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Information and strategyproofness in joint project selection
Journal of Public Economics 68 (1998) 207–221
Information and strategyproofness in joint project selection Edward C. Rosenthal* Department of Management Science and Operations Management, 006 -00, Temple University, Philadelphia, PA 19122, USA Received 30 April 1996; received in revised form 30 September 1997; accepted 21 October 1997
Abstract We introduce the following multi-activity public goods problem with excludables. A group of agents with limited budgets is to select a subset of new technologies to be developed. Each technology has a known cost and may provide different returns, which are private information, to the agents. Agents coordinate their activities through a central administrator. Once purchased, technologies are used freely by all contributors. Through examining information provided to the agents and the administrator, we create a bounded rationality environment and provide a strategyproof selection and cost allocation mechanism under a particular decision criterion. 1998 Elsevier Science S.A. Keywords: Excludable public goods; Mechanism design; Information; Strategyproof; Bounded rationality JEL classification: C72; D81; D82; H41; H43
1. Introduction Studying the mechanisms through which public goods may be acquired and their costs shared by a set of agents has been a long-standing issue of central concern to economists. An important subset of these problems treats public goods with excludables. In this category, an agent may desire not to receive one or more goods, in which case the agent will not pay toward their acquisition. Designing *Tel.: (1-215) 204-8177; e-mail: [email protected] 0047-2727 / 98 / $19.00 1998 Elsevier Science S.A. All rights reserved. PII S0047-2727( 97 )00089-3
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selection and allocation mechanisms for such situations is difficult on account of several endemic problems. One is the desire to elicit honest information from all agents. Related to this is the well-known free rider problem, i.e., the consequences of either imposing that any member who enjoys a good must contribute toward it, or the absence of this condition. Further obstacles to mechanism design include the determination of fair cost allocation, especially when the agents are heterogeneous, and, in this paper, issues of joint selection among competing alternatives in the presence of scarce individual resources and conflicting preferences. What we study is a situation in which a large set of indivisible goods is potentially available to the agents. These goods are considered to be projects, or technologies, that may be developed if sufficient funds can be raised. Examples in public economics are regional cooperation projects (such as cable connection), patenting, and technology development. The agents act as members of a consortium or as different divisions of an organization which pool funds toward joint project development, and their activities are coordinated through a central administrator, or headquarters. Their budgets for project development are limited. Because these agents have conflicting preferences and yet must cooperate privately in the efficient pooling of their resources, the technologies that they procure may be termed semi-public goods. We assume that the costs of the projects are known, at least to the administrator. The benefits, measured as monetary returns, of the projects are private information; each agent knows his or her own benefits from all of the potential projects, but knows none of the benefits to the other agents. Benefit information is not available, a priori, to the administrator. The technologies we consider are available separately to the agents. We primarily treat the case in which the technologies are independent, i.e., the selection of one project has no bearing on the potential returns from another project; the dependent case, when subsets of projects may generate synergies, is mentioned briefly. If a project is selected by the group of agents, its technology is made available only to contributing agents. This precludes ‘free riding’, and also serves to model situations in which members must not violate anti-trust laws. In addition, contributing agents enjoy unrestricted use of the technologies they purchase, as is often the case with public goods, or with receipt of private goods such as a software license, patent technology, or access to a telecommunication system. However, agents do not all have to share the same subset of technologies; certain projects will be developed by one subset of agents, and others, by a different subset. This defines the excludable nature of the problem, in that members can decline to contribute to the acquisition of a technology and are thereby excluded from its use. A primary contribution of our work is the study and partial solution of a complex problem of theoretical and practical importance, with multiple public goods competing for selection by associated agents with limited capital. We develop a bounded rationality information structure within which decision makers
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have incentive to report honestly. Without the structure, we argue that similar results cannot be obtained. There is a vast literature on public goods and mechanism design, from which we describe relevant articles below. Ferejohn and Noll (1976), and Ferejohn et al. (1981) consider a very similar group selection problem in which many proposed projects are available to choose among. Each excludable project has a known cost; the members of the group have differing (ordinal) preferences. Projects are undertaken only if sufficient funds are pooled by the members. The authors determine what subset of projects should be selected by the group, and how the costs of these projects should be shared among the members. The motivating application involved program selection for Public Broadcasting affiliates in the US. Ferejohn et al. stipulated five desirable properties for a selection and allocation procedure—truthful revelation of preferences, no bankruptcy, no deficit, feasibility and weak efficiency—and gave an impossibility result: that no mechanism can satisfy all of these properties. In addition they developed and implemented an iterative, interactive procedure, which worked rather well in practice, but is not Pareto optimal (i.e., efficient) in general, and may not feasibly terminate. The mechanism we present overcomes, at least partially, two of these drawbacks. Hoadley et al. (1993); Yoon and Sadrian (1992) describe similar problems in which members of telecommunications R&D consortia choose projects for development. However, they do not consider strategic reporting of members’ preferences. Aloysius and Rosenthal (1994) do consider the adverse selection case, and provide an incentive compatible, but inefficient, mechanism. Other relevant articles are Groves and Ledyard (1977); Gradstein (1994); Moulin (1994); Deb and Razzolini (1994); Kleindorfer and Sertel (1994), which rely on alternative assumptions to the present paper, such as varying costs to provide a good, different consumption levels for the good, requiring a sufficient number of people to produce a good, or the absence of budget balance. Deb and Razzolini, however, use an auctioning technique (for the provision of a single good) that we describe in Section 2 and generalize in Section 3. The present paper employs a bounded rationality environment with asymmetric information and myopic decision making. There is evidence from Ferejohn et al. and Hoadley et al. that their applications are consistent with this model. Ferejohn et al. noted that station managers were possibly not sophisticated or well-informed decision makers in their own markets; it is therefore unlikely that they could employ a sophisticated gaming strategy that anticipates others’ preferences. Similarly, it is likely in the Hoadley et al. setting, that member firms are not able to accurately measure the impact of various technologies on the other member firms—merely gathering such information is difficult—and, therefore, they would be unable to strategically misrepresent their own preferences accordingly. All of this contrasts sharply with the behavior seen in the series of recent radio spectrum auctions held by the FCC (see Cramton, 1997; Ausubel and Cramton,
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1996; Ausubel et al., 1997). In these auctions—for a single type of good, with similar industry competitors—a good deal of sophisticated gaming behavior took place. However, we emphasize that the present paper does not treat pure public goods, and, although we employ an auction-like procedure, our organizational setting is far from that of the FCC auctions. The remainder of this paper is organized as follows. In Section 2 we briefly give a background result. In Section 3 we present the main result, consisting of the development of a bounded rationality milieu for the problem we study, and, within it, a strategyproof mechanism under an expected value decision criterion. We also examine other properties the mechanism satisfies. Section 4 discusses cases in which the informational aspects of the model are modified, thus destroying the ability to construct a strategyproof procedure under these different conditions. Some conclusions and discussion follow in Section 5.
2. A background result Consider a set N 5 h1,...,nj of agents who may wish to contribute to the procurement of a single, indivisible, excludable good that has cost c . 0 and may be freely shared at any levels by the contributors. Each agent, who has a known private benefit from the good, reports a benefit publicly. Deb and Razzolini (1994) provide the following mechanism, called an English Auction Like Mechanism (EALM), which purchases the technology only if the sum of the reported benefits exceeds c, and which divides c ‘fairly’ among the agents (including exclusions). An administrator begins with the price of c /n. Each agent stays ‘active’ by accepting this price; if they do not, they are excluded. The active agents consider successively increasing prices c /(n 2 1), c /(n 2 2), and so on; when k agents each accept a price c /k, the bidding stops and the item is procured. Otherwise the good is not provided. This mechanism is strategyproof : honest revelation is a dominant strategy for all agents. In Section 3, we consider the broader problem with many goods that compete for selection among agents with limited resources, and provide a mechanism that generalizes the EALM.
