A Condorcet-Consistent Participatory Budgeting Algorithm

09/18/2017 ∙ by Ehud Shapiro, et al. ∙ Weizmann Institute of Science Ben-Gurion University of the Negev 0

The budget is the key means for effecting policy in democracies, yet its preparation is typically an opaque and arcane process. Participatory budgeting is making inroads in municipalities, but is usually limited to a small fraction of the total budget and the produced budgets usually do not provide axiomatic guarantees. Here we apply the Condorcet principle to a general participatory budgeting scenario that includes a budget proposal, a vote profile, and a budget limit. We devise a polynomial-time budgeting algorithm that, given such a scenario, produces the Condorcet winner if it exists, else a member of the Smith set. (A caveat -- our definition of dominance employs strict rather than relative majority.) Furthermore, we argue that if there is no Condorcet winner for this scenario, then the resulting budget would often be close to a weak Condorcet winner for a slightly smaller budget limit. Our algorithm allows items to be quantitative, indivisible, and have arbitrary costs and allows voters to specify weak orders as their preferences. Furthermore, our algorithm supports hierarchical budget construction, thus may be applied to entire high-stakes budgets.

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1 Introduction

Even in the best democracies the budgeting process is somewhat of a mystery. Various people and bodies are involved, officially and unofficially, in the limelight and behind the scenes. Arguments are made, pressures are mounted, and at the end of the process a budget proposal is placed in front of the governing body for approval. While the governing body may question the proposed budget and request amendments, there is presently no feasible method for turning the individual preferences of the members of the governing body into a budget that faithfully reflects their joint democratic will. Here we provide such a method, which can be used by a budget committee, the parliament, or the citizens at large.

Participatory budgeting (PB) [1] offers ways for a society to participate in constructing a budget. Initiated by Brazil workers’ party [2], various societies have adopted similar ideas [3], where usually only a small fraction of a municipality’s annual budget is decided directly by the residents. After a deliberative phase, participants vote on projects such as schools, bike roads, etc., usually by some variants of -Approval [4], where each resident specifies projects of highest priority (other elicitation methods are mentioned below).

Current methods of participatory budgeting may be limited not only in their application, but also in their foundations: (1) While the previous budget is typically the starting point of practical budgeting processes, current methods ignore it; (2) The elicitation methods are usually limited to approvals or simple pairwise comparisons; (3) The methods typically cannot handle budgeting of quantitative items (e.g., how many new school buses to buy?) with all their complexities (e.g., decreasing marginal costs); and (4) Current methods cannot deal with hierarchical budgeting.

Here we aim to address these issues. As our method supports the democratic conduct of budget committees and parliaments, we term our approach democratic budgeting. We consider one of the foundations of social choice, namely the Condorcet principle, and generalize it to our setting: In single winner elections, the Condorcet approach involves selecting a winner which dominates all others by a voter majority; a natural way to generalize it to budgets is to select a budget (i.e., a subset of the proposed items, the cost of which is within the budget limit) which dominates all other budgets.

We describe a polynomial-time budgeting algorithm that computes a Condorcet-winning budget, whenever such a budget exists; if no such budget exists, it finds a member of the Smith set, defined below. We first prove the correctness of our algorithm on a simple model, which is close to the standard model of participatory budgeting (voters specify rankings, items are not quantitative), but takes the previous budget into consideration; then, we enhance our model as discussed above and describe how our algorithm generalizes naturally and remains computationally efficient.

Our democratic budgeting scenario is natural and captures more situations than the usual model; it is powerful and allows voters to specify both very simple preferences (e.g., approvals) and very complex ones (e.g., partial orders); budgets computed by our algorithm satisfy the voters’ wishes, and cannot be argued against. Our method allows for the construction of hierarchical budgets, and thus can be applied to situations where a “high-level” budget (say, a city budget) consists of several “low-level” budgets (say, one for education, another for transportation, etc.).

1.1 Related Work

Our model differs from existing models of PB: (1) We directly incorporate last year’s budget, in accordance with Reality-aware Social Choice [5]; (2) We employ a powerful and flexible elicitation method, accommodating partial orders (contrasted to Approval voting [1, 6], Knapsack voting [7], Value voting, Value-for-money voting, and Threshold voting [8]); (3) We allow quantifiable and indivisible items of various costs, allowing to express, e.g., decreasing marginal costs of budgeting certain amounts of the same item; and (4) We provide for hierarchical budget construction.

