1. Introduction
Participatory budgeting gives members of a community power by allowing them to collectively decide how to divide a portion of the community’s budget among a set of proposed alternatives (Cabannes, 2004). While participatory budgeting was first introduced in Brazil (Schneider and Goldfrank, 2002), it has now been used in more than 3,000 cities around the world, including New York, Boston, Chicago, San Francisco, and Toronto (Participatory Budgeting Project, 2019). It has been used to determine how to allocate the budgets of states and cities as well as housing authorities and schools.
Many participatory budgeting elections are run using a variant of approval voting, in which each voter chooses up to projects to approve, and the projects with the highest number of approvals are funded, subject to budget constraints (Goel et al., 2016). Under such a voting scheme, each proposed project is either fully funded or not funded at all. This makes sense for welldelineated projects such as renovating a school or adding an elevator to a public library. For other kinds of projects, funding decisions need not be allornothing. For example, participatory budgeting could be used to decide how to divide a city’s tax surplus between its departments of health, education, infrastructure, and parks. A voter might propose a division of the tax surplus among the four departments into the fractions (30%, 40%, 20%, 10%). The city could invite each citizen to submit such a budget proposal, and they could then be aggregated by a suitable mechanism.
A first idea for aggregating the proposals would be to take the mean. But the mean has a serious flaw as an aggregator: it’s easily manipulated. A voter preferring a (60%, 40%) division across two alternatives may vote (100%, 0%) instead in order to distort the mean calculation and move the aggregate closer to his or her true preference if the first alternative has little community support.
In this paper, we seek mechanisms that are resistant to such manipulation. In particular, we require that no voter can, by lying, move the aggregate division toward his or her preference on one alternative without moving it away from his or her preference by an equal or greater amount on other alternatives. In other words, we seek budget aggregation mechanisms that are incentive compatible under preferences, with each voter’s disutility for a budget division equal to the distance between that division and the division she prefers most.
Goel et al. (2016) showed that choosing an aggregate budget division that maximizes the welfare of the voters—that is, a division that minimizes the total distance from each voter’s report—is both incentive compatible and Paretooptimal under this voter utility model. However, this utilitarian aggregate has a tendency to overweight majority preferences, creeping back towards allornothing allocations. For example, imagine that a hundred voters prefer (100%, 0%) while ninetynine prefer (0%, 100%). The utilitarian aggregate is (100%, 0%) even though the mean is close to (50%, 50%). In many participatory budgeting scenarios, the latter solution is more in the spirit of consensus. For example, imagine that each family votes for all education dollars to go to their own neighborhood school. The utilitarian aggregate would earmark the entire budget to the most populous school district, while we may prefer that funds are split in proportion to the districts’ populations. To capture this fairness constraint, we define a notion of proportionality, requiring that when voters are singleminded (as in this example), the fraction of the budget assigned to each alternative is equal to the proportion of voters who favor that alternative. Do there exist aggregators that are both incentive compatible and proportional?
For the special case of two alternatives, preferences are a specialcase of singlepeaked preferences, well studied in the voting literature. The seminal results of Moulin (1980) imply that, in this setting, all incentive compatible voting schemes correspond to inserting “phantom” proposals, where is the number of voters, and returning the median of the true proposals and the phantoms. We show that there exists a way of placing the phantoms that results in a proportional mechanism for two alternatives.
Generalizing Moulin’s phantom median mechanisms to allow for more than two alternatives is difficult. Existing proposals for such generalizations take a median in each dimension independently (Border and Jordan, 1983; Peters et al., 1992; Barberà and Jackson, 1994), but this strategy is doomed in our application with normalization constraints; unlike the mean, taking a coordinatewise median will by default fail to normalize. We address this problem by allowing the set of phantoms to continuously shift upwards, increasing the sum of the generalized medians until the aggregate becomes normalized. This idea allows us to define a very general class of moving phantom mechanisms. Although one might think that allowing the final phantom locations to depend on voters’ reports might give voters an incentive to misreport, we prove that every moving phantom mechanism is incentive compatible under preferences.
