# Funding Public Projects: A Case for the Nash Product Rule

We study a mechanism design problem where a community of agents wishes to fund public projects via voluntary monetary contributions by the community members. This serves as a model for participatory budgeting without an exogenously available budget, as well as donor coordination when interpreting charities as public projects and donations as contributions. Our aim is to identify a mutually beneficial distribution of the individual contributions. In the preference aggregation problem that we study, agents report linear utility functions over projects together with the amount of their contributions, and the mechanism determines a socially optimal distribution of the money. We identify a specific mechanism—the Nash product rule—which picks the distribution that maximizes the product of the agents' utilities. This rule is Pareto efficient, and we prove that it satisfies attractive incentive properties: the Nash rule spends an agent's contribution only on projects the agent finds acceptable, and it provides strong participation incentives. We also discuss issues of strategyproofness and monotonicity.

## Authors

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• ### Fair Division with Binary Valuations: One Rule to Rule Them All

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• ### Participatory Funding Coordination: Model, Axioms and Rules

We present a new model of collective decision making that captures impor...
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• ### Fair mixing: the case of dichotomous preferences

Agents vote to choose a fair mixture of public outcomes; each agent like...
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• ### Participatory Budgeting with Cumulative Votes

In participatory budgeting we are given a set of projects—each with a co...
09/06/2020 ∙ by Piotr Skowron, et al. ∙ 0

• ### Fair and Efficient Allocations with Limited Demands

We study the fair division problem of allocating multiple resources amon...
02/28/2021 ∙ by Sushirdeep Narayana, et al. ∙ 0

• ### FaRM: Fair Reward Mechanism for Information Aggregation in Spontaneous Localized Settings (Extended Version)

Although peer prediction markets are widely used in crowdsourcing to agg...
06/10/2019 ∙ by Moin Hussain Moti, et al. ∙ 0

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## 1. Introduction

Many cities run participatory budgeting programs, in which they set aside a fraction of their budget and allow their residents to vote for project proposals—such as school renovations, park construction, or accessibility additions—usually using an online voting interface. This process allows city governments to discover which projects the public cares most about, and thus can lead to a more efficient use of the budget (Cabannes, 2004).

In a city, there is an authority which provides an exogenous budget to be spent. For other communities without such a budget, a similar system could help determine the best public projects to implement, but these projects will need to be funded via voluntary monetary contributions from community members. Examples of such communities include residents of an apartment complex (who want to coordinate spending on gardening in a courtyard, or on cleaning services), homeowners on a city street (to coordinate tree care, snow removal, or security patrols), or student clubs in a university (to coordinate funding for events and meet-ups). A simple solution for these use cases is to set up an account for each proposed project and invite community members to contribute to projects as they wish. But this can lead to Pareto dominated outcomes. For example, in an apartment complex consisting of several buildings, residents might each give to projects specific to their own building (because for each individual, those projects provide the greatest benefit), when all residents would agree that it would be better to pool the money for projects benefiting the entire complex. To discover this kind of improvement, a voting-based system seems promising: if agents report their preferences to a mechanism, it can aggregate the preferences to identify a socially optimal funding scheme. Our aim is to develop the theory of such systems, and to show that a mechanism based on maximizing the Nash product, that is, the product of the agents utilities, provides remarkably strong incentives for community members to contribute financially.

The public projects funded by these voting-based funding systems could, in particular, be charities. Many companies run programs to encourage their employees to give charitably. For example, Microsoft runs an annual charity matching program, and in 2019, employees donated a total of $181 million to charities. Donations to charities have the characteristics of public projects, since each donor prefers more money being allocated to charities she perceives as impactful. A voting-based system like the one we will propose can help the employees of a company to coordinate their giving, and thereby identify a more efficient money distribution. The system can also be used in higher-stakes applications involving major philanthropic foundations. Notably, the Open Philanthropy Project, which grants more than$100 million a year to various organizations, has called on academics to develop mechanisms to combine different staff members’ views on the most effective giving opportunities, and also to help coordinate the giving of different philanthropic organizations (Muehlhauser, 2017). Interest in donor coordination mechanisms has also been expressed in the effective altruism community (Peters, 2019).