Within computational social choice [9], multiwinner elections [10] are receiving increased attention. We generalize multiwinner elections as (1) We allow partial orders (compare to, e.g., [11]; see also, e.g., [12]) and (2) We allow different costs (compare to, e.g., [10]; see also, e.g., [13]). Our generalization of the Condorcet criterion generalizes that of Fishburn [14, 15] (studied further in [16] and [17]; different, element-wise concepts exist [18, 19]). We concentrate on the more abstract minmax set extension [20, 21]; other set extensions are studied [16, 21]).

Our efficient algorithm iteratively uses Schwartz’s tournament solution. In this we mention the work of Bouysso [22] which study such algorithms abstractly.

2 Preliminaries

We provide preliminaries regarding rankings and tournaments. For an integer , we denote the set by . Given a set , a partial order on is a reflexive, antisymmetric, and transitive relation. A ranking is a total order. A weak ranking is a total order with ties, and corresponds to an ordered partition: Given a set , an ordered partition is a partition of into linearly ordered and disjoint sets, referred to as components, whose union is . E.g., given , the ordered partition , whose first, second, and third components are, correspondingly, , , and defines a weak order where, e.g., is preferred to while and tie. We interchange ordered partitions, weak orders, and weak rankings. A tournament is a directed graph. A tournament solution is a function which takes a tournament as an input and selects a nonempty subset of its vertices as output; i.e., , where is the set of all tournaments over . We use the following two tournament solutions.

Definition 1.

A Schwartz component of a tournament is a minimal set of vertices of such that for any there is no such that . The Schwartz set is the union of all Schwartz components.

Definition 2.

The Smith set of a tournament is the minimal set of vertices of such that for each and each , holds.

3 The Basic Model

We describe the basic model, where voters express preferences over distinct items, and, given the previous budget, we shall select a subset of these items whose total cost respects the predefined budget limit. Formally, a budget proposal is a set of distinct budget items, where the cost of a budget item is denoted by . A vote profile is a set of votes, where each vote is a linear order of the budget proposal , which specifies the preferences of the voter. A budget is a subset of the budget proposal (sometimes called bundle in the literature). Given a budget limit , a budget is feasible if its cost is within the budget limit, i.e., if . Members of are budgeted items while members of are unbudgeted items. A budget is exhaustive if it is feasible and for no item , the budget is feasible.

A budgeting algorithm gets as input the tuple , where is a budget proposal, is a vote profile, is a budget limit, and is the previous (e.g., last year’s) budget. Given this input, a budgeting algorithm computes a feasible budget . A budgeting algorithm is exhaustive if it computes only exhaustive budgets. We wish that the budget computed by a budgeting algorithm, in addition to being feasible and exhaustive, will also reflect the preferences of the votes of the vote profile , and we will use the previous budget as a guiding compass.

Example 1.

Consider the following simple budgeting scenario. The budget proposal contains the items: , , and . The cost of is , the cost of is , and the cost of is . The vote profile contains the votes: and . The budget limit is . Then, , , and are all feasible, while is the only exhaustive budget. An exhaustive budgeting algorithm, given the tuple , where is the previous budget, shall output the budget .

3.1 Condorcet-winning Budgets

We are interested in applying the Condorcet principle to budgets. In single-winner elections, the Condorcet principle states that if there is a candidate such that, for each other candidate , more voters prefer to , then shall be chosen.

To apply the Condorcet principle to budgets, a notion of voter preference among budgets shall be derived from voter preference among budget items. Fishburn et al. [14, 15] generalized the notion of Condorcet winner to multiwinner elections (where the aim is to select a set of unit-cost items); we further generalize it to sets of items of arbitrary costs. To achieve that, we apply the minmax set extension [20] to our budgeting setting: Intuitively, a voter prefers one budget over another if it ranks higher all the items budgeted by the first budget but not the second, compared to all items budgeted by the second budget but not by the first.

Definition 3 (Position).

Let be a vote over a proposal . We define , the positions of a budget in , to be .

E.g., for , .

