Combinatorial Agency of Threshold Functions

Transcription

1 Combnatoral Agency of Threshold Functons Shal Jan Computer Scence Department Yale Unversty New Haven, CT Davd C. Parkes School of Engneerng and Appled Scences Harvard Unversty Cambrdge, MA 0238 Abstract In ths paper, we study the combnatoral agency problem ntroduced by Babaoff, Feldman and Nsan [7] and resolve some open questons posed n ther orgnal paper. Our results nclude a characterzaton of the transton behavor for the class of threshold functons. Ths result confrms a conjecture of [7], and generalzes ther results for the transton behavor for the OR technology and the AND technology. In addton to establshng a (tght) bound of 2 on the socal Prce of Unaccountablty (POU) for the OR technology for the general case of n>2 agents (the ntal paper establshed ths for n = 2, an extended verson establshes a bound of 2.5 for the general case), we establsh that the POU s unbounded for all other threshold functons (the ntal paper establshed ths only for the case of AND technology). We also obtan a characterzaton result for certan compostons of anonymous technologes and establsh an unbounded POU for these cases. Introducton The classc prncpal-agent model of mcroeconomcs consders an agent wth unobservable, costly actons, each wth a correspondng dstrbuton on outcomes, and a prncpal wth preferences over outcomes [, 9]. The prncpal cannot contract on the acton drectly (e.g. the amount of effort exerted), but only on the fnal outcome of the project. The man goal s to desgn contracts, wth a payment from the prncpal to the agent condtoned upon the outcome, n order to maxmze the payoff to the prncpal n equlbrum wth a ratonal, self-nterested agent. The prncpal-agent model s a classc problem of moral hazard, wth agents wth potentally msalgned ncentves and prvate actons. A related theory has consdered the problem of moral hazard on teams of agents [5, 8, 7]. Much of ths work nvolves a contnuous acton choce by the agent (e.g., effort) and a contnuous outcome functon, typcally lnear or concave n the effort of the agents. Moreover, rather than consderng the desgn of an optmal contract that maxmzes the welfare of a prncpal, consderng the loss to the prncpal due to transfers to agents, t s more typcal to desgn contracts that maxmze the total value from the outcome net the cost of effort, and wthout consderaton of the transfers other than requrng some form of budget balance. Babaoff et al. [7] ntroduce the combnatoral agency problem. Ths a very specfc form of the moral hazard on team problem n whch the agents have bnary actons and the outcome s bnary, but where the outcome technology s a complex combnaton of the nputs of a team of agents. Each agent s able to exert hgh or low effort n ts own hdden acton, wth the success or falure of an overall project dependng on the specfc technology functon. In partcular, these authors consder the AND technology, n whch all agents must exert effort n order for the global project

2 to have some possblty of success. Other technologes nclude: the OR technology, the majorty technology, and nested models such as AND-of-ORs and OR-of-ANDs. Ths can be conceptualzed as a problem of moral hazard to teams where agents are stuated on a graph, each controllng the effort at a partcular vertex. The combnatoral agency framework consders the socal welfare, n terms of the cost to agents and the value to the prncpal, that can be acheved n equlbrum under an optmal contract where the prncpal seeks a contract that maxmzes payoff,.e. value net of transfers to the agents, n equlbrum. Thus the focus s on contracts that would be selected by a prncpal, not be a desgner nterested n fndng an equlbrum that maxmzes socal welfare. In partcular, Babaoff et al. suggest to consder the (socal) Prce of Unaccountablty (POU), whch s the worst case rato between the optmal socal welfare when actons are observable as compared to when they are not observable. The worst-case s taken over dfferent probabltes of success for an ndvdual agent s actons (and thus dfferent, uncertan technology functons), and over the prncpal s value for a successful outcome. The optmal socal welfare s obtaned by requestng a partcular set of agents to exert effort, n order to maxmze the total expected value to the prncpal mnus the cost ncurred by these agents. In the agency case, the socal welfare s agan ths value net cost, but optmzed under the contract that maxmzes the expected payoff of the prncpal. The man contrbuton of ths work s to characterze the transton behavor for the k-out-of-n (or threshold) technology, for n agents and k {,...,n}. The threshold technology s anonymous, meanng that the probablty of a successful outcome only depends on the number of agents contracted to for hgh effort, not the specfc set of agents. Because of ths, the transton behavor a characterzaton of the optmal contract, whch specfes whch agents to contract wth, as a functon of the prncpal s valuaton can be explaned n terms of the number of agents wth whom the prncpal contracts. We establsh that the transton behavor (n both the nonstrategc and agency cases) ncludes a transton from contractng between 0 and l agents for some l n, followed by all n l remanng transtons, for any 0 <α<β<, where α (resp. β) s the probablty that the acton of a low effort (resp. hgh effort) acton by an agent results n a successful local outcome. Ths generalzes the pror result of Babaoff et al. [7] for the AND gate (a sngle transton from zero agents contracted to all agents contracted) and the OR gate (all n transtons), and closes an mportant open queston. Ths result reles on the fact that a sngle functon can exhbt ncreasng returns to scale, or IRS, followed by decreasng returns to scale, or DRS, whereas Babaoff et al. only consdered the possblty that a functon exhbts ether IRS or DRS. In addton, we use propertes of (log) convex functons to establsh ths result. Consderng the POU, we establsh a tght bound of 2 for the OR technology, for all values of n, α and β = α. The ntal paper establshed ths POU for the case of n = 2 agents only, whle an extended verson of the paper provdes a bound of n =2.5 for the general n>2 case [8]. In addton, we establsh that the POU s unbounded for the threshold technology for the general case of k 2,n 2, ncludng Majorty. The ntal paper establshed ths result only for AND technology, and so our result closes ths for the more general threshold case for any 0 <α<β<. More specfcally, we observe that as α 0, the POU becomes unbounded. In addton, we consder non-anonymous technology functons such as the Majorty-of-AND, Majorty-of-OR, and AND-of-Majorty technologes, and study ther transton behavor. Our result regardng the majorty technology, and a techncal lemma of Babaoff et al., gve the transton behavor for the AND-of-majorty technology. In partcular, when a majorty gate has ts frst transton to l agents, then the frst transton as the prncpal s value ncreases under AND-of- Majorty s to l agents on each Majorty gate, and then follows the subsequent transtons, wth an addtonal agent contracted wth, n an ncrement deployed smultaneously on all Majorty gates. 2