We imagine the typical user experience in a funding system as follows. Users are presented with a list of proposed public projects. Among these, each user designates an arbitrary number of projects to be acceptable, meaning that the user will be satisfied with a distribution as long as the mechanism directs her contribution to these projects. Optionally, the user can specify fine-grained preferences among acceptable projects; in this paper we will allow cardinal utilities which are linear and weakly increasing in the amount spent on a project and are additive between projects. (Unlike some other works on participatory budgeting, we focus on divisible projects that can use an arbitrary amount of funding; this model is particularly suitable for charities.) Finally, the user is asked to commit a monetary amount to contribute to the funding system, and this money is transferred to the clearinghouse. The mechanism then aggregates the submitted preference information, decides how much money each project should receive, and then disburses the collected contributions to the projects accordingly.

As described, this system with pre-collected contributions requires the participants to trust that the system will in fact pay out the collected money (and not steal it). Our discussion is also applicable when this trust cannot be established (or when decentralization is desired); in such cases, we can interpret the mechanism output as recommendations to the agents about how they should distribute their contribution. However, this comes with the risk that participants will not follow these recommendations, and thereby jeopardize the efficiency and fairness properties of the mechanism’s overall distribution.

While many preference aggregation mechanisms are imaginable, we will focus on a specific mechanism called the Nash product rule and argue that it is remarkably well-adapted to the applications we have outlined. This mechanism performs a type of social welfare maximization. Specifically, it selects a distribution of the overall collected contributions which maximizes the product of voters’ utilities, with voters weighted by the size of their contribution. A voter ’s utility in distribution is defined as , where is the set of projects acceptable to , is the utility per unit of money that assigns to project , and is the amount spent on . (The formal definition implicitly sets the utility of unacceptable projects to .) Then the optimization objective will be , where is the size of ’s contribution. The idea of maximizing the product of voter utilities originates in the Nash bargaining solution, which is why we refer to this mechanism as the Nash product rule. The Nash product has recently become popular among researchers in various fields, including in the allocation of indivisible private items (Caragiannis et al., 2016), committee elections (Lackner and Skowron, 2018), and participatory budgeting (Fain et al., 2018, 2016). In all these areas, the Nash rule satisfies strong fairness and proportionality properties appropriate to the setting.

Since we are considering a model with voluntary contributions, certain incentive properties that have not been studied in other models become crucial for us. For starters, given the interpretation of voter preferences, it is important that the contribution of user is only spent on projects that are acceptable to —otherwise, the user can justifiably complain that the money is being misused. In the case where we interpret the mechanism output only as a recommendation, this property becomes essential, since a recommendation to send money to an unacceptable project is likely to be ignored. We say that a distribution is implementable if we can decompose it as such that spends exactly , and spends it only on projects acceptable to . In Theorem 4.1, we prove that the distribution selected by the Nash product is always implementable.

Our main result concerns participation incentives offered by the Nash product; we wish to assure agents that it is beneficial for them to contribute their money to the mechanism. This we achieve if the utility of an agent weakly increases with her contribution, where the utility of agent is the utility derives from the collective distribution minus the amount of money contributes. Thus, any additional money that contributes has to be spent so that it generates a higher utility for than keeping the money and the utility generated by the remaining contributions must not decrease. Consider for instance the mechanism that maximizes leximin welfare (see, e.g., Moulin, 2003) among all implementable distributions. This mechanism fails our condition, because it will take additional money as an opportunity to increase the welfare of low-utility agents, who may have interests different from . We introduce an axiom called pooling participation, requiring that ’s utility, upon contributing, is at least the utility would have derived from others’ money had not participated, plus the worst-case utility gets from spending her contribution on an acceptable project.222We also consider a version with best-case spending, but show that this version is incompatible with Pareto efficiency. We establish in Theorem 4.2 that the Nash product rule satisfies pooling participation. Our proof of this fact is rather involved, because we need to reason about the trajectory of the maximizer of the Nash product as agent

’s contribution continuously increases. Using the Taylor expansion of the Nash objective and a carefully chosen perturbation, we are able to estimate the derivative of

’s utility as ’s contribution changes, which after an integration yields the result.