Definition 4 (Prefers).

Let be a vote over a proposal . Let and be two budgets. Then, prefers over if , where .

Example 2.

Let be a proposal, where each item costs and let be a vote. Let and be two budgets. Then, while , thus, , and so prefers over .

Our definition is conservative as it may refrain from judging one budget as preferable to another even in cases when one might be intuitively inclined to make such a judgment. We find that it offers a good balance in being restrictive, and thus making only solid judgments, but powerful enough to provide a practical foundation for the following definitions and algorithm. Other options to define voter preference are discussed in Section 7. Next we describe what it means for one budget to dominate another. Note that we use strict rather than relative majority in defining this notion.

Definition 5 (Dominance).

Let be a profile and and budgets over a proposal . We say that (1) dominates if the number of votes preferring over  is larger than ; (2) weakly dominates , denoted by , if is not dominated by .

The weak dominance relation is reflexive ( for all ) and total ( or or both for all and ) but not transitive (see Example 3).

Definition 6 (Condorcet-winning budget).

Given a vote profile over a budget proposal and a budget limit , a feasible budget of is a Condorcet-winning budget if dominates all other feasible budgets.

A Condorcet-winning budget cannot be argued against, as there is no majority to support any change to it. Condorcet-winning budgets are also exhaustive. Such budgets, however, do not always exist.

Example 3.

Consider the budget proposal , where the cost of each item is , and the following vote profile over it: , , . Following the symmetry of the profile, we can assume, without loss of generality, that for the budget limit , any budget which is within limit equals to one of the following two options: ; or . Further, budget is not exhaustive, since, e.g., can fit within limit, and hence not a Condorcet winner (as Condorcet-winning budgets are exhaustive). Regarding budget , both and prefer the budget over and hence is not a Condorcet winner.

3.2 An Efficient Condorcet-consistent Budgeting Algorithm

A Condorcet-consistent budgeting algorithm is a budgeting algorithm that returns a Condorcet-winning budget whenever such exists. Consider the following naive algorithm: First, it creates a directed graph, referred to as the budgets graph, which contains a vertex for each feasible budget and an arc from vertex to vertex if dominates . Given this budgets graph, the algorithm returns the budget corresponding to a vertex with outgoing arcs to all other budgets, if such a vertex exists. Indeed, this budgeting algorithm is Condorcet-consistent. As there are exponentially-many feasible budgets, the naive algorithm described above runs in exponential time.

We describe a polynomial-time Smith-consistent budgeting algorithm (SBA): It always returns a budget from the Smith set of the budgets graph (see Preliminaries, Section 2). If a Condorcet-winning budget exists, then the Smith set is its singleton, thus our algorithm is also Condorcet-consistent. A pseudo-code of SBA is in Algorithm 1 (notice that we use the previous budget ; we defer a discussion on it to Section 3.4). it is composed of these procedures: A Ranking procedure which, given a vote profile over a budget proposal , aggregates the votes in into an ordered partition and outputs it; and a Pruning procedure which, given an ordered partition and a budget limit, outputs a feasible budget.

1:procedure Ranking
2:     input: a budget proposal
3:     input: a vote profile
4:     
5:     while  do
6:         
7:          ;      
8:     return  
9:
10:procedure Pruning
11:     input: a budget proposal
12:     input: an ordered partition
13:     input: a budget limit
14:     input: the previous budget
15:     ;
16:     for  do
17:          a maximal subset of closest to with
18:               
19:     return  
20:
21:procedure SBA
22:     input: a budget proposal
23:     input: a vote profile
24:     input: a budget limit
25:     input: the previous budget
26:     
27:     
28:     return
Algorithm 1 Smith-consistent budgeting (SBA)

The following notions play a key role in the description of the algorithm and its proof.

Definition 7 (Majority graph, weak majority graph).

Let be a budget proposal and ‘ be a vote profile. The majority graph of and has a vertex for each item and an arc between any two items if more than half of the votes rank higher than .

Its corresponding weak majority graph has, in addition, arcs and for every for which the majority graph has neither arc.

The Ranking procedure first creates the majority graph. Then, it iteratively construct the ordered partition , initiated to be empty. In each iteration, it identifies the Schwartz set (see Section 2), adds its vertices as the next component of and removes them from the majority graph. When no vertices remain in the majority graph it halts, at which point is indeed an ordered partition of .