Among this large family of incentive compatible mechanisms, we find one that satisfies our proportionality requirement. This moving phantom mechanism is obtained when phantoms are placed uniformly between 0 and a value which increases until the coordinatewise medians sum to one. To analyze this mechanism, we prove that the aggregate found by this mechanism can be interpreted as the clearing prices in a market system, and hence call it the independent markets mechanism. This reveals an unexpected connection between market prices and generalized medians that may be of broader interest. The independent markets mechanism can also be justified from a gametheoretic perspective as the unique Nash equilibrium of a natural voting game. Thus, this proportional moving phantom mechanism has two alternative interpretations:

Market interpretation: For each alternative, we set up a market in which units of a divisible good are sold. This amount is the same across all markets. Each voter has a value for the good in market that is equal to the fraction of the budget that the voter would prefer be allocated to alternative in the budget division setting, and has $1 to spend in each market. Increase until the point at which the market clearing prices across these independent markets sum to $1. At this point, the market clearing prices are exactly the aggregate division output by the independent markets moving phantom mechanism.

Voting game: Each agent receives one credit for each alternative and may choose any amount of that credit to spend on the alternative. The outcome of the game is a normalized vector proportional to the amount spent on each alternative. Agents choose their spending in such a way as to minimize the
distance between their preferred division in the budget division setting and the vector output by the game. The Nash equilibrium of this game is exactly the aggregate division output by the independent markets mechanism.
By analyzing the market and Nash equilibria of these systems, we can show that our mechanism satisfies several important social choice properties.
In contrast, the independent markets mechanism unfortunately fails to satisfy Pareto optimality. We show that this is unavoidable, as no proportional moving phantom mechanism is Paretooptimal. In fact, we prove that there is a unique moving phantom mechanism that is. In this mechanism, all phantoms start at 0 and then, one by one, transition to 1, with no two phantoms moving at the same time. This mechanism turns out to also have a phantomfree interpretation: it is equivalent to selecting the maximumentropy budget division out of all those that maximize social welfare—the same mechanism studied by Goel et al. (2016) up to the choice of tiebreaking rules.
While the motivation of our formal model is participatory budgeting, it applies to the division of other resources such as time. For example, one might imagine using such a mechanism as a way to reach consensus among a team of conference organizers who wish to divide a day between talks, poster sessions, and social activities. Another example would be a government that needs to decide on a target energy mix (that is, how much energy should come from fossil fuels, nuclear, or renewable sources) and wishes to aggregate expert proposals. In all these applications, our class of moving phantom mechanisms can be used to make better decisions.
Related Work
Several recent papers study voting rules for participatory budgeting, considering both axiomatics and computational complexity, but under the assumption that indivisible projects can either be fully funded or not funded at all (Goel et al., 2016; Benade et al., 2017; Lu and Boutilier, 2011; Aziz et al., 2018). The setting in which partial funding of alternatives is permitted has also been studied, but generally under a different utility model in which voters assign utility scores to the alternatives rather than having an ideal distribution (Fain et al., 2016; Bogomolnaia et al., 2005; Aziz et al., 2017). This body of work includes positive results for weak versions of incentive compatibility, but impossibilities for obtaining full incentive compatibility. Garg et al. (2019) perform a Mechanical Turk study exploring preference structure in a highdimensional continuous setting similar to ours.
Closest to our work is that of Goel et al. (2016). The primary focus of their paper is on knapsack voting, in which each voter submits her preferred set of projects to fully fund. However, they also consider the use of fractional knapsack voting in a setting in which partial funding of alternatives is permitted and voters have preferences. This corresponds exactly to our setting. They show that the mechanism that maximizes social welfare (with some fixed tiebreaking) is incentive compatible. We replicate this result by showing that the welfaremaximizing mechanism (with an arguably more natural way to break ties) is a member of the large class of moving phantom mechanisms, all of which are incentive compatible. Goel et al. do not consider other mechanisms for this setting.
