The multiple-volunteers principle *

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1 The multiple-voluteers priciple * Susae Goldlücke Thomas Tröger December 17, 2020 Abstract We cosider mechaisms for assigig a upleasat task amog a group of agets with heterogeous abilities. We emphasize threshold rules: every aget decides whether or ot to voluteer ; if the umber of voluteers exceeds a threshold umber, the task is assiged to a radom voluteer; if the umber is below the threshold, the task is assiged to a radom ovoluteer. We show that ay o-extreme threshold rule allows for a symmetric equilibrium i which every ability type is strictly better off tha i a radom assigmet. This holds for arbitrarily high costs of performig the task. Withi the class of biary-actio mechaisms, some threshold rule is utilitaria optimal. The first-best ca be approximated arbitrarily closely with a threshold rule as the group size teds to ifiity; that is, there exist threshold umbers such that with probability arbitrarily close to 1 the task is performed by a aget with a ability arbitrarily close to the highest possible ability. The optimal threshold umber goes to ifiity as the group size teds to ifiity. 1 Itroductio Voluteerig, accordig to Wilso (2000), is ay activity i which time is give freely to beefit aother perso, group or cause. Implicit here is the assumptio that there is little or o remueratio for the activity. Voluteerig plays a importat role i may differet areas of ay moder ecoomy. It cocers services as diverse as chairig a uiversity departmet, egagig i evirometal activities such as bird coutig, teachig the host * We would like to thak Johaes Abeler, Ted Bergstrom, Helmut Bester, Friedrich Breyer, Hedrik Hakees, Carl Heese, Stepha Lauerma, Meg Meyer, Adras Niedermayer, Alex Niemeyer, Georg Nöldeke, Adreas Roider, Ariel Rubistei, Klaus Schmidt, Urs Schweizer, Alex Smoli, Rolad Strausz, Robert Sugde ad may semiar participats for helpful commets ad discussios. Thomas Tröger gratefully ackowledges fiacial support from the Germa Research Foudatio (DFG) through CRC-TR 224, project B1. Uiversity of Kostaz, Departmet of Ecoomics, Kostaz, Germay. Phoe: Uiversity of Maheim, Departmet of Ecoomics, L7, 3-5, Maheim. Phoe:

2 coutry s laguage to a refugee, ad providig compay for termially ill patiets i hospices. 1 While may voluteerig activities are iformal, i the UK the ecoomic value of formal voluteerig aloe is estimated at 39 billio, accordig to the report by Low, Butt, Ellis, ad Davis Smith (2007). For may tasks, it is crucial to ot just fid ay voluteer, but to fid a well-qualified voluteer. As humas, we are heterogeous with respect to our abilities, whether the task is to lead a departmet, to spot a specime of a rare species i the field, to work with a refugee, or to talk to a dyig perso. If a highly able perso voluteers, everybody beefits. Our paper focusses o the ecoomic problem of assigig a task to the most able perso i a give group. No remueratio is possible. The task caot be delayed or avoided: oe of the group members must perform the task. Each aget is privately iformed about her ability, which is defied as the beefit that accrues to everybody if she performs the task. There is a free-rider problem because performig the task is costly. A simple task-assigmet rule that aturally comes to mid cosists i askig every aget whether or ot she voluteers. We imagie all agets beig asked simultaeously; if at least oe aget voluteers, the task is assiged radomly amog the voluteerig agets; if o aget voluteers, the task is assiged radomly amog all agets. This ay-voluteer rule ca, however, lead to rather poor voluteerig icetives. I particular, if performig the task is sufficietly costly, the the ay-voluteer rule leads, i ay symmetric equilibrium, to a purely-radom assigmet because obody will voluteer. I this paper, we preset alterative task-assigmet rules. The seed for our costructio ca be foud i the writigs of Thomas Schellig (Schellig, 2006). As a proposal i passig, he casts the idea of voluteerig if 20 others do likewise (p. 95). We may call this idea the multiple-voluteers priciple. Schellig s half-setece immediately raises may questios. What happes if the threshold umber of 20 is ot reached? Does the multiple-voluteers priciple lead to a welfare improvemet relative to the ay-voluteer rule? Ca we use the multiple-voluteers idea to costruct a mechaism that is optimal i some sese? Which threshold umber should be set? Our paper elaborates o these questios. We cosider task assigmet rules that are set by a social plaer. Thus, Schellig s advice to declare oe s coditioal willigess to voluteer is recast as follows. The rule allows each aget a choice betwee two actios that we call voluteerig ad ot voluteerig. If at least i (e.g., i = 20) agets voluteer, the all voluteers participate i a uiform lottery that determies the service provider. However, because the task caot be avoided, a fully specified task-assigmet rule must go beyod Schellig s advice: it must also specify who performs the task if the threshold umber of i voluteers is ot reached. We stipulate that the task is the assiged radomly amog the o-voluteers by a uiform lottery. We geeralize this costructio slightly: we allow that, if the umber of voluteers is equal to the threshold i, the, rather tha assigig the task to a voluteer for sure, a lottery may be used to decide whether the task is assiged to a voluteer or a o-voluteer. We 1 The Germa associatio of hospices reports that most of the idividuals workig uder their roof are voluteers who are ot remuerated, see hospizpalliativ ehreamt.html. 2