Since the Nash product rule maximizes a monotonic function of voter utilities, its outcome is guaranteed to be Pareto efficient. This feature is important in order to gain user acceptance, and to avoid sub-optimal outcomes such as those discussed earlier. While Pareto efficiency is often considered a minimal condition, it can be difficult to achieve by mechanisms that need to satisfy additional incentive properties. Implementation and pooling participation can very easily be satisfied by non-efficient mechanisms;333For example, by the uncoordinated “mechanism” that tells each user to direct her entire contribution to the project to which she assigned the highest utility, without doing any aggregation of preferences. the remarkable feature of the Nash product is that it provides these incentive properties while ensuring Pareto efficiency. The Nash product is the only rule known to us that is both Pareto efficient and satisfies pooling participation. While we can construct other rules that are Pareto efficient and implementable, these are artificial, and Nash is the only natural such rule that we are aware of.

While the Nash product rule is incentive-compatible in the sense of implementability and pooling participation, it still allows for other strategic behavior. In particular, agents may have an incentive to misrepresent their utility functions. Because the Nash product penalizes distributions in which some agents obtain very low utility, it can be beneficial for agents to pretend to like popular projects less, or even to mark them as unacceptable. This will make the Nash product rule worry that those agents will be underserved, and thus increase the funding of other projects acceptable to them. Unfortunately, by a result due to Hylland (1980, Thm. 2), every Pareto efficient mechanism will be vulnerable to misrepresentation of preferences. In Section 5, we sharpen this result by showing that every (symmetric) mechanism that is Pareto efficient and implementable can be manipulated by the same strategy effective in the case of the Nash product rule, namely by understating one’s utilities. We additionally note that the Nash product rule fails a natural monotonicity property, and we leave the intriguing open question whether there exists a mechanism that is monotonic and matches the other axiomatic virtues of the Nash rule.

In Section 6, we discuss strengthened versions of implementability and pooling participation that are appropriate for a utility model in which agents do not distinguish only between acceptable and unacceptable projects, and when agents have maximally attractive outside options. While these two stronger properties may be desirable in the context of donor coordination, we show that each of them is incompatible with Pareto efficiency.

Overall, our discussion suggests that the Nash rule is a prime candidate for deployment in participatory budgeting settings with voluntary contributions. It combines Pareto efficiency with strong incentive properties, and as we will mention in Section 2, it also satisfies important fairness and proportionality properties. Finally, the rule is simple to define, can be computed to arbitrary precision using convex programming, and because it is implementable, its distribution decisions can be easily understood by users.

## 2. Related Work

Participatory budgeting has mostly been studied under the assumption that the budget is provided by an outside source (such as the city government); the recent literature on aggregation mechanisms was surveyed by Aziz and Shah (2020). In the most common model, projects come with a fixed cost, and they can either be fully funded or not at all. For this model, researchers have studied issues of preference elicitation (Benade et al., 2017; Goel et al., 2019), monotonicity properties (Talmon and Faliszewski, 2019), and proportionality concepts (Aziz et al., 2018; Fain et al., 2018). In the case where each project has the same cost (the unit cost case), this model captures committee elections, which have received much recent attention. The literature on committee elections compares different voting rules, studies their winner determination problems, and considers axioms; for an overview, see Faliszewski et al. (2017).

In a related model, the projects can receive an arbitrary amount of funding (like in our paper) but the budget is still exogenous and of fixed size. In the computer science literature, the study of this setting was initiated by Fain et al. (2016), who argued that allocations in Lindahl equilibrium (Foley, 1970) are particularly desirable. The Lindahl equilibrium is a market equilibrium in an artificial market for public goods. In these markets, each agent faces personalized prices (usually interpreted as taxes) for the public goods, and in equilibrium each agent demands the same bundle of public goods. Under standard assumption, Foley (1970) showed that a Lindahl equilibrium exists (by reduction to the Arrow–Debreu private goods case), and is Pareto efficient. He also showed that equilibrium allocations are in the core: no coalition of agents can afford (using only a fraction of the budget proportional to their size) an allocation that each coalition member prefers to the equilibrium. For the case of additive linear utilities, Fain et al. (2016) proved that the Nash product rule yields an allocation in Lindahl equilibrium, and hence is in the core.444This mirrors the canonical result that the Nash product yields an equilibrium in Fisher markets for private goods under additive valuations (Eisenberg and Gale, 1959). The core can be interpreted as guaranteeing agents proportional representation: if a fraction of of voters assign positive utility only to some set of projects, then the Nash product rule will spend at least of the budget on projects in .

Our model is also closely related to probabilistic social choice (see, e.g., Brandt, 2017)

: a division of a fixed budget among projects is the same as a probability distribution over alternatives.