The truthful aggregation of preferences over numerical values (such as the temperature for an office) has been extensively studied. A famous result of Moulin (1980) characterizes the set of incentive compatible voting rules under the assumption that voters have singlepeaked preferences over values in . These voting rules are generalized median schemes. The bestknown example is the standard median, in which each voter reports her ideal point in and the median report is selected. Other voting rules in this class insert “phantom voters” who report a fixed top choice. Barberà et al. (1993) obtained a multidimensional analogue of this result for , and there are further generalizations that characterize truthful rules if other constraints are imposed on the feasible set (Barberà et al., 1997). Crucially, the constraints allowed by Barberà et al. (1997) do not include the normalization constraint that is fundamental to our setting. Several other papers (Border and Jordan, 1983; Peters et al., 1992; Barberà and Jackson, 1994) introduced multidimensional models in which one can achieve truthfulness by taking a generalized median in each coordinate, but such a strategy does not work with normalization constraints. We are not aware of results (prior to this work) that extend generalized medians to multiple dimensions without using a mechanism that decomposes into onedimensional mechanisms.
In the computer science literature, the abovementioned generalized median schemes have also been studied in the context of truthful facility location (Procaccia and Tennenholtz, 2009; Alon et al., 2009). In this context, the aim is to approximate social welfare subject to incentive compatibility.
One could apply our results to the aggregation of probabilistic beliefs. There is a large literature on probabilistic opinion pooling (Genest and Zidek, 1986; French, 1985; Clemen, 1989; Intriligator, 1973) which studies aggregators in this context. The main focus of that literature is to preserve stochastic and epistemic properties. To the best of our knowledge, strategic aspects have not been considered.
Finally, recently proposed rules for crowdsourcing societal tradeoffs (Conitzer et al., 2015, 2016) can also be used to aggregate budget divisions (with full support) after converting them into pairwise ratios of funding amounts, but this setting has also not been analyzed from a strategic viewpoint.
2. Preliminaries
Let be a set of voters and be a set of possible alternatives. Voters have structured preferences over budget divisions , with , where is the fraction of a public resource (such as money or time) allocated to alternative . Each voter has a most preferred division , with their preference over other divisions induced by distance from . Specifically, each voter has a disutility for division equal to , where denotes the distance between and . Note that a voter’s complete preference over all possible divisions can be deduced from their most preferred division .
A preference profile consists of a reported division for each voter . We use to denote the reports of all voters other than . A budget aggregation mechanism takes as input a preference profile , and outputs an aggregate division . A mechanism is continuous if it is continuous when considered as a function . We say that a mechanism is anonymous if its output is fixed under permutations of the voters, and neutral if a permutation of the alternatives in voters’ inputs permutes the output in the same way.
We are interested in mechanisms that satisfy incentive compatibility. Voters should not be able to change the aggregate division in their favor by misrepresenting their preference.
Definition 2.1 ().
A budget aggregation mechanism satisfies incentive compatibility if, for all preference profiles , voters , and divisions and , .
We are also interested in the basic efficiency notion of Pareto optimality. It should not be possible to change the aggregate so that some voter is strictly better off but no other voter is worse off.
Definition 2.2 ().
A budget aggregation mechanism satisfies Pareto optimality if, for all preference profiles , and all divisions , if for some voter , then there exists a voter for which .
Observe that the definitions of incentive compatibility and Pareto optimality depend only on the voters’ preference relations, not the exact utility model. Results pertaining to these properties therefore hold for any utility function that induces the same ordinal preferences as linear utilities.
We also consider a fairness property that we call proportionality: Suppose each voter is singleminded, in that they prefer a division in which the entire resource goes to a single alternative. Then it is natural to split the resource in proportion to the number of voters supporting each alternative. For example, if 6 voters report , 3 voters report , and 1 voter reports , then the aggregate should be . We call this property proportionality.