3 call ay such mechaism a threshold rule. Our mai results advertise the class of threshold rules, by demostratig three properties. First, i ay threshold rule that is o-extreme i a sese that will be defied, 2 there exists a equilibrium such that every ability type, icludig every o-voluteerig type, is strictly better off tha uder a purely-radom assigmet of the task. Thus, every type eve those with very low abilities ad those with very high abilities has a strict icetive to participate if the default is a purely-radom assigmet of the task. This property holds for arbitrarily high costs of performig the task. Thus, the existece of a improvemet over the purely-radom assigmet is a detail-free coclusio. Secodly, we show that, give the utilitaria welfare criterio, some threshold rule is optimal amog all biary-actio rules. I other words, i order to outperform the class of threshold rules, more complicated mechaisms with at least three actios would be eeded. Third, the first-best ca be approximated arbitrarily closely via a appropriate sequece of threshold umbers as the group size teds to ifiity. That is, i a large populatio a threshold rule (i particular, a biary-actio rule) is always good eough. While the utilitaria-optimal threshold umber teds to ifiity as the populatio size teds to ifiity, cosiderable welfare improvemets are ofte achieved already with very small threshold umbers. Cosider, for istace, a large populatio i which the average ability is equal to 1, the highest possible ability is equal to 5, ad the idividual cost of takig o the task is equal to 3. The followig equilibrium outcomes ca be computed. I equilibrium, oly agets with abilities close to 5 will voluteer, implyig that the expected ability of a voluteer is close to 5, ad the expected ability of a o-voluteer is close to 1. If a sigle voluteer is required (i.e., the ay-voluteer rule is used), the the task will be assiged to a voluteer with probability 45%; if two voluteers are required, the task will be assiged to a voluteer with probability 69%; if te voluteers are required, the task will be assiged to a voluteer with probability 89%. Related literature The allocatio problem cosidered i our paper is a (very) special case of a social-choice settig with iformatioal ad allocative exteralities (Jehiel ad Moldovau, 2001). However, i cotrast to the focus i that literature, we cosider here mechaisms without moetary trasfers ad with oly two actios, while maitaiig a cotiuum of types. I the absece of these restrictios, that is, with quasiliear prefereces, arbitrary moetary trasfers, ad arbitrary actio spaces, the first best could be obtaied i our settig by simply askig all agets for their ability types, assigig the task to the highest type, ad reimbursig the cost of performig the task, which is idetical across agets. 3 2 The set of o-extreme threshold rules icludes ay rule with a threshold umber i from 2 to 2, where is the group size. 3 The fact that the efficiet allocatio ca thus be implemeted reders our settig ogeeric from the poit of view of Jehiel ad Moldovau. 3

4 The earlier game-theoretic literature o voluteerig assumes that agets have idetical abilities, but are heterogeous with respect to the opportuity cost of providig the public good (i.e., performig the task) or, equivaletly, the persoal beefit from cosumig the public good. Moreover, i the literature it is usually allowed that the group of agets may fail to provide the public good (i.e., the task ca be avoided). Such a settig resembles the classic public-good provisio problem with private values (Clarke, 1971; Groves, 1973), except that o moetary trasfers are feasible. I a private-values settig, icetives for voluteerig arise from the threat that the public good is ot provided at all rather tha, as i our settig, the threat that the public good is provided i low quality. Rather tha takig a mechaism-desig approach, the voluteerig literature has focussed o two particular biary-actio game rules. The coordiated voluteer s game assigs the task radomly amog the voluteers if at least oe perso voluteers, ad otherwise avoids the task altogether. The ucoordiated voluteer s game is differet i the sese that ot just oe, but all voluteers pay the cost of performig the task. Olso (2009, first editio: 1965) cojectured that if a voluteer s game is played i a large populatio, the the probability that a voluteer is foud will be smaller tha i a small populatio. The first equilibrium aalysis of the (ucoordiated) voluteer s game is due to Diekma (1985). The subsequet literature has evaluated Olse s cojecture i various settigs (Makris, 2009; Bergstrom, 2017; Nöldeke ad Peña, 2020). I betwee our paper s assumptio that the public good must be provided ad the opposite assumptio that it ca be avoided lies the possibility that the provisio ca be delayed, leadig to discouted costs ad beefits. The possibility of delay aturally leads to a war-of-attritio game i which each aget waits, or egages i some other costly search process, util someoe agrees to provide the service. Withi the heterogeous-cost settig, such a game has bee aalyzed by Bliss ad Nalebuff (1984). 4 I equilibrium, it is ofte the right perso who voluteers first, e.g., the oe who has the lowest cost of providig the service, but it ca also be the oe who has the highest cost of waitig, ad substatial waitig costs may have to be icurred before a voluteer is foud. 2 Model A task of public iterest eeds to be allocated amog a group of agets 1,...,, where 2. Each aget is privately iformed about her ability at performig the task. Each aget s ability type is idepedetly distributed o a iterval [θ, θ] accordig to some strictly icreasig ad cotiuous cumulative distributio fuctio F. I a biary mechaism, each player chooses betwee two actios, deoted by Y ad N. Assumig aoymity, a biary mechaism is characterized 4 See also Bilodeau ad Sliviski (1996) for a related model with complete iformatio. See Klemperer ad Bulow (1999) for a geeral approach to war-or-attritio games, ad see LaCasse, Posati, ad Barham (2002) ad Sahuguet (2006) for more special extesios. 4

5 by a list p 1,..., p 1. For all j = 1,..., 1, the umber p j deotes the probability that the task is assiged to a radomly selected Y -player; with probability 1 p j, it is assiged to a radomly selected N-player. If the umber of Y -players is 0 or, the task is assiged radomly amog all agets, that is, each aget gets assiged the task with probability 1/. The purely-radom-assigmet mechaism (p 1,..., p 1 ) is give by p j = j/ for all j. These probabilities imply that the task is always assiged with equal probability to ay aget, idepedetly of the agets strategies. Give ay mechaism (p 1,..., p 1 ), a symmetric strategy profile is characterized by a fuctio σ that determies the strategy for each aget, where σ(θ) deotes the probability that type θ [θ, θ] chooses Y. The expected utility U a (σ, θ) of ay type θ takig actio a, who aticipates that the other agets will use the strategy σ, is deoted U a (σ, θ). The fuctio σ is a equilibrium if the followig implicatios hold for all θ: if σ(θ) > 0 the U Y (σ, θ) U N (σ, θ) 0, if σ(θ) < 1 the U Y (σ, θ) U N (σ, θ) 0. Mechaism-equilibrium combiatios (p 1,..., p 1, σ) ad (p 1,..., p 1, σ ) are equivalet if each type obtais the same expected utility i both combiatios. Notatio: selectio-probability fuctios Before cotiuig with the model descriptio, we itroduce four auxiliary fuctios, h Y, h N, q Y ad q N, ad discuss their basic properties. These fuctios will play a fudametal role throughout the paper. We call them selectio-probability fuctios. Takig the poit of view of a aget who has chose a actio (Y or N), the fuctios q Y ad q N describe the probability of persoally gettig assiged the task, ad the fuctios h Y ad h N describe the probability that ayoe i the set of agets who take a particular actio gets assiged the task. For the most part, i our computatios, biomial sums will remai hidde behid the selectio-probability fuctios. The argumet of the selectio-probability fuctios is the ex-ate probability that a give aget chooses Y, y = σ(θ)df (θ). (1) For ay y [0, 1], the probability that ayoe of the Y -playig agets is selected, coditioal o the evet that a give aget plays Y, is deoted h Y (y) = 1 By 1 (j)p j+1. (2) j=0 5