3 Our result for the Majorty-of-OR technology s a bt surprsng n that ts transton behavor s smlar to the case of the majorty technology where there s a sngle transton from 0 to l followed by all remanng transtons. Thus far, we have only been able to characterze the transton behavor for the non-strategc verson of Majorty-of-OR, but we conjecture a smlar transton behavor for the agency verson. Though we have been unable to characterze the transton behavor n the agency case, we show that the POU for the majorty-of-or technology s unbounded. We also consder the majorty-of-ands technology functon ntroduced by [7] and prove the transton behavor n the non-strategc case consstent wth a conjecture of Babaoff et al. [7] for the OR-of- ANDs. We are unable to prove the transton behavor for the agency case, but show that the POU s unbounded for the majorty-of-ands. We beleve that ths work s an nterestng step n extendng the combnatoral agency model n a drecton of nterest for crowd sourcng [23, 3,, 2]. In partcular, t s relevant n applcatons where nether the effort nor the ndvdual outcome of each worker s observable. All that s observable s the ultmate success or falure. One reason for ths s that the boundares between ndvdual contrbutons are hard to defne, or that the workers themselves preferred to anonymze or hde ndvdual contrbutons n some way (e.g., to protect ther prvacy.) Another motvaton s that t could be extremely costly, or even mpossble, to determne the qualty of the work performed by an ndvdual worker when studed n solaton. One can know whether or not the overall project was a success or falure (lots of ste traffc, or no ste traffc, an overall artfact that passes requred tests, or an artfact that crashes, etc.) but not know whether or not a counterfactual project outcome, where the work of any one worker was changed, would be dfferent. For software engneerng, the work of others to ntegrate ndvdual components has already been done. For a web ste, the opportunty to launch the ste has already passed. A threshold technology models a doman n whch a project only succeeds when enough agents provde hgh effort (e.g., Wkpeda or the development of open-source software.) For Majortyof-OR, consder domans such as TopCoder [4], where mn compettons (e.g. OR gates) are used for each module and then ultmate success occurs f enough ndvdual modules are judged to be successful. Many Games wth a Purpose [23, 22, 24] can be modeled wth a Majorty-of-AND technology, snce n an ndvdual game, both agents nvolved must succeed at ther task, however ths task s gven to more than one set of agents, and we need a majorty of these games (or AND gates) to succeed n order to verfy the qualty of the output.. Related Work A characterzaton of the transton behavor and the POU was frst conjectured for Majorty technology n Babaoff et al. [7], but almost all of the subsequent lterature s restrcted to readonce networks [9, 0, 5, 6]. A number of varatons of the basc combnatoral agency model have been studed. Consderng contracts that nduce mxed Nash equlbra, ths can sometmes mprove the POU over nsstng on a pure strategy NE, developng a number of upper and lower bound results on the relatve gan from mxed strateges, and dentfyng a suffcent condton under whch mxed strateges provde no advantage to the prncpal. These authors conjecture that for any technology functon, the relatve gan to the prncpal for nducng a mxed strategy Nash equlbrum s bounded above by a constant. Another varaton consders the cost of free labor, namely, f there are stuatons where the prncpal can beneft from havng certan agents reduce ther effort level, even when ths effort s free [0]. The prncpal s hurt by free labor under the OR technology, because free labor can lead to free rdng, whle for the AND technology (and any technology wth ncreasng 3

4 returns to scale), the prncpal s not hurt by free labor. A thrd varaton allows the prncpal to audt some fracton of the agents, and dscover ther ndvdual prvate acton [4]. Results provde the transton behavor for AND technology and also gve some consderaton to Majorty and OR technologes. Some computatonal complexty results for dentfyng optmal contracts have also been developed. Ths problem s NP-hard for OR technology [5], and the dffculty s later shown to be a property of unobservable actons [6]. Ths s n contrast to the AND technology, whch s shown to admt a polynomal tme algorthm for computng the optmal contract [5]. An FPTAS s developed for OR technology, and extended to almost all seres-parallel technologes [5]. A related topc n economc theory s that of contest desgn [20, 2, 6, 2]. Contests are stuatons n whch multple agents exert effort n order to wn a prze. All agents bear the cost of the effort exerted regardless of whether they wn a prze. Unlke the moral hazard on teams problem, or the combnatoral agency problem, the ndvdual outcome from each worker s observable. One agent s unable to hde behnd the success or falure of the overall project, snce the result from ts own effort are judged n solaton. Moreover, the contest desgn frameworks do not, to the best of our knowledge, consder combnatons of nputs from workers. Rather, the outcome depends on the maxmum qualty outcome generated ndvdually, by each agent. DPalatno and Vojnovc [3] analyze a varaton wth multple, smultaneous tasks and workers selectng the sngle task n whch they wll partcpate. 2 Model In the combnatoral agency model, a prncpal employs a set of n self-nterested agents. Each agent has an acton space A and a cost (of effort) assocated wth each acton c (a ) 0 for every a A.Weleta =(a,...,a,a +,...,a n ) denote the acton profle of all other agents besdes agent. Smlar to Babaoff et al. [7], we focus on a bnary-acton model. That s, agents ether exert effort (a = ) or do not exert effort (a = 0), and the cost functon becomes c f a = and 0 f a = 0. If agent exerts effort, she succeeds wth probablty β. If agent does not exert effort, she succeeds wth probablty α,where0<α <β <. We deal wth the case of homogenous agents (e.g. β = β, α = α and c = c for all ), though some of the pror work deals wth the case of heterogenous agents. Sometmes we use the addtonal assumpton of [7], that β = α, where0<α< 2. Completng the descrpton of the technology s the outcome functon f, whchdetermnesthe success or falure of the overall project as a functon of the success or falure of each agent. Let x =(x,...,x n ), wth x {0, } to denote the success or falure of the acton of agent gven ts selected effort level. Followng Babaoff et al. [7] we focus on a bnary outcome settng, so that the outcome s (= success) or 0 (= falure.) Gven ths, we study the followng outcome functons:. AND technology: f(x,x 2,...,x n )= N x. In other words, the project succeeds f and only f all agents succeed n ther tasks. 2. OR technology: f(x,x 2,...,x n )= N x. In other words, the project succeeds f and only f at least one agent succeeds n her task. 3. Majorty technology: f(x) = f a majorty of the x are. In other words, the project succeeds f and only f a majorty of the agents succeed at ther tasks. Note that ths s dfferent from the AND and OR technologes functon snce ths s not a read-once network. 4

5 4. Threshold technology: We can generalze the majorty technology nto a threshold technology, where f(x) = f and only f at least k of the x are, e.g. at least k of the n agents succeed n ther tasks. In fact, the threshold technology s a generalzaton of the OR, AND and majorty technologes, snce the k = case s equvalent to the OR technology, the k = n case s equvalent to the AND technology, and the k = n 2 case s equvalent to the majorty technology. It should be noted that the set of threshold technologes s exactly the set of threshold functons. It s easy to see that each of these outcome functons s anonymous, meanng that the outcome s nvarant to a permutaton on the agent denttes. Gven outcome functon f, and success probabltes α and β, then acton profle a nduces a probablty p(a ) [0, ] wth whch the project wll succeed. Ths s just p(a )=E x [f(x ) x a ] () where the local outcomes x are dstrbuted accordng to α, β and as a result of the effort a by agents. Snce p consders the combned effect of technology f, α and β, thenwerefertop as the technology functon. The prncpal has a value v for a successful outcome and 0 for an unsuccessful outcome. Lke [7], we assume that the prncpal s rsk-neutral and seeks to maxmze expected value mnus expected payments to agents. The prncpal s unable to observe ether the actons a or the (local) outcomes x. The only thng the prncpal can observe s the success or falure of the overall project. Based on ths, a contract specfes a payment t 0 to each agent when the project succeeds, wth a payment of zero otherwse. The prncpal can pay the agents, but not fne them. It s convenent to nclude n a contract the set of agents that the prncpal ntends to exert hgh effort; ths s the set of agents that wll exert hgh effort when the prncpal selects an approprate payment functon. The utlty to agent under acton profle a s u (a )=t p(a ) c f the agent exerts effort, and u (a )=t p(a ) otherwse. The prncpal s expected utlty s u(a )=v p(a ) N t p(a ). The prncpal s task s to desgn a contract so that ts utlty s maxmzed under an acton profle a that s a Nash equlbrum. We make the same assumpton as Babaoff et al. [7], that f there are multple Nash equlbra (NE), the prncpal can contract for the best NE. 2 The socal welfare for an acton profle a s gven by u(a )+ N u (a )=v p(a ) N c a, wth payments from the prncpal to the agents cancelng out. Throughout, we focus on outcome functons that are monotonc, so that f(x )= f(x,x ) = for x x. Based on ths, then the technology functon p s also monotonc n the amount of effort exerted, that s for all and all a {0, } n, p(,a ) p(0,a ). Smlarly, a technology functon p s anonymous f t symmetrc wth respect to the players. That s, t s anonymous f t only depends on the number of agents that exert effort and s ndfferent to permutatons of the jont acton profle a. Ths s true whenever the underlyng outcome functon s anonymous. In the non-strategc varant of the problem, the prncpal can choose whch agents exert effort and these agents need not be motvated, the prncpal can smply bear ther cost of exertng effort. Let S a and S ns denote the optmal set of agents to contract wth n the agency case and the A read-once network s a network that can be represented by a graph wth a labeled source and snk, where there s a unque player correspondng to each edge. The project succeeds f and only f there exsts a path, consstng of successful players, between the source and the snk [7]. Much of the prevous work on the combnatoral agency problem apples to read-once networks and thus the understandng of the majorty technology seems less well understood than the AND, OR, AND-of-ORs and OR-of-ANDs technologes. 2 Ths s reasonable, snce the prncpal can announce whch set of agents should exert effort and also desgn the payment to provde strct ncentve to exert effort for those contracted. 5

