analyzed the Tullock and the all-ay contest when the uninformed layer has a commonly known tye while the informed layer has two ossible tyes which are rivate information. We demonstrated that while in the Tullock contest the informed layer may have an incentive to send a costly signal to reveal his tye, in the all-ay contest he never has such an incentive. One of the reasons that there is a searating erfect Bayesian equilibrium in the Tullock contest and not in the all-ay contest is that the distributions of the layers revenues are comletely di erent. While in the all-ay contest only one of the layers has a ositive exected ayo, in the Tullock contest both layers have ositive exected ayo s and as such re-contest communication might be useful only in the latter form of contest. Because of the comlexity in analyzing multi-stage contests with signaling, we assumed the simlest case of two ossible tyes of layers. It would be of interest to examine whether our results hold when the set of tyes is larger or even continuous. 5 Aendix 5.1 Proof of Proosition 1 If layer j believes that he lays against layer i with tye H, by (7) his e ort will be x j = v ih v j (v ih + v j ) (3) We want to nd the otimal e ort for layer i with tye L who has the maximization roblem max x il eu s il = v il x j (x j + x il ) x il (4) The F.O.C. is v il x j (x il + x j ) 1 = 0 ) x il + x il x j + x j x j v il = 0 The solution of this quadratic equation is x il = q x j + (x j ) 4(x j x j v il ) 16 (5)
By substituting (3) in (5), we obtain that x il = v jv ih vih v il + vj vih v il v ih vj (v ih + v j ) (6) Substituting (6) in (4) yields that the exected ayo of layer i with tye L when layer j believes that he has tye H is eu s vil (v j + v ih ) il = v j + v ih v j vih Similarly, we obtain that the exected ayo of layer i with tye H when layer j believes that he has tye L is Q:E:D: eu s vih (v j + v il ) ih = v j + v il v j vil 5. Proof of Proosition By (7), when layer j believes that he lays against layer i with tye H, he will exert an e ort of x j H = v ihv j (v ih +v j) ; and when he believes that he lays against layer i with tye L he will exert an e ort of x j L = v ilv j (v il +v j) : It can be easily veri ed that x j H = v ih v j (v ih + v j ) v il v j (v il + v j ) = x j L (7) i v j v ih v il Now, by the Enveloe Theorem we obtain that if V = max x i v i x i x i + x j x i Then, dv dx j = v i x i (x i + x j ) < 0 In other words, layer i s exected ayo decreases in layer j s e ort. Thus, by (7), if v j < v ih v il, layer i with tye L refers that layer j will believe that he has tye H since then layer j will exert a lower e ort. Similarly, if v j > v ih v il, layer i with tye H refers that layer j will believe that he has tye L since then layer j will exert a lower e ort. Q:E:D: 17
5.3 Proof of Proosition 3 By (5) and (8), we have H s = v ih v ih (v ih + v j )! vil v ih + L v j (v ih vil v ih ) v il v ih + H v il v j + L v ih v j Since v ih > v il, we obtain that H s 0 i v ih (v il v ih + H v il v j + L v ih v j ) (v ih + v j )(v il v ih + L v j (v ih vil v ih )) The last inequality holds i Since 1 v j 1 r vih v il v ih ( v il vih q vih v il < 0, if we divide both sides by this term we obtain that v j v ih v il Thus, we obtain that layer i with tye H refers a searating equilibrium over a ooling equilibrium i v j v ih v il. Similarly, by (5) and (8), we have L s = v il v il (v il + v j )! vil v ih + H v j (v il vil v ih ) v il v ih + H v il v j + L v ih v j Since v ih > v il, we obtain that L s 0 i v il (v il v ih + H v il v j + L v ih v j ) (v il + v j )(v il v ih + H v j (v il vil v ih )) The last inequality holds i v j v il( v ih v il v ih ) vih v il + v il = v ih v il Thus, we obtain that layer i with tye L refers the searating equilibrium over the ooling equilibrium i v j v ih v il. Q:E:D: 6 Proof of Proosition 4 1. We rst need to show that if v j < v ih v il then e L s = eu s il u il > 0 18
