Budget-proposal aggregation (BPA) is a problem in social choice theory.
The problem is how to aggregate the different opinions into a single budget-distribution program.
BPA is a special case of participatory budgeting, with the following characteristics: It is also a special case of fractional social choice (portioning), in which agents express their preferences by stating their ideal distribution, rather than by a ranking of the issues.
Suppose a manager asks his worker to submit a budget-proposal for a project.
To mitigate this effect, it is possible to ask the worker for aggregate budget-proposals (for several projects at once).
The goal is to aggregate all distributions to a single probability-distribution, representing the probability that society should choose each candidate.
It has several advanatages: But the median rule may be considered unfair, as it ignores the minority opinion.
Forr every set of phantoms, the rule that chooses the median of the set of real votes + phantoms is strategyproof; see median voting rule for examples and characterization.
It has several characerizations: Aziz, Lam, Lee and Walsh[11] study the special case in which the preferences are single-peaked and symmetric, that is: each agent compares alternatives only by their distance from his ideal point, regardless of the direction.
In particular, they assume that each agent's utility is 1 minus the distance between his ideal point and the chosen allocation.
They consider several fairness axioms: The following is known about existing rules: They prove the following characterizations: Border and Jordan[12]: Cor.1 prove that the only rule satisfying continuity, anonymity, proportionality and strategyproofness is UPM.
In other words, if we pick the median on each issue separately, we might not get a feasible distribution.
In the multi-dimensional case, aggregation rules depend on assumptions on the utility functions of the voters.
Lindner, Nehring and Puppe[14] consider BPA with discrete amounts (e.g. whole dollars).
They define the midpoint rule: it chooses a budget-allocation that minimizes the sum of L1-distances to the voters' peaks.
Moreover, they prove that the possibility of strategic manipulation is limited: a manipulating agent cannot make the closest midpoint closer to his peak, nor make the farthest midpoint closer to his peak.
As a consequence, the midpoint rule is strategyproof if all agents have symmetric single-peaked preferences.
Goel, Krishnaswamy, Saskhuwong and Aitamurto[15] consider BPA in the context of participatory budgeting with divisible projects: they propose to replace the common voting format of approving k projects with "knapsack voting".
(1) If an agent changes his reported peak, but all the phantoms are fixed in place, then we have a median voting rule in each issue, so the outcome in each issue either stays the same or goes farther from the agent's real peak.
Note that the proof of (2) crucially relies on the assumption of L1 utilities, and does not work with other distance metrics.
A demo of the Independent Markets rule, and several other moving-phantoms rules, is available online.. Caragiannis, Christodoulou and Protopapas[2] extended the definition of proportionality from single-minded preference profiles to any preference profile.
Nehring and Puppe[17][18] aim to derive decision rules with as few assumptions as possible on agents' preferences; they call this the frugal model.
They study BPA allowing lower and upper bounds on the spending on each issue.
Fain, Goel and Munagala[21] assume that agents have additive concave utility functions, which represent convex preferences over bundles.
They study the Lindahl equilibrium of this problem, prove that it is in the core (which is a strong fairness property), and show that it can be computed in polynomial time.
For each agent i there is a coefficient ai,f that represents the utility of monetary gains and losses, and there is a function f which is strictly convex for negative values and strictly concave for positive values, and
For this utility model, they present a variant of the Vickrey–Clarke–Groves mechanism that is strategyproof, but requires side-payments (in addition to the tax).
Puppe and Rollmann present a lab experiment comparing the average voting rule and a normalized median voting rule in multidimensional budget aggregation setting.