3. A strategyproof mechanism under bounded rationality To formalize the situation described in Section 1, let A represent the administrator, who will organize and implement a procedure to select goods from a finite set S 5 h1,..., j,...,mj of available technologies and allocate their costs to a finite set N 5 h1,...,i,...,nj of agents. These technologies are indivisible, and each cost c j . 0, j 5 1,...,m, is known to A. If purchased by the group, the technologies can be received, and used at any levels by contributing agents; that is, all agents who pay
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toward an acquired technology may derive benefit from it without rivalry. Technologies are excludable: non-contributors to technology j do not receive j. Agent i, receiving technology j, will obtain a benefit of Bij $ 0. These benefits are private information (each i knows only her own benefits, and the administrator knows none of the benefits), and may not be offset by side payments among the agents. Each agent i [ N has a limited research and development fund Ui . 0 available for technology purchase. The budgets Ui are private information. The agents wish, in general, to contribute toward some or all of the candidate technologies, but may, given their tight budgets, prefer not to receive a particular technology, and thereby avoid paying toward its procurement. The administrator A is to solicit reports which represent agent i’s valuation for technology j. Based on these reports, A is to select an affordable subset of technologies of maximal benefit, and to fairly allocate the costs of those technologies to the agents. We note that ‘affordability’ can be enforced by requiring that agents place money ‘on the table’ to validate their reports. (If agents act honestly, then the set of projects selected will automatically be feasible, that is, within budget.) The technologies are first assumed to be independent, that is, receipt of one technology does not in general influence selection, benefits, or cost of another technology; we then outline the extension to the case where synergies are present and project benefits are dependent. For the administrator to solve the problem by collecting reports and selecting a subset of technologies to maximize collective welfare would require a cost allocation rule that satisfies the individual budgets. But use of a report-based rule would invite agents to misrepresent their benefits and budgets, so as to manipulate not only the cost allocation process, but also the technology selection. And encouraging the agents to embark on a cooperative procedure (by exchanging information among themselves, and perhaps utilizing side payments), again opens the door for adverse selection. The following procedure that we introduce is guided by three principles toward obtaining strategyproofness: that the agents’ contributions be independent of their reports; that budget balance be achieved; and that, perhaps counterintuitively, the less information agents receive, the better. The process is essentially an auction like mechanism, whereby, in the first stage, the administrator sets a price of c j /n for each technology j. Agents ‘bid’, or accept this price (and may be asked to pay ‘up front’), or else decline, in which case they are ineligible from any future bids for, or receipt of, the particular technology. If, at this stage, a technology j is such that all n agents have agreed to pay c j for it, the technology is slated for purchase by the administrator, the (equal) payments are collected, and the technology is eliminated from further consideration. Otherwise, some number n9 , n of the agents have accepted. If n9 5 0, then the project is not funded. If n9 . 0, then those who have accepted the price are still ‘active’, i.e.,
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eligible for future bids. Those who have declined are prohibited from re-entering bids for that technology. The next stages develop by incrementing the prices simultaneously for each technology j [ S according to the following schedule: c j /(n 2 1), c j /(n 2 2), and so on. At the last stage (if reached), the price for j is set to c j , and there is one eligible agent remaining who may accept or decline that price. In either event, the fate of the technology is finally determined at that stage, and the procedure will terminate. One accounting aspect needs to be managed. An agent i may, in general, be active on a number of technologies at a certain stage. When the prices are raised, i may, due to his budget or benefit information, decline one or more of the technologies. In this event, the funds that i has thus far put toward those technologies are returned, thereby expanding his budget for future bidding. A number of questions can be raised about the restrictiveness of this mechanism. We defer discussion of some of these issues to Section 4. At this point we formally present the mechanism. Let k index the bidding stages. We let pjk be the price that A sets for technology j at stage k; define Ejk to be the set of agents who accept this price, with e jk 5 uEj k u, C while E jk is the set of those agents who decline. We define an agent who accepted a price for technology j at stage (k 2 1) to be active for j at the beginning of stage 9 be the set of active agents for j at the beginning of stage k. We define k. Let N jk S9 # S, the set of active technologies at any point in the process, to be the set of projects with a nonempty set of associated active agents.
Initialize: let N j91 5 N for all j, and S9 5 S. For k 5 1 to n do For all j [ S9, set price pjk 5 c j /(n 2 k 1 1). 9 accepts or declines pjk for all j. Create Each agent i [ N jk Ejk accordingly. If e jk 5 (n 2 k 1 1), then technology j is purchased by A; each agent who accepts pays pjk toward the purchase, and adjusts her budget accordingly. Agents in E Cjk are reimbursed the funds that they had paid toward j. 9 : let N j,k11 9 5 Ejk . Update N jk 9 11 5 [, then S9 5 S9\h jj for Update S9: If N j,k all such j. end end Note that the reimbursement step is needed for agents to privately adjust their budgets for the next round.
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Our primary aim is to demonstrate, under this mechanism, that no agent has incentive to falsely accept or decline prices at any stage. There are two problems to overcome. One is for agents to evaluate preferences among many technologies. This is difficult, because the benefits and the ‘expected’ costs for an agent (which depend on how many other agents accept) may widely vary, and are difficult to evaluate within, or even without, a budget cap. In other words, agents would have difficulty developing a well-defined preference relation for technologies when benefits and cost shares vary (confounding the issue of what honest reporting would actually mean). Another problem, also due to the uncertainty regarding the others’ preferences, is that agents will not be able to anticipate which technologies will be selected. Therefore, an agent may end up ‘betting on the wrong horse’, a consequence of which is that for some realization, a strategy that truly represented her preferences may be outperformed by a different strategy; this may provide incentive to deviate from truthful action. In response, what we now present is a restricted environment and decision criterion—which we argue is a reasonable model of bounded rationality—within which the above mechanism is strategyproof. The decision criterion we require agents to adopt, is expected value maximization at each stage. We first consider the project benefits to be independent. Let P(i, j, k) be the prior probability that i assigns to technology j being purchased at stage k. Given a set S’ of active technologies, each agent i will solve, at each stage k, the following mathematical program:
O P(i, j, k)(B 2 p )x s.t.O p x # U for all i, k
max
ij
jk
ijk
(1)
j [S 9
jk ijk
ik
(2)
j
x ijk [ h0, 1j,
(3)
where x ijk 51 if i accepts price pjk at stage k, and 0 otherwise, and Uik is i’s available budget at stage k. We assume that agents will accept or decline prices at each stage according to the solution to the program in Eqs. (1)–(3). In the dependent case, evaluation of all possible subsets of projects by the agents would result in a combinatorial explosion of benefits information. However, certain applications do provide restrictions on the subset formations to make such evaluations tractable (see, for example, Rothkopf et al. (1995) for an auction application). In theory, then, it would certainly be possible to modify the program in Eqs. (1)–(3) for the more general case of dependent benefits. Such a modification would involve not only the benefits calculations, but also the probability updates for admissible subsets. For the sake of presentation, we will omit the details in the dependent case. However, we need to justify the model in this restricted context. All budget and
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benefit information is private. Therefore, it is reasonable to assume, a priori, the Laplacean notion that all active technologies are equally likely to be funded at a certain stage. Now, if all agents share this prior over all technologies, when the administrator moves from one stage to the next, the only information transmitted to the agents is whether technologies are active. Since agents have no other information by which to generate different posteriors, we argue that the posteriors are common knowledge (Aumann, 1976), and, therefore, must be identical. This serves to eliminate the possibility of, and the ability for, providing Bayesian updates of the funding probabilities (which also influence the likelihood of pjk being the ultimate price paid). To re-address whether our bounded rationality environment is consistent with what we know from the literature: in neither the public broadcasting nor the R&D applications, did the decision makers have information regarding the benefits of the programs or technologies to the other members. Conceptually, this seems to be true of many multi-divisional firms: shared technologies would impact on the different divisions’ productivities and outputs in ways that would be difficult for others to ascertain. At the same time, in our sequential bidding process, it would be equally difficult to perform a ‘rational’ update of the likelihood of a project’s getting funded, given the asymmetries present. The bounded rationality environment is not sufficient to guarantee strategyproofness. An additional necessity is the following information-limiting rule. At each stage, for all projects j, the price pj,k 11 is increased from c j /(n2k1 1) to c j /(n2k) regardless of the size e jk of the accepting set. The reason for this is as follows. If the number of active bidders for a technology were to decrease, say, from seven to two, communication of this to the agents (either directly, or indirectly via the magnitude of the price increase) would create an undesirable informational asymmetry. This would delegitimize the decision criterion, and tempt agents to reject it via alternative updates of P(i, j, k), which could result in strategic (mis)reporting. The mechanism therefore eliminates this possibility. Another operation in the mechanism is designed to eradicate ‘gaming’ by the agents. This is the prevention of ‘re-entry’ into the bidding. Were re-entry (or delayed entry) allowed, agents would have a variety of strategies available to mask their true preferences, and defer their commitments. Having discussed the information restrictions, we now examine the decision criterion. The key term in the program in Eqs. (1)–(3) is P(i, j, k). To mirror the argument above, we require that each agent i evaluate all active technologies as equiprobable, that is, P(i, j, k)5P(i, j9, k) for all i [N, and j, j9[S9, j ±j9. With this condition, the probabilities effectively disappear from Eq. (1), and the agents simply maximize their benefits minus costs, subject to the budget constraint. One could argue for a naive value of P(i, j, k)51 / 2 for all k, or for a more sophisticated approach. Exploration of the relaxation of this condition appears in Section 4. Also note that the budget constraint Eq. (2) neglects the probabilities of the technologies being funded at that stage, so as to avoid the possibility of an agent overcommitting himself.
