Participatory Funding Coordination: Model, Axioms and Rules
PParticipatory Funding Coordination:Model, Axioms and Rules
Haris Aziz , and Aditya Ganguly UNSW Sydney, Australia { haris.aziz,a.ganguly } @unsw.edu.au Data61 CSIRO
Abstract.
We present a new model of collective decision making thatcaptures important crowd-funding and donor coordination scenarios. Inthe setting, there is a set of projects (each with its own cost) and aset of agents (that have their budgets as well as preferences over theprojects). An outcome is a set of projects that are funded along withthe specific contributions made by the agents. For the model, we identifymeaningful axioms that capture concerns including fairness, efficiency,and participation incentives. We then propose desirable rules for themodel and study, which sets of axioms can be satisfied simultaneously.An experimental study indicates the relative performance of differentrules as well as the price of enforcing fairness axioms.
Consider a scenario in which a group of house-mates want to pitch in money tobuy some common items for the house but not every item is of interest or use toeveryone. Each of the items (e.g. TV, video game console, music system, etc.)has its price. Each resident would like to have as many items purchased that areuseful to her. She may have concerns about whether she is getting enough valuefor the contribution she makes. It is a scenario that is encountered regularly innumerous shared houses or apartments.As a second scenario, hundreds of donors want to fund charitable projects.Each of the projects (e.g. building a well, enabling a surgery, funding a schol-arship, etc.) has a cost requirement. Donors care about coordinating their do-nations in a way to fund commonly useful projects and they care about theamount of money that is used towards projects that they approve. How to coor-dinate the funding in a principled and effective way is a fundamental problem incrowdfunding and donor coordination. The model that we propose is especiallysuitable for coordinating donoations from alumni at our university.Both of the settings above are coordination problems in which agents con-tribute money and they have preferences over the social outcomes. A collectiveoutcome specifies which projects are funded and how much agents are charged. a r X i v : . [ c s . G T ] J a n Haris Aziz and Aditya Ganguly
Contributions.
We propose a formal model that we refer to as
ParticipatoryFunding Coordination (PFC) that captures many important donor coordinationscenarios. In this model, agents have an upper budget limit. They want as manyof their approved projects funded. We lay the groundwork for work on the modelby formulating new axioms for the model. The logical relations between theaxioms are established and the following question is studied: which sets of axiomsare simultaneously achievable. We propose and study rules for the problemsthat are inspired by welfarist concerns but satisfy participation constraints. Inaddition to an axiomatic study of the rules, we also undertake an experimentalcomparison of the rules. The experiment sheds light on the impact that variousfairness or participation constraints can have on the social welfare. This impacthas been referred to as the price of fairness in other contexts. In particular, weinvestigate the effects of enforcing fairness properties on instances that modelreal-world applications of PFC, including crowdfunding.
Our model generally falls under the umbrella of a collective decision making set-ting in which agents’ donations and preferences are aggregated to make fundingdecisions. It is a concrete model within the broad agenda of achieving effectivealtruism (MacAskill, 2015, 2017, Peters, 2019).The model we propose is related to the discrete participatory budgetingmodel (Aziz and Shah, 2020, Aziz, Lee, and Talmon, 2018, Goel, K., Sakshuwong,and Aitamurto, 2019, Fain, Goel, and Munagala, 2016, Talmon and Faliszewski,2019). In discrete participatory budgeting, agents do not make personal dona-tions towards the projects. They only express preferences over which projectsshould be funded. We present several axioms that are only meaningful for ourmodel and not for discrete participatory budgeting. Algorithms for discrete par-ticipatory budgeting cannot directly be applied to our setting because they donot take into account individual rationality type requirements.Another related setting is multi-winner voting (Elkind, Faliszewski, Skowron,and Slinko, 2017). Multi-winner voting can be viewed as a restricted versionof discrete participatory budgeting. The Participatory Funding Coordination(PFC) setting differs from multi-winner voting in some key respects: in ourmodel, each project (winner) has an associated cost, and we select projectssubject to a knapsack constraint as opposed to having a fixed number of winners.Our PFC model relies on approval ballots in order to elicit agents’ preferences.Dichotomous preferences have been considered in several important setting in-cluding committee voting (Lackner and Skowron, 2019, Aziz, Brill, Conitzer,Elkind, Freeman, and Walsh, 2017) and discrete participatory budgeting (Azizet al., 2018, Fluschnik, Skowron, Triphaus, and Wilker, 2019).Another related model that takes into account the contributions of agentswas studied by Brandl, Brandt, Peters, Stricker, and Suksompong (2020). Justlike in our model, an agent’s utilities are based on how much money is spent onprojects approved by the agent. However, their model does not have any costs articipatory Funding Coordination: Model, Axioms and Rules 3 and agents can spread their money over projects in any way. Our model hassignificant differences from the model of (Brandl, Brandt, Peters, Stricker, andSuksompong, 2019, Brandl et al., 2020): (1) in our setting, the projects are indi-visible and have a minimum cost to complete, (2) agents may not be charged thefull amount of their budgets. The combination of these features leads to chal-lenges in even defining simple individual rationality requirements. Furthermore,it creates difficulties in finding polynomial-time algorithms for some natural ag-gregation rules (utilitarian, egalitarian, Nash product, etc.). Our model is moreappropriate for coordinating donations where projects have short-term deadlinesand a target level of funding which must be reached for the project to be suc-cessfully completed. We show that the same welfarist rules that satisfy somedesirable properties in the model (Brandl et al., 2019, 2020), fail to do so in ourmodel. Just as the work of Brandl et al. (2019, 2020), Buterin, Hitzig, and Weyl(2019) consider donor coordination for the divisible model in which the projectsdo not have costs and agents do not have budget limits. They also assume quasi-linear utilities whereas we model charitable donors who are not interested inprofit but want their money being used as effectively as possible towards causesthat matter to them.The features of our PFC model enable the model to translate smoothly to anumber of natural settings. Crowdfunding, in particular, is a scenario in whichwe would like to capitalise upon commonalities in donors’ charitable preferences(Corazzini, Cotton, and Valbonesi, 2015). Furthermore, crowdfunding projects(e.g. building a well, funding a scholarship, etc.) often have provision points(see e.g. (Agrawal, Catalini, and Goldfarb, 2013, Chandra, Gujar, and Narahari,2016, Damle, Moti, Chandra, and Gujar, 2019)), and it can be critical for thesetargets to be met (for example, a project to raise funds for a crowdfundingrecipient to pay for a medical procedure would have to raise a minimum amountof money to be successful, otherwise all donations are effectively wasted).Crowdfunding projects have been discussed in a broader context with variouseconomic factors and incentive issues presented (Agrawal et al., 2013). Bagnoliand Lipman (1989) discuss additional fairness and economic considerations forthe related topic of the division of public goods. The discrete model that weexplore, where projects have finite caps, has potential to coordinate donors andincrease the effectiveness of a crowdfunding system. A Participatory Funding Coordination (PFC) setting is a tuple (