6 non-strategc case respectvely. That s, these sets of agents are those that maxmze the expected value to the prncpal net cost, frst where the sets must be nduced n a Nash equlbrum and second when they can be smply selected. Defnton 2.. [7] The Prce of Unaccountablty (POU) for an outcome functon f s defned as the worst case rato (over v, α and β) of the socal welfare n the non-strategc case and the socal welfare of the agency case: POU (f) = sup v>0,α,β p(sns(v)) v Sns(v) c p(sa(v)) v Sa(v) c, (2) where p s the technology functon nduced by f, α and β, wth0<α<β<. In studyng the POU, t becomes useful to characterze the transton behavor for a technology. The transton behavor s, for a fxed technology functon p, the optmal set of contracted agents as a functon of the prncpal s value v. We know that when v = 0 t s optmal to contract wth 0 agents and lkewse, as v, t s optmal to contract wth all agents. However, we would lke to understand what are the optmal sets of agents contracted between these two extreme cases. There are, n fact, two sets of transtons, for both the agency and the non-strategc case. For anonymous technologes, there can be at most n transtons n ether case, snce the number of agents n the optmal contract s (weakly) monotoncally ncreasng n the prncpal s value. We seek to understand how many transtons occur, and the nature of each jump (.e. the change n number of agents contracted wth at a transton.) We also consder compostons of these technologes such as majorty-of-and, Majorty-of- OR, and AND-of-Majorty. These technologes are no longer anonymous. For example, n the AND-of-Majorty case, one can magne that the probablty of success wll be dfferent when agents are contracted on the same majorty functon and when they are contracted on dfferent majorty functons. Wth non-anonymous technologes, one needs to specfy the contracted set of agents, n addton to the number of agents contracted. In consderng composton of anonymous technologes, we assume we are composng dentcal technology functons, e.g. each AND gate n the majorty-of-and technology conssts of the same number of agents. 3 Transton Behavor of the Optmal Contract Below we wll characterze the transton behavor of the threshold technology, whch gves us the transton behavor for the majorty technology. We show that there exsts an l {,...,n} such that the frst transton s from 0 to l agents followed by all remanng transtons. Ths result holds for any value of α, β such that 0 <α<β<. Our proof bulds on the framework of Babaoff et al. [7]. In Babaoff et al., t was shown that the AND technology always exhbts ncreasng returns to scale (IRS) and the OR technology always exhbts decreasng returns to scale (DRS). It was also shown that any anonymous technology that exhbts IRS has a sngle transton from 0 to n agents for the optmal contract n the nonstrategc case and that any anonymous technology that exhbts DRS exhbts all n transtons n the non-strategc case. Smlar to the non-strategc case, t was shown n Babaoff et al. that the AND technology always exhbts overpayment (OP), n the agency case, where the OP condton guarantees a sngle transton from 0 to n, and the OR technology always exhbts ncreasng relatve margnal payment (IRMP), n the agency case, where the IRMP condton guarantees all n transtons. 6

7 We show that the threshold technology exhbts IRS up to a certan number of agents contracted and DRS thereafter, whch gves the transton characterzaton for the non-strategc case. Lkewse, we show that the threshold functon exhbts OP to a pont and IRMP n the agency case, whch s suffcent to gve the transton characterzaton for the agency case. Our analyss s new, n the sense that we consder the possblty that a sngle technology can exhbt IRS up to a certan number of agents contracted, followed by DRS and lkewse, that t can exhbt OP up to a certan number of agents contracted, followed by IRMP. Babaoff et al. only consdered the possblty a functon exhbts ether IRS or DRS, and lkewse, ether OP or IRMP. In addton to ths nsght, we use propertes of (log) convex functons to establsh ths result. We state our man theorems that gve a complete characterzaton of the transton behavor of the majorty technology below: Theorem 3.2 For any threshold technology (any k, n, c, α and β) n the non-strategc case, there exsts an l ns n where, such that the frst transton s from 0 to l ns agents, followed by all remanng n l ns transtons. Theorem 3.7 For any threshold technology (any k, n, c, α and β) n the agency case, there exsts an l a l ns such that the frst transton s from 0 to l a agents, followed by all remanng n l a transtons. The followng observatons gve us the optmal payment rule for any technology and establsh a monotonc property for the optmal contract as a functon of v. Defnton 3.. [7] The margnal contrbuton of agent for a gven a s denoted by (a )= p(,a ) p(0,a ), and s the dfference n the probablty of success of the technology functon when agent exerts effort and when she does not. For anonymous technologes, f exactly j entres n a are, then = p j+ p j,wherep j s the probablty of success when exactly j agents exert effort. Snce p s strctly monotone, we have > 0 for all. Remark 3.2. [7] The best contracts (from the prncpal s pont of vew) that nduce the acton profle a {0, } n as a Nash equlbrum are t =0when the project s unsuccessful and t = c (a ) when the project succeeds and the prncpal requests effort a =from agent. The followng remark of Babaoff et al. [7] establshes that the optmal contract for an anonymous technology functon s (weakly) monotoncally ncreasng wth the prncpal s value and s used n establshng Lemma 3.6. Remark 3.3. [7] For any anonymous technology functon p, f contractng wth k agents s optmal for v, and contractng wth k 2 agents s optmal for v 2,andv >v 2, then k k 2. The followng two lemmas are from [7] and are used n the proof of Lemma 3.6, whch gves a suffcent condton for a technology functon to have a frst transton to l, followed by all remanng transtons. Let Q be the total expected payment when contractng wth agents, or n other words, p c p p Q = = p t. In the non-strategc case, let Q be the total sum of costs of the number of agents contracted, or n other words, Q = c. Note that the followng lemmas hold n both the non-strategc case and the agency case. Fnally, let v,j denote the specfc prncpal s value at whch he s ndfferent between contractng wth agents or j agents n the agency case. For the non-strategc case, v,j s the prncpal s value at whch he s ndfferent between agents exertng effort and j agents exertng effort. More formally, the pont v,j at whch the prncpal s ndfferent between contractng between agents and j agents can be expressed as p v,j Q = p j v,j Q j. Solvng for v,j, we get that v,j = Q j Q p j p. In what follows, we manly consder the value of v 0, for 7