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All of these measures are consistent with the objective of preventing rational Bayesian updating or Nash equilibration on the part of the agents. We feel that it is an advantage of this model not to require agents to develop expectations about others’ preferences. Furthermore, while not undesirable per se, such updating or response strategies in this environment create opportunities for strategic reporting, as we will argue in Section 4. Strategyproofness of the mechanism follows directly from the bounded rationality assumptions. The agents maximize expected profit at each stage, which amounts to selecting a subset of technologies to accept. As the prices increase, declining a stage-optimal subset in favor of a different one is a dominated strategy; also suboptimal is accepting a technology for too high a price. Therefore, by forward induction, they can do no better, given the set of technologies that they are active for at that stage, than accept or decline prices according to their preferences—as defined by their objective functions. This argument establishes Proposition 1. The mechanism is strategyproof in the bounded rationality environment. There are other advantageous properties of the mechanism that we want to address before discussion of alternatives in Section 4. The mechanism satisfies several desirable, normative conditions that appear in the public goods literature (see, for example, Moulin, 1994; Deb and Razzolini, 1994). These we give below; proof in some cases is omitted, but straightforward. To begin with, the mechanism is feasible, that is, for all selected technologies, the costs are covered, and agents are able to pay within their budgets (provided, say, that funds are ‘put on the table’ as they bid). Since no surpluses are gathered by the administrator, the mechanism is budget balancing. Another condition the mechanism satisfies is anonymity. Here, agents who send the same reports for a technology either receive the technology for the same price, or else do not receive it (and pay nothing). Since agents are excluded from technologies merely by their not accepting the administrator’s price, there is no envy of those who do receive the technology (since the denied agents were self-excluded according to their own budget caps and sets of preferences). This property is called voluntary non-participation in Deb and Razzolini (1994); roughly speaking, it states that if i 1 receives the technology at price p, and i 2 is willing to pay p also, then i 2 should not be denied the technology. Another desirable condition is nonbossiness, which stipulates that an agent cannot change her reports to affect others’ allocations, without affecting her own allocation. Deb and Razzolini (1994) partition this into nonbossiness for included agents (NBI) and nonbossiness for excluded agents (NBE). In our context, the mechanism trivially satisfies NBI, as seen from the following argument. For a funded technology, any included agent has a finite sequence of ‘yes’ reports, denoting acceptance of a price at each stage. The only way to deviate from this is to report ‘no’, which of course changes the agent’s own allocation.
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However, NBE is violated. To see this, consider a technology j that receives any ‘yes’ reports from only two agents, i 1 and i 2 . Suppose that they both have a sequence of r ‘yes’ reports, but at stage r11, i 1 reports ‘no’. Supposing that i 2 cannot afford the technology by himself, i 1 ’s report will remove j from consideration. At this point, i 2 will receive his committed funds back. Now suppose that some other technology j9 was sought after by i 2 , but i 2 preferred j to j9. If it was the case that i 2 had to decline j9 at stage r9,r, because of his budget cap and his commitment to j, and if j9 was funded by other agents at, say, stage r, then i 2 ’s allocation was altered by i 1 . Now, if i 1 ’s true preference was to report ‘no’ for j prior to stage r9, then i 1 ’s (eventual) allocation would not be changed. Therefore, the mechanism violates NBE. One addition desirable condition is individual rationality. In our setting, if we take the status quo to be the agents’ profiles before the allocation process, then it is clear that the mechanism leaves the agents at least as well off, since any funded technology that they contribute to yields a nonnegative profit under the optimization. Finally, we note that our assumptions of known project costs and benefits (as well as the mechanism process itself) rule out the possibility of the ‘winner’s curse’ that plagues auction design.
4. Circumscription of the model In this section, we justify our model by demonstrating that various relaxations of it affect strategyproofness. To begin with, we have scrupulously avoided a model where the agents have ‘look-ahead’ ability, that is, when they can, at a given stage, provide unequal posterior probabilities for technologies to be funded (thereby enhancing their selection abilities at each stage). Consider Example 1, in which technology A has benefit 150 to i, B has benefit 141; and cA 5120, c B 5100. If n55, then at stage 1, when i evaluates the pair, A has an expected profit of 126 (i.e., 1502120 / 5) as against 121 for B. Likewise, at stage 2, A is still preferred to B (by 120 to 116). (And A is preferred at stage 3 as well.) But at stage 4, with only two agents active, B is preferred to A (by 91 to 90); the same result holds at the last stage. With the objective Eq. (1), agents do not make use of this information (and reasonably so, since they have no ability to calculate how likely it is that they reach any future stage). But if an agent has the power to make Bayesian-style updates, she may generate unequal posteriors. This would, of course, allow her to assign likelihoods to the various future stages, yielding different selections along the way. Looking ahead in this sense could give rise to false reporting, since an agent i may, for example, send a ‘no’ report at an early stage for a technology to influence the updates and strategies of the others. This technology may consequently be
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dropped, with other agents then reallocating their returned funds to a different technology that i favored all along. Essentially, our model discounts future stages. That is, since agents have no look-ahead devices to assess the probability that technology j will be funded in stage r.k, given j is active in stage k, they are in effect not doing this calculation, and thereby are approximating the expected value from stage k onward, with that at stage k only. We note that this discounting yields a reasonable model, especially as k increases, since the prices set by A rise at ever-increasing rates, which, with finite budgets, would likely decrease the number of active bidders. In other words, generally speaking, the longer a technology remains active, (and not yet funded!), the less likely eventual funding will be. To summarize, though it is desirable to develop a broader class of strategyproof mechanisms by weakening our restrictions (in particular, the equiprobable conditions), it is not clear how to do this. A related issue is whether myopic optimization is, in general, strategyproof for a wide class of selection problems. This focuses on the term ‘myopic’: if agents are estimating future probabilities of events concerning various projects, they are not myopic. Our principle of limiting Bayesian updating by restricting information and predictive ability allows the agents no scope for accepting certain projects at a given stage over other, presently more profitable, ones, so the agents never have incentive to misreport their limited computations. Metaphorically speaking, true myopia reduces the agents to a state of naivete´ which strips them of strategic ability. This drives strategyproofness in our model and the same principle could be extended to competing mechanisms. But when different agents generate different future probabilities (on the same projects), they may decline currently profitable projects in favor of others less so—and, because of this, all projects may face a reduced chance for funding, resulting in further inefficiency. Another desirable weakening of our model would involve extending the equal payment shares to unequal shares (as in both Ferejohn et al. and Hoadley et al.), which could more closely approximate an agent’s ability to pay. However, it does not appear possible to use an ex ante fixed set of proportions to this end, since when certain agents drop out from stage to stage of the auctioning process, either the fixed proportions for the remaining agents will not yield budget balance, or, if the proportions are updated, the remaining agents will have thereby received information about which other agents remain. This could result in the sort of ‘look ahead’ problems discussed above, that yield strategic reporting. A way to fix this problem is to use unequal proportions based in part on a random element. That is, the administrator could, at each stage, devise for each project a set of random prices that, if accepted, are budget balancing. Randomness would preclude strategic reporting, but violate the anonymity and no envy conditions. To summarize: it is possible to extend the equal share mechanism to one that is arbitrary, unequal share, and budget balancing, while preserving strategyproofness, but at the expense of being ‘fair’.
