Combinatorial participatory budgeting, also called indivisible participatory budgeting or budgeted social choice, is a problem in social choice. There are several candidate projects, each of which has a fixed costs. There is a fixed budget, that cannot cover all these projects. Each voter has different preferences regarding these projects. The goal is to find a budget-allocation - a subset of the projects, with total cost at most the budget, that will be funded. Combinatorial participatory budgeting is the most common form of participatory budgeting.
Combinatorial PB can be seen as a generalization of committee voting: committee voting is a special case of PB in which the "cost" of each candidate is 1, and the "budget" is the committee size. This assumption is often called the unit-cost assumption. The setting in which the projects are divisible (can receive any amount of money) is called portioning, fractional social choice, or budget-proposal aggregation.
PB rules have other applications besides proper budgeting. For example:
Selecting validators in consensus protocols, such as the blockchain; Selecting web pages that should be displayed in response to user queries; Locating public facilities; Improving the quality of genetic algorithms.
Welfare-maximization rules One class of rules aims to maximize a given social welfare function. In particular, the utilitarian rule aims to find a budget-allocation that maximizes the sum of agents' utilities. With cardinal voting, finding a utilitarian budget-allocation requires solving a knapsack problem, which is NP-hard in theory but can be solved easily in practice. There are also greedy algorithms that attain a constant-factor approximation of the maximum welfare.
There are many possible utility functions for a given rated ballot:
Chamberlin-Courant satisfaction assumes that an agent's utility is 1 if at least one of his approved projects is funded, and 0 otherwise. Cardinality-based satisfaction assumes that an agent's utility is a function of the total score assigned to the projects that are funded. Cost-based satisfaction assumes that an agent's utility is the total cost of the projects funded, times the score assigned to said project. Decreasing-normalized-satisfaction (DNS) functions are satisfaction functions that are weakly-increasing with the cost, but the increase-rate is not faster than the cost. Cardinality-satisfaction is one extreme, in which satisfaction does not change with the cost; cost-based is another extreme, in which satisfaction grows exactly like the cost. In between, there are utility functions such as the square-root of the cost, or the logarithm of the cost. Approval-based satisfaction assumes that there is a function sat that maps a set of projects to a positive number, and an agent's utility equals sat(approved-funded-projects). All previous utilities are special cases of approval-based satisfaction functions. Sreedurga, Bhardwaj and Narahari study the egalitarian rule, which aims to maximize the smallest utility of an agent. They prove that finding an egalitarian budget-allocation is NP-hard, but give pseudo-polynomial time and polynomial-time algorithms when some natural paramerters are fixed. They propose an algorithm that achieves an additive approximation for restricted spaces of instances, and show that it gives exact optimal solutions on real-world datasets. They also prove that the egalitarian rule satisfies a new fairness axiom, which they call maximal coverage.
Annick Laruelle studies welfare maximization under weak ordinal voting, where a scoring rule is used to translate ranking to utility. She studies a greedy approximation to the utilitarian welfare and for the Chamberlin-Courant welfare. She tests three algorithms on real data from the PB in Portugalete in 2018; the results show that the algorithm including project costs in the ballot performs better than the other two.
Knapsack budgeting Fluschnik, Skowron, Triphaus and Wilker study maximization of utilitarian welfare, Chamberlin-Courant welfare, and Nash welfare, assuming cardinal utilities.
The budgeting method most common in practice is a greedy solution to a variant of the knapsack problem: the projects are ordered by decreasing order of the number of votes they received, and selected one-by-one until the budget is exhausted. Alternatively, if the number of projects is sufficiently small, the knapsack problem may be solved exactly, by selecting a subset of projects that maximizes the total happiness of the citizens. Since this method (called "individually-best knapsack") might be unfair to minority groups, they suggest two fairer alternatives:
Diverse knapsack - maximizing the number of citizens for whom at least one preferred item funded (similarly to the Chamberlin-Courant rule for multiwinner voting). Nash-optimal knapsack - maximizing the product of the citizens' utilities.
Unfortunately, for general utility domains, both these rules are NP-hard to compute. However, diverse-knapsack is polynomially-solvable in specific preference domains, or when the number of voters is small.
Proportional approval voting Proportional approval voting is a rule for multiwinner voting, which was adapted to PB by Pierczynski, Peters and Skowron. It chooses a rule that maximizes the total score, which is defined by a harmonic function of the cardinality-based satisfaction. It is not very popular, since in the context of PB it does not satisfy the strong proportionality guarantees as in the context of multiwinner voting.