Bogomolnaia et al. (2005) have studied the special case of dichotomous preferences in probabilistic social choice and introduced several rules (including the Nash product rule). Their results can be interpreted as results for participatory budgeting with projects that can receive an arbitrary amount of funding and where each voter assigns only utilities of either 0 or 1 to projects. Later, Duddy (2015) introduced another rule for this setting, and Aziz et al. (2019) undertook a more thorough axiomatic analysis. In particular, Aziz et al. (2019) showed that the Nash product rule guarantees average fair share: for any group of of the voters which is cohesive (there is a project that they all approve), for an average group member, the Nash product spends at least of the budget on approved projects. They also proved that Nash satisfies strict participation. This property, which was introduced by Brandl et al. (2015), makes sense for the fixed-budget setting, but in our variable-budget interpretation it is weak.555For dichotomous preferences, strict participation implies that if before contributing, of others’ money was spent on ’s approved projects, then strictly more than of ’s additional contribution will be spent on ’s approved projects, while others’ money is not spent in a worse way for . On the other hand, our pooling participation ensures that if agent contributes money, all of it will be spent on ’s approved projects, while again others’ money is not spent in a worse way for . Our main result that Nash satisfies pooling participation implies Aziz et al.’s (2019) result.

The key difference between all of the above literatures and our model is that in our model, the individual contributions to the endowment are owned by the agents. This suggests the definitions of the axioms of implementability and pooling participation, which—to the best of our knowledge—have not been considered in previous work. On the other hand, as we already mentioned, the Nash product rule has featured prominently in many streams of research, and has been shown to have desirable fairness and efficiency properties in many settings (e.g., Caragiannis et al., 2016; Fain et al., 2016; Conitzer et al., 2017; Aziz et al., 2019; Fain et al., 2018).

A major application of our model is for allocating donations to charities. Conitzer and Sandholm (2004, 2011) have also considered this application from a mechanism design perspective. They let agents incentivize other agents to donate more by devising “matching offers”, where a donation is made conditional on how much other agents donate, and to which charities. Rather than using matching offers, in our model, donations are incentivized by (implicitly) letting agents vote over how others donate their money, and weighting votes by the size of the voter’s contribution.

## 3. Model and Axioms

Let be a finite set of projects (e.g., charities or joint activities) and a finite set of agents. For each , agent ’s contribution is , giving rise to a total endowment of .

A distribution is a function that describes how some amount (e.g., an individual contribution or the entire endowment) is distributed among the projects, so such that . For convenience, we write distributions as linear combinations of projects, so that denotes for distribution with and . The set of all distributions of value is denoted by . Each distribution of the endowment can be divided into individual distributions such that . Clearly, the division of a distribution into individual distributions is not unique.

Apart from her contribution, each agent possesses a utility function , where describes how much utility an agent derives if one unit of money goes to project . So agent ’s utility for a distribution is

 ui(δ)=∑x∈Aδ(x)⋅ui(x).

A project is said to be acceptable by an agent if it gives her positive utility, and unacceptable otherwise. We assume that the utility assigned to unacceptable projects is .666If we allowed agents to differentiate between unacceptable projects, they could influence other agents’ distribution to those projects, while being guaranteed to never have their own contributions spent on unacceptable projects. This would lead to “bossiness”, so we prohibit this. We normalize utility functions such that the utility assigned to least-preferred acceptable projects is . (In the special case that an agent is completely indifferent, we label all projects as acceptable and so set all utilities to ). A utility function is dichotomous if we have for all , so that agent only distinguishes between acceptable and unacceptable projects without further discriminating between the acceptable ones. The type of an agent is and the set of all types . A profile assigns a type to every agent. We denote by the profile including all agents but . A mechanism is a function mapping a profile to a distribution .

We now discuss the main properties of distribution mechanisms that we are interested in. Recall that the purpose of such mechanisms is to identify a high-quality distribution and to exploit any synergies in the agents’ preferences. Thus, we aim for distributions that allocate a large portion of the endowment to projects that many agents value highly. On the other hand, we have to ensure that the distribution can be implemented even if agents are self-interested. In addition, it should be beneficial for agents to participate in the process of pooling their resources, since the presence of additional agents can only increase the potential for mutual benefits.

A minimal requirement for the first objective is Pareto efficiency (or Pareto optimality). Indeed, if a mechanism produces a distribution so that we could reshuffle money between projects and make every agent better off, it has not made full use of the potential for mutual gains.