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Indeed, one might question the emphasis that we place on strategyproofness. In the problem we study, the agents are organizationally allied. Revelation of true preferences is necessary for the common good; strategic ‘no’ reports, designed to lower the chance that a less preferred technology gets funded, in the aggregate serve only to reduce the number of technologies that are eventually funded— increasing inefficiency. We have assumed throughout that knowledge of others’ budgets and benefits is private information. Were this condition abandoned, agents would again engage in manipulation of their reports. Example 2 demonstrates this, as follows. There are two agents, 1 and 2. Technologies A and B each cost 90; benefit B1A 580, B2A 5200, B1B 5140, and B2B 50. Agent 1 has a budget of 90, while 2’s is 50. Since the two agents together cannot fund both projects, the best available outcome is to fund one of them (ex post, the better choice is technology A). The mechanism will begin at stage 1 with the administrator setting a price of 45 for each project. Agent 1 will accept this price for both projects, while 2, with his more severe budget restriction, will accept only A. Technology A is therefore funded. At stage 2, both agents have insufficient balances to fund B. However, if the players had information about others’ budgets (and benefits), the following scenario might occur. Suppose agent 1 knew (or had a high prior) that 2’s budget was relatively small. Agent 1 could therefore suspect that 2 would have little or no money left at the second stage, and, in light of this, protect her relative interest in B by declining the price of 45 for A at the first stage. This action would leave 1, who accepts B at both stages, better off than by reporting honestly; but 2 is left with nothing. Furthermore, the outcome is not group optimal in the sense of pooled profits. If the agents knew one another’s benefits, there would be even sharper incentives to report strategically, and our mechanism would be fruitless. Another important point to be made is the inefficiency of the mechanism. While, at each stage, the individual agents can do no better than to report honestly, in an ex post sense (i.e., with public information), the final outcome may not be first best. We provide two types of such inefficiency. In the first example, we show that the final outcome may not be group optimal. Suppose, from Example 2 above, that the benefit to 1 from project B were in fact 300, and not 140. Now the group optimal result is to fund B and not A. However, the mechanism, as seen above, selects A. Even worse, the mechanism may fail to fund even a single technology which has unbounded profit! Consider a single technology of cost 1, and n agents whose benefits, in descending order, are (12 ´), (1 / 22 ´), (1 / 32 ´) . . . (1 / n2 ´), for some ´ .0. (Budgets need not be binding here.) At the first stage, the price is set to 1 /n, and the first (n21) of the agents will accept. At stage two, the price is increased to 1 /(n21); only the first (n22) of the active agents accept this higher price. This will continue until the last stage, when the price is set to 1, and the first agent will decline. As n increases, and ´ →0, the sum of the benefits grows without bound, as does the aggregate benefit-to-cost ratio. This example, however,
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is contrived, and its pathology does not necessarily imply that the mechanism will perform badly in practice. A limited series of ten runs of our procedure, with four agents and 8–10 projects, using randomly generated costs and benefits, has produced an average efficiency, ex post, of 0.38. (That is, the procedure funded projects yielding, on the average, 38% of the benefits that would have been obtained with complete and honest revelation.) Note that it is unclear whether abandoning strategyproofness will increase efficiency. Finally, we comment on the ‘knife-edge’ specificity of the mechanism. Why can the administrator not simply raise the bids by, say, fixed increments, as was the case in the FCC auctions? Our more exact approach is required here to be budget balancing. Again, the provision of semi-public goods that we study is fundamentally different from auctions. Were our mechanism to raise bids by fixed (or convex) increments, we would introduce the possibility of generating surpluses. Our aim is not to maximize bids for projects, but to raise precisely the required amounts to fund them, which is more consistent with the limited budgets that consortium members and divisions of firms must cope with.
5. Conclusions In this paper we have described a new problem involving selection of potential goods for a set of individuals who are organizationally allied but have individual, and private, preferences over the items, as well as budget limitations. We elicit their individual preferences by means of establishing an auction-like pricing mechanism for the goods over a finite number of stages. This mechanism, which limits the information accessible to the members, is shown to be strategyproof under the imposition of reasonable bounded rationality conditions. In addition, we show that the mechanism satisfies several well-known, desirable properties. As in Deb and Razzolini (1994), our model provides a bridge between work on provision of public goods and that on private goods. In a tactical sense, we are simply adapting an auction technique to a public goods problem. But in a deeper sense, the excludable public goods we study actually serve to model technology development problems for research consortia and for partially decentralized multi-divisional companies. There are a variety of directions for further research into this model. One is to assign utility functions to the agents, and probability distributions to the project outcomes. Since selected technologies would all be developed at once, the appropriate notion from the literature would be that of joint receipt (see Kahneman and Lovallo, 1993), where a bundle of received technologies is treated as a pooled lottery. The implications are that even risk averse individuals may be willing to bid on a larger subset of technologies by framing them as pooled, and not as separate, risks. Another direction is to attempt to develop a more relaxed framework and
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mechanism, which utilizes transfer payments among the members. However, it should be noted that the problem of adverse selection we face is a very difficult one. Our approach has been to develop a justifiable, bounded rationality decision environment, within which group members will optimize their selection vectors stage by stage. It is a challenge to find a broader model.
Acknowledgements Most of this work was carried out while visiting the Department of Operational Research, London School of Economics, on study leave from Temple University. I am grateful to Temple for its generosity and LSE for its warm hospitality. I also want to thank John Aloysius for many comments on an earlier draft. Finally, I am very grateful to an anonymous referee for many insightful, provocative, and helpful remarks on two earlier versions, which have led to a much improved article.
References Aumann, R.J., 1976. Agreeing to disagree. The Annals of Statistics 4, 1226–1229. Aloysius, J.A., Rosenthal, E.C., 1994. The selection of joint projects by a consortium: A cost sharing mechanism, Working Paper, Temple University. Ausubel, L.M., Cramton, P.C., McAfee, R.P., McMillan, J., 1997. Synergies in wireless telephony: Evidence from the broadband PCS auctions, Journal of Economics and Management Strategy 6. Ausubel, L.M., Cramton, P.C., 1996. Demand reduction and inefficiency in multi-unit auctions, Working Paper No. 96-07, Department of Economics, University of Maryland. Cramton, P.C., 1997. The FCC spectrum auctions: An early assessment, Journal of Economics and Management Strategy 6. Deb, R., Razzolini, L., 1994. Auction like mechanisms for pricing excludable public goods, Center Working Paper No. 9504, Southern Methodist University. Ferejohn, J.A., Forsythe, R., Noll, R., 1981. Practical aspects of the construction of decentralized decision-making systems for public goods. In: Russell, C.S. (Ed.), Collective Decision Making, John Hopkins University Press, Baltimore, pp. 173–203. Ferejohn, J.A., Noll, R., 1976. An experimental market for public goods: The PBS station program cooperative. American Economic Review 66, 267–273. Gradstein, M., 1994. Efficient provision of a discrete public good. International Economic Review 35, 877–897. Groves, T., Ledyard, J., 1977. Optimal allocation of public goods: A solution to the ‘Free Rider’ problem. Econometrica 45, 783–809. Hoadley, B., Katz, P., Sadrian, A., 1993. Improving the utility of the Bellcore consortium. Interfaces 23, 27–43. Kahneman, D., Lovallo, D., 1993. Timid choices and bold forecasts: A cognitive perspective on risk taking. Management Science 39, 17–31. Kleindorfer, P.R., Sertel, M.R., 1994. Auctioning the provision of an indivisible public good. Journal of Economic Theory 64, 20–34.
