Volunteering and the strategic value of ignorance

Transcription

1 Volunteering and the strategi value of ignorane Florian Morath Max Plank Institute for Tax Law and Publi Finane November 10, 011 Abstrat Private provision of publi goods often takes plae as a war of attrition: individuals wait until someone else volunteers and provides the good. After a ertain time period, however, one individual may be randomly seleted. If the individuals are unertain about their ost of provision, but an nd out about this ost ahead of the volunteering game, a strategi value is attahed to the information, and individuals may prefer not to learn their ost of provision. If the time horizon is su iently short, in equilibrium only one individual may aquire information about his ost. For a long time horizon, aquiring information is stritly dominant. The time limit is an important instrument in in uening the e ieny of the volunteering game. Keywords: War of attrition; Volunteering; Disrete publi goods; Asymmetri information; Information aquisition JEL lassi ation numbers: H41; D44; D8; D83 Correspondene address: Max Plank Institute for Tax Law and Publi Finane, Department of Publi Eonomis, Marstallplatz 1, Munih, Germany. Phone: Fax: orian.morath@tax.mpg.de. I thank Kai Konrad, Mihael Mihael, Johannes Münster, partiipants of the CESifo Area Conferene 011 on Applied Miroeonomis, two anonymous referees, and an assoiate editor for valuable omments. Finanial support by the German Researh Foundation (DFG) through grant SFB/TR 15 is gratefully aknowledged. 1

2 1 Introdution Dragon-slaying and ballroom daning are two famous examples 1 of the provision of a publi good that indues a positive value for a ertain group of individuals. One of the individuals, however, has to pay some ost in order to provide the publi good. Suh situations are often best desribed by a war of attrition: one volunteer is needed for a ertain task, and everyone prefers someone else to volunteer rst and bear the ost of provision. Typially, there is a disutility or waiting ost attahed to the time until a volunteer is found. In this paper, we study the individuals inentives to obtain information about their own ost of provision of the publi good prior to a volunteering game or war of attrition. Wars of attrition are used to model a large number of appliations from di erent elds. Besides the (mythial) example of dragon-slaying, many unpleasant situations like intervening in a ght, alling the polie in ase of a re or rime, household hores, ghts between animals, or market exit exhibit properties of wars of attrition. Organizations typially rely on the voluntary performane of a large number of tasks. These tasks may have to be performed repeatedly, and the ost of performing the task may then be well-known. But often the individuals don t know exatly how ostly volunteering will turn out to be. They may, for instane, only have a guess about the time involved in hairing a university department or organizing a onferene, but an aquire information about this expenditure of time. In many ompanies or institutions, sta meetings take plae on a regular basis and are used to alloate tasks to individuals. Before volunteering to perform a task, employees typially have the possibility to nd out about their ost of performing this task, and they an do so by asking questions and olleting information. As one of the many examples, suppose that in a ompany the role of an ombudsman has to be assigned to one of the employees, and this is on the agenda of the upoming team meeting. What will be the individual opportunity ost of volunteering for this position? Ahead of the team meeting, there are de nitely possibilities for the 1 Cf. Bliss and Nalebu (1984). Many more examples are given, e.g., by Bilodeau and Slivinski (1996), LaCasse et al. (00), or Otsubo and Rapoport (008).

3 team members to nd out more about the duties involved. The question, however, is what impat information aquisition has on the volunteering game and whether individuals bene t from information aquisition. If suh information aquisition an be observed by the other individuals - for instane when employees ask questions - there is a strategi value attahed to the information: it an be used to ommit to a ertain behavior in the war of attrition. Similarly, international negotiations often have harateristis of a war of attrition (ompare Nalebu 1986 and Fearon 1994, 1998; examples are trade onessions or arms ontrol bargaining). Con it an arise when governments have to deide whether to provide an international publi good, in partiular when ost and bene t of the publi good are unertain. The governments an engage experts to provide them with a better estimate of the ost of provision, and sometimes, they make extensive use of the possibility to aquire information. But when suh investments in information are observable by other players, investments in information obtain a strategi harater: on the politial level, the plea of ignorane is often used to justify a ertain ation or ination or to deny responsibility (for instane, in the ontext of limate hange). Finally, labor strikes have been modeled as war of attritions (Kennan and Wilson 1989, 1990); here, not having too muh information about losses in sales and the onsequenes for employment helps the unions to ommit to a more aggressive behavior in the negotiations. These examples have in ommon that the players annot wait an in nite amount of time before volunteering, but that there is a time limit on their deision to onede. We analyze the individuals inentives to aquire information about their ost of provision of a publi good in a two-stage game with two individuals. In the rst stage, the individuals an obtain information about their ost of provision. In order to fous on the strategi onsiderations, we assume that the information is available at zero ost. Whether or not an individual deided to nd out about his ost an be observed by the rival before the volunteering game starts. The information that an individual has obtained, however, is only privately known to this individual. In the seond stage, a volunteering game or war of attrition takes plae: the individuals simultaneously hoose a maximum waiting time after whih they provide the publi 3