The Pruning procedure, given the ordered partition computed by the ranking procedure, gradually populates the set of budgeted items . It iterates over the components of , where the iteration considers the component . It chooses a maximal subset of such that the cost of remains within limit; if several such subsets exist, it chooses one which is the closest to the previous budget , where closeness between two budgets is measured by the total cost of the symmetric difference between them. In particular, observe that if then . If this is not the case, then some items of will remain unbudgeted, but budgeting any of them would cause to go over the limit. After considering all components of , it outputs the budget .

3.3 SBA is Condorcet-consistent

Here we prove that SBA is Smith-consistent, and thus in particular Condorcet-consistent (see Section 2). We refer to a budget as a Smith budget if it is in the Smith set and show that our algorithm computes only Smith budgets. To do so we introduce the following notions.

Definition 8 (Weak domination path).

There is a weak domination path (path for short) in the budgets graph, from a budget to a budget , denoted by , if there is a sequence of budgets , , such that , , and for each .

Consider a budget which is not in the Smith set and another budget which is in the Smith set. By definition, is dominated by , and thus does not weakly dominate . Further, there is no path from to , since does not dominate any budget in the Smith set, and this also holds for all other budgets which are, like , not in the Smith set. We conclude, by considering the counterpositive, that in order to prove that SBA is Smith-consistent, it is sufficient to show that any budget it computes has paths to all other budgets.

Corollary 1.

Given a vote profile over a budget proposal and a budget limit , a budget is a Smith budget if and only if holds for any feasible budget of .

A path in a directed graph is a sequence of arcs, where the second vertex of an arc in the path is the first vertex of the next arc in the path. It will be useful to consider paths in the weak majority graph, referred to as weak majority paths. We denote the existence of a weak majority path from one item to another by . The lemmas below relate the weak majority graph to the budgets graph (recall that the former is polynomial and has budget items as vertices, while the latter is exponential and has feasible budgets as vertices).

Lemma 1 (Weak majority arc implies domination).

Let and be two vertices in the majority graph computed by SBA for some profile over some proposal . Let and . If the arc is present in the weak majority graph then .

Proof.

Towards a contradiction, assume that does not weakly dominate , so dominates . Let be a set of more than half of the voters where each voter prefers to . By definition, these voters rank higher than they rank . Thus, the arc is present in the majority graph, in contradiction to the assumption that is in the weak majority graph. ∎

Lemma 2 (Weak majority path implies domination path).

Let and be two vertices in the majority graph computed by SBA for some vote profile over some budget proposal . Let and . If then .

Proof.

Since it follows that there is a set of vertices such that , , and the arc is present in the weak majority graph for each . From Lemma 1 it follows that holds for each , where and . We conclude that . ∎

Lemma 3 (Crucial lemma).

Let be a budget proposal, be a budget limit, two feasible budgets, and be a vote profile over . Then, if for some and , then .

Proof.

Let and such that . Let and notice that is feasible since and is feasible. Further, since in particular as . In order to show that it remains to show that . Since there is a set of vertices such that , , and is in the weak majority graph for each . Notice that and that . Let () be the smallest index for which but . (Indeed, it might be that ; this would happen if the weak majority path from to does not go through any vertex in .) Using Lemma 2 and since it follows that where , thus it remains to show that . Using Lemma 1, and since the arc is in the weak majority graph, it follows that where . Thus, for each set of more than half of the voters there exists at least one voter which ranks before it ranks . Since but , it then follows that . As such a voter exists in every set containing more than half of the voters, it follows that . ∎

Theorem 1.

SBA is Smith-consistent.

Proof.

Consider the ordered partition produced by the Ranking procedure (each is a maximum subset of the th Schwartz set). Consider the budget produced by the Pruning procedure, based on the ordered partition and let ; note that the ’s are exactly the maximal subsets selected for budgeting by the Pruning procedure. Let be a feasible budget and let . Below we show that which will finish the proof.

If then and thus cannot dominate , thus we assume that . Consider the smallest index () for which and consider some . By construction, . Since is a maximal subset of such that the budget is feasible (refer to Line 17 in Algorithm 1), it follows that . Let , let , and notice that is a feasible budget since and is feasible. Further, notice that since . It remains to show that which would finish the proof.