Definition 2.3 ().
A voter is singleminded if their preferred division is a unit vector. A budget aggregation mechanism is proportional if, for every preference profile consisting of only singleminded voters, and every alternative , , where is the number of voters that support alternative .
We note that proportionality is a fairly weak definition, only applying to a small subset of possible profiles. However, as we will see later, it is already strong enough to be incompatible with Pareto optimality within the class of moving phantom mechanisms that we introduce in this paper.
3. Two alternatives
To build intuition, we begin by considering the case in which . Due to the normalization of inputs and of the output, and with preferences, the problem is perfectly onedimensional in this case. This allows us to directly import Moulin’s (Moulin, 1980) famous characterization of generalized median rules as the only incentive compatible mechanisms for voters with singlepeaked preferences over a singledimensional quantity.^{1}^{1}1Our preference model using imposes slightly more structure than just singlepeakedness, namely that voters are indifferent between points that are equidistant to their peak. However, this restriction does not enlarge the class of incentive compatible mechanisms, at least if we impose continuity (Massó and de Barreda, 2011).
Theorem 3.1 ((Moulin, 1980)).
For , an anonymous and continuous budget aggregation mechanism is incentive compatible if and only if there are in such that, for all profiles ,
The numbers are known as phantoms. Each mechanism described by Theorem 3.1 can be understood as taking the coordinatewise median of the reported distributions, after inserting phantom voters (whose report is fixed and independent of the input profile).
One can check that define a neutral mechanism if and only if the phantom placements are symmetric, that is if and only if Note that there are phantoms but only voters, so that the phantoms can outweigh the voters. For example, when for all then the mechanism is just the constant mechanism returning . However, if we take and , then these two phantoms “cancel out” and there are only phantoms left. In fact, one can check that the mechanism is Paretooptimal if and only if and (Moulin, 1980).
A particularly interesting example is the uniform phantom mechanism, obtained when placing the phantoms uniformly over the interval , so that for each . This placement of phantom voters appears in a paper by Caragiannis et al. (2016). They were aiming for mechanisms whose output is close to the mean, and they prove that the uniform phantom mechanism yields an aggregate that is closer to the mean than that obtained from any other phantom placements, in the worst case over inputs. The uniform phantom mechanism has other attractive properties, including being proportional in the sense of Definition 2.3.
Proposition 3.2 ().
For , the uniform phantom mechanism is the unique (anonymous and continuous) budget aggregation mechanism that is both incentive compatible and proportional.
Proof.
Theorem 3.1 gives us that is incentive compatible if and only if it can be written in terms of phantom medians. We therefore need only to consider the additional requirement of proportionality. The uniform phantom mechanism is proportional, because if consists of voters reporting and voters reporting , then , as required.
For uniqueness, suppose are phantom positions that induce a proportional mechanism. Let . We show that . Let be a profile consisting of only singleminded voters with voters reporting . Then is the median, and proportionality requires that . ∎
Another natural way to place the phantoms is one that takes the coordinatewise median. When is even, this is achieved by placing half the phantoms at 0 and the other half at 1, outputting precisely the median of the reported values on each coordinate. When
is odd, we place
phantoms at 0, phantoms at 1, and we place a single phantom at to preserve neutrality. This mechanism outputs the point between the left and right medians that is closest to . The resulting mechanism returns an aggregate that minimizes the sum of distances between the reports and . We will generalize this mechanism for larger in Section 6.4. A Class of Incentive Compatible Mechanisms for Higher Dimensions
For , we have a complete picture of incentivecompatible mechanisms, thanks to Moulin’s characterization. For , it is less clear how to construct examples of incentivecompatible mechanisms. One could try to take a generalized median in each alternative independently, but the result of such a mechanism would not respect the normalization constraint.