By (5) and (11) e L vil (v j + v ih ) v j vih s = v j + v ih v il vil v ih + H v j (v il vil v ih ) v il v ih + H v il v j + L v ih v j Thus, we need to show that ( v il (v j + v ih ) v j vih )(v il v ih + H v il v j + L v ih v j ) vil (v il v ih + H v j (v il vil v ih ))(v j + v ih ) 0 The last inequality holds i vil v ih ( v ih vil) ) v j ( v ih vil) ) 0 Thus, e L s 0 i v j v ih v il :. Now we need to show that if v j > v ih v il then e H s = eu s ih u ih > 0 By (5) and (1) e H vih (v j + v il ) v j vil s = v j + v il v ih vil v ih + L v j (v ih vil v ih ) v il v ih + H v il v j + L v ih v j Thus, we need to show that ( v ih (v j + v il ) v j vil )(v il v ih + H v il v j + L v ih v j ) 0 (v il v ih + L v j (v ih vil v ih ))(v j + v il ) v ih The last inequality holds i v j ( v ih vil) ) vil v ih ( v ih vil) ) 0 Thus, e H s 0 i v j v ih v il : Q:E:D: 19
6.1 Proof of Theorem 5 By Proosition 3, if v j < v ih v il, layer i with tye H has a higher exected ayo under the searating equilibrium than under the ooling equilibrium, and layer i with tye L has a lower exected ayo under the searating equilibrium than under the ooling equilibrium. 1. If H s < L s, tye H is willing to ay less for the searating equilibrium than what tye L is willing to ay for the ooling equilibrium. In that case, by De nition, for any signal s ih there will be the same ooling equilibrium as without this signal. Thus, layer i with tye H has no incentive to send a costly signal and therefore the ooling equilibrium occurs.. If H s > L s, tye H is willing to ay more for the searating equilibrium than what tye L is willing to ay for the ooling equilibrium. In that case, if there is a searating equilibrium, by Proosition 4 layer i with tye L refers that layer j will believe that he has tye H. However, since layer i with tye H sends a signal of s ih = L s s > 0, tye L will not have an incentive to send this signal as well. Thus, since only tye H sends a signal, by De nition layer j can distinguish between layer i s tyes according to the signal. If layer i with tye L will send a signal s il < L s s, by De nition there will be the same ooling equilibrium as without this signal. Thus, layer i with tye L has no incentive to send a costly signal. In order to comlete the roof we need to show that a signal of L s s is not too exensive for tye H; namely, he refers the searating equilibrium with the signal ayment L s s over the ooling equilibrium without any signal ayment. Below, we show that lim ( H s L s H!0 s) > 0 By (5) and (8), lim H vih s = v ih H!0 (v ih + v j )! vil v ih + v j (v ih vil v ih ) v ih (v il + v j ) and by (5), (8) and (11), L vil (v ih + v j ) v j vih ) s s = v ih + v j v 3 il (v il + v j ) 0
Note that L s s does not deend on the value of H. Thus, we need to show that lim ( H s L vih s H!0 s) = v ih (v ih + v j )! vil vil v ih + v j (v ih vil v ih ) (v ih + v j ) v j vih ) v ih (v il + v j ) v ih + v j = = v3 ih (v ih vil v j ( v ih vil )) (v ih + v j ) + v3 il (v il vih + v j ( v ih vil )) (v il + v j ) 0 (v i It can be veri ed that the last inequality is satis ed i v j v ih v il which is exactly our assumtion here about the layers values. Therefore, in order to show that there is a searating erfect Bayesian equilibrium, it remains to show that lim ( H s ( L s H!0 )) = lim ( H s + L s H!0 ) > 0 By (5) and (8), lim H!0 L s vil v il = v il (v il + v j ) (v il + v j ) = 0 and since lim H!0 H s > 0, we obtain that lim H!0 ( H s + L s ) > 0: Q:E:D: 6. Proof of Theorem 6 By Proosition 3, if v j > v ih v il, layer i with tye L has a higher exected ayo under the searating equilibrium than under the ooling equilibrium and layer i with tye H has a lower exected ayo under the searating equilibrium than under the ooling equilibrium. Then we have two cases: 1. If H s > L s, tye L is willing to ay less for the searating equilibrium than what tye H is willing to ay for the ooling equilibrium. In that case, by De nition 3, for any signal s il there will be the same ooling equilibrium as without this signal. Thus, layer i with tye L has no incentive to send a costly signal and the ooling equilibrium occurs.. If H s < L s, tye L is willing to ay more for the searating equilibrium than what tye H is willing to ay for the ooling equilibrium. In that case, if there is a searating equilibrium, by Proosition 4 layer i with tye H refers that layer j will believe that he has tye L. However, if layer i with tye H will send a signal of s ih H s s > 0, tye L will not have an incentive to send this signal as well. Then, since only tye L sends a signal, by De nition 3 layer j could distinguish between layer i s tyes according to the signal sent in the rst stage. Any lower signal than H s s will give tye H the incentive to 1