4 good, given that nobody else has volunteered before. The waiting time until the publi good is provided involves a diret ost. As desribed above, individuals may not be able to wait for an in nite amount of time; therefore, we impose a nite time horizon after whih one of the individuals is randomly hosen to pay for the provision. 3 At some point in time, the dragon may itself deide to attak, or, in the ontext of a rm, one employee will be seleted by the team leader to perform the task. Similarly, in the example of labor strikes, there is often an arbitrator who, after a ertain time period, will make a deision. As we will show, the equilibrium of the volunteering game and the inentives to learn the own ost of provision ruially depend on the length of the time horizon. For a long time horizon, both individuals prefer to nd out about their ost of provision. If the time horizon of the volunteering game is su iently short, individuals without information about their provision ost prefer the random seletion when the time limit is reahed to an early onession. As a onsequene, an individual who found out that his ost would be low may prefer to onede immediately. Therefore, not knowing the own ost of provision an be advantageous in the volunteering game. For a su iently short time horizon, the set of equilibria of the information aquisition game onsists of two asymmetri equilibria where one individual nds out about his ost and the other does not, and one symmetri equilibrium where both individuals randomize their deision whether to learn their ost. The hoie of the time horizon is an important instrument in in uening the e ieny of the volunteering game. The literature on wars of attrition has its origin in appliations in biology, modeling ghts between animals (e.g., Maynard Smith 1974, Riley 1980). Further important appliations are industrial ompetition and market exit (Fudenberg and Tirole 1986, Ghemawat and Nalebu 1985, 1990). The seminal paper that studies the private provision of a publi good as a war of attrition is Bliss and Nalebu (1984). In their setup, the players are privately informed of their ost of provision, and the equilibrium is e ient in the sense that the player with the lowest ost provides the 3 The assumption of a purely random seletion ("fair oin ip") if the maximum waiting time is reahed is important for the following analysis. It is one of the rules that is most simple to implement and appropriate in partiular if the designer has to ommit to a rule ex ante and has no further information about the individuals. 4

5 publi good. The provision of multiple publi goods in the framework of a war of attrition is analyzed by LaCasse et al. (00) for the ase of omplete information, and by Sahuguet (006) in an environment with private information. 4 Bishop and Cannings (1978), Hendriks et al. (1988), Bilodeau and Slivinski (1996), and Myatt (005) study models that exhibit a nite time horizon. We add to this literature by studying the e ets of information on the individuals onession times in the private provision game, and the resulting inentives (not) to beome informed. The strategi onsiderations involved in the deision on information are similar to the strategi aspets identi ed in di erent settings suh as prinipal-agent relationships (e.g. Crémer 1995, Kessler 1998): by remaining uninformed, individuals preommit to a ertain behavior in the subsequent interation. 5 Closely related to this paper is work that onsiders information in autions. Whereas the war of attrition is, in fat, a seond-prie all-pay aution, Morath and Münster (010) study information aquisition in a rst-prie all-pay aution, but in their setup, there is no purely strategi value of remaining uninformed. In the ontext of winner-pay autions, inentives to aquire information when deisions are observable have been shown to depend on the exat aution format and on whether information is about a private or a ommon value. An early ontribution studying the value of information is Milgrom and Weber (198); reent work inludes Hernando-Veiana (009), Larson (009), and Hernando-Veiana and Tröge (010). The next setion desribes the setup of the model. We analyze in Setion 3 the three di erent situations that may arise in the volunteering game: no individual has private information about his provision ost, only one individual is informed, or both individuals are informed about their ost of provision. In Setion 4, we onsider the 4 Further papers onsidering wars of attrition with privately informed players are Bulow and Klemperer (1999), who analyze the ase of multiple prizes, and Krishna and Morgan (1997), who study the ase of a liated signals. Amann and Leininger (1996) onsider a general lass of all-pay autions with private information; the same lass of all-pay autions is analyzed in Riley (1999) for the ase of omplete information. Che and Gale (1998) study rst-prie all-pay autions with aps on bidding whih are similar to the nite time horizon of the volunteering game assumed here. 5 In the ontext of global warming, Morath (010) analyzes investments in information in a standard model of private provision of a ontinuous publi good; the strategi e ets that are present in this paper, however, are driven by the assumption that other ountries an observe what a ountry has learned. 5

6 inentives for information aquisition in a game de ned by the ontinuation payo s in the volunteering game, and we disuss some impliations from a designer s perspetive. Setion 5 assesses the robustness of our results. Finally, Setion 6 onludes. All proofs are in the appendix. Setup Consider the following game with two individuals, 1 and. One of the two individuals has to provide a publi good of xed quantity. (We assume that the ontribution that is needed for the provision is indivisible.) The individuals di er with respet to their ost of provision, denoted by 1 and. These ost parameters 1 and are independent draws from a probability distribution that is ommon knowledge and assumed to be a disrete funtion with i f L ; H g ; 0 < L < H ; and probabilities Pr ( i = L ) = p L ; Pr ( i = H ) = p H = 1 p L ; i = 1; : Moreover, := p L L + p H H is an individual s expeted ost of provision. At the beginning of the game, the individuals know neither their own ost of provision nor their rival s ost, but only that this ost an be high or low, and the orresponding probabilities (that is, only the distribution of ost is ommon knowledge, but the individual draws are unknown). 6 6 The assumption of a disrete distribution determines the struture of the equilibrium strategies in the war of attrition if at least one individual learned his ost. The result on inentives to beome informed qualitatively arries over to the ase where the individuals ost is drawn from a ontinuous distribution. See the disussion in the onluding setion. 6