However, there is a way of extending the idea of generalized medians to the higherdimensional setting. The basic idea is that if a coordinatewise generalized median violates the normalization constraint, then we can adjust the placement of the phantoms, increasing or decreasing the sum of the generalized medians as needed. Such a procedure might, in principle, give voters incentives to manipulate in order to affect the phantom placements. However, our class of moving phantom mechanisms manages to avoid this problem.
Definition 4.1 ().
Let be a family of functions, or phantom system, where is a continuous, weakly increasing function with and for each , and we have for all . Then, the moving phantom mechanism is defined so that for all profiles and all ,
(1) 
where is chosen so that .
For brevity, we write and abbreviate the median in (1) to .
Let us examine the definition. Each represents a phantom, and the phantom system represents a “movie” in which all phantoms continuously increase from 0 to 1, with the function argument defining an instantaneous snapshot of the phantom positions. The moving phantom mechanism defined by identifies a particular snapshot in time, , for which the sum of generalized medians over all coordinates is exactly 1. One can check that at least one such exists, and that the output of the mechanism is independent of which of these is chosen.
Proposition 4.2 ().
The moving phantom mechanism is welldefined for every phantom system satisfying the conditions of Definition 4.1.
Proof.
First note that the function is continuous and increasing in , because is continuous and increasing, and these properties are preserved under taking the median and sum. This implies that, provided the set is nonempty, the aggregate does not depend on the choice of .
When , , since all phantom entries are 0. When , , since all phantom entries are 1. By the Intermediate Value Theorem, using continuity, there exists with . ∎
To build intuition, we consider an example moving phantom mechanism in Figure 1. There are three alternatives, each occupying a column on the horizontal axis, and four voters. Voter reports are indicated by gray horizontal line segments, with their magnitude indicated by their vertical position. The phantom placements are indicated by the red lines and labeled . For each alternative, the median of the four agent reports and the five phantoms is indicated by a rectangle.
The four snapshots shown in Figure 1 display increasing values of . Observe that the position of each phantom (weakly) increases from left to right, as does the median on each alternative. Although the vertical axis is not labeled, for simplicity of presentation, normalization here occurs in the second image from the left. In the leftmost image, the sum of the highlighted entries is less than 1, while in the two rightmost images it is more than 1.
For simplicity, the definition of moving phantom mechanisms treats the number of voters as fixed. To allow to vary, it is necessary to define a family of phantom systems, one for each . In the next two sections, we give two examples of such families, but for this section we keep the presentation simple by considering only a fixed .
Moving phantom mechanisms satisfy some important basic properties. They are all anonymous and neutral. Here neutrality is a design choice—one could imagine defining moving phantom mechanisms for which the movement of the phantoms depends on the alternative. All moving phantom mechanisms are also, again by design, continuous.
Given a profile, we can efficiently approximate the output of a moving phantom mechanism, assuming oracle access to its defining functions , by performing a binary search on . In principle, the precise time at which the output of the mechanism is normalized may have many decimal digits, and for badlybehaved it may even be irrational. For the same reason, the mechanism may return an irrational division, so the precise computation of the output may not be possible. However, for the mechanisms studied in the following sections, we can show that has few digits and the output is always rational, so polynomialtime computation is possible.
We now show our main result in this section, that every moving phantom mechanism is incentive compatible. Before proving the result formally, we provide some intuition. If changes her report from to , the effect on the aggregate can be decomposed into two parts. First, we can think of holding the phantoms fixed at the snapshot dictated by the truthful instance, while changing ’s report to . Second, we can think of repositioning the phantoms to the snapshot required to guarantee normalization of the aggregate vector after reports . To prove incentive compatibility, we show that any change that the aggregate division undergoes in the first stage can only be bad for voter , pushing the aggregate away from . Change in the second stage can push the aggregate towards , helping voter , but the magnitude of this change is upper bounded by the magnitude of the harmful change in the first stage.
Theorem 4.3 ().
Every moving phantom mechanism is incentive compatible.
Proof.