send the same signal in order to retend that he has tye L. However, below we show that a signal of H s s is too exensive for tye L; namely, he refers the ooling equilibrium without any signal ayment over the searating equilibrium with the signal ayment H s s. Thus, below we show that L s H s s < 0 By (5) and (8), L s = v il v il (v il + v j )! vil v ih + H v j (v il vil v ih ) v il v ih + H v il v j + L v ih v j and by (5), (8) and (11), H vih (v il + v j ) v j vil ) s s = v il + v j v 3 ih (v ih + v j ) Note that vil v ih + (1 L )v j (v il vih v il ) = v il d L v il v ih + (1 L )v j v il + L v j v ih vj (v il vih v il )(v il v ih + (1 L )v j v il + L v j v ih ) v j (v ih v il )(v il v ih + (1 L )v j (v il vih v il ) (v il v ih + (1 L )v j v il + L v j v ih ) d L s Thus, dl s d L 0 i (v il vih v il )(v il v ih + (1 L )v j v il + L v j v ih ) + (v ih v il )(v il v ih + (1 L )v j (v il vih v il ) 0 It can veri ed that the last inequality holds i v ih v il < v j which is exactly our assumtion on the layers values. 4 Thus, L s decreases in L and therefore it is su cient to show that lim L!0 (L s H vil s s) = v il (v il + v j )! vih vil v ih + H v j (v il vil v ih ) (v il + v j ) v j vil ) v il v ih + H v il v j + L v ih v j v il + v j (9) (v ih v A comarison of equations (8) and (9) yields that lim L!0 (L s H s s) = lim H!0 ( H s L s s) Thus, the inequality in (9) is satis ed i v j v ih v il which is exactly our assumtion here about the layers values. Therefore a searating equilibrium is not ossible and we have only the ooling equilibrium in which no tye of layer i sends a signal. Q:E:D: 4 The comlete mathematical calculations are available uon request.
6.3 Proof of Theorem 7 By Proosition 3, if v j = v ih v il, layer i with either tye L or tye H has the same exected ayo under the searating equilibrium and under the ooling equilibrium. By De nition 4, any signal of layer i will not change the rior beliefs of layer j and therefore layer i, indeendent of his tye, has no incentive to send any signal. Q:E:D: 6.4 Proof of Proosition 5 We can see that the functions F ih (x); F j (x), given by (16) and (17), resectively, are well-de ned, strictly increasing on [0; v ih ], continuous, and that F j (0) = 0, F ih (0) = maxf0; Hv j H v j g, F ih (v ih ) = F j (v ih ) = 1. Thus, F ih (x); F j (x) are cumulative distribution functions of continuous robability distributions suorted on [0; v ih ]. In order to see that the above strategies are an equilibrium, note that when contestant j uses the mixed strategy F j (x), the exected ayo of contestant i with tyes L and H is zero for any e ort x [0; v ih ]. Since it can be easily shown that e orts above v ih would lead to a negative exected ayo for contestant i, any e ort in [0; v ih ] is a best resonse of contestant i with tye H to F j (x): Likewise, x = 0 is the best resonse of contestant i with tye L to F j (x): Similarly, when contestant i uses the mixed strategy v ih F ih (x) and F il (x); contestant j s exected ayo is v j v ih for any e ort x [0; v ih ]. Since it can be easily shown that e orts above v ih would result in a lower exected ayo for contestant j, any e ort in [0; v ih ] is a best resonse of contestant j to F ih (x) and F il (x): Hence, (F ih (x); F il (x); F j (x)) are the equilibrium strategies in the second stage. Q:E:D: 6.5 Proof of Proosition 6 We can see that the functions F il ; F ih (x); F j (x), given by (19), (0) and (1), resectively, are wellde ned, F il is strictly increasing on [0; L v j ], F ih (x) strictly increasing on [ L v j ; v j ], and F j (x) is strictly increasing on [0; v j ]. They are all continuous, satisfy F il (0) = F j (0) = 0, F il ( L v j ) = 1; F ih ( L v j ) = 0; and that F j (v j ) = F ih (v j ) = 1. Thus, F il ; F ih (x); F j (x) are cumulative distribution functions of continuous robability distributions suorted on [0; L v j ] ; [ L v j ; v j ] ; [0; v j ], resectively. In order to see that the 3
