Florian Morath. Volunteering and the Strategic Value of Ignorance. Max Planck Institute for Intellectual Property, Competition and Tax Law

Transcription

1 WISSENSCHAFTSZENTRUM BERLIN FÜR SOZIALFORSCHUNG SOCIAL SCIENCE RESEARCH CENTER BERLIN Florian Morath Volunteering and the Strategi Value of Ignorane Max Plank Institute for Intelletual Property, Competition and Tax Law SP II November 010 Researh Area Markets and Politis Researh Professorship & Projet "The Future of Fisal Federalism" Shwerpunkt Märkte und Politik Forshungsprofessur & Projekt "The Future of Fisal Federalism"

2 Florian Morath, Volunteering and the Strategi Value of Ignorane, Disussion Paper SP II , Wissenshaftszentrum Berlin, 010. Wissenshaftszentrum Berlin für Sozialforshung ggmbh, Reihpietshufer 50, Berlin, Germany, Tel. (030) Internet: ii

3 ABSTRACT Volunteering and the Strategi Value of Ignorane by Florian Morath * 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 find 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 suffiiently 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 influening the effiieny of the volunteering game. Keywords: War of attrition, volunteering, disrete publi goods, asymmetri information, information aquisition JEL lassifiation: H41, D44, D8, D83 ZUSAMMENFASSUNG Volunteering and the Strategi Value of Ignorane Die private Bereitstellung öffentliher Güter ähnelt häufig einem Zermürbungskrieg : Die Beteiligten warten, bis sih jemand anderes freiwillig meldet und das öffentlihe Gut bereitstellt. Nah einer gewissen Zeitperiode des Wartens kann jedoh ein Beteiligter zufällig dazu bestimmt werden, die Bereitstellung zu übernehmen. Wenn die Beteiligten ihre Bereitstellungskosten niht genau kennen, sih aber vor dem Bereitstellungsspiel Information über ihre Kosten beshaffen können, dann kommt dieser Information ein strategisher Wert zu; die Beteiligten könnten es vorziehen, ihre Bereitstellungskosten niht genau zu kennen. Wenn der Zeithorizont des Bereitstellungsspiels hinreihend kurz ist, entsheidet sih im Gleihgewiht lediglih ein Beteiligter, Information zu akquirieren. Bei einem längeren Zeithorizont ist es eine strikt dominante Strategie, sih Information zu beshaffen. Der Zeithorizont stellt ein wihtiges Instrument zur Beeinflussung der Effizienz des Bereitstellungsspiels dar. * I thank Kai Konrad, Mihael Mihael, and Johannes Münster for valuable omments. Finanial support by the German Researh Foundation (DFG) through grant SFB/TR 15 is gratefully aknowledged. iii

4 1 Introdution Dragon-slaying and ballroom daning are two famous examples 1 for 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 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. 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 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).

5 to ommit to a ertain behavior in the war of attrition. Similarly, on an international level, when governments have to deide whether to provide an international publi good, they an engage experts to provide them with a better estimate of the ost of provision. But when suh investments in information are observable by other players, investments in information obtain a strategi harater. 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 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. 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. 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 a 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 3

6 short time horizon, there are 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 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. 3 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. 4 Closely related to this paper is work that onsiders information in autions. 3 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. 4 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. 4

7 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 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; : 5

8 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. 5 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. 6 Stage is strategially equivalent to the war of attrition or seond-prie all-pay aution with a ap on bidding. Denoting by v an individual s utility from the provision of the publi good, the 5 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 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. 6

9 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 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. Whenever players are symmetri in the sense that both have (have not) aquired information, the analysis will fous on symmetri equilibria of the war of attrition. 7 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. 7 In the next 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. 7

10 Lemma 1 Consider the war of attrition for a given information struture. In any equilibrium of the war of attrition, there is zero probability that individual i with ost 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. Any t i ( i = + T; T ), however, 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 ( i = + T; T ). If T < i =, we have 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. 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 unique equilibrium of the war of attrition, both individuals wait until T independently of the stage 1 deisions and their true ost. 8

11 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 funtion F i. Moreover, q i (t) will be the probability that i onedes exatly at t, and it will be employed to desribe 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 type-ontingent pure strategies in ase i aquires information. 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. 9 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 to 9 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. 9