Let define a moving phantom mechanism . Consider some report , and fix the reports of all other voters . Let determine the phantom placement for reports and for reports .
Consider the effect of ’s misreport from to while holding the phantom placement fixed at . Then, because phantom placements are fixed on each alternative, any change that voter can cause on alternative by misreporting must be away from her preference . For each ,

[leftmargin=1em]

if , then we must have ;

if , then we must have ;

if and lie on the same side of , then .
Let denote the change caused on alternative by voter ’s misreport, subject to holding the phantom placement fixed at . By the above, the distance from ’s preferred division has increased by as a result of ’s misreport.
Next, we consider the change that results from moving the phantoms from to . Assume that (otherwise, a very similar argument applies). Then we have that , which implies that since the sum is monotonic in (see the proof of Proposition 4.2). This produces aggregate division with for all , and . That is, the distance between taking generalized medians with phantoms defined by and doing so with phantoms defined by , conditioned on voter reporting , is at most .
Therefore, . ∎
In addition to incentive compatibility, moving phantom mechanisms satisfy a natural monotonicity property that says that if some voter increases her report on alternative , and decreases her report on all other alternatives, then the aggregate weight on alternative should not decrease.
Definition 4.4 ().
A budget aggregation mechanism satisfies monotonicity if, for all , with for some and for all ,
Theorem 4.5 ().
Every moving phantom mechanism satisfies monotonicity.
Proof.
Let , be such that for some and for all . Let determine the phantom placement for reports and for reports .
Suppose that . We have
where the inequality holds because and for all .
Next, suppose that . Then
where the inequality holds because for all and for all . ∎
Before we move on to particular moving phantom mechanisms, let us end this section with a tantalizing open question: Does there exist an (anonymous, neutral, continuous) incentive compatible budget aggregation mechanism that is not a moving phantom mechanism? We have not been able to construct any example, and have found that some mechanisms that on first sight seem to have nothing to do with medians end up having an equivalent description as a moving phantom mechanism. For the simpler twoalternative case, we already have a characterization of all incentive compatible mechanisms (Theorem 3.1). This class can equivalently be described in terms of moving phantoms, and so the answer to our question for is no.
Theorem 4.6 ().
For , moving phantom mechanisms are the only budget aggregation mechanisms that satisfy anonymity, neutrality, continuity, and incentive compatibility.
Proof.
Certainly all moving phantom mechanisms satisfy these properties. For the other direction, we know from Theorem 3.1 that any mechanism satisfying these properties can be described as a generalized median with phantoms satisfying, due to neutrality, . We show that is equivalent to a moving phantom mechanism. Define using a phantom system for which there exists a with for every . Then, for every preference profile , we have that , and , matching the output of . ∎
5. The Independent Markets Mechanism
We have seen that uniform phantoms is uniquely proportional for . By a similar argument to the proof of Proposition 3.2, it is easy to see that any family of functions that generates uniform phantoms at some snapshot will be proportional, and will reduce to the uniform phantom mechanism for . However, this leaves a large class of moving phantom mechanisms to choose from. In this section, we identify a particular moving phantom mechanism that generalizes the uniform phantom mechanism for arbitrary . Its output can be interpreted as a market equilibrium.
Definition 5.1 ().
The independent markets mechanism () is the moving phantom mechanism defined by the phantom system for each .
To visualize the phantom placement, observe that for any , phantoms are being placed at . Once reaches , phantoms continue to grow in the same manner, but the higher phantoms get capped at 1.^{2}^{2}2As written, , but Definition 4.1 requires for all . This detail does not matter here, since normalization is always achieved without moving phantom , but one could write in a different form to satisfy Definition 4.1 without it changing the behavior of the mechanism. This is actually the mechanism that we displayed in Figure 1. Note that, when , the phantom placement is uniform on (as is the case in the third panel of Figure 1); thus, reduces to the uniform phantom mechanism for .
Example 5.2 ().