###### Definition 3.1 (Efficiency).

Let be a type profile. A distribution Pareto dominates another distribution if for all and for some . A distribution is efficient if no distribution dominates it.

We say that a mechanism is efficient if is efficient for all .

Depending on the concrete application, the mechanism might operate in a decentralized setting, and may not be able to directly control the use of the agents’ contributions (for example, when a donor coordination service does not actually collect money from its participants). In such cases, the mechanism’s output is better understood as a recommendation to the agents about how they should use their resources. We would then need to decompose into personalized recommendations , so that if every agent spends her money according to , we recover . In order for the distribution to be implementable, it should ask every agent to spend her contributions exclusively on acceptable projects.

###### Definition 3.2 (Implementability).

Let be a type profile. A distribution is implementable if it can be divided into individual distributions with for all and such that for all , we have only if .

We say that a mechanism is implementable if is implementable for all . Implementability thus refers to the implementation of payments in a decentralized setting rather than an implementation in truthful strategies as often considered in mechanism design. In Section 6, we discuss a strengthening of implementability which requires that only if has assigned maximum utility to , i.e., only if for all . However, this requirement turns out to be too strong; it clashes with efficiency.

We cannot force agents to participate in the mechanism. Even when not participating, agents still benefit from the projects funded by the participating agents; this fact reduces participation incentives (a kind of free-riding). However, there is a downside to not participating: the mechanism only considers the utility functions of those who contribute to the mechanism. Thus, agents who do not participate forego the opportunity to influence the spending of other agents.

To motivate the definition of our participation axiom, focus on agent , and recall that we have normalized ’s reported utilities so that the utility of each acceptable project is at least 1. Agent considers two possible actions:

1. contribute to the mechanism and instead spend the amount on some outside option that gives a utility of 1 per unit of money, for a total utility of ;

2. contribute to the mechanism, for a total utility of .

Pooling participation requires that obtains at least as much utility when choosing action (2) than when choosing (1).

###### Definition 3.3 (Pooling participation).

A mechanism satisfies pooling participation if for each and , .

While pooling participation is phrased as a choice between contributing or , it will also imply a similar property saying that it is beneficial for agents to increase their contribution from to ; this implication holds for all reasonable rules that treat agents with the same utility function as a single agent (like the Nash product rule does).

In Section 6, we again discuss a strengthening of pooling participation, where the quality of the outside option is higher, so that obtains utility when choosing action (1). Again, this stronger version is incompatible with efficiency.777Note that implementability and pooling participation are logically independent properties, even when utilities are dichotomous.

## 4. The Nash Product Rule

The Nash product, which refers to the product of agent utilities, is often seen as a compromise between utilitarian and egalitarian welfare (Moulin, 1988). Maximizing the Nash product has been found to yield fair and proportional outcomes in many preference aggregation settings, and it also turns out to be attractive in our context. Formally, it is defined as follows.

 NASH(θ)=argmaxδ∈Δ(C)∏i∈N(ui(δ))Ci=argmaxδ∈Δ(C)∑i∈NCilog(ui(δ)).

The distribution maximizes a sum of functions, namely , that are strictly increasing in the agents’ utilities. Thus, it is efficient (see, e.g., Moulin, 1988).

A simple example of for a type profile with two agents and two projects is shown in Table 1. We have . Pooling participation is satisfied in this example because and . On the other hand, simply maximizing utilitarian welfare—that is, the sum of individual utilities—in this example would result in , which violates both implementability, as project is unacceptable for agent 1, and pooling participation because agent 1 would prefer an outside option to participating in the mechanism.

In general, can be efficiently approximated using convex programming (see, e.g., Bogomolnaia et al., 2005). However, as can return distributions with irrational values (see in Figure 1 for such an example), its exact computation is not possible in general.

It is not obvious whether the distribution returned by is always implementable or if satisfies pooling participation. We will deal with these questions in turn and find that the answer in both cases is yes. The proof for pooling participation is rather involved.

The Nash distribution is the solution to an optimization problem, and thus satisfies the first-order conditions of optimality. By manipulating these conditions, we can show that is implementable.888This proof is similar to a result by Guerdjikova and Nehring (2014) who consider with dichotomous preferences, and establish an equivalent property in this restricted setting.

###### Theorem 4.1 ().

satisfies implementability.