6 Here, y deotes every other (i.e., ot the give) aget s probability of playig Y. Usig the biomial distributio, By 1 (j) = ( ) 1 j (1 y) 1 j y j deotes 5 the probability that, from the poit of view of the give aget, j other agets choose Y. We also use the otatio p = 1. The probability that ayoe of the Y -playig agets is selected, coditioal o the evet that the give aget plays N, is deoted h N (y) = 1 By 1 (j)p j, (3) j=0 where we use the otatio p 0 = 0. The probability that the give aget is selected if she chooses actio a = Y, N is deoted q a (y); i.e., q Y (y) = 1 By 1 (j) p j+1 j + 1 j=0 (4) ad q N (y) = 1 By 1 (j) 1 p j j. (5) j=0 Ofte we will omit the argumet y from h Y, h N, q Y ad q N. For a illustratio of the selectio-probability fuctios i a special case, cosider the purely-radom-assigmet rule. Here, a computatio that applies stadard properties of the biomial distributio to the defiitios (2) (5) shows that h Y (y) = 1 + y( 1), h N (y) = q Y (y) = 1, q N(y) = 1. y( 1), (6) To uderstad the umerators i the formulas for h Y ad h N, ote that, from the poit of view of a give aget, the expected umber of other Y -players is equal to y( 1); by playig Y, the aget adds i herself (1+). Next we establish several useful algebraic relatios betwee the selectioprobability fuctios. These relatios hold idepedetly of the uderlyig mechaism. 6 A particularly simple formula is available for expressig q Y ad q N i terms of h Y ad h N. To see this, suppose that all agets play Y with probability y. The the probability that the task is assiged to a Y -player ca be expressed i the form yh Y +(1 y)h N. Alteratively, the same probability ca be expressed i the form yq Y because every Y -playig aget is selected with the same probability: yq Y = yh Y + (1 y)h N. (7) 5 We use the covetio that 0 0 = 1. 6 Note that the Berstei polyomials y By 1 (j) (j = 0, 1,..., 1) form a basis of the vector space of polyomials of degree at most 1. Thus, from each of the four selectioprobability fuctios, the uderlyig mechaism (p 1,..., p 1) ca be recovered, implyig that each of the four fuctios determies the other three fuctios. 6

7 Similarly, the probability that a N-player is selected is give by (1 y)q N = y(1 h Y ) + (1 y)(1 h N ). (8) Addig up the equatios (7) ad (8) cofirms the ex-ate probability that ay give aget is selected: yq Y + (1 y)q N = 1. (9) We state oe other relatio betwee the selectio-probability fuctios; it refers to the derivatives of q Y ad q N. The proof, which relies o stadard properties of Berstei polyomials, is relegated to the Appedix. Lemma 1. Cosider ay mechaism ad ay 0 < y < 1. The q Y (y) = h Y h N q Y y, (10) q N(y) = h N h Y + q N. (11) 1 y To iterpret (10), take the poit of view of a aget who cosiders switchig from playig N to playig Y. Here, h Y h N q Y equals the chage i the probability that a Y -player other tha herself gets assiged the task. As log as this chage is positive, it holds that q Y (y) > 0, that is, a icrease of y icreases the probability that the aget herself gets assiged the task if she plays Y ; similarly if the chage is egative. A aalogous iterpretatio applies to (11). Expected utilities We assume the followig prefereces. Suppose the task is performed by a aget of ability θ. The every aget obtais the beefit θ. I additio, the performig aget bears a cost c 0, where c is commoly kow ad idepedet of the idetity of the aget. Agets are expected-utility maximizers. Cosider a strategy σ ad y defied via (1). Towards computig equilibria, it is crucial to evaluate a aget s expected-utility gai from playig Y versus playig N, assumig that all other agets use the strategy σ. We will establish a coveiet expressio for this utility gai. To this ed, we express the agets expected-utility fuctios i terms of the selectio-probability fuctios ad coditioal expected abilities. The coditioal expected ability of a aget who chooses Y is deoted E Y = 1 σ(θ)θdf (θ) if y > 0. y I other words, E Y is the expected beefit that accrues to every aget if the task is assiged to a Y -player. The coditioal expected ability of a aget who chooses N is deoted 1 E N = (1 σ(θ))θdf (θ) if y < 1. 1 y 7

8 That is, E N is the expected beefit that accrues to every aget if the task is assiged to a N-player. Usig this otatio, the agets expected-utility fuctios are U Y (σ, θ) = (h Y q Y )E Y + q Y (θ c) + (1 h Y )E N, (12) U N (σ, θ) = (1 h N q N )E N + q N (θ c) + h N E Y. (13) The iterpretatio of these expressios is straightforward. Cosider the expected utility (12) from playig Y : the first term captures the payoff that arises from the evet that the task is performed by a Y -player other tha the aget herself, which happes with probability h Y q Y ; the secod term captures the evet that the aget is selected herself, which happes with probability q Y, yieldig the utility θ c; the third term captures the payoff that arises from the evet that the task is performed by a N-player. The iterpretatio of the expressio (13) for the expected utility from playig N is aalogous. Combiig the expressios (12) ad (13) ad cacellig terms, the utility gai from playig Y versus playig N is U Y (σ, θ) U N (σ, θ) = (q Y q N )(θ c) (14) + (h Y h N q Y )E Y + (h N h Y + q N )E N. The three terms o the right-had side reflect that a aget s choice of actio affects three probabilities: to be selected herself (first term), the probability that a Y -player other tha herself is selected (secod term), ad the probability that a N-player other tha herself is selected (third term). The purely-radom assigmet is a atural bechmark for our aalysis. I the purely-radom-assigmet rule, every strategy is a equilibrium. To see this formally, ote that, from (6), the right-had side of (14) equals 0 for all σ. Moreover, all equilibria are equivalet: usig (12) ad the law of iterated expectatios (that is, ye y + (1 y)e N = E[θ]), U a (σ, θ) = (1 1 )E[θ] + 1 (θ c) for all σ, all θ, ad a = Y, N. (15) These purely-radom-assigmet payoffs i fact obtai ot oly if the radomassigmet rule is used. These payoffs obtai wheever the mechaism ad equilibrium are such that a aget s probability of gettig selected is idepedet of her actio. 7 Remark 1. Ay mechaism-equilibrium combiatio (p 1,..., p 1, σ) such that q Y (y) = 1/ = q N (y) (where y is give by (1)) is equivalet to a purelyradom assigmet. 7 The equilibrium assumptio i Remark 1 is idispesible. If a (possibly o-equilibrium) strategy leads to a y with the property q Y (y) = 1 ad qn (y) = 1, it does ot follow that hy (y) ad h N (y) are give as i the case of a purely-radom assigmet. For example, if = 3, (p 1, p 2) = (0, 1), ad y = 1/2, the q Y (y) = 1 = qn (y) ad hy (y) = 3/4, whereas we would obtai h Y (y) = 2/3 from usig the pure-radom-assigmet rule. 8