Let us consider a simple numerical example. Let , and suppose voter reports are , and . Consider the placement of the phantoms when . They are placed at . On the first alternative,
Similarly, it is easy to check that the generalized median on the second alternative is 0.4 and on the third alternative is 0.2. Because these are normalized, is a valid choice of , and the outcome .
5.1. Market Interpretation
Why do we call this mechanism the independent markets mechanism? To explain this, we first establish a connection between the market clearing price in a simple singlegood market and the median of some familiarlooking numbers.
Suppose we are selling a single divisible good, of which a total amount of is available. Each of voters has a budget of 1, and a value per unit of the good. At a price per unit of the good, the demand of voter , is given by the following function:
Thus, each voter demands as much of the good as their budget of 1 allows at price , as long as the price per unit is lower than their value per unit. The market clearing price is the price at which the supply of the good () equals the total demand. Formally,
(2) 
where the supremum is necessary because, due to discontinuities in the demand function, supply and demand may never be exactly equal.
It turns out that the market clearing price is equal to the median of the voter values and the
“phantom values” which are uniformly distributed on the interval
. To the best of our knowledge, this connection has not previously been appreciated in the literature.Lemma 5.3 ().
In the market defined above, the market clearing price equals
Proof.
We distinguish the cases that the median is a phantom entry or a voter entry. Suppose that the median is for some . Then we can partition the (real and phantom) entries, with the exception of the phantom at , into sets and with , where consists only of entries less than or equal to , and consists only of entries greater than or equal to .
The set contains phantoms, so voter reports. At any price , each voter has demand . The total demand of all voters in is therefore greater than . At price , each voter has demand (if ) or (if ), and each voter has demand 0. Therefore the total demand of all voters is at most , so the market clearing price is .
Next, suppose that the generalized median is for some (note that the generalized median cannot be greater than , because it cannot be higher than the largest phantom value). Then we can partition the (real and phantom) entries, not including a single voter with (one such voter must exist because the median coincides with some entry, and no phantom entry lies at ), into sets and each of size , where consists only of entries less than or equal to , and consists only of entries greater than or equal to .
Again, contains phantom reports, so voter reports. At all prices , each of these voters, as well as voter with , has demand . The total demand is thus greater than . At price , the total demand is at most (since the number of voters with is at most the number voter reports in set ). The market clearing price is therefore . ∎
The “market” connection to independent markets is now clear: For each alternative , we set up a market in which we sell an amount of a good; this amount is the same across markets. Voter has value for the good sold in market , and has a budget of 1 in each market. The markets are “independent” because, while each voter is engaged in every market, the budget of 1 for each market can only be used to buy the good sold in that market. Using Lemma 5.3, we can derive the market clearing prices in each of these markets. If we write , then these prices correspond exactly to the output of with the phantoms as placed at time . Changing the phantom placement by varying to normalize the output is equivalent to varying the amount of the good sold in each market until the clearing prices across markets sum to 1. While we prevent phantoms from moving above 1 in the definition of independent markets—complying with Definition 4.1—the exact positions of these phantoms do not affect the clearing price since all reports are at most 1.
Returning to Example 5.2, we can verify the outcome using the market interpretation, by setting the quantity of goods to be sold in each market to . In the market corresponding to alternative 1, the market clears at price , at which price voters 2 and 3 demand goods each, matching supply, and voter 1 demands nothing as . It can be checked that the market prices also match the independent markets outcome for alternatives 2 and 3.