###### Proof.

We have to show that there is a decomposition of into , , such that for all .

We consider the Karush–Kuhn–Tucker conditions and write the Lagrangian as

 L(δ,λ,μ1,…,μm)=∑i∈NCilog(ui(δ))+λ(C−∑x∈Aδ(x))+∑x∈Aμxδ(x),

where is the Lagrange multiplier for the constraint and is the multiplier for the constraint .

Suppose is an optimal solution. By complementary slackness, we must have whenever . Also, we must have , that is, . By case distinction based on whether , it follows that for all . Hence,

 λ⋅C=∑x∈Aλδ(x)=∑x∈A∑i∈NCiδ(x)ui(x)/ui(δ)=∑i∈NCiui(δ)/ui(δ)=∑i∈NCi=C.

So , and hence for all such that .

Now, for each , define an individual distribution with for all . Clearly, and , since . To see that , note that for with we have for all , and for with , we have

 ∑i∈Nδi(x)=∑i∈NCiδ(x)ui(x)/ui(δ)=δ(x)∑i∈NCiui(x)/ui(δ)=δ(x).\qed

By inspecting the proof, we see that the distribution of agent even satisfies a stronger notion of implementability: the fraction of her contribution that she gives to project is proportional to the utility she derives from in the Nash distribution (see also Guerdjikova and Nehring, 2014). So if for example, half of agent ’s utility is due to the amount spent on , then she transfers half of her contribution to . Thus, given , agents can easily compute their individual distributions without the need for a central clearinghouse to tell the agents their individual distributions. Agents do not even need to know the other agents’ utility functions or contributions.

The proof that satisfies pooling participation is technically involved and requires a number of lemmas, whose proofs we defer to the appendix. At a high level, we estimate the rate of change of an agent’s utility as her contribution increases, and integrate this quantity as she goes from not participating to participating in the mechanism to obtain the desired result. The estimation entails expressing the logarithm of the utilities as a Taylor expansion and analyzing the relationship between the change in an agent’s contribution and the change in these utilities at the distribution returned by .

###### Theorem 4.2 ().

satisfies pooling participation.

Recall that we normalized utilities so that the utility assigned to least-preferred acceptable projects is 1 and so that the utility assigned to unacceptable projects is 0. Formally, in order to prove Theorem 4.2, we must show that for all and , . Consider the function with for all , where denotes the sum of the contributions in . We will show that

 (1) ui(g(θ))≥1Cθ((Cθ−Ci)ui(g(θ−i))+Ci),

which is equivalent to the inequality above for .

Denote by the polytope of feasible utility profiles scaled by , i.e., . Note that is convex. For , let .

###### Lemma 4.3 ().

For all , has a unique maximizer . Moreover, for all .

Since by Lemma 4.3, has a unique maximizer for all , we can define the function that returns this unique maximizer. Observe that for all .

###### Lemma 4.4 ().

is continuous in on and is weakly increasing in for all .

###### Lemma 4.5 ().

For every and , there is such that for all with and , we have for all .

The next three lemmas will be useful for analyzing error terms obtained in the main analysis.

###### Lemma 4.6 ().

Let , , and such that . Then,

 ∑i∈NCidUiUi≤0.

If also , then equality holds.

###### Lemma 4.7 ().

Let , , and such that and for all . Then,

 ∑i∈NCix2i≤αβ∑i∈NCi.
###### Lemma 4.8 ().

For all there is with the following property: For any such that and

 αt−(1+ε)βt2≤Φ(t)≤αt−(1−ε)βt2,

for some and and all , it holds that .

###### Proof of Theorem 4.2.

We want to prove (1) with as the focal agent. The first step, and also the bulk of the proof, is to derive a lower bound on the derivative of as a function of whenever . Then, integrating this derivative and using monotonicity of (see Lemma 4.4) gives (1).

###### Step 1.

Let such that and . Moreover, let and let be such that the conclusion of Lemma 4.8 holds; let . Considering the Taylor expansion of the logarithm, exists such that for all and ,

 (2) ∣∣∣log(Ui+r)−logUi−rUi+12(rUi)2∣∣∣≤ε4(rUi)2.

Now let such that with and for all , and let . Consider the function defined on with , such that

 ϕ(dU):=Fθ′(U+dU)−Fθ(U)−dC1logU1=∑i∈NCidUiUi+dC1dU1U1−ψ(dU),

for some with

 (1−ε)12∑i∈NCi(dUiUi)2≤ψ(dU)≤(1+ε)12∑i∈NCi(dUiUi)2.