N, C, A, b, w )where N is the set of agents/voters, C is the set of projects (also generallyreferred to as candidates). The function w : C → R + specifies the cost w ( c ) ofeach project c ∈ C . The function b : N → R + specifies the budget b i of eachagent i ∈ C . The budget b i can be viewed as the maximum amount of moneythat agent i is willing to spend. For any set of agents S ⊆ N , we will denote (cid:80) i ∈ S b i by b ( S ). The approval profile A = ( A , . . . , A n ) specifies for each agent,her set of acceptable projects A i . An outcome is a pair ( S, x ) where S ⊆ C is the Haris Aziz and Aditya Ganguly set of funded projects and x is a vector of payments that specify for each i ∈ N ,the payment x i that is charged from agent i . We will restrict our attention tofeasible outcomes in which x i ≤ b i for all i ∈ N and only those projects getfinancial contributions that receive their required amount. Also, note that theprojects that are funded are only those that receive the entirety of their price inpayments from the agents. For any given PFC instance, a mechanism F returnsan outcome. We will denote the set of projects selected by F as F C and thepayments by F x . For any outcome (
S, x ), since x i ≤ b i , the money b i − x i caneither be kept by the agent i or it can be viewed as going into some commonpool. The main focus of our problem is to fund a maximal set of projects whilesatisfying participation constraints.We suppose that an agent’s preferences are approval-based . For any set offunded projects S , any agent i ’s utility is u i ( S ) = (cid:88) c ∈ S ∩ A i w ( c ) . That is, an agent cares about how many dollars are usefully used on his/herapproved projects. Our preferences domain is similar to the one used by Brandlet al. (2020) who considered a continuous model in which projects do not havetarget costs. In their model, agents also care about how much money is used fortheir liked projects.
Example 1.
The following is an instance of a PFC problem with 5 agents and6 projects. The costs of the projects is stated next to the project name. Thebudget of each agent is mentioned in front of the agent name. The plus signindicates the approval of an agent for a project.
A (7) B (6) C (1) D (1) E (8) F (7)BudgetAgent 1 3 + + +Agent 2 3 + + +Agent 3 3 + + +Agent 4 2 + + +Agent 5 1 + +
Table 1: Example of an PFC instance.
In this section, we design axioms for outcomes of the PFC setting. We consideran outcome (
S, x ). For any axiom Ax for outcomes, we say that a mechanismsatisfies Ax if it always returns an outcome that satisfies Ax . PFC can also be viewed as a matching problem in which the money of agents ismatched to projects.articipatory Funding Coordination: Model, Axioms and Rules 5
We first present three axioms for our setting that are based on the principleof participation: – Minimal Return (MR) : each agent’s utility is at least much as the moneyput in by the agent: u i ( S ) ≥ x i . In other words, the societal decision is asgood for each agent i as i ’s best use of the money x i that she is asked tocontribute. We will use this as a minimal condition for all feasible outcomes. – Implementability (IMP) : There exists a payment function y : N × C → R + ∪ { } such that (cid:80) c ∈ C y ( i, c ) = x i for all i ∈ N and (cid:80) i ∈ N y ( i, c ) ∈{ , w ( c ) } and there exists no i ∈ N and c / ∈ A i such that y ( i, c ) >
0. IMPcaptures the requirement that an agent’s contribution should only be usedon projects that are approved by the agent. – Individual Rationality (IR): the utility of an agent is at least as much asan agent can get by funding alone: u i ( S ) ≥ max S (cid:48) ⊆ A i ,w ( S (cid:48) ) ≤ b i ( w ( S (cid:48) )) . Notethat IR is easily achieved if the project costs are high enough: if for i ∈ N and c ∈ C , w ( c ) > b i , then every outcome is IR.We note that MR is specified with respect to the amount x i charged to theagent. It can be viewed as a participation property: an agent would only wantto participate in the market if she gets at least as much utility as the moneyshe spent. We will show IMP is stronger than MR. IMP can also be viewed asa fairness property: agents are made to coordinate but they only spend theirmoney on the projects they like. Remark 1.
If there is an IMP outcome where a set of projects are funded, thenthere is also an IMP outcome where any subset of these projects are funded. Inorder to find an IMP outcome for any subset, simply take the original outcomeand set the payments of agents to projects that are being “de-funded” to zero.Next, we present axioms that are based on the idea of efficiency. – Exhaustive (EXH) : There exists no set N (cid:48) ⊆ N and project c ∈ C \ S such that c ∈ ∩ i ∈ N (cid:48) A i ∩ ( C \ A ) such that w ( c ) ≤ (cid:80) i ∈ N (cid:48) ( b i − x i ). In words,agents in N (cid:48) cannot pool in their unspent money and fund another projectliked by all of them. – Pareto optimality (PO)-X : An outcome ( S, x ) is Pareto optimal withinthe set of outcomes satisfying property X if there exists no ( S (cid:48) , x (cid:48) ) satisfyingX such that u i ( S (cid:48) ) ≥ u i ( S ) for all i ∈ N and u i ( S (cid:48) ) > u i ( S ) for some i ∈ N .Note that Pareto optimality is a property of the set of funded projects S irrespective of the payments. • PO is Pareto optimal among the set of all outcomes. • PO-IMP: PO among the set of IMP outcomes. • PO-MR: PO among the set of MR outcomes. – Payment constrained Pareto optimality (PO-Pay) : An outcome isPO-Pay if it is not Pareto dominated by any outcome of at most the sameprice. Formally, there exists no ( S (cid:48) , x (cid:48) ) such that (cid:80) i ∈ N x (cid:48) i ≤ (cid:80) i ∈ N x i , u i ( S (cid:48) ) ≥ u i ( S ) for all i ∈ N and u i ( S (cid:48) ) > u i ( S ) for some i ∈ N . One can strengthen MR to a stronger version in which u i ( S, x ) > x i for each i ∈ N . Haris Aziz and Aditya Ganguly – Weak Payment constrained Pareto optimality (weak PO-Pay) : Anoutcome is weakly PO-Pay if it is not Pareto dominated by any outcome ofat most the same price. Formally, there exists no ( S (cid:48) , x (cid:48) ) such that x (cid:48) i ≤ x i and u i ( S (cid:48) ) ≥ u i ( S ) for all i ∈ N and u i ( S (cid:48) ) > u i ( S ) for some i ∈ N .A concept that can be viewed in terms of participation, efficiency, and fairnessis the adaptation of the principle of core stability for our setting. – Core stability (CORE) : There exists no set of agents who can pool intheir budget and each gets a strictly better outcome. In other words, anoutcome ( S ) is CORE if for every subset of agents N (cid:48) ⊆ N , for every subsetof projects C (cid:48) ⊆ C such that w ( C (cid:48) ) ≤ (cid:80) i ∈ N (cid:48) b i , the following holds for someagent i ∈ N (cid:48) : u i ( S ) ≥ w ( C (cid:48) ∩ A i ) . We describe a basic fairness axiom for outcomes and rules based on the ideaof proportionality. – Proportionality (PROP) : Suppose a set of agents N (cid:48) ⊆ N only approveof a set of projects C (cid:48) ⊆ C such that (cid:80) i ∈ N (cid:48) b i ≥ w ( C (cid:48) ). In that case, all theprojects in C (cid:48) are selected.Finally, we consider an axiom that is defined for mechanisms rather thanoutcomes. We say that a mechanism satisfies strategyproofness if there existsno instance under which some agent has an incentive to misreport her preferencerelation.We conclude this section with some remarks on computation. The followingproposition follows via a reduction from the Subset Sum problem. Proposition 1.