7 In stage 1 of the game, the individuals an nd out about their own provision ost: if an individual deides to beome informed, he privately observes his provision ost. Information aquisition does not involve any diret ost, and the deisions whether or not to obtain information are made simultaneously and beome ommonly known at the end of stage 1. In stage, the individuals i = 1; simultaneously hoose a time of onession t i, i.e., individual i plans to provide the publi good in t i if individual j 6= i has not volunteered before t i. As soon as one individual volunteers, the game ends. However, there is a maximum waiting time T whih is exogenously given and ommon knowledge. Thus, the strategy spae is restrited to t i [0; T ]. If both individuals volunteer exatly at the same time, the provision of the publi good is alloated with equal probability to the individuals. Waiting involves a diret ost to both individuals, whih is assumed to be linear in the waiting time. 7 Stage is strategially equivalent to the war of attrition or seond-prie all-pay aution with a ap on bidding. As it will beome lear in the next setion, this ap on bidding (or maximum waiting time) and the tie-breaking rule if both individuals hoose to wait until T are important for the equilibria of the war of attrition and onsequently for the results on equilibrium information aquisition. Denoting by v an individual s utility from the provision of the publi good, the payo funtions are given by 8 >< v t j ; t i > t j i (t i ; t j ) = v i t i ; t i = t j ; i = 1; : (1) >: v i t i ; t i < t j For all possible t 1 and t, the publi good is provided, and its value v to the individuals is assumed to be the same for both individuals and independent of the provision time. The idiosynrasies are aptured by the provision ost. The individual who hooses the lower waiting time has to bear the provision ost, and both individuals have to pay the ost of waiting, determined by the minimum of t 1 and t. If both individuals 7 If the individuals have idential and stritly inreasing ost funtions b (t i ) for the waiting time t i, the analysis an be arried out in a similar way by employing k i = b (t i ) as hoie variable. 7

8 deide not to onede before T, that is t 1 = t = T, one of them is randomly seleted to provide the publi good, and their expeted payo in this ase is equal to v i = T. 3 The volunteering game This setion analyzes the war of attrition in isolation, xing the deisions on information aquisition. The equilibrium onept is Bayesian Nash equilibrium. In the following setion, deisions on information aquisition will be onsidered in a game de ned by the payo s in the war of attrition for the respetive information struture. Whenever players are symmetri in the sense that both have (have not) aquired information, the analysis of information aquisition will mainly draw upon on symmetri equilibria of the war of attrition. In the war of attrition, the individuals hoose their time of onession t i, knowing the deisions on information. The time horizon T a ets the properties of the equilibrium of the war of attrition for all possible stage 1 deisions. Compared to a provision in t i < T, individuals an redue their expeted ost of provision by waiting until T and then possibly being subjet to a random seletion. This trade-o between lower expeted provision ost and higher ost of waiting generates a time interval before T in whih, in equilibrium, there is zero probability that an individual volunteers. Lemma 1 Consider the war of attrition for a given information struture and denote by i individual i s (expeted or true) ost of ontribution. In any equilibrium of the war of attrition, there is zero probability that individual i provides the publi good in i + T; T. For a large T, it will always be an equilibrium of the volunteering game that an individual j volunteers immediately. In this ase, the equilibrium strategy of i is not uniquely determined, and he may hoose a onession time t i ( i = + T; T ), given that in equilibrium he will not provide the publi good. Lemma 1, however, shows that there is no equilibrium where i provides the publi good (i.e. onedes 8

9 rst) with positive probability in an interval just before T. Any t i ( i = + T; T ) is weakly dominated, and whenever there is positive probability that j waits until T, individual i (with ost i ) stritly prefers t i = T to any t i ( T < i =, we have i = + T; T ). If i = + T < 0, and i prefers the random seletion in T to a ontribution in any t i < T. Lemma 1 holds independently of i being i s true or expeted ost of provision; therefore, it an also be employed if individual i deides not to beome informed. In what follows, we will fous on the ase of an intermediate time limit T : Assumption 1 L < T < H : As will beome lear in the remainder of this setion, Assumption 1 implies that an individual with high ost will nd it optimal to wait until T, aepting the onsequene that he might be randomly hosen to ful ll the task. An individual with low ost will prefer an early onession if the rival waits su iently long. 8 Building on this assumption, we rst determine the equilibria of the volunteering game onditional on the deisions in stage 1, and we then analyze the inentives to beome informed in a game de ned by the ex ante expeted payo s in the war of attrition. Ex ante expeted payo s are de ned as the individuals expeted payo s given the deisions on information, but before they nd out about their provision ost. In the analysis of the war of attrition, if individual i knows his ost of provision, we will denote by i L (i H ) player i with low (high) ost of provision. Moreover, we will have to allow individuals to randomize their onession time. Consequently, a mixed strategy of an uninformed individual i f1; g will be a umulative distribution 8 This assumption ensures the strategi role of the information aquisition beause the equilibrium of the volunteering game will ruially depend on the individuals deisions whether or not to nd out about their ost of provision. If T H =, there is always an equilibrium of the war of attrition where one individual onedes immediately, independently of the deisions in stage 1 and the individuals true provision ost. If T L =, in the war of attrition, both individuals wait until T independently of the stage 1 deisions and their true ost, and the deisions to aquire information beome irrelevant. (To be preise, if T = L = and j - informed or uninformed - waits until T, an informed player i is indi erent between oneding immediately and waiting until T if he has low ost. In this ase, there is a ontinuum of equilibria where i randomizes between 0 and T if he learned that his ost is low and waits until T if his ost is high.) 9