9 Here is a sketch of the proof. The coclusio is straightforward if the mechaism-equilibrium combiatio is such that all types prefer Y to N or vice versa, or the coditioal expected quality of the task is the same across the two actios. Suppose ow that some type is idifferet betwee the actios N ad Y, ad the coditioal expected quality of the task is ot the same across the two actios. Cosider a aget who chages her actio from N to Y. By assumptio, this chage has o impact o the probability of gettig selected. Thus, the chage icreases the probability that a Y - player other tha herself is selected by the same amout as it decreases the probability that a N-player other tha herself is selected. The a type who is idifferet betwee the actios ca exist oly if the chage of the aget s actio does ot actually icrease (or decrease) the probability that Y -player other tha herself is selected. Thus, ay type s expected utility is as i a purely-radom assigmet. The formal proof is relegated to the Appedix. Threshold equilibria We ow itroduce a special class of equilibria, threshold equilibria. Our aalysis will focus o this class. We show i Lemma 2 that this focus is without loss of geerality. A strategy σ has the threshold form if there exists ˆθ [θ, θ] such that σ(θ) = 0 for all θ < ˆθ ad σ(θ) = 1 for all θ > ˆθ. Igorig probability-0 evets, ay strategy i threshold form is characterized by the playig-y - probability y = 1 F (ˆθ). Wheever we deal with a threshold strategy y, we will use the otatio E Y (y) = E[θ θ F 1 (1 y)] ad E N (y) = E[θ θ F 1 (1 y)] for the expected ability of a Y -player ad a N-player, respectively; we defie the cotiuous extesios E Y (0) = θ ad E N (1) = θ. Similarly, we will use the otatio U a (y, θ) for the expected payoff of type θ from takig actio a = Y, N if all others use the strategy y. Give ay threshold strategy y, E Y (y) > E N (y). (16) The strategies y = 0 ad y = 1 imply that oe actio is chose with probability 1, so that the purely-radom-assigmet payoffs obtai. I ay equilibrium y i which both actios are chose with positive probability (i.e., 0 < y < 1), the type ˆθ = F 1 (1 y) is idifferet betwee the two actios, that is, (y) = 0, where (y) = U Y (y, F 1 (1 y)) U N (y, F 1 (1 y)). (17) Moreover, usig (14) ad the equilibrium coditio, if 0 < y < 1 the a aget s switch from the actio N to the actio Y caot reduce the probability that she gets selected, q Y (y) q N (y). (18) Lemma 2 shows that focussig o threshold equilibria is without loss of geerality, ad the properties (17) ad (18) ca be maitaied eve if y = 0 or y = 1. 9

10 Give the property (18), from ow o we iterpret the actio Y as voluteerig ad the actio N as o-voluteerig. Lemma 2. For ay mechaism-equilibrium combiatio, there exists a equivalet mechaism-threshold-equilibrium combiatio, (p 1,..., p 1, y), such that the properties (17) ad (18) hold. The ituitio is that we ca costruct a equivalet mechaism-equilibrium combiatio by switchig the labels of the actios Y ad N. The formal proof is relegated to the Appedix. For later use, we establish a simple property of those threshold equilibria i which voluteerig actually icreases the probability of gettig selected: i such a equilibrium it caot be true that all types voluteer. 8 Remark 2. Ay threshold equilibrium y with q Y (y) > q N (y) satisfies y < 1. The ituitio behid this result is simple: if a aget expects that with probability y = 1 somebody else will voluteer, the by voluteerig herself she will reduce the expected ability of the selected aget if she herself is edowed with the lowest ability θ or a ability close to that. The formal proof is relegated to the Appedix. Because the iequality (18) ad its strict versio will occur frequetly i the subsequet aalysis, it is useful to ote that these iequalities ca be expressed i a alterative form if the strategy is such that some types do ot voluteer. The proof is straightforward from (9). Remark 3. Cosider ay threshold strategy y < 1. The the iequality (18) holds if ad oly if q Y (y) 1/. The iequality q Y (y) > q N (y) holds if ad oly if q Y (y) > 1/. The plaer s (biary-secod-best) problem We cosider the utilitaria welfare objective. Give our focus o symmetric equilibria, this objective is equivalet to maximizig ay aget s ex-ate expected utility. Because the task caot be avoided, each aget pays the cost c/ i ay mechaism-equilibrium combiatio. Thus, the plaer s objective boils dow to assigig the task such that the expected ability of the selected aget is maximized. Without loss of geerality, we restrict the allowed mechaism-equilibrium combiatios i lie with the result of Lemma 2. Give ay strategy y, a voluteer is selected with probability yq Y ad a o-voluteer is selected with probability (1 y)q N. Thus, the expected ability of the selected aget is E = yq Y E Y + (1 y)q N E N. (19) 8 The assumptio q Y (y) > q N (y) i Remark 2 caot be replaced by the weaker coditio (18); this coditio would leave ope the possibility of a purely-radom-assigmet rule i which y = 1 is i fact a equilibrium. 10

11 Hece, the plaer s (biary-secod-best) problem is to max E p 1...,p 1,y s.t. 0 p j 1 (j = 1,..., 1), 0 y 1, (y) = 0, q Y (y) q N (y) 0. Usig (9), we ca alteratively write the objective purely i terms of q Y, as We will solve this problem i Sectio 3.2. E = yq Y (E Y E N ) + E N. (20) The biary-first-best problem I this sectio, we solve, as a bechmark, the problem of a plaer who is ot restricted by equilibrium costraits. Secodly, we show how the cost of performig the task creates a coflict betwee the solutio to the first-best problem ad the equilibrium coditio. Fially, we characterize the largepopulatio limit of the biary-first-best solutio. The plaer s biary-first-best problem is as follows: max E p 1...,p 1,y s.t. 0 p j 1 (j = 1,..., 1), 0 y 1. The iterpretatio is that, by settig ay y, the plaer has the power to make the types i [F 1 (1 y), θ] play Y ad to make the types i [θ, F 1 (1 y)] play N. The biary-first-best problem maitais the restrictio to biary mechaisms ad threshold strategies. The stadard first best, i cotrast, is defied without these restrictios. The solutio to the stadard first-best problem is to always assig the task to the aget with the highest ability amog all agets. Give that a cotiuum of types exist, this solutio ca obviously ot be reached exactly with a biary mechaism. The mechaism is called the ay-voluteer rule. (p 1..., p 1 ) = (1,..., 1) Propositio 1. The solutio to the biary-first-best problem ivolves usig the ay-voluteer rule. Deotig by y b the voluteerig rate i a solutio, we have 0 < y b < 1 ad de/dy y=y b = 0. Proof. Give ay 0 < y < 1, (16) together with (20) shows that the optimal mechaism maximizes q Y (y). Thus, from (4) the ay-voluteer rule is the 11