Recall that and consider the following two cases. The cases differ in whether is in the same component as (Second case) or in an earlier component , (First case). In each case we show that and thus conclude that indeed and thus SBA is Smith-consistent.

First case.  If for some , then the arc cannot be in the majority graph, since is a Schwartz set in the majority graph not containing the vertices in but containing . Thus, the arc is present in the weak majority graph, thus . Applying Lemma 3 we conclude that .

Second case.  If , then and are in the same Schwartz set . If and are in different Schwartz components, then the arcs and are not present in the majority graph, thus they are present in the weak majority graph. In particular, , and by applying Lemma 3 we conclude that . Otherwise, if and are in the same Schwartz component then since each Schwartz component is a cycle in the majority graph. Again, applying 3 we conclude that . ∎

SBA is efficient, as the majority graph has linearly-many vertices and computing a Schwartz set can be done in polynomial time (Schwartz components correspond to strongly connected components; this is folklore). Furthermore, choosing a maximal subset of a Schwartz set which is the closest to the previous budget can be done efficiently (e.g., greedily choosing items budgeted by ).

Corollary 2.

SBA is a polynomial-time Smith-consistent budgeting algorithm.

3.4 Reality-aware Budgeting and Hysteresis

Notice how SBA uses the previous budget to break ties when selecting a maximal subset of each Schwartz set (in the pruning phase). This is a special case of the Reality-aware voting rules considered in [5] and is motivated, e.g., by Arrow’s claim that “The status quo does have a built-in edge over all alternative proposals” [23, Page 95]. More generally, considering the previous budget is both natural and useful, as budgeting scenarios fit very well in the framework of Reality-aware Social Choice [5]: (1) Votes may be provided as a simple amendment to the previous budget, e.g., by adding/removing items or changing their quantities (indeed, recall that we can incorporate partial orders and also refer to the UI considerations at Section 7); and (2) The distance of a new budget to the previous budget can be computed, e.g., as the total cost of the items in the symmetric difference between the two (see the Distance-based Reality-aware Social Choice model [5]).

Formally, the previous budget can be seen as a dichotomous ordered partition of the budget proposal, where the first indifference class contains the previously-budgeted items and the second contains those which are not. Then, the previous budget can be used as follows: (1) Limit voters’ flexibility, requiring them to submit preferences which are not too far from the previous budget; (2) Orthogonally, limit the distance of the new budget from the previous budget. This might be a form of hysteresis (as discussed, e.g., in [24]), limiting the rate of change of budgets year-to-year.

This perspective allows for the budget limit itself to be decided upon democratically, e.g., by taking the median of the proposed budget limits. This view of the previous budget as an ordered partition motivates the generalization described next.

4 Incorporating Partial Orders

In this section we adapt SBA to voters specifying partial orders. This would be especially useful for considering quantitative items (Section 5). The main observation is that we shall only care for the definition of when a voter prefers some budget over another . For linear orders, we had Definition 4. Next we consider ordered partitions. Ordered partitions (linear orders with ties) are particularly interesting in the context of budgets, as they can be understood as preference classes.

Example 4.

Consider a budget proposal and a vote . Such a vote can be understood as a voter wishing to first budget ; then, if possible, budget and , and only if the budget is not exhausted yet, budget and .

The definition of when a voter prefers one budget to another extends as is from linear orders (i.e., Definition 4) to ordered partitions. Notice that, in particular, it means that a voter is indifferent to two budgets having a nonempty symmetric difference in at last one component of her ordered partition.

Example 5.

Let be an ordered partition over the budget proposal . Let the cost of all items be , and let and . Then, , thus neither prefer to nor to .

It seems that ordered partitions provide enough practical expressive power for voters (see Example 4). Still, the following definition is a generalization, wrt. to the minmax extension, for general partial orders.

Definition 9 (Prefers for partial orders).

Let be a partial order over some budget proposal . Then, prefers a budget over a budget if the following hold: (1) For each item there is an item which comes before in the partial order; (2) For each item there is no item which comes before in the partial order.

Importantly, the proof of Theorem 1 follows through with this definition, by adapting Lemma 3 to account for this adapted notion of preference among budgets.