10 funtion F i. Moreover, q i (t) will be the probability that i onedes exatly at t, and it will be employed to desribe both mass points in mixed strategies and pure strategies. If i aquires information, we denote by F il (F ih ) the distribution funtion that orresponds to the mixed strategy i hooses when his ost is low (high). Again, we will use q il (t) and q ih (t) to desribe mass points and type-ontingent pure strategies in ase i aquires information. 9 Mixed strategies that individuals hoose in the di erent ontinuation games will exhibit a ommon struture. For this purpose, we de ne a funtion as 8 >< 1 (1 q 0 ) e t ; 0 t < t t; ; t; q 0 = 1 (1 q >: 0 t ) e ; t t < T 1; t T : () (t; ; t; q 0 ) desribes a umulative distribution funtion of onession times t with positive mass in the interval (0; t), no mass in (t; T ), and possibly a mass point at zero (of size q 0 ) and/or a mass point at T. No individual knows his ost of provision. If neither of the individuals knows his true provision ost, both hoose their waiting time based on their expeted ost, and the volunteering game is strategially equivalent to the war of attrition with omplete information. 10 Consider individual i and suppose that j waits until T with probability one. If i onedes in t i < T, his expeted payo is v t i. For t i = T, he gets a payo of v = T. Thus, if T < =, t i = T is stritly preferred to any t i < T, and there is an equilibrium where both wait until T with probability one, whih is the unique equilibrium. If, however, T > =, i s best response to t j = T is 9 The distribution funtions F i, F il and F ih as well as the pure strategies will of ourse depend on both individuals information aquisition deision. Additional indexes to denote the respetive ontinuation game are omitted for simpliity, as the distribution funtions are derived in partiular subsetions. 10 This holds beause individuals are assumed to be risk-neutral and the payo s are linear in the provision ost. Thus maximizing expeted payo s is equivalent to the maximization based on the expeted ost. For a detailed analysis of the war of attrition with omplete information in nite time and results on the set of equilibria see Hendriks et al. (1988). 10

11 to onede immediately, and there are two equilibria, eah with one individual i hoosing q i (0) = 1 and the other individual j hoosing q j (0) = If T > =, there are also equilibria in mixed strategies. Lemma (No individual is informed.) Suppose that Assumption 1 holds. a) If T < =, in equilibrium, q 1 (T ) = q (T ) = 1. (If T = =, there is a ontinuum of equilibria where where q i (0) + q i (T ) = 1 and q j (T ) = 1, i = 1; ; j 6= i.) b) If T > =, the set of equilibria ontains two types of equilibria: (i) a set of payo -equivalent equilibria where q i (0) = 1 and q j (0) = 0, i = 1; ; j 6= i; (ii) a ontinuum of mixed strategy equilibria where F i (t) = t; ; and F j (t) = t; ; + T; q0 j, q 0 j [0; 1), i = 1; ; j 6= i. In any mixed strategy equilibrium (ase T > =), for any t j (0; marginal ost of waiting is one, multiplied by the probability (1 + T; 0 = + T ), j s F i (t j )) that this waiting ost has to be paid. The marginal gain of waiting slightly longer is equal to Fi 0 (t j ), i.e. the expeted provision ost multiplied by the additional probability that this ost an be saved. Individual j is indi erent between all t j (0; = + T ) if ost and bene t of inreasing t j (i.e. of waiting slightly longer) are equal. This leads to F i and F j. The only di erene to the standard war of attrition with omplete information is that, due to the time limit, no individual onedes in ( = + T; T ), but instead both hoose a onession in T with stritly positive probability. In addition to the mass point at T, one individual j an plae a mass point of size q 0 j [0; 1) at t = 0, i.e. onedes immediately with positive probability. 1 Obviously, there an t be an equilibrium where both individuals have a mass point at zero, beause then waiting an in nitesimally small amount of time would, at a negligibly 11 To be preise, there are two lasses of payo -equivalent equilibria where i f1; g onedes immediately with probability one and j 6= i waits su iently long with su iently high probability (for instane, q j (T ) = 1), to make q i (0) = 1 optimal for i. In all of these equilibria, it has to hold that q j (0) = 0: 1 Note that if q 0 [0; 1), F j (t) < 1 for all t < T ; thus j also plaes a mass point at T. 11

12 higher expeted waiting ost, stritly inrease the probability that the rival provides the publi good. The xed time limit has an important impat on the individuals equilibrium behavior if T > =. At the beginning of the game, the individuals are willing to onede, and they play a mixed strategy for a ertain time period (t (0; = + T )). As the time limit approahes, it beomes less ostly to wait until the end, and thus there is a point in time after whih the individuals are inative (for all t ( = + T; T )) beause they prefer the random seletion at T. Finally, they put the remaining probability mass on a onession at T. As mentioned before, the analysis of information aquisition will mainly fous on the symmetri equilibrium. From Lemma, we an ompute the individuals expeted payo in the (unique) symmetri equilibrium, whih is equal to ( v = T if T = E ( i ) =, i = 1; : (3) v if T > = If T > =, in any asymmetri mixed strategy equilibrium, individual j with qj 0 > 0 gets an expeted payo of v, while i gets v 1 qj 0 > v. One individual knows his ost of provision. Suppose that only individual j has beome informed about his provision ost, while i 6= j remained uninformed. j s strategy is now ontingent on his type (denoted by j L or j H ), and i s optimal strategy is to hoose his onession time as if his ost was. Lemma 3 (One individual is informed.) Suppose that Assumption 1 holds. a) If T =, in equilibrium, q i (T ) = q jh (T ) = 1, and q jl (0) = 1. (If T = =, there is an additional equilibrium where q i (0) = 1 and q jl (T ) = q jh (T ) = 1.) b) If = < T < = ln p H, the set of equilibria ontains two types of equilibria: (i) a set of payo -equivalent equilibria where q i (0) = 1 and q jl (0) = q jh (0) = 0; 1