12 uique optimal mechaism if 0 < y < 1. Moreover, the ay-voluteer rule is a optimal mechaism if y = 0 or y = 1. Cosequetly, a biary-first-best optimal y = y b is foud by solvig the problem max E s.t. 0 y 1, y where E is evaluated specifically for the case of the ay-voluteer rule (p 1..., p 1 ) = (1,..., 1). I the case of the ay-voluteer rule, (2) ad (3) imply that, for all y, Thus, (7) implies that Also, (5) implies that h Y (y) = 1 ad h N (y) = 1 (1 y) 1. (21) q Y (y) = 1 y (1 (1 y) ) if y > 0. (22) q N (y) = 1 B 1 y (0) = 1 (1 y) 1. (23) Thus, usig (19), i the case of the ay-voluteer mechaism, E = (1 (1 y) ) E Y + (1 y) E N. The first-order effect of icreasig y is de dy = (1 y) 1 (E Y E N ) + (1 (1 y) ) de Y dy + (1 y) de N dy, where de Y dy = d dy ( ) 1 θ θdf (θ) y F 1 (1 y) = F 1 (1 y) E Y y (24) ad de N dy ( = d 1 dy 1 y F 1 (1 y) θ θdf (θ) ) = E N F 1 (1 y). (25) 1 y Clearly, ay biary-first-best y b satisfies 0 < y b < 1 because otherwise the purely-radom assigmet would obtai. To cofirm, oe ca verify that de/dy y=1 = θ E[θ] < 0 ad de/dy y=0 = ( 1)(θ E[θ]) > 0. This completes the proof of Propositio 1. I the proof above we have evaluated the welfare effect of margially icreasig the voluteerig rate y whe the ay-voluteer rule is used. If a geeral biary-actio rule is used, there is a surprisigly simple ad useful formula that coects this welfare effect to the margial type s utility gai from playig Y versus playig N. This formula, stated i Lemma 3, captures how the coflict betwee the plaer s welfare goal ad a aget s equilibrium coditio depeds o the cost of performig the task. 12

13 Lemma 3. Cosider ay mechaism ad ay (ot ecessarily equilibrium) threshold strategy y. The de dy = (y) + (q Y (y) q N (y))c. Proof. Applyig the product differetiatio rule to (19), we fid 1 de dy = q Y E Y q N E N + y dq Y dy E Y + (1 y) dq N dy E N + yq Y de Y dy + (1 y)q de N N dy. (26) Pluggig (24), (25), (10), ad (11) ito (26), we get 1 de dy = (E Y E N )(h Y h N ) + q N E N q Y E Y + F 1 (1 y)(q Y q N ). Usig (14) ad the defiitio of (y) i (17), the claimed formula follows. A immediate implicatio of Lemma 3 is that, if the cost of performig is positive (c > 0) ad a aget s task-assigmet probability is ot idepedet of her actio (i.e., q Y (y) > q N (y)), the at ay equilibrium voluteerig rate y (i.e., (y) = 0), the welfare is strictly icreasig i the voluteerig rate. This free-rider problem vaishes if the voluteerig cost is equal to 0. Corollary 1. Assume c = 0. The the ay-voluteer rule, together with ay biary-first-best voluteerig probability, solves the plaer s biary-secodbest problem. Proof. If c = 0, the ay biary-first-best voluteerig probability y = y b is a equilibrium i the ay-voluteer rule. This is because de dy y=y = 0 by b biary-first-best optimality, so that Lemma 3 implies (y b ) = 0. Moreover, as show above, ay solutio to the biary-first-best problem ivolves the ay-voluteer rule because 0 < y b < 1. The biary-first-best expected ability of the selected aget is E b = y b q y (y b )(E Y (y b ) E N (y b )) + E N (y b ). Remark 4 states that, i the biary first-best i a large populatio, the idividual voluteerig probability teds to 0, the probability that at least oe aget voluteers teds to 1, ad the expected ability of the selected aget teds to the highest possible ability. This follows from the fact that i a large populatio, a aget with a ability close to the highest possible ability exists with a probability close to 1. A detailed proof ca be foud i the Appedix. Remark 4. As, y b 0, y b q Y (y b ) 1, ad E b θ. A immediate implicatio of Remark 4 is that the stadard first-best is approximated by the biary first best if the populatio is large. 13

14 Threshold rules A mechaism (p 1,..., p 1 ) is called a threshold rule if there exists a umber i (1 i 1) such that p j = 1 for all j > i ad p j = 0 for all j < i. Our mai results will cocer threshold rules. The ay-voluteer rule is a threshold rule; set i = 1 ad p i = 1. More geerally, a threshold rule captures the idea of what we call the multiplevoluteers priciple. Each aget aticipates that playig Y puts her i a lottery box together with the other Y -playig agets if altogether more tha i players play Y, ad releases her from the task if altogether fewer tha i players play Y. If the threshold umber i is reached exactly, the decisio whether or ot she will be i the lottery box may itself be radomized (via the probability p i ). Stipulatios are aalogous if the aget plays N. If the umber of other agets who play Y equals i 1 or i, the the aget ca be pivotal, that is, her ow actio choice ca have a impact o whether the task is assiged via a lottery amog the Y -players or via a lottery amog the N-players. From a give aget s poit of view, the pivotality of her actio choice may be measured i terms of the differece betwee the selectio probabilities defied i (2) ad (3). Whe applied to a threshold rule, this differece simplifies to h Y (y) h N (y) = By 1 (i )(1 p i ) + By 1 (i 1)p i. (27) This differece will play a importat i our aalysis. I particular, a very useful property is its quasicocavity: as the voluteerig rate icreases, the pivotality first icreases ad the decreases. More precisely, the followig holds. Lemma 4. If = 2, the the threshold rule with i = 1 ad p i = 1/2 satisfies h Y (y) h N (y) = 1/2 for all y [0, 1]. For ay other threshold rule if = 2, ad for ay threshold rule if 3, y m [0, 1] y (0, 1) : (28) (h Y h N ) (y) > 0 if y < y m, ad (h Y h N ) (y) < 0 if y > y m. Note that formula (28) is immediate from stadard properties of biomial probabilities if p i = 1 or p i = 0. The complete proof, i which we also cosider the mixed cases where 0 < p i < 1, ad the special case = 2, is relegated to the Appedix. For later referece, we restate the other two selectio-probability fuctios as specialized for a threshold rule: q Y (y) = 1 By 1 j=i 1 (j) j B 1 y (i 1 1)p i i (29) ad q N (y) = i 1 j=0 B 1 y 1 (j) j + B 1 y (i 1 )(1 p i ) i. (30) 14