8 all and the value of v,+ for all. Wedefneu(, v) as the utlty to the prncpal of contractng wth agents when hs value s v. In other words, u(, v) =p v Q,whereQ s the total expected payment, ether n the non-strategc case or the strategc case. Lemma 3.4. [7] u(l, v 0,l ) >u(, v 0,l ) for all = l f and only f Q Q l > p p 0 for all = l. Lemma 3.5. >l. [7] u(, v, ) >u( +,v,+ ) for all >lf and only f Q + Q t + t > Q Q t t for all The followng lemma gves a set of suffcent condtons for an anonymous technology to have a frst transton from 0 to l, for some l {,...,n}, followed by all remanng n l transtons. Ths lemma holds for both the non-strategc case (by settng Q = c) and the agency case (by settng Q = c ). We vew ths lemma as a generalzaton of Theorem 9 from [7] and t follows a smlar proof structure. Ths lemma states that as long as a technology functon exhbts OP up to a certan number of agents contracted followed by IRMP, then the transton behavor nvolves a frst transton from 0 to l, for some l {,...,n}, followed by all remanng n l transtons. Lemma 3.6. Any anonymous technology functon that satsfes:. Q Q l > p p 0 2. Q l+ Q l p l+ p l > Q l 3. Q + Q p + p for all = l > Q Q p p for all >l for some l {,...,n} has a frst transton from 0 to l and then all n l subsequent transtons, where Q s defned approprate for the non-strategc case or the agency case. Now that we have establshed a set of suffcent condtons for an anonymous technology to exhbt a frst transton from 0 to l, followed by all remanng transtons (for ether the nonstrategc case or the agency case), we nterpret what these condtons are for the non-strategc case. Lemma 3.7. Any anonymous technology that has a probablty of success functon that satsfes:. p p 0 > p p 0 for all 2 l and p p 0 < p p 0 2. p + p > p p for all >l for all >l for some l {,...,n} has a frst transton from 0 to l and then all n l subsequent transtons for the nonstrategc verson of the problem. In establshng that the threshold technology satsfes the condtons outlned n Lemma 3.7, t becomes useful to defne a property of the probablty of success functon. Defnton 3.8. We say that a probablty of success p for a partcular technology s unmodal f t satsfes one of three alternatves:. p p >p p 2 for all 2 j and p p <p p 2 for all >j 2. p p >p p 2 for all 2 n 3. p p <p p 2 for all 2 n 8

9 Let f() = p p 0. Ths functon s useful to consder, because n order to establsh the frst condton of Lemma 3.7, we need to show that f() s unmodal. Lemma 3.9. If the probablty of success functon s unmodal over the set {,...,n}, then we know that f() s also unmodal. Corollary 3.0. For any anonymous technology functon (p, c) that has a unmodal probablty of success, there exsts an l n such that the frst transton n the non-strategc case s from 0 to l agents (where l s the smallest value that satsfes p l p 0 l > p l+ p 0 l+ ) followed by all remanng n l transtons. Therefore, t suffces to show that p s unmodal n order to establsh that the technology (p, c) exhbts a frst transton from 0 to l, for some l {,...,n}, followed by all remanng n l transtons, n the non-strategc case. Lemma 3.. The probablty of success functon for any threshold technology s unmodal. The characterzaton of the transton behavor of the threshold technology n the non-strategc case follows from Lemmas 3.7, 3.9, and 3.. Theorem 3.2. For any threshold technology (any k, n, c, α and β) n the non-strategc case, there exsts an l ns n where, such that the frst transton s from 0 to l ns agents, followed by all remanng n l ns transtons. Now that we have characterzed the transton behavor of the threshold technology, for any k, n the non-strategc case, we focus on establshng the condtons of Lemma 3.6, for the agency case. The followng lemma s used to show that the frst condton n Lemma 3.6 s satsfed by the threshold technology. Lemma 3.3. The dscrete valued functon, Q, s convex. Lemma 3.4. There exsts a value of l a n such that Q Q la > p p 0 p la p 0 for all = l a. Snce there exsts an l a such that Q l a <n, we have the followng corollary. > Q + p + p 0 for all <l a and Q < Q + p + p 0 Corollary 3.5. We have Q la+ Q la p la+ p la > Q la p la p 0, where l a n satsfes Q Q la > p p 0 p la p 0 = l a. Lemma 3.6. We have Q + Q Q la p la p 0 < Q la+ p la+. p + p > Q Q p p for all for all for all >l a where l a s the smallest value such that Lemmas 3.6, 3.4, 3.6 and 3.8 and Corollary 3.5 establsh the followng result. Theorem 3.7. For any threshold technology (any k, n, c, α and β) n the agency case, there exsts an l a l ns such that the frst transton s from 0 to l a agents, followed by all remanng n l a transtons. Fnally we show that the frst transton n the agency s at most the value of the frst transton n the non-strategc. Lemma 3.8. For any threshold technology, we get l a l ns. Below we gve the trend n transton behavor as a functon of β, whenα = 0. Remark 3.9. For any threshold technology wth fxed k 2, n, c and α =0, we have that l = k for β close enough to and l = n for β close enough to 0. 9

10 4 Prce of Unaccountablty In ths secton, we provde results regardng the Prce of Unaccountablty for OR and threshold technologes. Lemma 4.. [7] For any technology functon, the prce of unaccountablty s obtaned at some value v whch s a transton pont, of ether the agency or the non-strategc cases. We are able to mprove slghtly upon ths result, for the OR technology, whch s needed to establsh Theorem 4.5. We suspect that ths result can be mproved further, n that the POU occurs at the frst transton n the agency case. Lemma 4.2. For the OR technology, the prce of unaccountablty occurs at a transton n the agency case, as opposed to a transton n the non-strategc case. The followng theorem s a result of Babaoff et al. [7], where they derve the prce of unaccountablty for AND technology where β = α. (In fact, they gve the prce of unaccountablty for any anonymous technology wth a sngle transton n both the agency and non-strategc cases.) It s easy to see from the closed form expresson of the POU that POU as α 0. Theorem 4.3. [7] For the AND technology wth α = β, the prce of unaccountablty occurs at the transton pont of the agency case and s POU =( α )n +( α α ). Remark 4.4. [7] The prce of unaccountablty for the AND technology s not bounded. More specfcally, POU as α 0 and POU as β 0. In ther orgnal paper, Babaoff et al. [7] show that the Prce of Unaccountablty for the OR technology s bounded by 2 for exactly 2 agents and gve an upper bound of 2.5 for the general case [8], when β = α. We extend these results for the β = α case and show that the Prce of Unaccountablty s bounded above by 2 for any OR technology (.e. for all n). Ths result s tght, namely, as α 0, POU 2. We suspect that these results hold for the more general 0 <α<β< case, but we have been unable to prove t for all values of α, β. Theorem 4.5. The POU for the OR technology s bounded by 2 for all α, β = α and n. The followng remark follows from the proof of Theorem 4.5. Remark 4.6. For any n, asα 0, POU 2 for the OR technology. In contrast to the OR technology, we show that the POU for the threshold technology wth k 2 s unbounded. Ths result holds for any 0 <α<β<. Theorem 4.7. The Prce of Unaccountablty for the threshold technology s not bounded for all values of k 2 and n. More specfcally, when α 0, POU. Lemma 4.8. As α 0, we know that k l a l ns, where l a s the frst transton n the agency case and l ns s the frst transton n the non-strategc case. It should be noted that there s nterestng structure to the socal welfare rato as a functon of the prncpal s value v. For a fxed number of agents contracted n the agency case, the socal welfare rato s ncreasng. However, at a transton n the agency case, the socal welfare rato drops sgnfcantly such that the maxmum rato for each successve agency contract never reaches the maxmum rato for the prevous agency contract. Provng ths behavor could be useful n studyng the Prce of Unaccountablty for restrcted of v and α. 0