Even for one agent, computing an IR, PO, PO-MR, or PO-IMP outcome if NP-hard.Proof.
Consider the Subset Sum problem in which there is a set of items M = { , . . . , m } with corresponding weights w , . . . , w m , and a real value W . Theproblem is to find a subset S with maximum weight (cid:80) j ∈ S w j such that (cid:80) j ∈ S w j ≤ W . The problem is well-known to be NP-hard. We reduce it our setting for asingle agent by taking an item for each project that our agent approves of, andchoosing the item weights to be the corresponding project costs. Then any set ofprojects S satisfies the axioms in the proposition if and only if the correspondingset of items is the solution to the Subset Sum problem.Note that IMP is a property of an outcome not a set of projects. We say thata set of projects S is IMP if there exists a feasible vector of charges to agents x such that the outcome ( S, x ) is IMP. The propoerty IMP can be tested inpolynomial time via reduction to network flows.
Proposition 2.
For a given set of projects S , checking whether there exists avector of charges x such that ( S, x ) is implementable can be done in polynomialtime. Similarly, we can also check whether a particular outcome (
S, x ) is imple-mentable with a variation of the above linear program, where the upper boundon the sum of payments for each agent is x i instead of b i . articipatory Funding Coordination: Model, Axioms and Rules 7 In this section, we study the compatibility and relations between the axiomsformulated.
Remark 2.
Note that IR and MR are incomparable. Any outcome in which anagent does not pay any money trivially satisfies MR. However, it may not satisfyIR. On the other hand, an IR outcome may not be MR. Consider the case inwhich an agent’s utility is at least as high as by funding alone. However, theagent may have been asked to pay more than the utility she gets which violatesMR.Next, we point out that that PO-Pay is equivalent to weak PO-Pay.
Proposition 3.
PO-Pay is equivalent to weak PO-Pay.Proof.
Suppose an outcome (
S, x ) is not weakly PO-Pay. Then, it is trivially notPO-Pay. Now suppose (
S, x ) is not PO-Pay. Then, there exists another outcome( S (cid:48) , x (cid:48) ) such that (cid:80) i ∈ N x (cid:48) i ≤ (cid:80) i ∈ N x i , u i ( S (cid:48) ) ≥ u i ( S ) for all i ∈ N and u i ( S (cid:48) ) >u i ( S ) for some i ∈ N . Note that S (cid:48) can be funded with total amount (cid:80) i ∈ N x (cid:48) i irrespective of who paid what. So S (cid:48) is still affordable if x (cid:48) i ≤ x i .The next proposition establishes further logical relations between the axioms. Proposition 4.
The following logical relations hold between the properties.1. IMP implies MR.2. PO implies PO-Pay.3. PO- X implies PO- Y if Y implies X .4. PO-IMP implies EXH.5. PO-IR implies EXH.6. CORE implies IR.7. The combination of PO-IMP and IMP imply PROP. Next, we show that MR is compatible with PO-Pay.
Proposition 5.
Suppose an outcome is MR and there is no other MR outcomethat Pareto dominates it. Then, it is PO-Pay.Proof.
Suppose the outcome (
S, x ) is MR and PO constrained to MR. We claimthat (
S, x ) is PO-Pay. Suppose it is not PO-Pay. Then there exists anotheroutcome ( S (cid:48) , x (cid:48) ) such that (cid:80) i ∈ N x (cid:48) i ≤ (cid:80) i ∈ N x i , u i ( S (cid:48) ) ≥ u i ( S ) for all i ∈ N and u i ( S (cid:48) ) > u i ( S ) for some i ∈ N . Note that S (cid:48) is afforadable with totalamount (cid:80) i ∈ N x (cid:48) i irrespective of who paid what. So S (cid:48) is still affordable if x (cid:48) i ≤ x i . Therefore, we can assume that x (cid:48) i ≤ x i for all i ∈ N . Note that since S (cid:48) Pareto dominates S and since ( S, x ) is MR, u i ( S (cid:48) ) ≥ u i ( S ) ≥ x i ≥ x (cid:48) i for all i ∈ N . Hence ( S (cid:48) , x (cid:48) ) also satisfies MR. Since ( S (cid:48) , x (cid:48) ) is MR and since S Paretodominates S (cid:48) , it contradicts the fact that ( S, x ) PO constrained to MR.
Haris Aziz and Aditya Ganguly
Proposition 6.