above strategies are an equilibrium, notice that when contestant i uses the mixed strategy F il (x) or F ih (x), the exected ayo of contestant j is zero for any e ort x [0; v j ]. Since it can be easily shown that e orts above v j would lead to a negative exected ayo for contestant j, any e ort in [0; v j ] is a best resonse of contestant j: Likewise, when contestant j uses the mixed strategy F j (x); the exected ayo of contestant i with tye H is v ih v j for any e ort x [ L v j ; v j ], and the exected ayo of contestant i with tye L is v il v j ( Hv il v ih + L ) for any e ort x [0; L v j ]. Since it can be easily shown that e orts above L v j would result in a lower exected ayo for contestant i with tye L, and e orts below L v j or above v j would result in a lower exected ayo for contestant i with tye H, any e ort in [0; L v j ] is a best resonse of contestant i with tye L, and any e ort in [ L v j ; v j ] is a best resonse of contestant i with tye H to F j (x): Hence, (F ih (x); F il (x); F j (x)) are the equilibrium strategies in the second stage. Q:E:D: 6.6 Proof of Proosition 7 1. Assume rst that v j > v ih > v il : Then, by (15), if there is a searating erfect Bayesian equilibrium, the layers exected ayo s in the second stage are u j = v j v i u i = 0 where v i = v ih if layer i has tye H and v i = v il if layer i has tye L. Since, indeendent of his tye, layer i has an exected ayo of zero in the second stage he has no incentive to send a costly signal in the rst stage.. Assume that v j < v il < v ih : Then, by (15), if there is a searating erfect Bayesian equilibrium, the layers exected ayo s in the second stage are u j = 0 u i = v i v j where v i = v ih if layer i has tye H and v i = v il if layer i has tye L. If, on the other hand, there is a 4
ooling erfect Bayesian equilibrium, by () the layers exected ayo s in the second stage are u il = v il v j ( H v il v ih + L ) u ih = v ih v j u j = 0 Since layer i with tye H has the same exected ayo in the second stage in both tyes of equilibrium he has no incentive to send a costly signal in the rst stage. However, Player i with tye L has an incentive to send a signal i or alternatively, i v il v j > v il v j ( H v il v ih + L ) H v il v ih + L > 1 But since v ih > v il, the last inequality does not hold. Thus, layer i with either tye H or L has no incentive to send a costly signal in the rst stage. Q:E:D: References [1] Akerlof, G.A.: The market for "lemons": quality uncertainty and the market mechanism. Quartery Journal of Economics 84(3), 488-500 (1970) [] Amegashie, J. A.: Information transmission in elimination contests. Working aer, University of Guelh (005) [3] Amann, E., Leininger,W.: Asymmetric all-ay auctions with incomlete information: the two-layer case. Games and Economic Behavior 14, 1-18 (1996) [4] Baye, M., Kovenock, D., de Vries, C.: The all-ay auction with comlete information. Economic Theory 8, 91-305 (1996) [5] Che, Y-K., Gale, I.: Cas on olitical lobbying. American Economic Review 88, 643-651 (1998) [6] Clark, D., Riis, C.: Contest success functions: an extension. Economic Theory 11, 01-04 (1998) 5
[7] Fey, M.: Rent-seeking contests with incomlete information. Public Choice 135(3), 5-36 (008) [8] Hilman, A., Riley, J.: Politically contestable rents and transfers. Economics and Politics 1, 17-39 (1989) [9] Krishna, V., Morgan, J.: An analysis of the war of attrition and the all-ay auction. Journal of Economic Theory 7(), 343-36 (1997) [10] Moldovanu, B., Sela, A.: The otimal allocation of rizes in contests. American Economic Review 91(3), 54-558 (001) [11] Moldovanu, B., Sela, A.: Contest architecture. Journal of Economic Theory 16(1), 70-97(006) [1] Siegel, R.: All-ay contests. Econometrica 77 (1), 71-9 (009) [13] Skaerdas, S.: Contest success functions. Economic Theory 7, 83-90 (1996) [14] Sence, M.: Job market signaling. Quarterly Journal of Economics 87(3), 355-374 (1973) [15] Szidarovszky, F., Okuguchi, K.: On the existence and uniqueness of ure Nash equilibrium in rentseeking games. Games and Economic Behavior 18, 135-140 (1997) [16] Tullock, G.: E cient rent-seeking, in J.M. Buchanan, R.D. Tollison and G. Tullock (Eds.), Toward a theory of the rent-seeking society. College Station: Texas A.&M. University Press (1980) [17] Wasser, C.: Incomlete information in rent-seeking contests. Economic Theory 53(1), 39-68 (013) [18] Zhang, J., Wang, R.: The role of information revelation in elimination contests. The Economic Journal 119(536), 613-641 (009) 6