11 5 Composton of Anonymous Technologes In ths secton, we study the composton of varous technology functons. For the majorty-of-and and majorty-of-or technology, we are unable to provde the characterzaton of transton behavor for the agency case, but we provde the characterzaton of transton behavor n the non-strategc case and we provde a result regardng the Prce of Unaccountablty. 5. Majorty-of-ANDs We prove the transton behavor for the majorty-of-and technology n the non-strategc case. These results for the more general threshold-of-ors case. For the followng assume that n the majorty-of-and technology, the majorty gate contans q AND gates, each wth m agents. Ths bulds on a conjecture of Babaoff et al. who conjecture the followng behavor for both the nonstrategc and agency cases. We are unable to prove the transton behavor for the agency case. Lemma 5.. If the prncpal decdes to contract wth j m + a agents for some j Z + and some 0 a<m, the probablty of success s maxmzed by fully contractng j AND gates and contractng wth a remanng agents on the same AND gate. Lemma 5.2. For any prncpal s value v, the optmal contract nvolves a set of fully contracted AND gate. Theorem 5.3. The transton behavor for the majorty-of-and technology n the non-strategc case has a frst transton to l fully contracted AND gates, where l n, followed by each subsequent transton of fully contracted AND gates. Whle we are unable to characterze the transton behavor for the majorty-of-and technology n the agency case, we know that the frst transton n the agency case must nvolve contractng at most l m agents (proof smlar to that of Lemma 4.8). Ths allows us to prove that the Prce of Unaccountablty s unbounded. The proof of Theorem 5.4 s omtted but has a vrtually dentcal proof as Theorem 4.7. Theorem 5.4. The Prce of Unaccountablty s unbounded for the majorty-of-and technology. 5.2 Majorty of ORs We wll characterze the transton behavor for the non-strategc case of the majorty of ORs below. In what follows, we assume that each OR gate has j agents and there are m of them comprsng a majorty functon (.e. n = j m). We also assume that k = m 2.3 Snce the followng lemma s a statement regardng the probablty of success, t holds for both the non-strategc and agency cases, because the probablty of success s the same n both. In consderng the majorty-of-or case, we further assume that β = α and 0 <α< 2. Lemma 5.5. Consder an nteger such that = a m + b, where 0 b<m. Fxng, the probablty of success for a majorty-of-ors functon s maxmzed when a + agents are contracted on each of b OR gates and a agents are contracted on each of n b OR gates. The followng lemma gves the complete transton behavor n the majorty-of-or technology n the nonstrategc case. 3 It should be noted that these results do not hold for the more general threshold-of-ors case. In fact we can construct a settng where ths transton behavor wll not occur.

12 Lemma 5.6. The frst transton for the non-strategc case of the majorty-of-or technology jumps from contractng wth 0 agents to l agents, where l k, followed by all remanng transtons, where the transtons proceed n such a way so that no OR gate has more than more agent contracted as compared to any other OR gate. We conjecture that a smlar transton behavor holds n the agency case, but we have thus far been unable to prove t. Although we have been unable to characterze the transton behavor n the agency case, we do know that as α 0, the frst transton jumps to k. Whle we omt the proof of ths lemma, t s very smlar to Lemma 4.8. Ths s enough to determne that the POU s unbounded. Lemma 5.7. In the agency case of the majorty-of-or technology, as α 0, thefrsttranston occurs to a value k. The followng theorem has a smlar proof to Theorem 4.7. Theorem 5.8. The Prce of Unaccountablty s unbounded for the majorty-of-or technology. 5.3 AND of Majorty In what follows, we wll also characterze the transton behavor of AND-of-majortes. Smlar to the prevous case, these results hold for the more general AND-of-threshold s. We gve a result from [7] that allows for the characterzaton of the transton behavor of AND-of-majorty. Let g and h be two Boolean functons on dsjont nputs wth any cost vectors, and let f = g h. An optmal contract S for f for some v s composed of some agents from the g-part (denoted by the set R) and some agents from the h-part (denoted by the set T ). Lemma 5.9. [7] Let S be an optmal contract for f = g h on v. Then, T s an optmal contract for h on v t g (R), andr s an optmal contract for g on v t h (T ). The prevous lemma gves us a characterzaton of the transton behavor n the AND-ofmajortes technology. The statement of ths result s analogous to the result gven n [7] for the AND-of-ORs technology. Snce the prevous lemma holds for both the non-strategc and agency varatons of the problem, the followng theorem holds for both the non-strategc and agency varatons of the problem. Theorem 5.0. Let h be an anonymous majorty technology and let f = n c j= be the AND of majorty technology that s obtaned by a conjuncton of n c of these majorty technology functons on dsjont nputs. Then for any value v, an optmal contract contracts wth the same number of agents n each majorty component. Theorem 5.0 gves us a complete characterzaton of the transton behavor n the AND-ofmajortes technology for both the non-strategc and the agency cases. Snce we know that the frst transton n both the agency and non-strategc cases for the AND-of-majorty technology occurs to a value greater than, we have the followng result. The proof structure s smlar to that of Theorem 4.7. Theorem 5.. The Prce of Unaccountablty s unbounded for the AND-of-majorty technology. 2

13 6 Conclusons In ths work, we advance the understandng of the combnatoral agency model. We prove the transton behavor for the threshold technology for general α, β. We study the majorty technology, the majorty-of-or technology, and the AND-of-majorty technology and observe the connecton between these technologes and crowdsourcng systems. Babaoff et al. [7] showed that the POU was not bounded for the AND technology, for any n. We strengthen ths result, and prove that the POU s not bounded for the threshold technology for all k 2, any n 2, and any 0 <α<β<. More specfcally, the POU for the threshold technology (wth k 2) approaches as α 0. Babaoff et al. [7] showed that the POU was bounded by 2 for the OR technology wth 2 agents and bounded by 2.5 n the general case [8]. We show that the POU s bounded by 2 for the OR technology for all values of α, β = α and n and ths bound s tght. Whle we do not study the entre class of anonymous functons, we do study a natural class n the k-out-of-n (or threshold) technology. The entre class of anonymous functons s easy to characterze usng the set of exact-value functons [25]. The exact value functon E k s f and only f exactly k agents succeed and 0 otherwse. The set of exact-value functons form a bass for the class of anonymous functons, or n other words, any anonymous functon f can be wrtten as follows: f(x) = 0 k n E k(x) v k,where(v 0,...,v n ) {0, } n+ and x s the success vector of the agents [25]. We leave studyng the entre class of anonymous functons as a drecton for future work. In fact, we suspect that the threshold technology has a more well-behaved transton behavor than other anonymous functons. Ths would mply that the threshold functon s a desrable technology for crowdsourcng work and s the most sgnfcant open drecton. References [] [2] [3] [4] [5] A. A. Alchan and H. Demsetz. Producton, nformaton costs, and economc organzaton. The Amercan Economc Revew, 62(5): , December 972. [6] N. Archak and A. Sundararajan. Optmal desgn of crowdsourcng contests. In ICIS, [7] M. Babaoff, M. Feldman, and N. Nsan. Combnatoral agency. In ACM Conference on Electronc Commerce, pages 8 28, [8] M. Babaoff, M. Feldman, and N. Nsan. Combnatoral agency. Full Verson, [9] M. Babaoff, M. Feldman, and N. Nsan. Mxed strateges n combnatoral agency. In WINE, pages , [0] M. Babaoff, M. Feldman, and N. Nsan. Free-rdng and free-labor n combnatoral agency. In SAGT, pages 09 2, [] P. Bolton and M. Dewatrpont. Contract Theory. MIT Press,