15 3 Results 3.1 Improvemet over the purely-radom assigmet I this sectio, we defie o-extreme threshold rules ad show that ay such rule always has a equilibrium such that every type of aget is strictly better off tha i a purely-radom assigmet (Propositio 3). The ay-voluteer rule, i geeral, does ot have this strict-improvemet property (Propositio 4). We begi by showig that ay biary mechaism provides a weak improvemet compared to a purely-radom assigmet, ad formulate coditios for a strict improvemet (Propositio 2). The weak-improvemet property justifies our formulatio of the desiger s problem without a participatio costrait: if upo rejectio of the plaer s rule a purely-radom assigmet obtais, all types fid it weakly optimal to participate i the rule. Propositio 2. Cosider ay mechaism-threshold-equilibrium combiatio (p 1..., p 1, y). The all types are at least as well off as i the purely-radom assigmet. If 0 < y < 1 ad q Y (y) > 1/, the all types are strictly better off tha i the purely-radom assigmet. The proof of the at-least-as-well part follows from Remark 1. The proof of the strictly part of Propositio 2 is as follows. Cosider a equilibrium y ad cosider ay aget with a give type. Suppose that this type of the aget deviates from the equilibrium by voluteerig with probability y ad ot voluteerig with probability 1 y. Because the aget mimics the average behavior of ay other aget, she will be selected with probability 1/; i this evet, her payoff is the same as i a purely-radom assigmet. I the complemetig evet that the aget is ot selected, her payoff is the same as her ex-ate expected payoff whe she follows the equilibrium strategy, coditioig o the same evet. This payoff equals the equilibrium expected ability of the selected aget, which is higher tha the expected ability i a radom assigmet if q Y > q N. The formal proof ca be foud i the Appedix. Propositio 2 does ot aswer the questio whether or ot a strict improvemet over the purely-radom assigmet is possible at all. Propositio 3 gives a affirmative aswer for all group sizes > 2. A threshold rule is called o-extreme if the assigmet probability to a voluteer is below pure radomess if there is a sigle voluteer (i.e., p 1 < 1/), ad the assigmet probability to a o-voluteer is below pure radomess if there is a sigle o-voluteer (i.e., p 1 > 1 1/). This coditio is satisfied for all threshold rules with 2 i 2. A o-extreme threshold rule exists if ad oly if > 2. Propositio 3 shows that ay o-extreme threshold rule has a equilibrium that satisfies the strict-improvemet coditios stated i Propositio 2 ad, thus, is ot equivalet to a purely-radom assigmet. 9 This coclusio 9 Amog the extreme threshold rules are the ay-voluteer rule (where p 1 = 1) ad the allvoluteer rule (where p 1 = 0). These rules lead to fudametally differet icetives from the o-extreme threshold rules. We discuss the ay-voluteer rule at the ed of this sectio. I the all-voluteer rule, a threshold equilibrium with a positive level of voluteerig does ot exist because q Y (y) < 1/ for all y < 1. 15

16 h Y (y) h N (y) q Y (y) 1 ˇy y m1 ŷ 1 y y m (y) Figure 1: A example of a task assigmet problem. There are = 5 agets. The ability of each aget is uiformly distributed o [0, 1]. The performace cost is c = 1. The diagram shows several fuctios of the voluteerig rate y, for the case of the threshold rule that requires two voluteers (i.e., i = 2 ad p i = 1). The fuctio h Y (y) h N (y) captures the impact of a aget s switch from o-voluteerig to voluteerig o the probability (computed from the switchig aget s poit of view) that the task gets assiged to a voluteer. The fuctio q Y (y) captures the probability a aget assigs to the evet of beig selected if she voluteers. The fuctio (y) captures a aget s payoff gai from voluteerig. The voluteerig rate ŷ is a equilibrium that satisfies the strictimprovemet coditios stated i Propositio 2. holds o matter how large the cost c is. Propositio 3. For ay o-extreme threshold rule, the set {y [0, 1] (y) = 0} is o-empty ad its maximal elemet, ŷ, is a equilibrium with 0 < ŷ < 1 ad q Y (ŷ) > 1/. The mai step towards provig Propositio 3 is Lemma 5. This result cocers the impact of a aget s switch from o-voluteerig to voluteerig o the probability h Y h N q Y that a voluteer other tha herself gets assiged the task. Suppose this impact is strictly positive if the voluteerig rate is small, is strictly egative if the voluteerig rate is large, ad the quasicocavity coditio (28) is satisfied. The Lemma shows that there is oly oe voluteerig rate such that the impact equals 0, that is, the impact fuctio chages its sig oly oce. Lemma 5. Cosider ay biary mechaism such that (28) holds. Assume that h Y (y) h N (y) q Y (y) > 0 for all y > 0 sufficietly close to 0, (31) ad h Y (y) h N (y) q Y (y) < 0 for all y < 1 sufficietly close to 1. (32) The there exists a uique y m1 (0, 1) such that h Y (y m1 ) h N (y m1 ) q Y (y m1 ) = 0. The proof of Lemma 5 ca be foud i the appedix. Here is a sketch. We have to show that the fuctios q Y ad h Y h N itersect oly oce (cf. 16