The existence of is guaranteed by (2) and the bound on .

Now let and . Note that, since the only term in that depends on is , maximizes among all with . By Lemma 4.5, there is such that, for all with and , we have for all . Since is continuous in by Lemma 4.4, will be small if is small and we can choose to be even smaller if necessary so that . Then, the function with is well-defined and satisfies the prerequisites of Lemma 4.8 with

 α=∑i∈NCidU′iUi+dC1dU′1U1andβ=12∑i∈NCi(dU′iUi)2.

Hence, it follows from Lemma 4.8 that

 ∑i∈NCidU′iUi+dC1dU′1U1≥μΦ(1).

Since maximizes , by Lemma 4.6, . It follows that

 (3) dC1dU′1U1≥μΦ(1).

Next, let . Let be the fraction spent on agent 1’s acceptable projects, i.e., those that agent 1 assigns positive utility. Recall that , and so . Since satisfies the positive share property, i.e., it gives each agent positive utility, we have . From , we get that for all . Thus, for small enough, take the distribution with for all with and for all with . One can check that :

 ∑a∈Aδt(a) =∑a∈Au1(a)>0(1+t)δ(a)+∑a∈Au1(a)=0(1−H11−H1t)δ(a) =(1+t)∑a∈Au1(a)>0δ(a)+(1−H11−H1t)∑a∈Au1(a)=0δ(a) =(1+t)H1+(1−H11−H1t)(1−H1)=1.

Let . For small enough, we have that and . Indeed, , and for the second statement we can perturb infinitesimally in the opposite direction. This is a valid perturbation because for all , and for such that we have . Thus, by Lemma 4.6, we have

 ∑i∈NCidUtiUi=0.

So for sufficiently small , we have

 ϕ(dUt)=dC1dUt1U1−ψ(dUt)≥dC1dUt1U1−(1+ε)12∑i∈NCi(dUtiUi)2.

Since , we have that . Similarly, it follows that for all .

Now, by definition of , we have . Thus . Hence . Thus, applying Lemma 4.7 with , , and , it follows that

 ϕ(dUt)≥dC1t−(1+ε)12U1Cθ1−U1t2.

Now let . If is small enough, then is also small enough and, recalling that maximizes among all with , we get

 Φ(1)=ϕ(dU′)≥ϕ(dUt)≥12(1−ε)1−U1U1Cθ(dC1)2.

Thus, by (3), we get

 dC1dU′1U1≥μ2(1−ε)1−U1U1Cθ(dC1)2,

from which it follows from that

 dU′1≥μ2(1−ε)1−U1CθdC1.

Since was arbitrary and can be chosen arbitrarily small, it follows that

 dU′1≥1−U1CθdC1.
###### Step 2.

Now let be arbitrary. We show (1). Let such that and for all , , and . Let .

###### Case 1.

Consider the case . By Lemma 4.4, is monotonic in , and so

 ~U1(C1)≥~U1(0)=1Cθ((Cθ−C1)~U1(0)+C1~U1(0))≥1Cθ((Cθ−C1)~U1(0)+C1)

which proves (1).

###### Case 2.

If , let be arbitrary. By creftypecap 1, the lower right derivative of at is at least . Integrating this estimate from to yields

 −∫C′1ϵ∂~U1(s)∂s1−~U1(s)ds≤−∫C′1ϵ1Cθ−C1+sds

from which we get

 log(1−~U1(C′1))−log(1−~U1(ϵ))≤−(log(Cθ−C1+C′1)−log(Cθ−C1+ϵ)).

Exponentiation yields . Since was arbitrary and is monotonic, we get . Rewriting this equation gives us

 ~U1(C′1)≥1Cθ−C1+C′1((Cθ−C1)~U1(0)+C′1)

Finally, we note that if , then . Together with monotonicity, this gives us

 ~U1(C1) =1Cθ((Cθ−C1+C′1)~U1(C′1)+(C1−C′1)~U1(C′1)) ≥1Cθ((Cθ−C1)~U1(0)+C′1+(C1−C′1)~U1(C′1))=1Cθ((Cθ−C1)~U1(0)+C1)

which is (1). ∎

is not the only efficient rule that satisfies implementability. Whether a distribution is efficient only depends on its support (see, e.g., Aziz et al., 2014). So we can, for example, take the support of the Nash distribution and let every agent assign her contribution to one of her most preferred projects within the support. Since the Nash distribution is implementable, this support contains at least one acceptable project for every agent. The allocation rule we obtain in this way is thus implementable, although it is artificial.