The market system we have described yields an incentivecompatible aggregator, since it corresponds to a moving phantom mechanism. There are other marketbased aggregation mechanisms described in the literature, most famously the parimutuel consensus mechanism of Eisenberg and Gale (1959). That mechanism differs from ours in that voters have only a single budget of 1 which they can use in all of the markets. (The supply of goods can be fixed at , which guarantees that prices are normalized, because total spending is fixed.) For the case , it does not matter whether markets are independent or not, and our mechanism is equivalent to the one of Eisenberg and Gale (1959). It follows that the parimutuel consensus mechanism is incentive compatible for (in our sense). However, for , the mechanism is manipulable,^{3}^{3}3Let , . Parimutuel consensus yields prices , at distance from . If voter 1 instead reports , the price vector is , at distance from . and hence cannot be represented as a moving phantom mechanism. We point the reader to the work of Garg et al. (2018) for a detailed overview of other settings in which market mechanisms have been used in the context of public decision making.
5.2. Voting Game Interpretation
We have seen descriptions of as a moving phantom mechanism and as clearing prices of a market system. We next give a gametheoretic description: the independent markets mechanism can be seen as the unique Nash equilibrium outcome of a voting game inspired by range voting (Smith, 2000). The game works as follows. Each voter receives one “credit” per alternative, and chooses how much of that credit to place on the alternative. That is, each voter chooses a vector . The outcome of the game is the division where is proportional to , the amount of credits spent on alternative . Voters choose their spending so as to obtain an outcome that is as close to their ideal point as possible, according to distance.
Theorem 5.4 ().
The voting game defined above has a unique outcome that can be obtained in Nash equilibrium, and it is equal to the output of the Independent Markets mechanism.
The idea of the proof is to set equal to the amount that agent spends in the market for alternative , under the market setup that we described earlier. We show that when every agent casts “vote” , the system is at (its unique) equilibrium, and that the spending is proportional to the independent markets outcome.
Proof.
Let denote the spending profile of all voters other than , and let denote the aggregate division in which the weight on an alternative is proportional to the number of credits spent on that alternative. Suppose that is a best response for voter . We first show that for every alternative , implies and implies . That is, voters will always prefer to increase their spending on alternatives that they consider “undervalued”, and decrease their spending on “overvalued” alternatives.
To prove this, suppose that and . Since and are both normalized, there must exist some alternative for which . By marginally increasing while holding constant for all , voter can move the aggregate division from to , where , and for all , with and .
We can now show that , which implies that is not a best response for .
Next, we show that there exists an equilibrium of the voting game that produces the same outcome as independent markets. To this end, consider the market interpretation. For every alternative, some amount of a divisible good is sold to voters with a budget of 1 credit each. The number of credits that each voter spends in each market defines some spending profile .^{4}^{4}4This spending profile is not unique, since for voters with equal to the clearing price for alternative , there is some flexibility as to which voters pay for and get assigned goods, and which do not. Importantly, whenever , voter spends their full budget on alternative (i.e. ), and whenever , voter spends nothing on alternative (i.e. ). Further, the amount spent on each alternative is – the amount of goods sold multiplied by the price per unit – which is proportional to the aggregate division, . Therefore, the induced spending profile does produce the same aggregate division as independent markets when aggregated under the rules of the voting game. It remains to show that this spending profile is an equilibrium.
To do so, consider the spending vector of some voter , and suppose they have a better response . Denote the aggregate division when spends by (this division is , but we use for short), and the aggregate division when spends by . There must exist either some alternative with for which , and/or with for which . Suppose without loss of generality that the former case holds (if instead only the latter case holds, a very similar argument applies). Because , we know that . Therefore, the only way for
is for . But, because for all with , the reduction in ’s spending must come from alternatives with . That is, for all alternatives with . Therefore
which contradicts that is a better response than for voter .
Next we show that the voting game has a unique equilibrium aggregate division.^{5}^{5}5In contrast, the exact spending profile is clearly not unique. For instance, in an instance with only a single voter, that voter can enforce their belief exactly as long as they spend on each outcome in proportion to their belief, with no restriction on the magnitude of their spending. We know that the independent markets aggregate is an equilibrium with spending profile . For contradiction, suppose that there is some other equilibrium aggregate with spending profile . Then there is an alternative for which and an alternative for which . Thus, there are weakly fewer voters with than with . Because only voters with can have , and all voters with
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