14 [2] S. Chawla, J. Hartlne, and B. Svan. Optmal crowdsourcng contests. In Workshop on Socal Computng and User Generated Contest, 20. [3] D. DPalatno and M. Vojnovc. Crowdsourcng and all-pay auctons. In EC, [4] R. Edenbenz and S. Schmd. Combnatoral agency wth audts. In IEEE Internatonal Conference on Game Theory for Networks (GameNets), [5] Y. Emek and M. Feldman. Computng optmal contracts n seres-parallel heterogeneous combnatoral agences. In WINE, pages , [6] Y. Emek and I. Hatner. Combnatoral agency: The observable acton model. Manuscrpt, [7] B. E. Hermaln. Toward an economc theory of leadershp: Leadng by example. The Amercan Economc Revew, 88(5):88 206, 998. [8] B. Holmstrom. Moral hazard n teams. The Bell Journal of Economcs, 3(2): , 982. [9] J.-J. Laffont and D. Martmort. The Theory of Incentves: The Prncpal-Agent Model. Prnceton Unversty Press, 200. [20] B. Moldovanu and A. Sela. The optmal allocaton of przes n contests. Amercan Economc Revew, 9: , June 200. [2] B. Moldovanu and A. Sela. Contest archtecture. Journal of Economc Theory, 26():70 96, January [22] L. von Ahn. Games wth a purpose. IEEE Computer, 39(6):92 94, [23] L. von Ahn and L. Dabbsh. Labelng mages wth a computer game. In Proceedngs of the 2004 Conference on Human Factors n Computng Systems (CHI), pages , [24] L. von Ahn and L. Dabbsh. Desgnng games wth a purpose. Communcatons of the ACM, 5(8):58 67, [25] I. Wegener. The Complexty of Boolean Functons. New York: Wley, 987. A Proofs from Secton 3 Lemma 3.6 Any anonymous technology functon that satsfes:. 2. Q Q l > p p 0 Q l+ Q l p l+ p l > Q l for all = l 3. Q + Q p + p > Q Q p p for all >l for some l {,...,n} has a frst transton from 0 to l and then all n l subsequent transtons, where Q s defned approprate for the non-strategc case or the agency case. 4

15 Proof. From Lemma 3.4, we know that f Q Q l > p p 0 for all = l, thenu(l, v 0,l ) >u(, v 0,l ) for all = l. By Remark 3.3, snce l s the optmal contract at v 0,l, for any v>v 0,l, t must be the case that the optmal contract nvolves contractng wth at least l agents. Lkewse, snce 0 s optmal at v 0,l, by Remark 3.3, f were optmal for any v<v 0,l, then 0 could not be optmal at v 0,l. Therefore we know that for all v<v 0,l, contractng wth 0 agents s the only optmal contract. Snce at v 0,l, the only optmal contracts are 0 and l, there s no value of v for whch t s optmal to contract wth, 2,...,l agents. Thus the technology functon exhbts a jump between contractng between 0 agents and contractng wth l agents. From Lemma 3.5, we know that f Q + Q p + p,v,+ ) for all >l. Also, we know that the statement Q + Q p + p > Q Q p p for all >l,thenu(, v, ) >u( + > Q Q p p for all >l,s equvalent to v,+ >v, for all >l. Snce Q l+ Q l p l+ p l > Q l, we also know that v l,l+ >v 0,l. In what follows, we show that for any v (v,,v,+ ) for some >l, contractng wth exactly agents s the only optmal contract. If we combne ths wth the fact that 0 s optmal for all v v 0,l and the fact that n s optmal for all v v n,n we get that the frst transton occurs from 0tol and all remanng n l transtons occur. Now consder any value v (v 0,l,v l,l+ ), we know that contractng wth l agents yelds hgher utlty to the prncpal than contractng wth j<lagents from above. Lkewse, consder any value v (v l,l+,v l+,l+2 ), we know know by the defnton of v l,l+, that contractng wth l + agents s strctly better than contractng wth l agents for all v>v l,l+ and we know that contractng wth l + agents s strctly better than contractng wth j agents, where j<l, snce contractng wth l s strctly better than contractng wth j<lagents for all v>v 0,l. Now we wll proceed nductvely (much lke the proof of Theorem 9 n [7]), as follows: consder any value v (v,+,v +,+2 ) for any >l, we know that contractng wth + s strctly better than contractng wth for all v>v,+. We know that contractng wth + agents s strctly better than contractng wth j agents, where j <, because the nducton hypothess gves us that contractng wth agents s strctly better than contractng wth j<agents. Now we nduct backwards as n [7]. Consder the v>v n,n, we know that contractng wth n agents has strctly greater utlty than contractng wth j>nagents (trvally true). Now consder v (v,,v,+ ) for all l<<n, contractng wth agents s strctly better than contractng wth + agents by the defnton of v,+ and by the nducton hypothess, we know that contractng wth agents s strctly better than contractng wth j>+ agents. Now consder v (v 0,l,v l,l+ ), we know that contractng wth l agents s strctly better than contractng wth l + agents and all j > l + agents, by the nducton hypothess. Fnally consder v (0,v 0,l ), we know that contractng wth 0 agents s strctly better than contractng wth j {, 2,...,l} agents from above. The nducton hypothess gves us that contractng wth 0 agents s strctly better than contractng wth j>lagents. Combnng the two nductve arguments gves us that contractng wth 0 agents s optmal for v (0,v 0,l ), contractng wth l agents s optmal for v (v 0,l,v l,l+ ) and contractng wth + agents s optmal for v (v,+,v +,+2 ) for all l. Lemma 3.7 Any anonymous technology that has a probablty of success functon that satsfes:. > p p 0 for all 2 l and p p 0 < p p 0 2. p + p > p p for all >l for all >l for some l {,...,n} has a frst transton from 0 to l and then all n l subsequent transtons for the nonstrategc verson of the problem. 5