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Moulin, H., 1994. Serial cost sharing of excludable public goods. Review of Economic Studies 61, 305–325. Rothkopf, M.H., Pekec, A., Harstad, R.M., 1995. Computationally manageable auctions, Working Paper, Rutgers University. Yoon, Y.S., Sadrian, A., 1992. An integer linear programming model to increase project participation in R&D consortia, Working Paper, Bell Communications Research.
Information and strategyproofness in joint project selection Edward C. Rosenthal* Department of Management Science and Operations Management, 006 -00, Temple University, Philadelphia, PA 19122, USA Received 30 April 1996; received in revised form 30 September 1997; accepted 21 October 1997
Abstract We introduce the following multi-activity public goods problem with excludables. A group of agents with limited budgets is to select a subset of new technologies to be developed. Each technology has a known cost and may provide different returns, which are private information, to the agents. Agents coordinate their activities through a central administrator. Once purchased, technologies are used freely by all contributors. Through examining information provided to the agents and the administrator, we create a bounded rationality environment and provide a strategyproof selection and cost allocation mechanism under a particular decision criterion. 1998 Elsevier Science S.A. Keywords: Excludable public goods; Mechanism design; Information; Strategyproof; Bounded rationality JEL classification: C72; D81; D82; H41; H43
1. Introduction Studying the mechanisms through which public goods may be acquired and their costs shared by a set of agents has been a long-standing issue of central concern to economists. An important subset of these problems treats public goods with excludables. In this category, an agent may desire not to receive one or more goods, in which case the agent will not pay toward their acquisition. Designing *Tel.: (1-215) 204-8177; e-mail: [email protected] 0047-2727 / 98 / $19.00 1998 Elsevier Science S.A. All rights reserved. PII S0047-2727( 97 )00089-3
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selection and allocation mechanisms for such situations is difficult on account of several endemic problems. One is the desire to elicit honest information from all agents. Related to this is the well-known free rider problem, i.e., the consequences of either imposing that any member who enjoys a good must contribute toward it, or the absence of this condition. Further obstacles to mechanism design include the determination of fair cost allocation, especially when the agents are heterogeneous, and, in this paper, issues of joint selection among competing alternatives in the presence of scarce individual resources and conflicting preferences. What we study is a situation in which a large set of indivisible goods is potentially available to the agents. These goods are considered to be projects, or technologies, that may be developed if sufficient funds can be raised. Examples in public economics are regional cooperation projects (such as cable connection), patenting, and technology development. The agents act as members of a consortium or as different divisions of an organization which pool funds toward joint project development, and their activities are coordinated through a central administrator, or headquarters. Their budgets for project development are limited. Because these agents have conflicting preferences and yet must cooperate privately in the efficient pooling of their resources, the technologies that they procure may be termed semi-public goods. We assume that the costs of the projects are known, at least to the administrator. The benefits, measured as monetary returns, of the projects are private information; each agent knows his or her own benefits from all of the potential projects, but knows none of the benefits to the other agents. Benefit information is not available, a priori, to the administrator. The technologies we consider are available separately to the agents. We primarily treat the case in which the technologies are independent, i.e., the selection of one project has no bearing on the potential returns from another project; the dependent case, when subsets of projects may generate synergies, is mentioned briefly. If a project is selected by the group of agents, its technology is made available only to contributing agents. This precludes ‘free riding’, and also serves to model situations in which members must not violate anti-trust laws. In addition, contributing agents enjoy unrestricted use of the technologies they purchase, as is often the case with public goods, or with receipt of private goods such as a software license, patent technology, or access to a telecommunication system. However, agents do not all have to share the same subset of technologies; certain projects will be developed by one subset of agents, and others, by a different subset. This defines the excludable nature of the problem, in that members can decline to contribute to the acquisition of a technology and are thereby excluded from its use. A primary contribution of our work is the study and partial solution of a complex problem of theoretical and practical importance, with multiple public goods competing for selection by associated agents with limited capital. We develop a bounded rationality information structure within which decision makers
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have incentive to report honestly. Without the structure, we argue that similar results cannot be obtained. There is a vast literature on public goods and mechanism design, from which we describe relevant articles below. Ferejohn and Noll (1976), and Ferejohn et al. (1981) consider a very similar group selection problem in which many proposed projects are available to choose among. Each excludable project has a known cost; the members of the group have differing (ordinal) preferences. Projects are undertaken only if sufficient funds are pooled by the members. The authors determine what subset of projects should be selected by the group, and how the costs of these projects should be shared among the members. The motivating application involved program selection for Public Broadcasting affiliates in the US. Ferejohn et al. stipulated five desirable properties for a selection and allocation procedure—truthful revelation of preferences, no bankruptcy, no deficit, feasibility and weak efficiency—and gave an impossibility result: that no mechanism can satisfy all of these properties. In addition they developed and implemented an iterative, interactive procedure, which worked rather well in practice, but is not Pareto optimal (i.e., efficient) in general, and may not feasibly terminate. The mechanism we present overcomes, at least partially, two of these drawbacks. Hoadley et al. (1993); Yoon and Sadrian (1992) describe similar problems in which members of telecommunications R&D consortia choose projects for development. However, they do not consider strategic reporting of members’ preferences. Aloysius and Rosenthal (1994) do consider the adverse selection case, and provide an incentive compatible, but inefficient, mechanism. Other relevant articles are Groves and Ledyard (1977); Gradstein (1994); Moulin (1994); Deb and Razzolini (1994); Kleindorfer and Sertel (1994), which rely on alternative assumptions to the present paper, such as varying costs to provide a good, different consumption levels for the good, requiring a sufficient number of people to produce a good, or the absence of budget balance. Deb and Razzolini, however, use an auctioning technique (for the provision of a single good) that we describe in Section 2 and generalize in Section 3. The present paper employs a bounded rationality environment with asymmetric information and myopic decision making. There is evidence from Ferejohn et al. and Hoadley et al. that their applications are consistent with this model. Ferejohn et al. noted that station managers were possibly not sophisticated or well-informed decision makers in their own markets; it is therefore unlikely that they could employ a sophisticated gaming strategy that anticipates others’ preferences. Similarly, it is likely in the Hoadley et al. setting, that member firms are not able to accurately measure the impact of various technologies on the other member firms—merely gathering such information is difficult—and, therefore, they would be unable to strategically misrepresent their own preferences accordingly. All of this contrasts sharply with the behavior seen in the series of recent radio spectrum auctions held by the FCC (see Cramton, 1997; Ausubel and Cramton,
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1996; Ausubel et al., 1997). In these auctions—for a single type of good, with similar industry competitors—a good deal of sophisticated gaming behavior took place. However, we emphasize that the present paper does not treat pure public goods, and, although we employ an auction-like procedure, our organizational setting is far from that of the FCC auctions. The remainder of this paper is organized as follows. In Section 2 we briefly give a background result. In Section 3 we present the main result, consisting of the development of a bounded rationality milieu for the problem we study, and, within it, a strategyproof mechanism under an expected value decision criterion. We also examine other properties the mechanism satisfies. Section 4 discusses cases in which the informational aspects of the model are modified, thus destroying the ability to construct a strategyproof procedure under these different conditions. Some conclusions and discussion follow in Section 5.