17 Figure 1). By assumptio (32), q Y lies above h Y h N at y = 1. Because q Y lies below h Y h N at small values of y, ad lies above at large values, there is a maximal itersectio poit y m1. What we have to show is that aother, earlier, itersectio poit caot exist. We prove this i two steps. First, there exists o iterval bouded by itersectio poits y 1 ad y 1 such that at the poits i the iterior of the iterval q Y lies above. Secod, q Y actually lies below h Y h N at all poits smaller tha y m1. The crucial tool for both steps is (10). As for the first step, by costructio of the supposed iterval, h Y h N is at most as steep as q Y at the left boudary poit y 1, ad is at least as steep as q Y at the right boudary poit y 1. Because both poits are itersectio poits, the fuctio q Y has a horizotal taget at these poits by (10). Therefore, h Y h N must have a o-positive derivative at y 1 ad a oegative derivative at y 1, cotradictig the quasiliearity assumptio (28). Thus, the supposed iterval caot exist, showig that h Y h N is at least as large as q Y at all poits to the left of y m1. As for the secod step, suppose that there exists a itersectio poit smaller tha y m1. At this poit, by the first step, the fuctios h Y h N ad q Y must have the same slope, which by (10) equals 0. Now usig the quasiliearity assumptio (28), this itersectio poit must be the poit y m. Agai usig (10), the fuctio q Y is strictly icreasig o the iterval [y m, y m1 ], cotradictig the fact that o this same iterval the fuctio h Y h N is strictly decreasig by the quasiliearity assumptio (28). This completes the proof of Lemma 5. The proof of Propositio 3 begis by showig that every o-extreme threshold rule satisfies the assumptios of Lemma 5. By Lemma 4, (28) holds. To get the ituitio for why (31) is satisfied, cosider a threshold rule with p i = 1. Cocerig both fuctios, the pivotality h Y (y) h N (y) ad the idividual assigmet probability q Y, the relevat evet is that at least i 1 other agets choose to voluteer. Coditioig o this evet, ad cosiderig a small voluteerig rate y, it is the extremely likely that the threshold i is reached exactly, so that the aget is almost certaily pivotal, but she herself gets assiged the task oly with probability 1/i. Thus, for all i 2 the fuctio h Y (y) h N (y) lies above the fuctio q Y if y is small; i the case i = 1 this argumet breaks dow ad the o-extremeess assumptio becomes relevat. To see why (32) is satisfied, cosider ay aget who believes that everybody else voluteers (y = 1). The switchig her actio from ovoluteerig to voluteerig chages the probability that a voluteer is selected from p 1 to 1. Thus, h Y (1) h N (1) = 1 p 1, whereas the idividual assigmet probability is q Y (1) = 1/, idepedetly of the uderlyig rule. Hece, (32) is immediate from the o-extremeess assumptio. Give that the coditios of Lemma 5 are satisfied, the ext observatio is that q Y (y m1 ) > 1/; this follows because q Y (1) = 1/ ad q Y is strictly decreasig o the iterval [y m1, 1] by (10). Lowerig y further below y m1, we reach a poit ˇy < y m1 where q Y (ˇy) = 1/ (because q Y (0) < 1/ by oextremeess). Note that q Y lies strictly above 1/ o the ope iterval (ˇy, 1). Usig Remark 3, q N (ˇy) = 1/. That is, at the poit ˇy, the aget s actio has o impact o the probability that she gets assiged the task. O the other 17

18 had, Lemma 5 implies that h Y h N lies above q Y at the poit ˇy. That is, switchig from o-voluteerig to voluteerig icreases the probability that the task gets assiged to a voluteer other tha herself. Thus, the payoff gai from switchig is strictly positive at the poit ˇy. Fially, the payoff gai is clearly strictly egative at the poit where everybody else voluteers (y = 1). Thus, the maximal poit ŷ where the payoff gai equals 0, lies strictly betwee ˇy ad 1, implyig that the strict-improvemet coditios stated i Propositio 2 are satisfied at this equilibrium. The formal proof of Propositio 3 is relegated to the Appedix. The followig result is immediate from Propositio 2 ad Propositio 3. Corollary 2. Suppose that there are 3 agets. The the purely-radom assigmet does ot solve the biary secod-best problem. I the case = 2, it is straightforward to verify that q Y (y) q N (y) = p 1 1/2 for all y [0, 1] ad ay mechaism p 1 [0, 1], ad (14) simplifies to U Y U N = (p 1 1/2)(θ c E[θ]). Thus, a threshold equilibrium y with 0 < y < 1 ad q Y (y) q N (y) > 0 (or, equivaletly, q Y (y) > 1/2) exists if ad oly if p 1 > 1/2 ad c < θ E[θ]. Thus, i the case = 2 the purely-radom assigmet is optimal if ad oly if c θ E[θ]. We ed this sectio with a discussio of the ay-voluteer rule. We show that there always exists a equilibrium i threshold form. If the cost is low, the a strict improvemet over the purely-radom assigmet is achieved i equilibrium; if the cost is high, the the threshold will be such that obody voluteers ad the purely-radom assigmet obtais. Thus, the icetives i the ay-voluteer rule differ fudametally from the icetives i a oextreme threshold rule: a o-extreme threshold rule always allows for a improvemet over the purely-radom assigmet, while the ay-voluteer rule does ot. Propositio 4. If c < θ E[θ], the the ay-voluteer rule has a threshold equilibrium y such that 0 < y < 1 ad q Y (y) > q N (y). If c θ E[θ], the the uique equilibrium of the ay-voluteer rule is y = 0, so that the purely-radom allocatio obtais. If c < θ E[θ], the ay equilibrium y satisfies 0 < y < 1. The reaso the ay-voluteer rule ca lead to the breakdow of voluteerig ca be uderstood if we cosider a aget of highest ability who believes that obody else will voluteer. Switchig her actio from o-voluteerig to voluteerig raises the probability that she herself gets assiged the task by 1 1/. At the same time, the switch reduces, by the same amout, the probability that a o-voluteer other tha herself is selected. Thus, the aget faces a equal-probability tradeoff betwee the payoff from voluteerig herself, θ c, ad the payoff from lettig somebody else do the job, E[θ]. Thus, she will ot voluteer if the cost is high. This argumet shows that i case c θ E[θ] there exists o equilibrium y 0 i the viciity of 0. To provide a complete proof of Propositio 4, we must also exclude equilibria y arbitrarily far away from 0. All the remaiig steps ca be foud i the Appedix. 18