16 Proof. We show that the condtons of Lemma 3.6 are satsfed. Snce Q = c for the nonstrategc case, the condton that Q Q l > p p 0 for all = l s equvalent to l > p p 0 for all = l or p l p 0 l >, for all = l. The latter s clearly satsfed by condton of ths Lemma. The condton Q + Q p + p > Q Q p p for all >ls equvalent to p + p > p p for all >l, whch s equvalent to condton 3 of ths Lemma. The condton Q l+ Q l p l+ p l > Q l s equvalent to p l+ p l > l.snce condton of ths Lemma gves us p l p 0 l > p l+ p 0 l+,weknow p l p 0 l >p l+ p l, whch gves us the desred result. Lemma 3.9 If the probablty of success functon s unmodal over the set {,...,n}, thenwe know that f() s also unmodal. Proof. If p p > p p 2 for all 2 n, then p p 0 > p p 0 as well. Lkewse, f p p < p p 2 for all 2 n, then p p 0 for all 2 n < p p 0 for all 2 n as well. Fnally, consder the case that p p >p p 2 for all 2 j and p p <p p 2 for all >j.sncep p >p p 2 for all 2 j, we know that > p p 0 for all 2 j. Now consder the smallest value of l for whch p l p 0 l < p l p 0 l. Note that f p l p l >p l p l 2 >...>p p 0, t must be the case that p l p 0,so l > p l p 0 l therefore we know that p l p l <p l p l 2. We also know that p l p l < p l p 0 l p l+ p l <p l p l < p l p 0 l, we know that p l+ p l < p l p 0 l, and therefore p l+ p 0 Applyng ths reasonng nductvely, we get the desred result.. Snce l+ < p l p 0 l. Corollary 3.0 For any anonymous technology functon (p, c) that has a unmodal probablty of success, there exsts an l n such that the frst transton n the non-strategc case s from 0tol agents (where l s the smallest value that satsfes p l p 0 l > p l+ p 0 l+ ) followed by all remanng n l transtons. Proof. It suffces to show that the condtons of Lemma 3.7 are met. We know from Lemma 3.9, that f() s unmodal, so condton s satsfed. We also know from the proof of Lemma 3.9, that f p p 0 < p p 0,thenp p <p p 2. Snce p p 0 < p p 0 for all >l, p p < p p 2 for all >land condton 2 s satsfed. Lemma 3. The probablty of success functon for any threshold technology s unmodal. Proof. Denote the probablty of success when contractng wth j agents as P (n, j, k). More specfcally, let P (n, j, k) denote the probablty of success when you contract wth j agents out n and at least k succeed. Note that, j+ = P (n, j +, k) P (n, j, k) =(β α) (P (n,j, k ) P (n,j, k)) = (β α) P (n,j,= k ), where P (n,j,= k ) s the probablty that exactly k agents succeed when j agents are contracted out of the n. Note that: P (n,j+, = k ) P (n,j,= k ) = (β α) (P (n 2,j,= k 2) P (n 2,j,= k )). Note that the dscrete dstrbuton: P (n 2,j,0),P(n 2,j,),...,P(n 2,j,n 2) s the convoluton of two bnomal random varables. Snce bnomal random varables are strongly unmodal and the convoluton of any two strongly unmodal functon s also strongly unmodal, we know that the dstrbuton: P (n 2,j,0),P(n 2,j,),...,P(n 2,j,n 2) s strongly unmodal. Note that f (P (n 2,j,= k 2) <P(n 2,j,= k )), ths means the mode of ths dstrbuton s greater than k. Therefore the mode of the dstrbuton, P (n 2,j+, 0),P(n 2,j+, ),...,P(n 2,j+,n 2), s also greater than k, so we know (P (n 2, j+, = k 2) < P(n 2, j+, = k )). Hence we know f P (n,j+, = k ) <P(n,j,= k ), P (n,j+2, = k ) <P(n,j+, = k ), whch gves us that j s a unmodal functon. 6

17 Lemma 3.3 The dscrete valued functon, Proof. Snce s log-convex (Lemma A.) and p Q, s convex. p s log-convex (Lemma A.3), we know that ( ) s also log-convex. Snce a log convex functon s also convex, we know that p + p ( ) > p ( ) p ( ) ( ) p p (p p 0 ) p (p p 0 ) (p p 0 ). Addng, as desred. for all. Therefore we know that p p ( ) to both sdes, p + (p + p 0 ) + (+) + (p + p 0 ) p (+) ( ) > + (+) + (p + p 0 ) p ( ) > p ( ) Lemma 3.4 There exsts a value of l a n such that Q Q la > p p 0 p la p 0 for all = l a. Q Proof. Snce Q s convex, + p + p 0 Q > Q Q p p 0 for all. Therefore f 0, then Q + Q > 0. Let l a be the smallest l such that l+ that p + p 0 Q > Q + all = l a. If =. Q p + p 0 for all <l a and Q Q < Q + p + p 0 p l+ p 0 Q Q p p 0 > Q l, therefore we know for all l a <n,so Q Q la > p p 0 p la p 0 Q p p 0 < 0 for all, then Q p p 0 > Q 2 p 2 p 0 >...> Qn p n p 0,so Q Q > p p 0 p p 0 for for all Q la p la p 0 Lemma 3.6 We have Q + Q < Q la+ p la+. p + p > Q Q p p for all >l a where l a s the smallest value such that Proof. We know for all l a, Q < Q + p + p 0 or n other words p ( ) < p + (+) + (p + p 0 ). Snce p > p + p + p 0 for any value of, we know that < + +. Note that f p p 0 > p + p 0 be that + <,sncep s unmodal, so p + p > + p + p or n other words, other words,sncep s log-concave. p + > p > p. Note that f p + + > p. If p p 0 p + p 0 > p,then p + + +,tmust +, then t must be that p p + > p p p or n p Snce > p + p + p 0, t must be that + + > p f ( ) < p + (+) + + (p + p 0 ).Thus + > for all l a. We know from Lemma A. that s a log-convex functon so therefore + > for all. Addng to both sdes we get for all. Therefore we know that > that + + > for all. Combnng ths wth the fact that p + + and + + > p for all l a, we get that + ( + + )+ + + > p l a as desred. Lemma A.. s log-concave. > p ( )+ for all l for all Proof. = p p = P (n,, k) P (n,, k), where P (n,, k) s the probablty that at least k agents succeed when succeed wth probablty β and n succeed wth probablty of α. Note that P (n,, k) P (n,, k) =(β α) (P (n,, k ) P (n,, k)) = (β α) P (n,, = k ), where P (n,, = k ) s the probablty that exactly k agents succeed when agents succeed wth probablty β and n succeed wth probablty α. We abbrevate the followng: f + = P (n,+, = k ), f = P (n,,= k ) and f = P (n,, = k ). It suffces to show that: f 2 f + f. We can wrte: f + = β P (n 2,,= k 2) + ( β) P (n 2,,= k ) 7

18 f = α P (n 2,,= k 2) + ( α) P (n 2,,= k ) f = β P (n 2,, = k 2) + ( β) P (n 2,, = k ) f = α P (n 2,, = k 2) + ( α) P (n 2,, = k ) Note that: f + f = αβp (n 2,,= k 2)P (n 2,, = k 2)+α( β)p (n 2,,= k )P (n 2,, = k 2)+β( α)p (n 2,,= k 2)P (n 2,, = k )+( α)( β)p (n 2,,= k )P (n 2,, = k ) Also note that: f f = αβp (n 2,,= k 2)P (n 2,, = k 2) + α( β)p (n 2,,= k 2)P (n 2,, = k )+β( α)p (n 2,,= k )P (n 2,, = k 2)+( α)( β)p (n 2,,= k )P (n 2,, = k ) f 2 f f + =(β α)(p (n 2,,= k )P (n 2,, = k 2) P (n 2,, = k )P (n 2,,= k 2)). Note that we can then wrte: P (n 2,,= k ) = βp(n 3,, = k 2) + ( β)p (n 3,, = k ) P (n 2,, = k ) = αp (n 3,, = k 2) + ( α)p (n 3,, = k ) P (n 2,,= k 2) = βp(n 3,, = k 3) + ( β)p (n 3,, = k 2) P (n 2,, = k 2) = αp (n 3,, = k 3) + ( α)p (n 3,, = k 2) So we can wrte: P (n 2,,= k )P (n 2,, = k 2) = (βp(n 3,, = k 2) + ( β)p (n 3,, = k ))(αp (n 3,, = k 3)+( α)p (n 3,, = k 2)) = αβp (n 3,, = k 2)P (n 3,, = k 3) + β( α)p (n 3,, = k 2)P (n 3,, = k 2) + α( β)p (n 3,, = k )P (n 3,, = k 3) + ( α)( β)p (n 3,, = k )P (n 3,, = k 2) And: P (n 2,, = k )P (n 2,,= k 2) = (αp (n 3,, = k 2) + ( α)p (n 3,, = k ))(βp(n 3,, = k 3)+( β)p (n 3,, = k 2)) = αβp (n 3,, = k 2)P (n 3,, = k 3) + β( α)p (n 3,, = k 3)P (n 3,, = k ) + α( β)p (n 3,, = k 2)P (n 3,, = k 2) + ( α)( β)p (n 3,, = k )P (n 3,, = k 2) So (P (n 2,,= k )P (n 2,, = k 2) P (n 2,, = k )P (n 2,,= k 2)) = (β α)(p (n 3,, = k 2)P (n 3,, = k 2) P (n 3,, = k )P (n 3,, = k 3)) Note that P (n, j, k) for fxed n, k s strongly unmodal snce t s the convoluton of two bnomal random varables, whch are also strongly unmodal. Therefore we know that P (n 3,, = k 2)P (n 3,, = k 2) P (n 3,, = k )P (n 3,, = k 3) > 0, so f 2 j f j f j+ > 0 for all n>3 and all n>k>2. Now we address the k = 2 case. When k = 2: f + = β ( β) ( α) n 2 +( β) P (n 2,,= k ) f = α ( β) ( α) n 2 +( α) P (n 2,,= k ) f = β ( β) ( α) n +( β) P (n 2,, = k ) f = α ( β) ( α) n +( α) P (n 2,, = k ) Note that: f + f = β ( β) ( α) n 2 α ( β) ( α) n + α( β)p (n 2,,= k )( β) ( α) n + β( α)( β) ( α) n 2 P (n 2,, = k ) + ( α)( β)p (n 2,,= k )P (n 2,, = k ) Also note that: f f = α ( β) ( α) n 2 β ( β) ( α) n + α( β)( β) ( α) n 2 P (n 2,, = k ) + β( α)p (n 2,,= k )( β) ( α) n +( α)( β)p (n 2,,= k )P (n 2,, = k ) 8