2. A background result Consider a set N 5 h1,...,nj of agents who may wish to contribute to the procurement of a single, indivisible, excludable good that has cost c . 0 and may be freely shared at any levels by the contributors. Each agent, who has a known private benefit from the good, reports a benefit publicly. Deb and Razzolini (1994) provide the following mechanism, called an English Auction Like Mechanism (EALM), which purchases the technology only if the sum of the reported benefits exceeds c, and which divides c ‘fairly’ among the agents (including exclusions). An administrator begins with the price of c /n. Each agent stays ‘active’ by accepting this price; if they do not, they are excluded. The active agents consider successively increasing prices c /(n 2 1), c /(n 2 2), and so on; when k agents each accept a price c /k, the bidding stops and the item is procured. Otherwise the good is not provided. This mechanism is strategyproof : honest revelation is a dominant strategy for all agents. In Section 3, we consider the broader problem with many goods that compete for selection among agents with limited resources, and provide a mechanism that generalizes the EALM.
3. A strategyproof mechanism under bounded rationality To formalize the situation described in Section 1, let A represent the administrator, who will organize and implement a procedure to select goods from a finite set S 5 h1,..., j,...,mj of available technologies and allocate their costs to a finite set N 5 h1,...,i,...,nj of agents. These technologies are indivisible, and each cost c j . 0, j 5 1,...,m, is known to A. If purchased by the group, the technologies can be received, and used at any levels by contributing agents; that is, all agents who pay
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toward an acquired technology may derive benefit from it without rivalry. Technologies are excludable: non-contributors to technology j do not receive j. Agent i, receiving technology j, will obtain a benefit of Bij $ 0. These benefits are private information (each i knows only her own benefits, and the administrator knows none of the benefits), and may not be offset by side payments among the agents. Each agent i [ N has a limited research and development fund Ui . 0 available for technology purchase. The budgets Ui are private information. The agents wish, in general, to contribute toward some or all of the candidate technologies, but may, given their tight budgets, prefer not to receive a particular technology, and thereby avoid paying toward its procurement. The administrator A is to solicit reports which represent agent i’s valuation for technology j. Based on these reports, A is to select an affordable subset of technologies of maximal benefit, and to fairly allocate the costs of those technologies to the agents. We note that ‘affordability’ can be enforced by requiring that agents place money ‘on the table’ to validate their reports. (If agents act honestly, then the set of projects selected will automatically be feasible, that is, within budget.) The technologies are first assumed to be independent, that is, receipt of one technology does not in general influence selection, benefits, or cost of another technology; we then outline the extension to the case where synergies are present and project benefits are dependent. For the administrator to solve the problem by collecting reports and selecting a subset of technologies to maximize collective welfare would require a cost allocation rule that satisfies the individual budgets. But use of a report-based rule would invite agents to misrepresent their benefits and budgets, so as to manipulate not only the cost allocation process, but also the technology selection. And encouraging the agents to embark on a cooperative procedure (by exchanging information among themselves, and perhaps utilizing side payments), again opens the door for adverse selection. The following procedure that we introduce is guided by three principles toward obtaining strategyproofness: that the agents’ contributions be independent of their reports; that budget balance be achieved; and that, perhaps counterintuitively, the less information agents receive, the better. The process is essentially an auction like mechanism, whereby, in the first stage, the administrator sets a price of c j /n for each technology j. Agents ‘bid’, or accept this price (and may be asked to pay ‘up front’), or else decline, in which case they are ineligible from any future bids for, or receipt of, the particular technology. If, at this stage, a technology j is such that all n agents have agreed to pay c j for it, the technology is slated for purchase by the administrator, the (equal) payments are collected, and the technology is eliminated from further consideration. Otherwise, some number n9 , n of the agents have accepted. If n9 5 0, then the project is not funded. If n9 . 0, then those who have accepted the price are still ‘active’, i.e.,
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eligible for future bids. Those who have declined are prohibited from re-entering bids for that technology. The next stages develop by incrementing the prices simultaneously for each technology j [ S according to the following schedule: c j /(n 2 1), c j /(n 2 2), and so on. At the last stage (if reached), the price for j is set to c j , and there is one eligible agent remaining who may accept or decline that price. In either event, the fate of the technology is finally determined at that stage, and the procedure will terminate. One accounting aspect needs to be managed. An agent i may, in general, be active on a number of technologies at a certain stage. When the prices are raised, i may, due to his budget or benefit information, decline one or more of the technologies. In this event, the funds that i has thus far put toward those technologies are returned, thereby expanding his budget for future bidding. A number of questions can be raised about the restrictiveness of this mechanism. We defer discussion of some of these issues to Section 4. At this point we formally present the mechanism. Let k index the bidding stages. We let pjk be the price that A sets for technology j at stage k; define Ejk to be the set of agents who accept this price, with e jk 5 uEj k u, C while E jk is the set of those agents who decline. We define an agent who accepted a price for technology j at stage (k 2 1) to be active for j at the beginning of stage 9 be the set of active agents for j at the beginning of stage k. We define k. Let N jk S9 # S, the set of active technologies at any point in the process, to be the set of projects with a nonempty set of associated active agents.
Initialize: let N j91 5 N for all j, and S9 5 S. For k 5 1 to n do For all j [ S9, set price pjk 5 c j /(n 2 k 1 1). 9 accepts or declines pjk for all j. Create Each agent i [ N jk Ejk accordingly. If e jk 5 (n 2 k 1 1), then technology j is purchased by A; each agent who accepts pays pjk toward the purchase, and adjusts her budget accordingly. Agents in E Cjk are reimbursed the funds that they had paid toward j. 9 : let N j,k11 9 5 Ejk . Update N jk 9 11 5 [, then S9 5 S9\h jj for Update S9: If N j,k all such j. end end Note that the reimbursement step is needed for agents to privately adjust their budgets for the next round.