13 (ii) one mixed strategy equilibrium where F i (t) = t; L ; + T; 0, F jl (t) = 1 p L t; ; + T; p Lqj 0 L with q 0 jl = 1 p L 1 p H e 1 + T, and q jh (T ) = 1. ) If T = ln p H, in equilibrium, q i (0) = 1 and q jl (0) = q jh (0) = 0. If T < =, both i and j H prefer a random seletion at T to any onession before T, and this makes it optimal for j L to onede immediately. Sine there is positive probability that the time limit T is reahed, the equilibrium strategies of i and j H are uniquely pinned down. If T > =, the struture of the equilibrium reverses, and there is a pure strategy equilibrium where i onedes immediately and both j L and j H wait until T. To be preise, there is a ontinuum of payo -equivalent equilibria where the uninformed individual i onedes immediately and the informed individual j hooses a (su - iently) high waiting time for eah of the two possible provision osts he ould have been informed of (su iently high to make it optimal for i to onede immediately). Given that Assumption 1 holds, by Lemma 1, j H will never provide the publi good with stritly positive probability before T. Thus, there is no further pure strategy equilibrium. To see why, suppose that i onedes in t 0 > 0 with probability one. j L s best response is either t jl = 0, or t jl > t 0, and i stritly prefers a onession in t 0 = over a onession in t 0 sine in both ases this doesn t hange his probability of ontribution, but stritly redues the expeted waiting ost. There an, however, be an additional equilibrium whih is in mixed strategies. In fat, if = < T < = ln p H, there is a mixed strategy equilibrium where i and j L randomize their onession time. By Assumption 1 and Lemma 1, j H will never provide the publi good before T. Thus, in any equilibrium in mixed strategies, only i and j L ontribute before T with stritly positive probability, and the equilibrium strategies exhibit similar properties as in the ase of omplete information. Contrary to the ase where no individual knows his ost, this mixed strategy equilibrium is uniquely determined by the ondition that there is zero probability that any individual onedes in ( = + T; T ) and that therefore j L onedes before 13

14 = + T with probability one (see Appendix). 13 This requires that F jl has a mass point at zero, and thus i s payo in the mixed strategy equilibrium is stritly higher than v, whih is i s payo from oneding immediately. The mixed strategy equilibrium haraterized in Lemma 3b(ii) has several interesting properties. Whenever p H and/or T are large, this equilibrium does not exist: as it is likely that (the informed) individual j has a high ost and the waiting time until T is ostly, waiting beomes too ostly for (the uninformed) individual i; thus i prefers to volunteer immediately. When T! = ln p H (from below), the probability that individual j L onedes immediately onverges to zero, and i s expeted payo onverges to v, whih is equal to his payo in the pure strategy equilibrium. On the other hand, when T! = (from above), the probability that j L onedes immediately onverges to one, and the probability that i onedes before T onverges to zero. The equilibrium strategies in the mixed strategy equilibrium and the individuals expeted payo s onverge to the equilibrium for T < =. Sine individuals are not symmetri in this ontinuation game and there is no partiular reason to fous on one or the other equilibrium, 14 for T > = the analysis of the individuals inentives to beome informed will distinguish whih equilibrium is seleted in ase exatly one individual learned his ost of provision. Given that the pure strategy equilibrium is seleted (Lemma 3b(i)), ex ante expeted payo s of (the uninformed individual) i and (the informed individual) j are ( v p H E ( i ) = + T if T < (4) v if T > ( v p L L p H H + T if T < E ( j ) = (5) v if T > 13 If T = = ln p H, there is a ontinuum of equilibria where i and j L randomize on 0; + T, F i exhibits a mass point qi 0 [0; 1) at zero and a mass point at T, and F j L exhibits no mass points. i s equilibrium payo is the same in all of these equilibria, and fousing on one of these equilibria does not hange the analysis of information aquisition. 14 In partiular, the two equilibria annot be Pareto-ranked. 14

15 In ase the mixed strategy equilibrium is seleted (Lemma 3b(ii)), ex ante expeted payo s equal E ( i ) = E ( j ) = 8 >< >: 8 >< >: v p H + T if T < v p H e 1 + T if < T < ln p H v if T ln p H (6) v p L L p H H + T if T < T p v H L e L ( H + L ) if < T < ln p H (7) v if T ln p H For T < =, j L onedes immediately; therefore, the expeted payo of the uninformed individual i inreases with the probability that j has a low ontribution ost. Note that in this ase E ( i ) > E ( j ), i.e. the individual who does not know his ost of provision has a higher expeted payo than the informed individual. For a large T, however, the uninformed individual may onede immediately and gets a lower expeted payo than the informed individual. Both individuals know their ost of provision. Suppose that both individuals have deided to aquire information about their provision ost. By Lemma 1 together with Assumption 1, there an t be an equilibrium where a type of i with high ost, i H, provides the publi good in t ih < T with stritly positive probability. Hene, if i H plaes positive probability over some interval (t 0 ; T ) with t 0 < T, then j H must onede before t 0 with probability one, ontraditing Lemma 1. Therefore, in any equilibrium, q ih (T ) = q jh (T ) = 1. It remains to haraterize the individuals equilibrium strategies for a low provision ost. As before, denote by i L an individual i with low ost. There an t be an equilibrium where i L hooses a pure strategy. In partiular, there an t be an equilibrium where an individual with low ost volunteers immediately. To see why, suppose that i L hooses t = 0 with probability one. j L s best response is to onede in t 0 = ", " in nitesimally small, knowing that i H will wait until T. But then, i L is stritly better o by hoosing t 00 = ". 15