5 Quantitative Budgets

We extend our model to deal with quantitative budgets, and show that SBA remains efficient even when quantities are given in binary encodings. To this end, we adapt the definition of a budget proposal to account for quantitative budgets.

Definition 10 (Quantitative representation of budget proposals).

In the quantitative representation of budget proposals, a budget proposal is a set of tuples , where is the quantity of the budget item . Each budget item , having copies of it in the budget proposal, is associated with a cost function, where , , is the cost of copies of . The definition of cost is generalized to budgets in a natural way: The cost of a budget containing copies of budget item , is .

Example 6.

Consider a proposal containing submarines, boats, and airplane. Each submarine costs million, but there is a discount of million for each submarines; each boat costs million; and an airplane costs million. Then, denoting submarines by , boats by , and airplanes by , the proposal is with cost functions: For submarines: , , , , ; For boats: , , ; For airplanes: . A voter can then specify: “my first priority is 2 submarines; my second priority is 1 airplane and 3 boats; my last priority is another 3 submarines”; by voting: .

Remark 1.

A voter, specifying an ordered partition over a quantitative budget proposal , can be represented as , where each contains several tuples of the form , where ; the sum of those quantities sums to . E.g., in the vote considered in the example above, corresponds to ; and there are, e.g., submarines in all of these components together.

5.1 Prefers with Quantities

To adapt SBA to quantitative budget proposals, first we adapt Definition 4 to define when does a voter prefer one budget over another. We introduce the concepts of remainders and ranked difference. The remainder of a budget with respect to a vote consists of all the items ranked by a vote but not budgeted by the budget, keeping their original ranking in the vote.

Definition 11 (Remainder).

Let be a quantitative budget proposal, a budget of , and an ordered partition over . The unbudgeted remainder of with respect to is , with components , , defined via the following equations: , and for each , , and .

Intuitively, the component of the remainder of a budget of with respect to contains exactly the elements of the component of which are unbudgeted by . Given a vote and a budget over , the union of the elements of equals , which are all items unbudgeted by . We use remainders to compute the ranking of the elements in the symmetric difference between two budgets.

Definition 12 (Ranked difference between budgets).

Let be a quantitative budget proposal, a ranking over , and and budgets over . The ranked difference between the unbudgeted remainders of the two budgets with respect to is , where the set difference operator is applied component-wise. We define as the set of indices of the components for which is non-empty; that is, denoting as , we define .

That is, while collects and ranks items unbudgeted by according to their ranking by , collects and ranks items budgeted by but not by according to their ranking, and each index in names a component of that has at least one item budgeted by but not by .

Next, intuitively, a vote prefers one budget over another if it ranks higher all the items budgeted by the first budget but not the second, compared to all items budgeted by the second budget but not by the first.

Definition 13 (Prefers for quantitative budgets).

Let be a proposal, a ranking over , and and budgets over . We say that prefers over if it holds that , where we set .

E.g., if , then is preferred over (by any vote ) since while is not. In addition, prefers is irreflexive since . This definition indeed generalizes the simple definition of the basic model, described in Section 3. To illustrate the notions defined above, consider the following example.

Example 7.

Let be a quantitative budget proposal, where the cost of each item is . Let be a profile, where (each occurrence of is a different ‘‘-item). Let and be two budgets. Then, , , , and . Thus, prefers over .

5.2 SBA is Efficient for Quantitative Proposals

A straightforward adaptation of SBA to quantitative budget proposals would be to rename the items such that the quantity of each item would be one, and then to use the description of SBA from Section 3; this, however, results in pseudo-polynomial time.

It is possible to slightly modify SBA to run in polynomial time for quantitative budget proposals. To explain our modifications to SBA, it is useful to identify the cause of the computational inefficiency of the straightforward adaptation of SBA: The constructed majority graph would have a vertex for each single copy of each item, possibly resulting in a super-polynomial-sized majority graph. To remedy this inefficiency, it seems natural to construct a majority graph with one vertex for each item, representing all copies of that item: The resulting majority graph would indeed have polynomial number of vertices. However, we would not be able to represent all majority relations between the budget items, as such graph would be, roughly speaking, too “coarse”. To overcome this difficulty, we will first initiate the majority graph as described above, followed by splitting its vertices according to the way the corresponding copies of each item are split in the vote profile.