12 onede immediately, and there are two equilibria, eah with one individual hoosing q i (0) = 1, i = 1;. In the latter ase, there are also equilibria in mixed strategies. 10 As players are symmetri, we fous on the (unique) symmetri equilibrium. Lemma (No individual is informed.) a) If T =, in the symmetri equilibrium, q 1 (T ) = q (T ) = 1. b) If T > =, in the symmetri equilibrium, individual i f1; g randomizes his onession time aording to F i (t) = t; ; + T; 0. In the mixed strategy equilibrium (ase T > =), for any t j (0; marginal ost of waiting is one, multiplied by the probability (1 = + 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 (t) = t; ; + T; 0. 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 the symmetri equilibrium, no individual onedes immediately with positive probability (that is, q 0 = 0). There are asymmetri mixed strategy equilibria where one of the individuals plaes a mass point at t = 0, i.e. onedes immediately with stritly positive probability. 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 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 )). 10 For a detailed analysis see Hendriks et al. (1988). 10

13 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. From Lemma, we an ompute the individuals expeted payo in the symmetri equilibrium, whih is equal to ( v = T if T = E ( i ) =, i = 1; : (3) v if T > = 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. Reall that we still assume that Assumption 1 holds. Lemma 3 (One individual is informed.) 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 ) = 1.) b) If = < T < = ln p H, (i) there is a pure strategy equilibrium where q i (0) = 1; (ii) there is a mixed strategy equilibrium where F i (t) = t; L ; F jl (t) = 1 p L t; ; + T; 1 (1 p L) e 1 + T, and q jh (T ) = 1. + T; 0, ) If T = ln p H, in equilibrium, q i (0) = 1. 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 11

14 be preise, there is a ontinuum of payo -equivalent equilibria where i onedes immediately and 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, the 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 = + T with probability one (see Appendix). 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 j has a high ost and the waiting time until T is ostly, waiting beomes too ostly for 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 1

15 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, 11 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 and T > =. Given that the pure strategy equilibrium is seleted (Lemma 3b(i)), ex ante expeted payo s are ( E ( i ) = E ( j ) = ( v p H if T < v if T > v p L L p H H + T if T < v if T > 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. 11 In partiular, the two equilibria annot be Pareto-ranked. (4) (5) 13

16 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. If i H hooses a time of onession t ih < T with stritly positive probability, then j H must onede before t ih 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 = ". 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 may be a ontinuum of equilibria whih di er in the size of the mass point at zero. Sine the individuals are symmetri ex ante, we fous on the symmetri equilibrium. Lemma 4 (Both individuals are informed.) In the symmetri equilibrium, q ih (T ) = 1 and F il (t) = 1 p L (t; L ; t; 0) where t = min L + T; L ln p H, i = 1;. If the probability p H that the other individual has a high ost is large, it is more attrative for an individual with low ost to volunteer early. For su iently high p H, i L and j L onede before T with probability one. This holds if L + T L ln p H or T L L ln p H : 14

17 Otherwise, the low types put stritly positive probability on a onession in T, as waiting until T is less ostly. Again, up to a point in time t, 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 to wait until T if the ost of the additional waiting time is su iently low. Ex ante expeted payo s are E ( i ) = ( 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 for i = 1; : 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. 1 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 an be de ned as V j i = E (I; j) i E (N; j) i : 1 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). 15

18 For the analysis of the optimal deision on information aquisition, we have to distinguish whether or not T > =. This distintion does not in uene the equilibrium of the war of attrition in ase both individuals know their provision ost, but it is ruial for the nature of the equilibrium if at least one individual does not know his ost of provision. 13 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. Yet the following proposition holds independently of whih equilibrium is seleted in ase only one individual deides to learn his provision ost We still assume that Assumption 1 holds. 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. 14 Due to the possible multipliity of equilibria of the war of attrition in ase of T > =, departing from the analysis of the redued form game makes the equilibrium analysis more omplex in 16

19 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, there are 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. If 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 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. this ase. Then, players an ondition their strategies in the war of attrition on the information aquisition. As in our analysis for the pure strategy equilibrium in ase (N; I), this an support information aquisition of both players in equilibrium if T > =. 17