16 Hene, individuals randomize their waiting time if they have a low provision ost. By Lemma 1, there must be zero probability that an individual volunteers in the interval ( L = + T; T ), and at most one individual an have a mass point at zero. As it is a typial feature of the war of attrition, there an be a ontinuum of equilibria whih di er in the size of the mass point at zero. Lemma 4 (Both individuals are informed.) Suppose that Assumption 1 holds. a) If T < L L ln p H, the set of equilibria onsists of a ontinuum of equilibria where i L and j L randomize aording to F il (t) = 1 p L t; L ; + T; 0 and h F jl (t) = 1 p L t; L ; L + T; p L qj 0 L with q 0 jl 0; 1 p L 1 p H e 1 + T L i, and i H and j H hoose q ih (T ) = q jh (T ) = 1, i = 1; ; j 6= i. b) If T L L ln p H, in the unique equilibrium, F il (t) = 1 p L (t; L ; L ln p H ; 0) and q ih (T ) = 1, i = 1;. If the time horizon is su iently short (T < L L ln p H ), individuals with low ost randomize on the interval 0; L + T, and they put stritly positive probability on a onession in T, as waiting until T is not too ostly. Similarly as in the ase where no individual is informed, one individual j L an plae a mass point at zero, and there is a ontinuum of asymmetri equilibria. This mass point, however, annot be "too large", in order to guarantee that j L indeed randomizes up to L = + T. For T L L ln p H, however, there is a unique equilibrium where neither i L nor j L plaes a mass point at zero or at T. The reason is that, even without plaing a mass point at zero, low types will have oneded before some t L + T with probability one. Mass points at zero an hene be ruled out beause i L and j L must randomize with the same support and waiting until T annot be part of a low type s equilibrium strategy anymore. determined. Therefore, the equilibrium strategies are uniquely To summarize, up to a point in time t = min L + T; L ln p H, there is a positive probability that an individual onedes in ase he has a low ost, and there is a time period just before T where both individuals are inative, sine they prefer L 16

17 to wait until T if the ost of the additional waiting time is su iently low. Low types plae a mass point at T if the time horizon is short and/or the probability p H of faing a high type is small. Otherwise, if the probability p H that the other individual has a high ost is large (or T is large), it is more attrative for an individual with low ost to volunteer early, and i L and j L onede before T with probability one. The main analysis of information aquisition will build on the (unique) symmetri equilibrium, where ex ante expeted payo s are E ( i ) = for i = 1; : ( p v H L ( H L ) e 1 T L if T < L L ln p H v L p H T + H L (1 ln p H ) (8) if T L L ln p H 4 The value of beoming informed This setion onsiders the deisions on information aquisition in a game de ned by the payo s in the war of attrition that have been determined in the previous setion. 15 The main analysis fouses on the symmetri equilibrium in ontinuation games where both or none of the players aquired information. At the end of this setion, we will disuss the impliation for inentives to aquire information when we onsider asymmetri equilibria of symmetri ontinuation games. Let i fn; Ig be an individual i s deision on information where I refers to information aquisition and N to a deision not to learn one s own provision ost. Moreover, denote by E individual i s ex ante expeted payo in the war ( i; j ) i of attrition given the deisions ( i ; j ). In ase (I; I), for instane, both individuals have learned their ost of provision, whereas ase (N; I) refers to a situation where exatly one individual has deided to learn his ost. Given j, i s value of information 15 This approah is employed to simplify the exposition, and it shows that in the equilibrium of the game, one player may remain uninformed. The equilibria of the redued game an also be supported as perfet Bayesian equilibria in the analysis of the two-stage game, assuming beliefs about the rival s type that do not hange with the information aquisition deision (players have no private information when deiding whether to aquire information). 17

18 an be de ned as V j i = E (I; j) i E (N; j) i : For the analysis of the optimal deision on information aquisition, we have to distinguish whether or not T > =. While the war of attrition in ases (N; N) and (I; I) always has a unique symmetri equilibrium, this distintion is ruial for the nature of the equilibrium if at least one individual does not know his ost of provision. 16 Lemma 5 Suppose that Assumption 1 holds. (i) V j=n i is stritly positive for all T. (ii) V j=i i is stritly negative if T is su iently small and stritly inreasing in T for T ( L =; =). (iii) Suppose in ase (N; I) the pure strategy equilibrium is seleted. Then V j=i i is stritly positive for all T > =. (iv) Suppose in ase (N; I) the mixed strategy equilibrium is seleted. Then V j=i i is ontinuous and stritly inreasing in T for T ( L =; = ln p H ). Provided that the rival does not learn his ost of provision ( j = N), learning one s own ost always inreases one s expeted payo as the value of information is positive (Lemma 5 part (i)). T is small, this result is reversed. If instead the rival deides to learn his ost and However, as long as T < =, an inreasing time limit makes waiting more ostly in ase the rival has a high provision ost, whih inreases one s own value of information (part (ii)). If T > =, the value of information depends on whih equilibrium is seleted in ase (N; I). For the pure strategy equilibrium, i s value of information given that j learns his ost of provision, Vi I, exhibits a disontinuity at T = = and is stritly positive for all T > = (part (iii)). For the mixed strategy equilibrium, however, Vi I is ontinuous at T = =. This ontinuity in T makes the analysis for the seleted equilibrium more appealing. 16 We still assume that Assumption 1 holds. Reall that, if T < L =, then deisions on information are irrelevant, sine both individuals never onede before T ; if T > H =, the war of attrition always has equilibria where one of the individuals onedes immediately, independent of the deisions on information. 18