There always exists an outcome that satisfies IMP, IR, PO-IMP and hence also MR and EXH.Proof. Existence of an outcome that satisfies IMP, IR, PO-IMP : For each i ∈ N compute ( S i , y i ) that is an IR outcome. This can be computed by finding a max-imum total weight set of projects that has weight at most b i . Then consider theoutcome ( (cid:83) i ∈ N S i , ( y , . . . , y n )). In such an outcome, we also keep track of whichagent contributed to which project. Note that if c ∈ S i , then i contributed w ( c )to that project. Note that w ( (cid:83) i ∈ N S i ) ≥ (cid:80) i ∈ N y i . If w ( (cid:83) i ∈ N S i ) > (cid:80) i ∈ N y i ,we need to return w ( (cid:83) i ∈ N S i ) − (cid:80) i ∈ N y i , back to the agents to ensure that nomore money is charged than needed to pay for (cid:83) i ∈ N S i . We return the moneyas follows. Recall that we know the amount paid by each agent to each project,i.e., agent i paid w ( c ) to project c if and only if c ∈ S i . Some projects may havereceived more money than needed. For each project c ’s surplus, we uniformly al-locate it among the agents who paid for it. Suppose the outcome satisfying IMP,IR and EXH does not satisfy PO-IMP. Then there exists another outcome thatsatisfies IMP that Pareto dominates the outcome. Such a Pareto improvementstill satisfies IR because the utility of each agent is at least as high.Note that PO-Pay and IMP are both satisfied by an empty outcome with zerocharges. PO-IMP and IMP are easily satisfied by computing a PO outcome fromthe set of IMP outcomes. PO-Pay and PO-IMP are easily satisfied by computinga PO outcome which may not necessarily satisfy IMP. Proposition 7.
There always exists an outcome that satisfies MR, IR, PO-MRand hence also EXH.Proof. Existence of an outcome that satisfies MR, IR, PO-MR : From the proofof part (i) , we know that an IMP and IR outcome always exists. Also, fromProposition 4 we know that every IMP outcome is MR, so there always exists anMR and IR outcome. Now suppose the outcome satisfying MR and IR does notsatisfy PO-MR. Then there exists another outcome satisfying MR that Paretodominates the original outcome, which is still IR. There cannot exist an infinitenumber of Pareto improvements because the budgets of the agents are finite.Hence we can reach a PO-MR outcome that is also IR and MR.We note that if no agent can individually fund a project, then every out-come is IR. In crowdfunding settings in which projects have large costs, the IRrequirement is often easily satisfied.
In this section, we take a direct welfarist view to formalize rules that maxi-mize some notion of welfare. We consider three notions of welfare: utilitarian,egalitarian, and Nash welfare; and we define the following rules. – UTIL: define the utilitarian welfare derived from an outcome (
S, x ) as (cid:80) i ∈ N u i ( S ) . Then, UTIL returns an outcome that maximises the utilitarian welfare. articipatory Funding Coordination: Model, Axioms and Rules 9 – EGAL: given some output (
S, x ), write the sequence of agents’ utilities fromthat outcome as u ( S ) = ( u i ( S )) i ∈ N , where u is sorted in non-decreasing or-der. Then, EGAL returns an outcome ( S, x ) such that u ( S ) is lexicographi-cally maximal among the outcomes. – NASH: maximises the Nash welfare derived from an output (
S, x ), i.e. (cid:81) i ∈ N ( u i ( S )) . Proposition 8.
UTIL, EGAL, and NASH satisfy PO and hence PO-MR, PO-IMP, PO-Pay, and EXH.
One notes that the rules UTIL, EGAL, and NASH do not satisfy minimalguarantees such as MR. The reason is that an agent may donate her budget to awidely approved project even though she may not approve any of such projects.Given that the existing aggregation rules do not provide us with guarantees thatthe outcomes they produce will satisfy our axioms, we can instead define rulessuch that optimize social welfare within certain subsets of feasible outcomes.For a property X, we can define UTIL-X, EGAL-X, and NASH-X as rules thatmaximise the utilitarian, egalitarian and Nash welfare respectively among onlythose outcomes that satisfy property X. Next, we analyse the properties satisfiedby rules EGAL/UTIL/NASH constrained to the set of MR or IMP outcomes.In the continuous model introduced by Brandl et al. (2020), there is no needto consider the rule NASH-IMP, as the NASH rule in the case where projectscan be funded to an arbitrary degree (given there is sufficient budget) alreadysatisfies IMP.Before we study the axiomatic properties, we note that most meaningfulaxioms and rules are NP-hard to achieve or compute. The following propositionfollows from Proposition 1.
Proposition 9.
Even for one agent, computing an UTIL, UTIL-MR, UTIL-MR, EGAL, EGAL-MR, EGAL-IMP, NASH, NASH-MR, NASH-IMP outcomeis NP-hard.
Similarly, the following proposition follows from Proposition 5.
Proposition 10.
UTIL-MR, EGAL-MR, and NASH-MR satisfy PO-Pay.
From Proposition 5, it follows that UTIL-MR, EGAL-MR, and NASH-MRsatisfy PO-Pay. In contrast, we show that UTIL-IMP, EGAL-IMP, and NASH-IMP do not satisfy PO-Pay. In order to show this, we prove that it is possible insome instances for the set of jointly IMP and PO-IMP outcomes to be disjointfrom the set of PO-Pay outcomes.
Proposition 11.
UTIL-IMP, EGAL-IMP and NASH-IMP do not satisfy PO-Pay. In fact it is possible that no IMP and PO-IMP outcome satisfies PO-Pay.Proof.
Consider the following instance in Table 2.
Claim.
Observe that no implementable outcome can fund project E since it istoo expensive to be funded solely by its supporters. Table 2: Example instance for proof of UTIL/EGAL/NASH-IMP not satisfyingPO-Pay.
Claim.
Any implementable outcome funds a subset of the following project sets: { A, B, C } , { B, D } . Note that for an implementable outcome, if D is funded, thenonly B can also be funded (there is not enough money for agents who approveof A or C to fund these projects after funding D ). A,B,C B,D D,E (not IMP)Agent 1 4 4 7Agent 2 7 0 7Agent 3 7 0 7Agent 4 7 7 7Agent 5 3 7 7
Table 3: Utilities provided to each agent by outcomes that fund the project sets { A, B, C } , { B, D } and { D, E } .Note that when we are looking for the optimal outcome under a certain rule,we can ignore those project sets that are subsets of other project sets. Then,from Table 3, we see that the unique UTIL-IMP, EGAL-IMP and NASH-IMPoutcome is the outcome that funds { A, B, C } where each agent sends all theirmoney to the only project of those three that they approve of. But clearly, thisoutcome is Pareto dominated by any outcome that funds { D, E } , which has thesame total cost. Thus we have shown that UTIL-IMP, EGAL-IMP and NASH-IMP do not satisfy PO-Pay in general. In fact the set of IMP and PO-IMPoutcomes can be disjoint from the set of PO-Pay outcomes.The most striking aspect of Proposition 11 is that in the continuous domainwithout project costs, NASH-IMP is equivalent to NASH and the rule satisfiesPareto optimality and hence PO-Pay. In our context, NASH-IMP fails to satisfyPO-Pay.We also considered the issue of strategyproofness and found examples thatshow that none of the UTIL/EGAL/NASH rules are strategyproof whether theyare unconstrained or constrained to MR or IMP outcomes. In contrast, there areseveral natural rules such as UTIL that are strategyproof in the continuoussetting as well in the multi-winner voting setting. articipatory Funding Coordination: Model, Axioms and Rules 11 Proposition 12.