20 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 =. 15 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. Example Consider the following example where L =, and H = 10. Assumption 1 requires that 1 < T < 5. (a) Suppose that p H = 0:75. If T! = = 4 from below, the value of information V I i 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, 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 = 3:56. () 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. 15 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 i = 0. 18

21 Figure 1: Equilibrium information aquisition (for L = ; H = 10). 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 designer might want to minimize the expeted waiting time. 16 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 )) 16 In a framework of a ontest, a designer may want to indue long times of ghting, i.e. high waiting times. 19

22 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 )). 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. Another objetive ould be to fous on the expeted waiting time. 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. 17 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 17 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. 0

23 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. 5 Extensions Sequential deisions on information. Whenever there is an inentive to remain uninformed, this an ause a oordination problem. When individuals randomize their information aquisition deision, they may aquire too muh or too little information from their own point of view. Considering sequential hoies on information an mitigate this oordination problem, and it will re et, for instane, situations where individuals an, one after the other, ask questions about a task that has to be performed. Suppose that deisions on information take plae sequentially: individual 1 deides rst, and individual moves seond. 18 As stated in Proposition 1, information 18 We do not disuss the question of endogenous timing of information aquisition deisions. 1

24 aquisition is stritly dominant for T > T ~, and both individuals will aquire information. For T < T ~, however, if individual 1 aquires information, will remain uninformed, and vie-versa. Proposition Suppose that deisions on information take plae sequentially and Assumption 1 holds. If the time horizon is su iently small, the rst mover will deide to remain uninformed, and the seond mover will aquire information. Whenever T < = and exatly one individual has aquired information, the payo of the uninformed individual is higher than the payo of the informed individual. Thus, individuals prefer to be the uninformed player. If T > =, we have to distinguish whih equilibrium is seleted in ase (N; I). For the mixed strategy equilibrium, a strategi inentive to remain uninformed exists, and an inreasing time horizon T makes it less attrative to remain uninformed. There is, however, a range of parameters T where the strategi advantage from being uninformed is su iently high suh that a rst mover would hoose to remain uninformed. Information about a ommon value. In the previous setion, we have identi ed a strategi value of ignorane in situations where information about a private value an be obtained. If the information is about some omponent whih is ommon to all individuals, a similar strategi inentive is present. Consider the extreme ase of a pure ommon value and suppose that the individuals osts of provision are perfetly orrelated. Thus, if an individual has aquired information, he knows not only his own type, but also his rival s type. In the war of attrition, if no individual has aquired information, the analysis does not hange. Moreover, if both individuals have aquired information, they randomize their onession time if they both have a low ost, and they wait until T if they both have a high ost. If exatly one individual knows the ost of provision and T < =, the equilibrium of the war of attrition is similar to the one haraterized in Lemma 3a. Here, if the informed individual j does not onede immediately, the uninformed individual i

25 knows that his ost is high and will nd it optimal to wait until T. 19 If T > =, the mixed strategy equilibrium of Lemma 3b(ii) does not exist. The intuition is as follows. If there were suh an equilibrium, the uninformed individual i would update his beliefs about his ost following the ation of this rival, and if the game reahes a point in time + T, positive but small, i would know almost with ertainty that his ost is high. But then, i would not onede in + T ; + T, but instead wait until T. For T =, there is an equilibrium where the uninformed individual onedes immediately (q i (0) = 1). Moreover, ontrary to the ase of private values, there is an equilibrium where q jl (0) = 1 and q jh (T ) = q i (T ) = 1. Here, i knows that his ost is high if there is no immediate onession of j and thus nds it optimal to wait until T. In turn, j annot pro tably deviate given that q i (T ) = 1. As in the private values ase, we onsider the game of information aquisition de ned by the payo s in the war of attrition. Proposition 3 Consider the game of information aquisition with ommon values and suppose that Assumption 1 holds. (i) If T < =, only one individual aquires information in equilibrium; (ii) if T =, dependent on the equilibrium seletion in the war of attrition (ase (N; I)), only one individual or both individuals aquire information in equilibrium. In ase of T =, the war of attrition in ase (N; I) has two diametrially opposed equilibria, and deisions on information ruially depend on whih of the equilibria is played. For a small T, however, as in the ase of private values, one individual strategially hooses to remain uninformed of the ost of provision, and in turn the informed individual onedes immediately if the (ommon) ost is low There is no further pure strategy equilibrium beause, even if q jl (T ) = q jh (T ) = 1, i s best response is q i (T ) = 1. Moreover, there is no mixed strategy equilibrium: intuitively, if j L randomized and i provided the good at some t > 0, i would know that his expeted ost would be higher than (it beomes more likely that the ost is high); thus, i prefers to wait until T. 0 As in Proposition 1, there are two asymmetri equilibria where exatly one individual aquires information and one symmetri equilibrium where both individuals randomize their information aquisition deision. 3