19 3.2 Optimality of a threshold rule I this sectio, we show that the solutio to the plaer s problem always ivolves a threshold rule (Propositio 5). Towards provig this, it is useful to kow that the equilibrium coditio ca be relaxed so that it becomes a iequality. Lemma 6. Ay solutio to the biary secod-best problem also solves the relaxed problem i which the costrait (y) = 0 is replaced by the iequality (y) 0. Proof. Let (p 1,..., p 1, y) be a solutio to the relaxed problem. Suppose first that q Y (y) = q N (y). The q Y (y) = q N (y) = 1/ by (9), implyig E = E[θ] by the law of iterated expectatios. Thus, the value at the optimum of the relaxed problem equals the value at the optimum of the plaer s problem. This implies the desired coclusio. Now cosider cases i which q Y (y) > q N (y). Suppose that y = 1. Applyig the equatio (45) at θ = θ = F 1 (1 y), (y) = U Y (y, F 1 (1 y)) U N (y, F 1 (1 y)) = ( 1 1+p 1) (θ c E[θ]). }{{} <0 The right-had side is < 0 because the costrait q Y > q N implies p 1 > 0, yieldig a cotradictio to the relaxed iequality costrait. We coclude that y < 1. Suppose that (y) > 0. Applyig Lemma 3, de/dy > 0. This is a cotradictio to optimality because oe of the costraits o y is bidig. Propositio 5. Ay solutio to the plaer s problem ivolves a threshold rule. Proof of Propositio 5. Cosider ay solutio (p 1..., p 1, y). If = 2, the we have othig to prove because ay biary mechaism is a threshold rule. Assume that 3. Corollary 2 implies that 0 < y < 1 because otherwise (p 1..., p 1, y) would be equivalet to a purely-radom assigmet. Similarly, q Y (y) > q N (y) because otherwise q Y (y) = 1/ = q N (y) from (9), implyig purely-radom-assigmet payoffs by Remark 1. By Lemma 6, (p 1..., p 1, y) solves the relaxed problem. Fixig y, the remaiig relaxed maximizatio problem over (p 1,..., p 1 ) is a liear problem. Hece the Lagrage coditios are ecessary ad sufficiet, without ay qualificatio. Let λ 0 deote the Lagrage multiplier for the costrait U Y (y, F 1 (1 y)) U N (y, F 1 (1 y)) 0. Due to q Y (y) > q N (y), the Lagrage multiplier for the costrait q Y (y) q N (y) 0 equals 0. 19

20 Let ˆθ = F 1 (1 y). For all j = 1,..., 1, cosider ( ) ŝ j = y j (1 y) j (E Y E N ) j ( ( ( 1 j 1 + )y j 1 (1 y) j λ E Y + 1 )) j 1 j j (ˆθ c) E N ( ( ( 1 )y j (1 y) 1 j λ E Y j 1 E N 1 )) j j j (ˆθ c). The Lagrage coditios require: Moreover, if ŝ j > 0, the p j = 1, if ŝ j < 0, the p j = 0. (33) if U Y (y, ˆθ) > U N (y, ˆθ), the λ = 0. (34) The sig of ŝ j is preserved if istead of ŝ j we cosider the variable s j = ŝ j ( j) y j 1 (1 y) j 1 for all j. Thus, s j = y(1 y)(e Y E N ) + λ j ( j 1 (1 y) E Y + 1 ) j j (ˆθ c) E N λ j ( y E Y j 1 E N 1 ) j j (ˆθ c) = y(1 y)(e Y E N ) + λ 1 ( ) (1 y) (j 1)E Y + (ˆθ c) je N λ 1 ) (( y j)e Y ( j 1)E N (ˆθ c). (35) Cosider the case that λ > 0. s j = λ 1 (E Y E N ) j + [terms idepedet of j]. }{{} >0 If s j < 0 for all j, the (33) implies that (p 1..., p 1 ) = (0,..., 0), a threshold rule. Otherwise let i be the smallest iteger such that s j 0. The (33) implies that (p 1..., p 1 ) is a i -threshold rule. It remais to cosider the case λ = 0. The (35) implies s j = y(1 y)(e Y E N ). That is, s j is idepedet of j ad s j > 0, implyig p j = 1 for all j, that is, the solutio etails the ay-voluteer rule. The remaiig questio is which threshold mechaism ad threshold equilibrium solves the plaer s problem. We have already see (Corollary 1) that the ay-voluteer rule is uiquely optimal at c = 0. We will show below that threshold rules with arbitrarily large i ca be optimal as the group size becomes large. 20

21 3.3 Voluteerig i a large populatio I this sectio, we characterize equilibrium voluteerig levels of threshold rules whe the populatio is large (Propositio 6) ad demostrate how the first best ca be approximated i a large populatio (Propositio 7). To simplify the otatio, we oly cosider pure threshold rules, that is, we assume that p i = 1, where i 1 is the threshold. Propositio 6 cosiders sequeces of equilibria that are idexed by the populatio size. We show that ay pure i -threshold rule with a sufficietly large threshold i has a sequece of equilibria alog which the expected umber of voluteers remais bouded away from 0 as the populatio becomes arbitrarily large; we derive a formula for the large-populatio limit of the expected umber of voluteers. Furthermore, we obtai a formula for the limit probability that the task is assiged to a voluteer, which i tur yields a formula for the limit expected-ability of the selected aget. Propositio 6 cosiders thresholds i so high that the iequality c/i < θ E[θ] is satisfied. 10 This iequality is crucial towards provig the existece of equilibria with voluteerig rates that stay bouded away from 0 as the populatio grows large. To uderstad why, assume a large populatio ad cosider a aget who believes that the margially voluteerig type amog the other agets is close to θ; the expected ability amog the other voluteers is the close to θ as well. Coditioal o the evet that the required umber of other voluteers i 1 is ot reached, the aget is essetially idifferet betwee voluteerig or ot because the populatio is large ad most likely she will ot be selected. Now cosider the evet that the required umber of other voluteers i 1 is reached. The, for a type close to θ, the beefit from voluteerig is approximately equal to the quality chage from the job beig doe at average quality (E[θ]) to the job beig doe at top quality (θ). O the other had, if the umber of other voluteers is exactly equal to i 1, the the cost of voluteerig is c/i because the aget because will be selected with probability 1/i. A cetral role is played by the Poisso distributio. For ay z > 0, let Pois(z)(i) = z i e z /i! deote the probability of the realizatio i = 0, 1,... accordig to the Poisso distributio with expectatio z. The correspodig hazard-rate fuctio, 11 h Pois(z) (i) = Pois(z)(i) j=i Pois(z)(j) = 1 i!, (36) j=i zj i j! will be used i the characterizatio of equilibrium voluteerig. 10 Bergstrom ad Leo (2015) obtai formulas similar to those i Propositio 6 i the case i = 1, i a settig without private iformatio. They defie the coordiated voluteer s dilemma as the game i which, similar to the ay-voluteer mechaism, the task is performed by a radomly selected voluteer if ad oly if at least oe voluteer comes forward; if obody voluteers, the the task is ot performed at all. The task has a commoly kow beefit b to each aget; thus, equilibria are i mixed strategies. Deotig by r the large-populatio-limit probability that at least oe idividual voluteers i equilibrium, formulas aalogous to those i Propositio 6 hold with i = 1 ad θ E[θ] replaced by b. 11 The deomiator i this defiitio ca also be writte by usig the upper icomplete gamma fuctio, which is give i terms of a itegral istead of a ifiite sum. The ifiite sum is the more useful represetatio for our aalysis. 21

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