UTIL, UTIL-MR, UTIL-IMP, NASH, NASH-MR and NASH-IMP are not strategyproof.Proof.
Consider the instance given in Table 16. Note that we only include project Z for the purpose of making the NASH welfare of outcomes non-zero. Now,observe that it is impossible to fund all three projects, so our possible candidateproject sets to be funded by the above rules are those where two projects getfunded. X (10) Y (4) Z (9)BudgetAgent 1 8 + +Agent 2 1 + +Agent 3 10 + + +
Table 4: Example instance where rules are not strategyproof. { X, Y } {
X, Z } {
Y, Z } Utilitarian Welfare 22 37 39Nash Welfare 224 1539 2197
Table 5: Utilitarian and Nash welfares of certain project sets to be funded.We check that there is an implementable outcome that funds { Y, Z } , andfind that the outcome where Agents 1 and 2 pay for Z and Agent 3 pays for Y is implementable. Hence, { Y, Z } is the result of UTIL, UTIL-MR, UTIL-IMP,NASH, NASH-MR, NASH-IMP. Note that the utility for Agent 3 is 13.Now, suppose Agent 3 were to misrepresent her preferences as in Table 18.Again, according to this new (perceived) instance, it is impossible for all projectsto be funded, so in Table 19 we check the welfares produced by funding any twoof the projects. X (10) Y (4) Z (9)BudgetAgent 1 8 + +Agent 2 1 + +Agent 3 10 + +
Table 6: Instance where Agent 3 is misrepresenting her preferences. { X, Y } {
X, Z } {
Y, Z } UTIL 18 37 35NASH 160 1539 1521
Table 7: Perceived welfares of certain project sets to be funded if Agent 3 mis-represents her preferences.Since { X, Z } can be funded by an implementable outcome where Agents 1and 2 paying for Z and Agent 3 paying for X , { X, Z } is the result of UTIL, UTIL-MR, UTIL-IMP, NASH, NASH-MR, NASH-IMP. With this outcome, Agent 3sees her utility rise to 19.Then, by misrepresenting her preferences, Agent 3 can cause the choice of theaforementioned rules to change from funding { Y, Z } to funding { X, Z } , hence in-creasing her own utility. Therefore, UTIL, UTIL-MR, UTIL-IMP, NASH, NASH-MR and NASH-IMP are not strategyproof.Similarly, the following also holds. Proposition 13.
EGAL, EGAL-MR and EGAL-IMP are not strategyproof.
Table 8 shows the axioms that are satisfied by restricting the aggregationrules to optimising within the space of MR or IMP outcomes.
UTIL-MR EGAL-MR NASH-MR UTIL-IMP EGAL-IMP NASH-IMPMR (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88)
IMP – – – (cid:88) (cid:88) (cid:88)
PROP – – – (cid:88) (cid:88) (cid:88)
IR – – – – – –PO – – – – – –PO-MR (cid:88) (cid:88) (cid:88) – – –PO-IMP (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88)
PO-Pay (cid:88) (cid:88) (cid:88) – – –EXH (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88)
CORE – – – – – –SP – – – – – –
Table 8: Properties satisfied by UTIL-MR, EGAL-MR, NASH-MR, UTIL-IMP,EGAL-IMP and NASH-IMP.
In addition to the axiomatic study of the welfare-based rules, we undertake asimulation-based experiment to gauge the performance of different rules with articipatory Funding Coordination: Model, Axioms and Rules 13 respect to utilitarian and egalitarian welfare. Our study shows the impact offairness axioms such as MR and IMP on welfare.We generate random samples of profiles in order to simulate two potentialreal-world applications of PFC.1. Share-house setting: In this example, we can imagine a group of house-matespooling their resources to fund communal items for their house. We operateunder the following assumptions: – Number of agents from 3-6: this represents a reasonable number of house-mates in a share-house. – Number of projects from 5-12: projects may include buying items suchas tables, chairs, sofas, televisions, lights, kitchen appliances, washingmachines, dryers, etc. – Agent budgets from 300-600 and project costs from 50-1000. We basethese costs on typical rent and furniture costs in Australia as well ascosts of the above items in first and second-hand retailers. We expectthat each agent brings some money to the communal budget, and wouldspend around one or two weeks’ worth of rent on one-time communalexpenses.2. Crowdfunding setting: In this example, we imagine a relatively small numberof expensive projects to be funded, and a large number of philanthropicdonors, and make the following assumptions. – Number of agents from 20-50: A review of crowdfunding websites such asKickstarter and GoFundMe shows that the most promoted projects aretypically funded by thousands of donors, and smaller projects can attracttens of donors. For the purposes of our simulation, we use between 20-50donors, which is still relatively large to the number of available projects. – Number of projects from 3-8: In crowdfunding, there are far more projectsavailable than a donor actually sees. However, we can estimate that in abrowsing session, a donor might view the top 3-8 promoted projects. – Agent budgets from 0-400 and project costs from 1000-10000: Projects inreal-life crowdfunding can have vastly varying costs. For our simulation,we want for the agents with all their money combined to be able to affordsome, but not all of the available projects in order to create instancesthat are not trivially resolved by funding all or none of the projects.Imposing MR on a rule seems to have a significant impact on both utilitarianand egalitarian welfare on average. Of course, since IMP implies MR, we expectthat imposing IMP as a constraint will have an even greater cost on welfare,but from our experiment, this cost is a relatively small increase on the the costof imposing MR. It is worth noting that in worst-case scenarios, it is alwayspossible that there are no non-trivial outcomes that satisfy the constraints, andso there is a risk that a rule subject to a constraint could produce an outcomethat gives all agents zero utility.When considering average performance, rules are more resilient to the impo-sition of fairness constraints for instances that simulate crowdfunding scenarios compared to share-house scenarios. When the number of agents is large andthe number of projects is small, and project costs are large compared to agentbudgets, it seems to be easier to achieve fairness properties.We typically expect the NASH rule to be a compromise between UTIL andEGAL. This manifests in the results, where the performance losses for NASHwith respect to utilitarian welfare are considerably less than those for EGAL.Likewise, NASH loses considerably less with respect to egalitarian welfare thanUTIL.