26 Lost opportunity of provision at T. The provision of many publi goods is alloated on a voluntary basis, but it is ompulsory in the sense that one individual has to ontribute. In ompanies, for instane, a team leader may selet one individual if no one volunteers. Other publi goods an only be provided within a ertain period, after whih the opportunity of provision disappears. Instead of assuming that at T one individual is randomly seleted, suppose that the investment opportunity disappears if no individual has oneded before T. In this ase, the inentive to wait until T in the war of attrition is weakened; the analysis, however, qualitatively arries over from the previous setion if we modify Assumption 1 on the time limit suh that individuals with a high ost do not want to provide the publi good and individuals with a low ost prefer to onede. This requires the time limit to be suh that: Assumption 1 0 < (v L ) < T < (v H ) : Hene, high types have a (weakly) dominant strategy to wait until T. Moreover, if T is su iently small (T < (v )) and only individual j knows his ontribution ost, the uninformed individual i waits until T (as in Lemma 3a). For a larger T, there is an equilibrium where i and j L randomize on some interval [0; t][ft g (similar to Lemma 3b(ii)) and j L has a mass point at zero. Proposition 4 Consider the game of information aquisition and suppose that at T the opportunity of provision disappears. If Assumption 1 holds and T is su iently small, only one individual aquires information in equilibrium. We do not provide a omplete analysis of equilibria of the war of attrition 1 and inentives to aquire information, but we show that, whenever T is small, there is a strategi value of ignorane: as in the previous setion, remaining uninformed an be used as a ommitment not to onede if an individual s expeted ost of provision is su iently high in relation to the payo from waiting until T. 1 A detailed analysis of equilibria of the war of attrition would build on Theorems 1-3 in Hendriks et al. (1988). 4

27 6 Conlusion The private provision of a disrete publi good is likely to end up in a war of attrition: individuals prefer to wait until someone else volunteers and provides the publi good. But they may not be able to wait for an in nite amount of time. This an be due to time onstraints or to a nite time horizon imposed by a third party. In many appliations, suh as alloating tasks in rms or ommunities, time limits are a typial feature of the volunteering game. In this paper, we analyzed inentives to obtain information ahead of a war of attrition. The information that is available to the individuals has an important impat on the equilibrium outome of the volunteering game. This suggests that individuals have an inentive to use information aquisition strategially when they antiipate the private provision game. We assumed that initially the individuals do not know exatly their own ost of provision of the publi good, but that they an nd out about this ost prior to the volunteering game. Indeed, there an be an inentive for one individual not to beome informed of his ost of provision even if the information is available without ost. For a su iently short time horizon, being uninformed indues an informed individual to volunteer immediately in ase he has a low ost of provision, whereas not knowing the own ost of provision onstitutes a ommitment to delay the own onession. For a su iently long time horizon, however, nding out about the own ost is a stritly dominant strategy. Sine the time horizon has a ruial impat on information aquisition as well as on the equilibrium outome of the volunteering game, it may be used as an instrument to in uene the e ieny of the publi good provision. Our model assumed that the individuals osts of provision follow a two-point probability distribution. For ontinuous distribution funtions, similar results an be obtained. The equilibrium properties hange in the sense that an individual with private information about his ost of provision hooses his onession time as an inreasing funtion of his provision ost. In the ase where exatly one individual has learned his ost, we get a similar result for a small time limit T : the informed individual volunteers immediately if he has a low ost of provision, whih reates 5