19 Yet the following proposition holds independently of whih equilibrium is seleted in ase only one individual deides to learn his provision ost. Proposition 1 Consider the game of information aquisition and suppose that Assumption 1 holds. There exists a threshold ~ T > L = suh that (i) if T < ~ T, the set of equilibria onsists of two asymmetri equilibria where exatly one individual aquires information and one symmetri equilibrium where both individuals randomize their information deision; (ii) if T > ~ T, it is stritly dominant to aquire information. If both individuals remained uninformed, this would ause a high ine ieny in the volunteering game and lead to the lowest expeted payo s. Therefore, it is bene ial for at least one individual to nd out about his provision ost even if information aquisition leads to a higher ex ante probability of being the one who onedes rst. As a onsequene, there is never an equilibrium where both individuals deide not to learn their ost of provision. If, however, T is su iently small and only individual j aquires information, then j onedes immediately with high probability, and i prefers to remain uninformed. Being uninformed onstitutes a strategi advantage in the volunteering game, being a ommitment not to volunteer too early. This, in turn, indues the rival to onede immediately, whih outweighs i s waiting ost in ase j has a high provision ost. For a higher T, this waiting ost inreases, and, in the ase of the mixed strategy equilibrium in (N; I), the probability that j onedes immediately dereases. There exists a threshold ~ T suh that, for T > ~ T, i is better o if he nds out about his provision ost as well. 17 the value of information Vi I is negative for all ( L =; =), the loation of T ~ depends on whih equilibrium is seleted in ase (N; I). In both ases, the threshold T ~ is uniquely determined suh that Vi I T > T ~. is negative for all T < ~ T and positive for all 17 If T = ~ T and j aquires information, i is indi erent between aquiring and not aquiring information. Thus, if T = ~ T, there is a ontinuum of equilibria where i aquires information with probability [0; 1], j aquires information with probability 1, i = 1; ; j 6= i. (j stritly prefers to beome informed whenever < 1 and hene there is some hane that i will be uninformed.) If 19

20 Corollary 1 (i) If in ase (N; I) the pure strategy equilibrium is seleted, ~ T =. (ii) If in ase (N; I) the mixed strategy equilibrium is seleted and p H is small, ~ T is stritly larger than =. Then, there may be no equilibrium where both individuals aquire information with probability one for all T ful lling Assumption 1. If T > = and, in ase (N; I), the pure strategy equilibrium is seleted, learning the own provision ost is stritly dominant, and thus the threshold ~ T is (weakly) smaller than =. 18 However, if we fous on the mixed strategy equilibrium, ~ T > = for a small p H, and the value of information V I i an even be negative for all T ( L =; H =). Thus, the strategi value of remaining uninformed is not only present in the ase where an uninformed individual i has a dominant strategy not to onede before T (as in Lemma 3a), but also when the individuals randomize their onession time (as in Lemma 3b(ii)). The su iently high probability that the rival has a low ost and volunteers immediately with positive probability makes it optimal for i to disregard information that is available without ost. This strategi value disappears only if the probability of having a high ontribution ost, p H, is large, beause, from the point of view of the rival, an early onession of the individual who knows his provision ost is less likely. Numerial example Consider the following numerial example where L =, and H = Assumption 1 requires that 1 < T < 5. (a) Suppose that p H = 0:75. If T! = = 4 from below, the value of information is positive. Hene, the ritial threshold T ~ < =. Setting Vi I (T ) = 0 yields ~T = 1:94. Thus, for all T < 1:94, only one individual learns his ost of provision, V I i and for all T > 1:94, both individuals learn their ost of provision. (b) Now suppose that p H = 0:5. Vi I is negative if T approahes = = 3 (from below). Hene, if in ase (N; I) the pure strategy equilibrium is seleted, T ~ = = = 3, and if the mixed strategy equilibrium is seleted, ~ T > =. In the latter ase, ~ T = 3: Conretely, if V j=i i is negative for all T < =, T ~ = =. Otherwise, T ~ is de ned as the solution to V j=i ~T = 0. i 19 Details on this example are in Appendix B. 0

21 Figure 1: Equilibrium information aquisition (for L = ; H = 10). () If p H = 0:5, again Vi I is negative if T approahes = =, and T ~ = = if in ase (N; I) the pure strategy equilibrium is seleted. If the mixed strategy equilibrium is seleted, Vi I is negative for all T satisfying Assumption 1, and thus there is no equilibrium where both individuals nd out about their ost of provision with probability one. Figure 1 shows the equilibrium outome for di erent ombinations of T and p H. The 45-degree line desribes the ondition T = =. In the areas B and D, nding out about the own ost of provision is stritly dominant; in area A, the individuals prefer to remain uninformed if the rival aquires information, and in equilibrium only one individual learns his ost (or both individuals randomize their information aquisition deision). In area C, the outome depends on whih equilibrium is seleted in ase (N; I). Here, T > =, and for the pure strategy equilibrium, information aquisition is stritly dominant. For the mixed strategy equilibrium, however, only one individual aquires information. A designer s perspetive. There are several dimensions along whih e ieny an be de ned. On the one hand, a designer ould be interested in the individual with the lowest ost (highest ability) providing the publi good. On the other hand, the 1