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Our primary aim is to demonstrate, under this mechanism, that no agent has incentive to falsely accept or decline prices at any stage. There are two problems to overcome. One is for agents to evaluate preferences among many technologies. This is difficult, because the benefits and the ‘expected’ costs for an agent (which depend on how many other agents accept) may widely vary, and are difficult to evaluate within, or even without, a budget cap. In other words, agents would have difficulty developing a well-defined preference relation for technologies when benefits and cost shares vary (confounding the issue of what honest reporting would actually mean). Another problem, also due to the uncertainty regarding the others’ preferences, is that agents will not be able to anticipate which technologies will be selected. Therefore, an agent may end up ‘betting on the wrong horse’, a consequence of which is that for some realization, a strategy that truly represented her preferences may be outperformed by a different strategy; this may provide incentive to deviate from truthful action. In response, what we now present is a restricted environment and decision criterion—which we argue is a reasonable model of bounded rationality—within which the above mechanism is strategyproof. The decision criterion we require agents to adopt, is expected value maximization at each stage. We first consider the project benefits to be independent. Let P(i, j, k) be the prior probability that i assigns to technology j being purchased at stage k. Given a set S’ of active technologies, each agent i will solve, at each stage k, the following mathematical program:
O P(i, j, k)(B 2 p )x s.t.O p x # U for all i, k
max
ij
jk
ijk
(1)
j [S 9
jk ijk
ik
(2)
j
x ijk [ h0, 1j,
(3)
where x ijk 51 if i accepts price pjk at stage k, and 0 otherwise, and Uik is i’s available budget at stage k. We assume that agents will accept or decline prices at each stage according to the solution to the program in Eqs. (1)–(3). In the dependent case, evaluation of all possible subsets of projects by the agents would result in a combinatorial explosion of benefits information. However, certain applications do provide restrictions on the subset formations to make such evaluations tractable (see, for example, Rothkopf et al. (1995) for an auction application). In theory, then, it would certainly be possible to modify the program in Eqs. (1)–(3) for the more general case of dependent benefits. Such a modification would involve not only the benefits calculations, but also the probability updates for admissible subsets. For the sake of presentation, we will omit the details in the dependent case. However, we need to justify the model in this restricted context. All budget and
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benefit information is private. Therefore, it is reasonable to assume, a priori, the Laplacean notion that all active technologies are equally likely to be funded at a certain stage. Now, if all agents share this prior over all technologies, when the administrator moves from one stage to the next, the only information transmitted to the agents is whether technologies are active. Since agents have no other information by which to generate different posteriors, we argue that the posteriors are common knowledge (Aumann, 1976), and, therefore, must be identical. This serves to eliminate the possibility of, and the ability for, providing Bayesian updates of the funding probabilities (which also influence the likelihood of pjk being the ultimate price paid). To re-address whether our bounded rationality environment is consistent with what we know from the literature: in neither the public broadcasting nor the R&D applications, did the decision makers have information regarding the benefits of the programs or technologies to the other members. Conceptually, this seems to be true of many multi-divisional firms: shared technologies would impact on the different divisions’ productivities and outputs in ways that would be difficult for others to ascertain. At the same time, in our sequential bidding process, it would be equally difficult to perform a ‘rational’ update of the likelihood of a project’s getting funded, given the asymmetries present. The bounded rationality environment is not sufficient to guarantee strategyproofness. An additional necessity is the following information-limiting rule. At each stage, for all projects j, the price pj,k 11 is increased from c j /(n2k1 1) to c j /(n2k) regardless of the size e jk of the accepting set. The reason for this is as follows. If the number of active bidders for a technology were to decrease, say, from seven to two, communication of this to the agents (either directly, or indirectly via the magnitude of the price increase) would create an undesirable informational asymmetry. This would delegitimize the decision criterion, and tempt agents to reject it via alternative updates of P(i, j, k), which could result in strategic (mis)reporting. The mechanism therefore eliminates this possibility. Another operation in the mechanism is designed to eradicate ‘gaming’ by the agents. This is the prevention of ‘re-entry’ into the bidding. Were re-entry (or delayed entry) allowed, agents would have a variety of strategies available to mask their true preferences, and defer their commitments. Having discussed the information restrictions, we now examine the decision criterion. The key term in the program in Eqs. (1)–(3) is P(i, j, k). To mirror the argument above, we require that each agent i evaluate all active technologies as equiprobable, that is, P(i, j, k)5P(i, j9, k) for all i [N, and j, j9[S9, j ±j9. With this condition, the probabilities effectively disappear from Eq. (1), and the agents simply maximize their benefits minus costs, subject to the budget constraint. One could argue for a naive value of P(i, j, k)51 / 2 for all k, or for a more sophisticated approach. Exploration of the relaxation of this condition appears in Section 4. Also note that the budget constraint Eq. (2) neglects the probabilities of the technologies being funded at that stage, so as to avoid the possibility of an agent overcommitting himself.
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All of these measures are consistent with the objective of preventing rational Bayesian updating or Nash equilibration on the part of the agents. We feel that it is an advantage of this model not to require agents to develop expectations about others’ preferences. Furthermore, while not undesirable per se, such updating or response strategies in this environment create opportunities for strategic reporting, as we will argue in Section 4. Strategyproofness of the mechanism follows directly from the bounded rationality assumptions. The agents maximize expected profit at each stage, which amounts to selecting a subset of technologies to accept. As the prices increase, declining a stage-optimal subset in favor of a different one is a dominated strategy; also suboptimal is accepting a technology for too high a price. Therefore, by forward induction, they can do no better, given the set of technologies that they are active for at that stage, than accept or decline prices according to their preferences—as defined by their objective functions. This argument establishes Proposition 1. The mechanism is strategyproof in the bounded rationality environment. There are other advantageous properties of the mechanism that we want to address before discussion of alternatives in Section 4. The mechanism satisfies several desirable, normative conditions that appear in the public goods literature (see, for example, Moulin, 1994; Deb and Razzolini, 1994). These we give below; proof in some cases is omitted, but straightforward. To begin with, the mechanism is feasible, that is, for all selected technologies, the costs are covered, and agents are able to pay within their budgets (provided, say, that funds are ‘put on the table’ as they bid). Since no surpluses are gathered by the administrator, the mechanism is budget balancing. Another condition the mechanism satisfies is anonymity. Here, agents who send the same reports for a technology either receive the technology for the same price, or else do not receive it (and pay nothing). Since agents are excluded from technologies merely by their not accepting the administrator’s price, there is no envy of those who do receive the technology (since the denied agents were self-excluded according to their own budget caps and sets of preferences). This property is called voluntary non-participation in Deb and Razzolini (1994); roughly speaking, it states that if i 1 receives the technology at price p, and i 2 is willing to pay p also, then i 2 should not be denied the technology. Another desirable condition is nonbossiness, which stipulates that an agent cannot change her reports to affect others’ allocations, without affecting her own allocation. Deb and Razzolini (1994) partition this into nonbossiness for included agents (NBI) and nonbossiness for excluded agents (NBE). In our context, the mechanism trivially satisfies NBI, as seen from the following argument. For a funded technology, any included agent has a finite sequence of ‘yes’ reports, denoting acceptance of a price at each stage. The only way to deviate from this is to report ‘no’, which of course changes the agent’s own allocation.
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However, NBE is violated. To see this, consider a technology j that receives any ‘yes’ reports from only two agents, i 1 and i 2 . Suppose that they both have a sequence of r ‘yes’ reports, but at stage r11, i 1 reports ‘no’. Supposing that i 2 cannot afford the technology by himself, i 1 ’s report will remove j from consideration. At this point, i 2 will receive his committed funds back. Now suppose that some other technology j9 was sought after by i 2 , but i 2 preferred j to j9. If it was the case that i 2 had to decline j9 at stage r9,r, because of his budget cap and his commitment to j, and if j9 was funded by other agents at, say, stage r, then i 2 ’s allocation was altered by i 1 . Now, if i 1 ’s true preference was to report ‘no’ for j prior to stage r9, then i 1 ’s (eventual) allocation would not be changed. Therefore, the mechanism violates NBE. One addition desirable condition is individual rationality. In our setting, if we take the status quo to be the agents’ profiles before the allocation process, then it is clear that the mechanism leaves the agents at least as well off, since any funded technology that they contribute to yields a nonnegative profit under the optimization. Finally, we note that our assumptions of known project costs and benefits (as well as the mechanism process itself) rule out the possibility of the ‘winner’s curse’ that plagues auction design.