Theorem 2.

SBA can be adapted to quantitative budget proposals to run in polynomial time.

Proof.

We describe a modification to SBA, denoted by ESBA, which differs from SBA in the way it constructs the majority graph. Specifically, first construct a majority graph having a vertex for each tuple in the quantitative budget proposal. Then, iteratively process each tuple in the quantitative budget proposal, initially corresponding to a vertex in , as follows. For each vote , represented over the quantitative representation of budget proposals, and where appears in the components (, ) with corresponding quantities , such that (, ), notice that , and create the following sequence of numbers, denoted by : . Intuitively, collects ’s splitting points of . E.g., consider a vote : Notice that splits the occurrences of into two components, where in the first component has two of its occurrences while in the second component has the third occurrence of . Correspondingly, .

After computing for each vote , merge those sequences into a single sequence denoted by ; that is, , where is the given vote profile. Intuitively, collects the splitting points of across all voters. Recall that currently the majority graph has one vertex corresponding to the budget item ; first, we delete from the majority graph. Then, denote and create the following vertices . Intuitively, a vertex stands for the , , , , and occurrence of .

The above process is done for each tuple in the quantitative budget proposal. Importantly, the number of vertices in the resulting majority graph is polynomially bounded by the input, as any splitting point in , and thus each vertex in the majority graph, can be charged to at least one vote, and each vote can be responsible only for polynomially-many vertices. Furthermore, the constructed majority graph can represent all majority relations, as no vote splits a budget item “finer” than . ∎

6 Hierarchical Budgeting

Budgets of complex organizations and societies are constructed hierarchically; here we use our approach, specifically the fact that SBA works in two independent phases, to achieve that.

Hierarchical budgets are constructed in two phases. First, each section of the organization performs an independent budgeting process that decides upon the priorities within the section, but without a given budget limit. Then, a consolidating budgeting process determines the allocation of funds to the different sections by providing each with a section budget limit, the sum of which is the consolidated budget limit. Each section’s ordered partition is pruned by its section budget limit. The resulting section budgets are combined into a consolidated feasible budget. E.g., consider a government with several ministries: each ministry conducts its own budgeting process and then the government conducts its overall budgeting process by allocating resources to each of the ministries.

An important feature of our algorithm is that it does not require fixing an a priori section budget limits, as the ranking procedure does not depend on the limit. This flexibility turns out to be crucial for the algorithmic construction of hierarchical budgets. It also has important practical implications: The allocation of section budget limits need not be done a priory, but can use the ranking of the section to take into account and deliberate the implications of different section budget limits on the budgeting of various items. So while the government might not intervene in the ranking of the section (perhaps respecting the subsidiary principle), it has some control of what will be funded and what will not by choosing a specific section budget limit.

Our hierarchical budgeting method can be applied in two scenarios: One, where each section budget is created by votes of experts/stakeholders, and then the consolidated budget is created by the votes of the sovereign body. Another, when the same body decides on both section budgets and the consolidated budget. This method helps the sovereign body to focus in turn on each section budget independently, and then to consolidate the results.

Our next description has only two levels of hierarchy, however the process naturally generalizes to further levels. Given sections, we begin by creating budget proposals , one for each section. Then, the voters of each section vote independently on each section proposal. We separately apply only the ranking procedure to the votes of each section to produce the section rankings . We arbitrarily linearize the unary rankings; denote the resulting sequences and define to be the set of the first items of . We then define “artificial” derived budget items , one for each section, and associated cost functions . The consolidated budget proposal is then , where is the number of of items in . Then, each vote on the consolidated budget proposal is a ranking of the consolidated proposal , which consists of only ’s. (Practically, however, when a voter chooses, say, to rank first six ’s, this would be after the voter has looked “under the hood” and knows what are the first six items of the budget of the section.) Given a budget limit on the consolidated budget, all votes on the consolidated budget proposal are then treated as usual: We apply the ranking procedure and then the pruning procedure with the limit , and produce a high-level budget.