22 designer might want to minimize the expeted waiting time. In the example of a rm where a team leader alloates tasks by asking for a volunteer, the team leader will prefer sta meetings to be short and therefore he ares about the expeted waiting time. But the team leader might put even a higher weight on alloating the task to the employee with the highest ability (this will ertainly depend on the task to be ful lled and on individual heterogeneity). 0 To apture these di erent dimensions, onsider the following objetive funtion W = v 1 E (min ft 1 ; t g) E (k (t 1 ; t )) where 8 >< 1 if t 1 < t k (t 1 ; t ) = ( 1 + ) = if t 1 = t >: if t 1 > t is the (expeted) ost of providing the publi good and 1 and are the weights given to the expeted waiting time and the expeted provision ost. We assume that the designer does not know the individuals ost of provision and annot hange the struture of the game. Suppose rst that 1 = 0 and > 0, that is, maximizing W is equivalent to minimizing the expeted ost of provision, E (k (t 1 ; t )): the team leader does not are about the waiting time, but only about individual ost/ability. Here, W is highest if both individuals aquire information (ase (I; I)) and an individual with low ost volunteers with probability one before the time limit is reahed. This implies that T > L = L ln p H (by Lemma 4) and T > ~ T (by Proposition 1). In this ase, information aquisition is e ient. Remark 1 If the designer wants to minimize the expeted ost of provision, a suf- iently high time limit ensures both e ient information aquisition and e ient provision of the publi good. 0 In a framework of a ontest, a designer may want to indue long times of ghting, i.e. high waiting times.

23 Another objetive ould be to fous on the expeted waiting time, in partiular if individual ability is not very important for a suessful ompletion of the task. Let = 0. Obviously, if 1 > 0, the time horizon should be as short as possible, and W is maximized for T = 0. In this ase, the deisions on information beome irrelevant. 1 If the designer takes into aount both the expeted ost of provision and the expeted waiting ost, a benevolent designer may want to maximize the individuals expeted payo s, whih is equivalent to 1 = and = 1. Then, T = 0 need not be optimal: if T is only slightly larger than L =, one individual aquires information, and he onedes immediately in ase he has a low ost. The gain from the derease in the expeted provision ost (due to information aquisition) outweighs the higher waiting ost if p H is su iently small and/or H is large, and it an be optimal to hoose an intermediate time limit suh that individuals have an inentive to aquire information and to hoose an early onession if they have a low provision ost. Similarly, it an be desirable that both individuals aquire information. In the latter ase (ase (I; I)), the sum of expeted payo s is highest if T = L = L ln p H suh that individuals with low ost onede before T with probability one. Higher T do not hange the e ieny of the provision (aptured by k (t 1 ; t )), but inrease the waiting ost given that both individuals have a high ost. In general, the optimal hoie of T depends on the balaning of expeted waiting time and ost of provision and on the probability of faing individuals with a high ost of provision. Remark If the designer wants to maximize the sum of expeted payo s, the tradeo between e ieny of the provision and ost of waiting makes an intermediate time limit optimal whenever p H is su iently small and/or H is large. 1 If T > = and in ase (N; I) the pure strategy is seleted, W would also be maximized if exatly one individual aquires information. This, however, does not our in equilibrium if the individuals deide on information aquisition, but only if information aquisition is forbidden for one individual. In this sense, there an be too muh information aquisition in equilibrium if 1 > 0 and = 0. If instead 1 < 0 and the designer wants to maximize the expeted waiting time, the waiting times are highest if T > = and none of the individuals aquires information. Thus, it would be optimal to prohibit information aquisition. 3

24 Disussion of the equilibrium seletion. The previous analysis has foused on symmetri equilibria of the war of attrition whenever the players are symmetri. While this may be a natural equilibrium seletion, the analysis an hange if asymmetri equilibria are taken into aount. Moreover, onsidering information aquisition in a redued form game ould have an impat on the set of equilibria obtained. These issues should be disussed in what follows. Symmetri vs. asymmetri equilibria While in the asymmetri war of attrition where exatly one individual has aquired information (ase (N; I)), all possible equilibria have already been onsidered for the analysis of information aquisition, the symmetri equilibrium has been seleted in symmetri ontinuation games. Suppose instead that, in ases (N; N) and (I; I), asymmetri equilibria are played in the war of attrition. Obviously, (N; N) an never be an equilibrium of the game of information aquisition beause at least one individual gets an expeted payo of v in this ase and stritly prefers to aquire information. Moreover, suppose that T < T ~ (from Proposition 1). Then, (I; I) annot be an equilibrium beause, in any equilibrium in ase (I; I), the expeted payo of at least one individual j (with possibly q jl (0) > 0) is equal to his expeted payo in the symmetri equilibrium; thus j prefers not to beome informed (as analyzed in the ontext of Proposition 1). Correspondingly, if T > T ~, (I; I) is the unique equilibrium of the information aquisition game; all possible deviation payo s in ase (N; I) make the uninformed individual worse o. Now, in ase of T < T ~, the set of equilibria an still be di erent from Proposition 1(i) if we allow asymmetri equilibria to be played. To see why, suppose rst that, in ase (N; N), q (0) > 0 and su iently large (this requires that T > =). Then (I; N) (where player 1 aquires information but does not) annot be an equilibrium beause player 1 prefers to remain uninformed and to have player oneding immediately with high probability. Thus, (N; I) (where aquires information but 1 does not) an be the unique equilibrium of the information aquisition game. Seond, and similarly, if in ase (I; I), q L (0) > 0 and su iently large (note that this requires T < L = L ln p H ), then it an be that (N; I) (where 1 remains uninformed) will 4