We proposed a concrete model for coordinating funding for projects. A formalapproach is important to understand the fairness, participation, and efficiencyrequirements a system designer may pursue. We present a detailed taxonomy ofsuch requirements and clarify their properties and relations. We also analyse nat-ural welfarist rules both axiomatically and experimentally. Our model is not justa rich setting to study collective decision making. We feel that the approachesconsidered in the paper go beyond academic study and can be incorporated inportals that aggregate funding for charitable projects. We envisage future workon online versions of the problem.In practical applications of PFC, it is important to balance welfare demandswith fairness conditions. Our experiment investigated the cost of fairness whenimposing MR or IMP on UTIL, EGAL and NASH rules over instances thatmodel crowdfunding and share-house scenarios. We find that imposing MR alonesignificantly reduces welfare on average, but imposing IMP as well produces arelatively small additional cost on welfare. The costs of imposing any fairnesscondition are much more pronounced on instances that model a share-housesetting than a crowdfunding setting, suggesting that for a large number of agentsand large project costs, fairness conditions are more easily met.
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A Remaining Proofs
Proof of Proposition 1
Proof.
Consider the Subset Sum problem in which there is a set of items M = { , . . . , m } with corresponding weights w , . . . , w m , and a real values W . Theproblem is a subset S with maximum weight (cid:80) j ∈ S w j such that (cid:80) j ∈ S w j ≤ W .The problem is well-known to be NP-hard. We reduce it our setting by takinga project for each corresponding item and project weight for each item weight.Then any set of projects S satisfies the axioms in the proposition if and only ifthe corresponding set of items S is the solution to the Subset problem. Proof of Proposition 2
Proof.
In order to check whether a given set of projects W is implementable, wejust need to check whether the following linear program has a feasible solutionor not. The following can also be checked via network flows. x i,j = 0 for all i ∈ N, j ∈ [ m ] s.t. p j / ∈ A i (cid:88) i ∈ N x i,j = w ( p j ) for all p j ∈ W (cid:88) i ∈ N x i,j = 0 for all p j / ∈ W (cid:88) j ∈ C x i,j ≤ b i for all i ∈ Nx i,j ≥ i ∈ N, j ∈ [ m ] Proof of Proposition 3
Proof.
Suppose an outcome (
S, x ) is not weakly PO-pay. Then, it is trivially notPO-pay. Now suppose (
S, x ) is not PO-pay. Then, there exists another outcome( S (cid:48) , x (cid:48) ) such that (cid:80) i ∈ N x (cid:48) i ≤ (cid:80) i ∈ N x i , u i ( S (cid:48) ) ≥ u i ( S ) for all i ∈ N and u i ( S (cid:48) ) >u i ( S ) for some i ∈ N . Note that the S (cid:48) is afforadable with total amount (cid:80) i ∈ N x (cid:48) i irrespective of who paid what. So S (cid:48) is still affordable if x (cid:48) i ≤ x i . Proof of Proposition 4
Proof.
We distinguish between the cases.1.
IMP implies MR : Suppose an outcome (
S, x ) satisfies IMP. Then there existsa set of vectors { y c } c ∈ S where y c is a vector of payments from each agentto project c such that (cid:80) c ∈ S y c = x and y c,i = 0 if c / ∈ A i . Examining anyrow i of the vector x , which denotes the money charged from agent i , we see articipatory Funding Coordination: Model, Axioms and Rules 17 that x i = (cid:80) c ∈ S y c,i . But since y c,i = 0 if c / ∈ A i , we have x i = (cid:80) c ∈ S ∩ A i y ij .Now, u i ( S ) = (cid:88) c ∈ S ∩ A i w ( c ) = (cid:88) c ∈ S ∩ A i (cid:88) j ∈ N y c,j ≥ (cid:88) c ∈ S ∩ A i y c,i = x i . Hence, (
S, x ) satisfies MR.2.
PO implies PO-Pay : Suppose an outcome is PO. Then, it is not Paretodominated by any other outcome. Hence, it is not Pareto dominated by anyoutcome of lesser total cost, and so it is PO-Pay.3.
PO- X implies PO- Y if Y implies X : Suppose some condition Y impliesanother condition X . Now, suppose some outcome ( S, x ) is PO- X . Then,( S, x ) is not Pareto dominated by any outcome that satisfies X . Since Y implies X , the set of all outcomes satisfying Y is a subset of the set of alloutcomes satisfying Y . Thus, ( S, x ) is not Pareto dominated by any outcomethat satisfies Y , and so ( S, x ) is PO- Y .4. PO-IMP implies EXH : Suppose for a contradiction that (
S, x ) is an outcomethat satisfies PO-IMP but not EXH. Let an implementable payment functionfor this outcome be y : N × C → R + ∪ { } . Since this outcome is notexhaustive, there is a set of agents N (cid:48) who can pool together their unspentmoney to fund another commonly-liked project c (cid:48) . We can construct a newoutcome ( S ∪ { c (cid:48) } , x (cid:48) ) with a payment function y (cid:48) : N × C → R + ∪ { } such that for all agents i ∈ N and projects c ∈ S , y (cid:48) ( i, c ) = y ( i, c ), and forall agents j ∈ N (cid:48) , y (cid:48) ( j, c (cid:48) ) = δ j , where δ j is the contribution of each agent j ∈ N (cid:48) to the new project c (cid:48) . Note that ( S ∪ { c (cid:48) } , x (cid:48) ) is implementable sinceits payment function has agents only funding projects they approve of. Now,( S ∪ { c (cid:48) } , x (cid:48) ) Pareto dominates ( S, x ) which is a contradiction since (
S, x ) isPO-IMP. Therefore, any PO-IMP outcome must satisfy EXH.5.