Efficiency and pooling participation seem to be harder to satisfy simultaneously. We are not aware of any rule other than that satisfies these properties.

## 5. Strategyproofness and Monotonicity

Even when agents do participate in the mechanism, they may be strategic and try to misrepresent their preferences in a way that induces the mechanism to choose a distribution that gives them higher utility. Mechanisms that are immune to strategic misrepresentation are called strategyproof.

###### Definition 5.1 (Strategyproofness).

A mechanism is strategyproof if for all with except , it holds that .

In other words, strategyproofness disincentivizes an agent from misreporting her utility function . The other part of her type is the contribution , which she might also misreport (more precisely, underreport), but this worry is already captured by pooling participation

In many mechanism design settings, only degenerate mechanisms that ignore most of the information, such as dictatorships, are strategyproof. Our setting is no exception: Hylland’s (1980) Theorem 2 implies that only mechanisms that allocate the entire endowment to the most preferred projects of a single fixed agent are efficient and strategyproof, when . If we look at specifically, it can be manipulated by agents who understate their utility for projects, i.e., by reporting a lower utility for some project, or by labeling a project as unacceptable when it really is acceptable. As we now show, this misrepresentation strategy can be used against every efficient and implementable rule. For technical reasons, we additionally add a mild symmetry axiom.

###### Definition 5.2 ().

A mechanism is symmetric if it assigns the same amount of funding to projects and whenever they are symmetric, in the sense that swapping and leaves the type profile as a whole unchanged, possibly after also renaming the agents.

###### Proposition 5.3 ().

Every symmetric mechanism which is efficient and implementable can be manipulated by understating utilities, when .

###### Proof.

We prove the incompatibility for and , and this proof (and also the following proofs) can be adapted to larger values by adding agents who are indifferent between all projects or by adding projects which no one finds acceptable.

Assume that is a symmetric mechanism which is efficient and implementable. Now consider a profile with uniform contributions ( for all agents ) and the sets of acceptable projects for each agent

 ({a},{abc},{abd},{ace},{de}).

We assume that each agent is indifferent between acceptable projects, so whenever is acceptable to . Let be the distribution returned by the mechanism. By implementability for 5, we must have . Since the profile is symmetric under the permutation , we can assume without loss of generality that . It follows that because a positive amount is spent on project , which is not one of agent 4’s acceptable projects.

Now, suppose that the fourth agent changes her utility for from 1 to 0, i.e., marks as unacceptable. Then we get the profile with the following sets of acceptable projects for each agent, with agents again being indifferent between all acceptable projects:

 ({a},{abc},{abd},{ce},{de}).

Let be the distribution now returned by the mechanism. By efficiency, we must have since otherwise we can redistribute resources from to to get a Pareto improvement. Next, suppose that both and are positive, say and for some . Then is Pareto dominated by the distribution obtained from by moving from to and from to . This contradicts efficiency of , so or . Since projects and are symmetric in , we must have . Hence, distributes the entire endowment between projects and , and so .

Thus, agent 4 has successfully manipulated by understating her utilities. ∎

Each use of the implementability axiom in this proof can easily be replaced by pooling participation. Hence, every efficient and symmetric mechanism that satisfies pooling participation can also be manipulated by understating utilities.

Proposition 5.3 is related to a result of Bogomolnaia et al. (2005, Prop. 6) which shows that no mechanism can satisfy efficiency, strategyproofness, and a condition called “fair welfare share” for dichotomous utility functions. Their proof works for cases with at least 17 agents and 5 projects.

We have focused on incentive issues for the voters, but the projects may also be strategic. The proposers or managers of specific projects wish to acquire funding, and charities hope for large donations. From this point of view, it is important that a mechanism satisfies monotonicity: a project should not receive less funding when it becomes more popular. A failure of monotonicity may result in perverse incentives for a project manager, such as deliberately making her own project seem unattractive to agents in order to acquire more funding.

###### Definition 5.4 (Monotonicity).

A mechanism is monotonic if for every agent , project , and type profiles with