19 f 2 f f + =(β α)(p (n 2,,= k )( β) ( α) n P (n 2,, = k )( β) ( α) n 2 )=(β α) 2 ( β) ( α) n 2 P (n 2,,= k ) > 0. Fnally we consder the case that n = 3 (and k = 2 necessarly). For the n = 3 case: P (3,, 2) P (3, 0, 2) = (β α)p (2, 0, = ) = 2α( α) P (3, 2, 2) P (3,, 2) = (β α)p (2,, = ) = β( α)+α( β) P (3, 3, 2) P (3, 2, 2) = (β α)p (2, 2, = ) = 2β( β) We know that: (β( α) α( β)) > 0 β 2 ( α) 2 +2αβ( α)( β)+α 2 ( β) 2 > 4αβ( α)( β) (P (3, 2, 2) P (3,, 2)) 2 > (P (3,, 2) P (3, 0, 2))(P (3, 3, 2) P (3, 2, 2)) Now consder the case that k =. When k =, p p =( ( β) ( α) n ) ( ( β) ( α) n + )=( β) ( α) n (β α. Therefore / = β α for any and s a log-concave functon. Now consder the case that k = n. When k = n, p p = β α n β αn + = β α n (β α). Therefore / = β α for any and s a log-concave functon. Lemma A.2. p s log-concave. Proof. Snce s a dscrete functon, we know that < 0so < 0. In other words, we know + ( ) ( ) +, or + p p + < 0 or p 2 > (p ++ )(p )=p + p, as desred. Lemma A.3. p s log-convex. Proof. Snce p s log concave, we know that p 2 >p +p =(p ++ )(p ) or n other words + + p + p > 0, whch gves us that p 0 (p p 0 )( + + p + p > 0) and that p 0 ((2p p 0 ) + (p p 0 )( p + p )) > 0, whch means that p 2 (p p 0 ) 2 ) + p 0 p (p p 0 )( p + p ) > 0. Therefore we have: (p p 0 ) 2 (+ p p ) p 2 (+(p p 0 ) (p p 0 )) + p 2 (p p 0 ) 2 ) + > 0, so (p p 0 ) 2 (p 2 ++p + ) >p 2 (p p 0 ) ( )p 2 ( )p 2, or (p p 0 ) 2 (p ++ )(p ) >p 2 (p p 0 ++ )( ), whch gves us ( p p 0 p ) 2 > p + p 0 p p 0 p +, as desred. p B Proofs from Secton 4 Lemma 4.2 For the OR technology, the prce of unaccountablty occurs at a transton n the agency case, as opposed to a transton n the non-strategc case. Proof. It suffces to show that for a fxed agency contract, the socal welfare rato s ncreasng as v ncreases. Frst consder the OR technology. We know that for all <j, p j p 0. Therefore we have that for all <j, p j j < p.ifp j <jp,wehave: j < p p 0 9

20 jp (v v) >p j (v v) for any v >v jp v p j v > jp v p j v jcp v cp j v > jcp v cp j v p vp j v jcp v cp j v + cjc > p v p j v jcp v cp j v + cjc (p j v jc)(p v c) > (p j v jc)(p v c) p j v jc p v c > p jv jc p v c for any v >v Therefore, we know for fxed non-strategc and fxed agency contracts, the socal welfare rato ncreases and v ncreases. Fnally, we know that p j+v (j+)c p v c = p jv jc p v c,wherev s the pont at whch a prncpal s ndfferent between contractng between j agents and contractng wth j + agents. Therefore we know, for fxed agency contract, the socal welfare rato s ncreasng as v ncreases. Theorem 4.5 The POU for the OR technology s bounded by 2 for all α, β = α and n. Proof. To establsh ths result, t suffces to show that the socal welfare rato s bounded everywhere by 2. Gven Lemma 4.2, t suffces to consder only transton ponts n the agency case, so let us consder the socal welfare rato at a transton n the agency case. Let us consder the socal welfare rato at v,+, where the prncpal s ndfferent between contractng wth agents and + agents n the agency case. Also suppose at v,+, the optmal non-strategc contract s j, wheren j>. Therefore, we can wrte the socal welfare rato as: p jv,+ j p v,+. We want to show that p jv,+ j p v,+ 2. If 2p p j > 0, then ths statement s equvalent to v,+ 2 j 2p p j. If 2p p j < 0, then ths j 2 p j 2p. statement s equvalent to v,+ Frst we consder the case that 2p p j > 0. Frst suppose that 2 j 0, then we know that 2 j 2p p j < 0, so v,+ 2 j 2p p j. Therefore t suffces to consder the case that 2 j>0. Snce the optmal non-strategc contract s j at v,+, we know that j+ v,+ j. Therefore, t suffces to show that j 2 j 2p p j.snce2 j<jand j < j <...<, we know that j < p 2 j 2 j = p (p p 2 j ) 2 j = p ( j+ ) 2 j < p ( j + j ) 2 j = p (p j p ) 2 j = 2p p j 2 j. Now we consder the case that 2p p j < 0. Snce the optmal non-strategc contract s j at v,+, we know that j+ v,+ j. Therefore t suffces to show that j+ j 2 p j 2p p j 2p (j 2)(p j+ p j ). We can wrte p j = θ j q 0,whereθ = β α and q 0 =( α) n. Therefore t suffces to show that ( θ j q 0 ) 2( θ q 0 ) (j 2)( θ j+ q 0 ( θ j q 0 )) or equvalently +2θ q 0 θ j q 0 (j 2)θ j q 0 ( θ) or equvalently, θ q 0 (2 θ j (j 2)θ j ( θ)). Therefore t suffces to show that θ q 0 2. θ q 0 =( β) ( α) n. Snce β = α< 2,we know that ( β) ( α) n < 2 for all >0. Therefore the only remanng case s = 0 and 2p 0 p j < 0. We note that when = 0, 2 j<0, for any value of j. We also notce that f 2p 0 > 0, then 2p 0 p j > 0. In other words f α n 2,then2p 0 p j > 0. For n 3, the RHS s at most 0.207, therefore f α>0.207 and n 3, we know from above that p jv 0, j p 0 v 0, 2. (The n = 2 case s establshed n [7]). Now suppose that α Ths means that α< If α 5 5 0,then: 0 5 α 2 5 α + or 20