PO-IR implies EXH : Suppose for a contradiction that ( S, x ) is an outcomethat satisfies PO-IR but not EXH. Then there is a set of agents N (cid:48) who canpool together their unspent money to fund another commonly-liked project c (cid:48) . Since no agent’s utility decreases by funding this project, a new outcome( S ∪ { c (cid:48) } , x (cid:48) ) is still IR, where x (cid:48) is any valid vector of payments. Alsonote that this ( S ∪ { c (cid:48) } , x (cid:48) ) Pareto dominates ( S, x ) since no agent’s utilitydecreases and at least one agent’s utility increases. This is a contradictionas ( S, x ) is PO-IR by our initial assumption. Hence, any PO-IR outcome isEXH.6. Suppose an outcome ( S, x ) is CORE. Then for every subset of agents N (cid:48) ⊆ N , for every subset of projects C (cid:48) ⊆ C such that w ( C (cid:48) ) ≤ (cid:80) i ∈ N (cid:48) b i , for someagent i ∈ N (cid:48) u i ( S ) ≥ w ( C (cid:48) ∩ A i ) . Now consider the case where | N (cid:48) | = 1,i.e. N (cid:48) is a subset of one agent. We now have that for every agent i , for all C (cid:48) ⊆ C such that w ( C (cid:48) ) ≤ b i , u i ( S ) ≥ w ( C (cid:48) ∩ A i ) . Equivalently, for everyagent i , u i ( S ) ≥ max S (cid:48) ⊆ A i ,w ( S (cid:48) ) ≤ b i ( w ( S (cid:48) )) and so ( S, x ) is IR.7. The combination of PO-IMP and IMP imply PROP. Suppose an outcomedoes not satisfy PROP. Then this means that there is set of agents N (cid:48) ⊆ N only approve of a set of projects C (cid:48) ⊆ C such that (cid:80) i ∈ N (cid:48) b i ≥ w ( C (cid:48) ) but not all projects in C (cid:48) are selected. Then one of the two cases occurs: (1) eitherthe money of agents in N (cid:48) is used for projects not approved by them whichviolates IMP (2) the agents in N (cid:48) can pool in unspent money to fund anadditional project in C (cid:48) that is not funded, which means that the outcomeis not PO-IMP. Proof of Proposition 13
Proof.
Consider the instance given in Table 20. Due the total budget constraint,at most two of the projects can be funded, so we check the egalitarian welfarederived by funding any two projects in Table 21.
X (3) Y (2) Z (1)BudgetAgent 1 1 + +Agent 2 1 + +Agent 3 3 + +
Table 9: Example instance where EGAL rules are not strategyproof. { X, Y } {
X, Z } {
Y, Z } Egalitarian Welfare (2, 2, 5) (1, 1, 3) (2, 3, 3)
Table 10: Egalitarian welfares of certain project sets to be funded.Observe that it is possible for an implementable outcome to fund { Y, Z } byhaving Agents 1 and 2 pay for them, and so { Y, Z } is funded by each of theabove rules. Then, the utility for Agent 3 is 2.Now, suppose Agent 3 misrepresents her preferences to suppress the fact thatshe approves of project Y . The new perceived instance is shown in Table 22 andagain, we compute the egalitarian welfare produced by funding any two projectsin Table 23.Note that there is an implementable outcome that funds { X, Y } , where Agent3 pays for X and Agents 1 and 2 pay for Y . Hence, { X, Y } is funded by each ofthe rules. The new utility for Agent 3 is 3.Thus, by misrepresenting her preferences, Agent 3 is able to increase theutility she receives when egalitarian rules are used. Therefore, EGAL, EGAL-MR and EGAL-IMP are not strategyproof. articipatory Funding Coordination: Model, Axioms and Rules 19X (3) Y (2) Z (1)BudgetAgent 1 1 + +Agent 2 1 + +Agent 3 3 + Table 11: Example instance where rules are not strategyproof. { X, Y } {
X, Z } {
Y, Z } Egalitarian Welfare (2, 2, 3) (1, 1, 3) (0, 3, 3)
Table 12: Egalitarian welfares of certain project sets to be funded.
B Experiments
The results of the experiments are depicted in Figures 1, 2, 3, 4, 5, 6, 7, and 8.
C Additional Propositions
Proposition 14.
UTIL, EGAL and NASH satisfy PO.Proof.
Suppose there was an outcome that was Pareto dominant over the out-come returned by any of these rules. Then, it would also have a strictly greaterutilitarian/egalitarian/Nash welfare to this outcome, which is a contradiction.
Proposition 15.
UTIL and NASH do not satisfy MR (or IMP by corollary) orIR (or CORE by corollary).Proof.
For the profile in Table 14, UTIL and NASH would require that onlyproject X is funded by using all of both agents’ money. But then, Agent 2 couldhave left the mechanism and derived better utility on her own (IR). Also, herreturn is less than her contribution, so UTIL does not satisfy MR. X (20) Y (10)BudgetAgent 1 10 +Agent 2 10 +
Table 13: Example profile where UTIL and NASH outcomes do not satisfy MR.
Proposition 16.
EGAL does not satisfy MR (or IMP by corollary).
Fig. 1: Average performance of ruleswith respect to utilitarian welfare inshare-house simulations as a percent-age of the maximum achievable utili-tarian welfare. Fig. 2: Average performance of ruleswith respect to utilitarian welfare incrowdfunding simulations as a percent-age of the maximum achievable utili-tarian welfare.Fig. 3: Worst-case performance of ruleswith respect to utilitarian welfare inshare-house simulations as a percent-age of the maximum achievable utili-tarian welfare. Fig. 4: Worst-case performance of ruleswith respect to utilitarian welfare incrowdfunding simulations as a percent-age of the maximum achievable utili-tarian welfare.Fig. 5: Average performance of ruleswith respect to egalitarian welfare inshare-house simulations as a percent-age of the maximum achievable egali-tarian welfare. Fig. 6: Average performance of ruleswith respect to egalitarian welfare incrowdfunding simulations as a percent-age of the maximum achievable egali-tarian welfare.Fig. 7: Worst-case performance of ruleswith respect to egalitarian welfare inshare-house simulations as a percent-age of the maximum achievable egali-tarian welfare. Fig. 8: Worst-case performance of ruleswith respect to egalitarian welfare incrowdfunding simulations as a percent-age of the maximum achievable egali-tarian welfare. articipatory Funding Coordination: Model, Axioms and Rules 21
Proof.
For the profile in Table 14, the maximally egalitarian output is to imple-ment both projects. However, in this case, Agent 1 is charged 25 but receives autility of 20, so this outcome does not satisfy MR.
X (20) Y (10)BudgetAgent 1 25 +Agent 2 5 +
Table 14: Example profile where EGAL outcomes do not satisfy MR.
Proposition 17.
EGAL does not satisfy IR.Proof.