Sequential purchase rules In sequential purchase rules, each candidate project should be "bought" by the voters, using some virtual currency. Several such rules are known.
Phragmen's rule This method Phragmen's rules for committee elections. Los, Christoff and Grossi describe it for approval ballots as a continuous process:
Each voter starts with 0 virtual money, and receives money in a constant rate of 1 per second. At each time t, we define a not-yet-selected project x as affordable if the total money held by voters who approve x is at least the cost of x. At the first time in which some project is affordable, we choose one affordable project y arbitrarily. We add y to the budget, and reset the virtual money of voters who approve y (as they "used" their virtual money to fund y). Voters keep earning virtual money and funding projects, until the next fundable project would bring the total cost over the total available budget; at that point, the algorithm stops.
Maximin support rule This rule is an adaptation of the sequential Phragmen rule, which allows a redistribution of the loads in each round. It was first introduced as a multiwinner voting rule by Sanchez-Fernandez, Fernandez-Garcia, Fisteus and Brill. It was adapted to PB by Aziz, Lee and Talmon (though they call it 'Phragmen's rule'). They also present an efficient algorithm to compute it.
Method of equal shares This method generalizes the method of equal shares for committee elections. The generalization to PB with cardinal ballots was done by Pierczynski, Peters and Skowron.
Each voter starts with B/n virtual money, where B is the available budget and n is the number of voters. A project x is called r-affordable if it is possible to cover its cost by taking money from agents such that each agent i pays min(current-money-of-i, r*ui(x)). That is: each agent participates in funding x in proportion to ui(x). The number r represents the "price per unit of utility" (note that the utilities are normalized to the range [0,1]). In the special case of approval ballots, the utilities are 0 or 1, so a project is r-affordable if it is possible to cover its cost by taking money from agents who approve x, such that each agent i pays min(current-money-of-i, r). Agents with less than r money pay only their current balance. We iteratively add to the budget, an r-affordable project for the smallest possible value of r, and decrease the virtual balance of agents who support this project. When no more r-affordable projects can be found, for any r, the process stops.
Other rules Shapiro and Talmon present a polynomial-time algorithm for finding a budget-allocation satisfying the Condorcet criterion: the selected budget-allocation should be at least as good as any other proposed budget, according to a majority of the voters (no proposed change to it has majority support among the votes). Their algorithm uses Schwartz sets.
Skowron, Slinko, Szufa and Talmon present a rule called minimal transfers over costs that extends the single transferable vote to participatory budgeting.
Aziz and Lee present a rule called the expanding approvals rule that uses the . Pierczynski, Peters, and Skowron present a variant of the method of equal shares for weak-ordinal ballots, and show that it is an expanding approvals rule.
Fairness considerations An important consideration in budgeting is to be fair to both majority and minority groups. To illustrate the challenge, suppose that 51% of the population live in the north and 49% live in the south, and that every possible budget . suppose there are 10 projects in the north and 10 projects in the south, each of them costs 1 unit, and the available budget is 10. Voting on budgets by simple majority rule will result in the 10 projects in the north being selected, with no projects in the south, which is unfair to the southerners.
To partially address this issue, many municipalities perform a separate PB process in each district, to guarantee that each district receives a proportional representation. But this introduces other problems. For example, projects on the boundary of two districts can be voted only by one district, and thus may not be funded even if they are supported by many people from the other district. Additionally, projects without a specific location, that benefit the entire city, cannot be handled. Moreover, there are groups that are not geographic, such as: parents, or bike-riders.
The notion of fairness to groups is formally captured by extending the justified representation criteria from multiwinner voting. The idea of these criteria is that, if there is a sufficiently large group of voters who all agree on a sufficiently large group of projects, then these projects should receive a sufficiently large part of the budget. Formally, given a group N of voters and a set P of projects, we define:
N can afford P if , that is: N is sufficiently large to fund the projects in P with their proportional share of the budget. The potential utility of N from P is . In particular, in case of approval ballots and cardinal satisfaction, the potential utility is simply the number of projects in P approved by all members in N.
Based on these definitions, many fairness notions have been defined; see Rey and Maly for a taxonomy of the various fairness notions. Below, the chosen budget-allocation (the set of projects chosen to be funded) is denote by X.
Strong extended justified representation Strong extended justified representation (SEJR) means that, for every group N of voters that can afford a set P of projects, the utility of every member of N from X is at least as high as the potential utility of N from P. In particular, with approval ballots and cardinal satisfaction, if N can afford P and all members in N approve P, then for each member i in N, at least |P| projects approved by i should be funded.