4. Circumscription of the model In this section, we justify our model by demonstrating that various relaxations of it affect strategyproofness. To begin with, we have scrupulously avoided a model where the agents have ‘look-ahead’ ability, that is, when they can, at a given stage, provide unequal posterior probabilities for technologies to be funded (thereby enhancing their selection abilities at each stage). Consider Example 1, in which technology A has benefit 150 to i, B has benefit 141; and cA 5120, c B 5100. If n55, then at stage 1, when i evaluates the pair, A has an expected profit of 126 (i.e., 1502120 / 5) as against 121 for B. Likewise, at stage 2, A is still preferred to B (by 120 to 116). (And A is preferred at stage 3 as well.) But at stage 4, with only two agents active, B is preferred to A (by 91 to 90); the same result holds at the last stage. With the objective Eq. (1), agents do not make use of this information (and reasonably so, since they have no ability to calculate how likely it is that they reach any future stage). But if an agent has the power to make Bayesian-style updates, she may generate unequal posteriors. This would, of course, allow her to assign likelihoods to the various future stages, yielding different selections along the way. Looking ahead in this sense could give rise to false reporting, since an agent i may, for example, send a ‘no’ report at an early stage for a technology to influence the updates and strategies of the others. This technology may consequently be
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dropped, with other agents then reallocating their returned funds to a different technology that i favored all along. Essentially, our model discounts future stages. That is, since agents have no look-ahead devices to assess the probability that technology j will be funded in stage r.k, given j is active in stage k, they are in effect not doing this calculation, and thereby are approximating the expected value from stage k onward, with that at stage k only. We note that this discounting yields a reasonable model, especially as k increases, since the prices set by A rise at ever-increasing rates, which, with finite budgets, would likely decrease the number of active bidders. In other words, generally speaking, the longer a technology remains active, (and not yet funded!), the less likely eventual funding will be. To summarize, though it is desirable to develop a broader class of strategyproof mechanisms by weakening our restrictions (in particular, the equiprobable conditions), it is not clear how to do this. A related issue is whether myopic optimization is, in general, strategyproof for a wide class of selection problems. This focuses on the term ‘myopic’: if agents are estimating future probabilities of events concerning various projects, they are not myopic. Our principle of limiting Bayesian updating by restricting information and predictive ability allows the agents no scope for accepting certain projects at a given stage over other, presently more profitable, ones, so the agents never have incentive to misreport their limited computations. Metaphorically speaking, true myopia reduces the agents to a state of naivete´ which strips them of strategic ability. This drives strategyproofness in our model and the same principle could be extended to competing mechanisms. But when different agents generate different future probabilities (on the same projects), they may decline currently profitable projects in favor of others less so—and, because of this, all projects may face a reduced chance for funding, resulting in further inefficiency. Another desirable weakening of our model would involve extending the equal payment shares to unequal shares (as in both Ferejohn et al. and Hoadley et al.), which could more closely approximate an agent’s ability to pay. However, it does not appear possible to use an ex ante fixed set of proportions to this end, since when certain agents drop out from stage to stage of the auctioning process, either the fixed proportions for the remaining agents will not yield budget balance, or, if the proportions are updated, the remaining agents will have thereby received information about which other agents remain. This could result in the sort of ‘look ahead’ problems discussed above, that yield strategic reporting. A way to fix this problem is to use unequal proportions based in part on a random element. That is, the administrator could, at each stage, devise for each project a set of random prices that, if accepted, are budget balancing. Randomness would preclude strategic reporting, but violate the anonymity and no envy conditions. To summarize: it is possible to extend the equal share mechanism to one that is arbitrary, unequal share, and budget balancing, while preserving strategyproofness, but at the expense of being ‘fair’.
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Indeed, one might question the emphasis that we place on strategyproofness. In the problem we study, the agents are organizationally allied. Revelation of true preferences is necessary for the common good; strategic ‘no’ reports, designed to lower the chance that a less preferred technology gets funded, in the aggregate serve only to reduce the number of technologies that are eventually funded— increasing inefficiency. We have assumed throughout that knowledge of others’ budgets and benefits is private information. Were this condition abandoned, agents would again engage in manipulation of their reports. Example 2 demonstrates this, as follows. There are two agents, 1 and 2. Technologies A and B each cost 90; benefit B1A 580, B2A 5200, B1B 5140, and B2B 50. Agent 1 has a budget of 90, while 2’s is 50. Since the two agents together cannot fund both projects, the best available outcome is to fund one of them (ex post, the better choice is technology A). The mechanism will begin at stage 1 with the administrator setting a price of 45 for each project. Agent 1 will accept this price for both projects, while 2, with his more severe budget restriction, will accept only A. Technology A is therefore funded. At stage 2, both agents have insufficient balances to fund B. However, if the players had information about others’ budgets (and benefits), the following scenario might occur. Suppose agent 1 knew (or had a high prior) that 2’s budget was relatively small. Agent 1 could therefore suspect that 2 would have little or no money left at the second stage, and, in light of this, protect her relative interest in B by declining the price of 45 for A at the first stage. This action would leave 1, who accepts B at both stages, better off than by reporting honestly; but 2 is left with nothing. Furthermore, the outcome is not group optimal in the sense of pooled profits. If the agents knew one another’s benefits, there would be even sharper incentives to report strategically, and our mechanism would be fruitless. Another important point to be made is the inefficiency of the mechanism. While, at each stage, the individual agents can do no better than to report honestly, in an ex post sense (i.e., with public information), the final outcome may not be first best. We provide two types of such inefficiency. In the first example, we show that the final outcome may not be group optimal. Suppose, from Example 2 above, that the benefit to 1 from project B were in fact 300, and not 140. Now the group optimal result is to fund B and not A. However, the mechanism, as seen above, selects A. Even worse, the mechanism may fail to fund even a single technology which has unbounded profit! Consider a single technology of cost 1, and n agents whose benefits, in descending order, are (12 ´), (1 / 22 ´), (1 / 32 ´) . . . (1 / n2 ´), for some ´ .0. (Budgets need not be binding here.) At the first stage, the price is set to 1 /n, and the first (n21) of the agents will accept. At stage two, the price is increased to 1 /(n21); only the first (n22) of the active agents accept this higher price. This will continue until the last stage, when the price is set to 1, and the first agent will decline. As n increases, and ´ →0, the sum of the benefits grows without bound, as does the aggregate benefit-to-cost ratio. This example, however,
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is contrived, and its pathology does not necessarily imply that the mechanism will perform badly in practice. A limited series of ten runs of our procedure, with four agents and 8–10 projects, using randomly generated costs and benefits, has produced an average efficiency, ex post, of 0.38. (That is, the procedure funded projects yielding, on the average, 38% of the benefits that would have been obtained with complete and honest revelation.) Note that it is unclear whether abandoning strategyproofness will increase efficiency. Finally, we comment on the ‘knife-edge’ specificity of the mechanism. Why can the administrator not simply raise the bids by, say, fixed increments, as was the case in the FCC auctions? Our more exact approach is required here to be budget balancing. Again, the provision of semi-public goods that we study is fundamentally different from auctions. Were our mechanism to raise bids by fixed (or convex) increments, we would introduce the possibility of generating surpluses. Our aim is not to maximize bids for projects, but to raise precisely the required amounts to fund them, which is more consistent with the limited budgets that consortium members and divisions of firms must cope with.
5. Conclusions In this paper we have described a new problem involving selection of potential goods for a set of individuals who are organizationally allied but have individual, and private, preferences over the items, as well as budget limitations. We elicit their individual preferences by means of establishing an auction-like pricing mechanism for the goods over a finite number of stages. This mechanism, which limits the information accessible to the members, is shown to be strategyproof under the imposition of reasonable bounded rationality conditions. In addition, we show that the mechanism satisfies several well-known, desirable properties. As in Deb and Razzolini (1994), our model provides a bridge between work on provision of public goods and that on private goods. In a tactical sense, we are simply adapting an auction technique to a public goods problem. But in a deeper sense, the excludable public goods we study actually serve to model technology development problems for research consortia and for partially decentralized multi-divisional companies. There are a variety of directions for further research into this model. One is to assign utility functions to the agents, and probability distributions to the project outcomes. Since selected technologies would all be developed at once, the appropriate notion from the literature would be that of joint receipt (see Kahneman and Lovallo, 1993), where a bundle of received technologies is treated as a pooled lottery. The implications are that even risk averse individuals may be willing to bid on a larger subset of technologies by framing them as pooled, and not as separate, risks. Another direction is to attempt to develop a more relaxed framework and
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mechanism, which utilizes transfer payments among the members. However, it should be noted that the problem of adverse selection we face is a very difficult one. Our approach has been to develop a justifiable, bounded rationality decision environment, within which group members will optimize their selection vectors stage by stage. It is a challenge to find a broader model.
Acknowledgements Most of this work was carried out while visiting the Department of Operational Research, London School of Economics, on study leave from Temple University. I am grateful to Temple for its generosity and LSE for its warm hospitality. I also want to thank John Aloysius for many comments on an earlier draft. Finally, I am very grateful to an anonymous referee for many insightful, provocative, and helpful remarks on two earlier versions, which have led to a much improved article.
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