In the below example, given the total budget, the choice is to implementeither X or Y or neither. The egalitarian outcome with be to implement Y , butthen, by leaving the system, Agent 1 could pay for X herself and get a betteroutcome. X (20) Y (10)BudgetAgent 1 20 + +Agent 2 5 +
Proposition 18.
UTIL-IMP, EGAL-IMP and NASH-IMP do not satisfy PO-MR.Proof.
In Table 15, the only implementable outcomes are those in which noprojects are funded or Agents 1 and 2 pay for project X . But the outcomewhere all three projects are funded by all agents spending all of their moneysatisfies MR and Pareto dominates any of the implementable outcomes. X (10) Y (12)BudgetAgent 1 10 +Agent 2 10 +Agent 3 2 +
Table 15: Example profile where UTIL-IMP, EGAL-IMP and NASH-IMP out-comes do not satisfy PO-MR.
Proposition 19.
UTIL-MR and NASH-MR do not satisfy IMP.Proof.
Consider the example below. UTIL-MR and NASH-MR will require thatboth projects are funded. However, at least one of Agents 1 and 2 must givesome money to project Y , as Agent 3 cannot afford it by herself. Thus, thisoutcome is not implementable. X (30) Y (10)BudgetAgent 1 20 +Agent 2 20 +Agent 3 5 +
Proposition 20.
UTIL-MR and UTIL-IMP do not satisfy IR (or CORE bycorollary).Proof.
Consider the example below. The overall (unique) utilitarian outcome isachieved by funding projects Y and Z with x = 4 and x = 11. Observe thatthis is an implementable outcome, and so this would be the result of UTIL-MRand UTIL-IMP. However, if Agent 1 left the system, she could have individuallyfunded projects W and X which would have returned to her a greater utility.Therefore, this outcome does not satisfy IR. W (3) X (3) Y (5) Z (10)BudgetAgent 1 6 + + +Agent 2 11 + +
Proposition 21.
NASH-MR, MASH-IMP, EGAL-MR and EGAL-IMP do notsatisfy IR (or CORE by corollary).Proof.
From the example above, we see that any egalitarian distribution alsowill fund only projects Y and Z . For this output to be implementable (and alsosatisfy MR), we have x = 4 and x = 11 or x = 5 and x = 10. In either case,we have seen from above that this outcome will not satisfy IR. – NASH-MR and NASH-IMP do not satisfy IR (or CORE by corollary): Sameargument as above.
Proposition 22.
UTIL, UTIL-MR, UTIL-IMP, NASH, NASH-MR and NASH-IMP are not strategyproof. articipatory Funding Coordination: Model, Axioms and Rules 23
Proof.
Consider the instance given in Table 16. Note that we only include project Z for the purpose of making the NASH welfare of outcomes non-zero. Now,observe that it is impossible to fund all three projects, so our possible candidateproject sets to be funded by the above rules are those where two projects getfunded. X (10) Y (4) Z (9)BudgetAgent 1 8 + +Agent 2 1 + +Agent 3 10 + + +
Table 16: Example instance where rules are not strategyproof. { X, Y } {
X, Z } {
Y, Z } Utilitarian Welfare 22 37 39Nash Welfare 224 1539 2197
Table 17: Utilitarian and Nash welfares of certain project sets to be funded.We check that there is an implementable outcome that funds { Y, Z } , andfind that the outcome where Agents 1 and 2 pay for Z and Agent 3 pays for Y is implementable. Hence, { Y, Z } is the result of UTIL, UTIL-MR, UTIL-IMP,NASH, NASH-MR, NASH-IMP. Note that the utility for Agent 3 is 13.Now, suppose Agent 3 were to misrepresent her preferences as in Table 18.Again, according to this new (perceived) instance, it is impossible for all projectsto be funded, so in Table 19 we check the welfares produced by funding any twoof the projects. X (10) Y (4) Z (9)BudgetAgent 1 8 + +Agent 2 1 + +Agent 3 10 + +
Table 18: Instance where Agent 3 is misrepresenting her preferences.Since { X, Z } can be funded by an implementable outcome where Agents 1and 2 paying for Z and Agent 3 paying for X , { X, Z } is the result of UTIL, UTIL- { X, Y } {
X, Z } {
Y, Z } UTIL 18 37 35NASH 160 1539 1521
Table 19: Perceived welfares of certain project sets to be funded if Agent 3misrepresents her preferences.MR, UTIL-IMP, NASH, NASH-MR, NASH-IMP. With this outcome, Agent 3sees her utility rise to 19.Then, by misrepresenting her preferences, Agent 3 can cause the choice of theaforementioned rules to change from funding { Y, Z } to funding { X, Z } , hence in-creasing her own utility. Therefore, UTIL, UTIL-MR, UTIL-IMP, NASH, NASH-MR and NASH-IMP are not strategyproof. Proposition 23.
EGAL, EGAL-MR and EGAL-IMP are not strategyproof.Proof.
Consider the instance given in Table 20. Due the total budget constraint,at most two of the projects can be funded, so we check the egalitarian welfarederived by funding any two projects in Table 21.
X (3) Y (2) Z (1)BudgetAgent 1 1 + +Agent 2 1 + +Agent 3 3 + +
Table 20: Example instance where EGAL rules are not strategyproof. { X, Y } {
X, Z } {
Y, Z } Egalitarian Welfare (2, 2, 5) (1, 1, 3) (2, 3, 3)
Table 21: Egalitarian welfares of certain project sets to be funded.Observe that it is possible for an implementable outcome to fund { Y, Z } byhaving Agents 1 and 2 pay for them, and so { Y, Z } is funded by each of theabove rules. Then, the utility for Agent 3 is 2.Now, suppose Agent 3 misrepresents her preferences to suppress the fact thatshe approves of project Y . The new perceived instance is shown in Table 22 andagain, we compute the egalitarian welfare produced by funding any two projectsin Table 23. articipatory Funding Coordination: Model, Axioms and Rules 25X (3) Y (2) Z (1)BudgetAgent 1 1 + +Agent 2 1 + +Agent 3 3 + Table 22: Example instance where rules are not strategyproof. { X, Y } {
X, Z } {
Y, Z } Egalitarian Welfare (2, 2, 3) (1, 1, 3) (0, 3, 3)
Table 23: Egalitarian welfares of certain project sets to be funded.Note that there is an implementable outcome that funds { X, Y } , where Agent3 pays for X and Agents 1 and 2 pay for Y . Hence, { X, Y }}