With respect to the allocation between sections, the properties of this consolidated budget are the same as the properties of the non-hierarchical budgets produced by our algorithm. Votes on the consolidated proposal can prioritize items among the different sections, and know what are they prioritizing, but cannot affect, at the consolidation stage, the relative priority of the underlying “true” items within each section.

To compare constructing a budget to constructing it hierarchically, we assume that the voters have preferences over all items, of all sections, and that, to ease the budgeting process, they decide to construct it hierarchically. Then, we compare the two options: (1) Using SBA directly on the preferences of the voters; and (2) Using SBA hierarchically, first on each section independently, and then consolidating the budget at a high level, over the sections, as described above. The resulting budget might be different.

Example 8.

Consider two sections, Section A, containing items and , and Section B, containing item . Let all items cost , and set the budget limit to . Assume the following voters with preferences over all items of all sections together: , , . The budget is a Condorcet-winning budget (notice that both and prefer it to while both and prefer it to ); thus, using SBA directly, disregarding hierarchy, would result in being the winning budget, while using SBA hierarchically would result in a different winning budget: Using the ranking phase of SBA on Section A would give the ordered partition , and using the ranking phase of SBA on Section B would give the ordered partition . Then, considering those ordered partitions and employing SBA in the high-level, would result in the budget .

There are certain cases in which using SBA directly (disregarding hierarchy) and using it hierarchically is guaranteed to give the same results. One simple, extreme such case is where all voters agree unanimously on the order of the low-level section items of all sections.

7 Discussion

We identified certain aspects, generally disregarded in the growing literature on PB, which are nevertheless useful in employing such methods to produce budgets in a democratic way. Accordingly, we provided a formal treatment of a general setting of democratic budgeting which (a) Takes into account the previous (last year’s) budget; (b) Allows voters to specify partial orders or special cases of such; (c) Incorporates quantitative, indivisible items of arbitrary costs, capturing, e.g., items with decreasing marginal costs; and (d) Allows for hierarchical construction of budgets. We generalized the Condorcet criterion to democratic budgeting and described a polynomial-time Condorcet-consistent participatory budgeting. Our algorithm is designed in such a way that it supports hierarchical budget construction, making it applicable to entire, high-stakes budgets. Next, we discuss avenues for further research.

Complementarities.  Our elicitation method allows for quite complex user preferences, but does not incorporate complementarities between budget items. Such inter-relations between budget items have not been considered in the literature on PB. Having efficient ways for users to express such, while avoiding exponential blow-up, is an interesting future research direction.

More Axiomatic Considerations.  Here we concentrated on Condorcet-consistency. This makes our methods suitable for situations where majority decisions are morally justified and not to situations where proportionality is desired (where one can use, e.g., the methods of Aziz et al. [6]; notice that usually majoritarianism and proportionality are somewhat mutually exclusive). On a different note, one might consider other axiomatic properties, such as monotonicity.

Aspects of Democratic Budgets.  One might consider other methods of democratic budgeting, e.g., investigating how to generalize existing methods of participatory budgeting to be applied to the more general setting of democratic budgeting.

Breaking Ties.  SBA picks a maximal subset of each Schwartz set which is the closest to the previous budget. We suggest further to (1) Sort items by cost, favoring cheaper items, resulting in budgets with more items; and (2) Sort items by indices, favoring an item which was never budgeted to another item of a type which was already budgeted several times, resulting in more diverse budgets. One might also aim at solving the underlying Knapsack problem, by computing (e.g., using dynamic programming) the maximum-sized subset of each Schwartz set.

User Interface Considerations.  For deployment, and if a vote is a weak order, a convenient UI is needed to specify it. Two simple user interfaces are a sequence of pairs (item, quantity) and a pie chart. We interpret the former as an ordered partition where each partition consists of one pair. A pie chart might be interpreted as one component with multiple pairs, but we propose to interpret it differently, as follows: If not all items can be funded, then items should be funded proportionally according to the cost ratios specified by the chart. As our budgeting model refers to discrete items, each with a specific cost, such proportional allocation can only be approximated. To exploit the full power of ordered partitions, a user interface should cater for a sequence of pie charts. Furthermore, the user interface may directly incorporate the previous budget, viewed as a dichotomous ordered partition, and allow voters to move items from the two indifference classes, and also to provide for the possibility of refining them to more classes.

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