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Social Choice and Welfare

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18-11-2020 | Original Paper

Markets for public decision-making

Authors: Nikhil Garg, Ashish Goel, Benjamin Plaut

Published in: Social Choice and Welfare | Issue 4/2021

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Abstract

A public decision-making problem consists of a set of issues, each with multiple possible alternatives, and a set of competing agents, each with a preferred alternative for each issue. We study adaptations of market economies to this setting, focusing on binary issues. Issues have prices, and each agent is endowed with artificial currency that she can use to purchase probability for her preferred alternatives (we allow randomized outcomes). We first show that when each issue has a single price that is common to all agents, market equilibria can be arbitrarily bad. This negative result motivates a different approach. We present a novel technique called pairwise issue expansion, which transforms any public decision-making instance into an equivalent Fisher market, the simplest type of private goods market. This is done by expanding each issue into many goods: one for each pair of agents who disagree on that issue. We show that the equilibrium prices in the constructed Fisher market yield a pairwise pricing equilibrium in the original public decision-making problem which maximizes Nash welfare. More broadly, pairwise issue expansion uncovers a powerful connection between the public decision-making and private goods settings; this immediately yields several interesting results about public decisions markets, and furthers the hope that we will be able to find a simple iterative voting protocol that leads to near-optimum decisions.

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An alternative interpretation is that the issues themselves are divisible: for example, in the case of a city choosing how much money to a particular project, any amount of money is a valid outcome.

This can also be thought of as a private goods market with externalities: each agent’s utility depends not only on her own bundle, but also other agents’ bundles.

The concept of Nash welfare is due to Nash (1950) and Kaneko and Nakamura (1979).

These utility classes will be defined and discussed later.

A nested Leontief-Leontief function is still a Leontief function. Incidentally, this also implies that for a public decision making problem where agent utilities are Leontief, we get a Fisher market which has exactly the same form, i.e. with Leontief utilities.

In public goods, all agents have nonnegative utility for every good, and the question is how to allocate their money between the goods. In contrast, in public decision-making, agents have opposing preferred alternatives and are in direct competition on each issue. With a careful modification, Foley’s work does carry over to the public decision-making setting.

This discussion is thanks to Kamesh Munagala via private correspondence.

An outcome is Pareto optimal if there is no way to improve the utility of any agent without hurting another agent.

A similar result holds for indivisible goods Klaus and Miyagawa (2002).

In our model, divisibility and randomization are equivalent, but this is not always the case: for example, if there were a some sort of global budget constraint, randomization could lead to a non-viable deterministic outcome.

Although the entire supply is typically allocated, it is standard in the private goods literature to allow for outcomes where this does not occur, i.e. \(\sum _{i \in N} x_{ij} < 1\). This will be discussed in Sect. 2.2.

The weights \(w_{ij}\) have different interpretations for Leontief utilities vs other CES utilities. For example, if there is only a single good, the CES utility form reduces to \(w_{i1} x_{i1}\) and the Leontief utility form reduces to \(x_{i1}/w_{i1}\). When we say that taking the limit as \(\rho \rightarrow - \infty\) yields Leontief utilities, we mean that we obtain the form of Leontief utilities (i.e., a minimization over all the goods).

This is technically the “budget-weighted” Nash welfare, but we will omit “budget-weighted” throughout the paper.

This correspondence still holds under slightly weaker assumptions that our six assumptions Jain and Vazirani (2010).

Whenever we maximize over outcomes of a public decisions instance, i.e., \(\max _{\mathbf {z}} NW(\mathbf {z})\), we implicitly assume that only valid outcomes are considered, meaning that \(z^{j,0} + z^{j,1} \le 1\) for all \(j\in M\). The same is true when we maximize over outcomes of a private goods instance, and we adopt these conventions throughout the paper.

As in the Fisher market setting, agents are allowed to demand more than 1 unit of an issue if the cost is less than their budget. The interpretation of demanding more than unit probability is difficult, but the prices will ensure that this never occurs in equilibrium.

If some issue j is not sold completely and so \(z^{j,0} + z^{j,1} < 1\), one can think of the remaining \(1 - z^{j,0} - z^{j,1}\) being allocated to some third option that has no value for any agent.

Throughout most of the paper, we use \(\mathbf {z}\) to refer to the outcome of a public decisions instance and \(\mathbf {x}\) to refer to the outcome of a private goods instance. In this discussion, the welfare functions are the same for both public and private instances, so we will use \(\mathbf {z}\) to denote outcomes for both.

For completeness, we mention an additional mild condition required for this result: the polytope containing the set of feasible utilities of the two agents must be able to be described via a combinatorial LP.

With strictly concave utility functions, each agent’s demand at a given price is unique. With linear utilities, the demand set can be expressed as the set of goods that are equally desirable at the given prices. Also, we assume that agents are price-taking, meaning that they honestly report their demand given a set of prices, and do not anticipate how prices will change as a result of their actions.

functions that meet the five conditions from Sect. 2.

The outcome will not be well-defined for a list of private bundles not at equilibrium, since agents may have incompatible demands. This will not be important; we mention it only for completeness.

We assume that agents truthfully report their demands according to this definition: recall that we do not consider strategic behavior.

In the proof of Theorem 4.2, the definition of \(z^{j,1}\) was simply \(1 - z^{j,0}\). It is necessary to use \(\max (1 - z^{j,0}, 0)\) here instead: because this is an approximate equilibrium, it is possible that \(z^{j,0} > 1\), which would make \(1 - z^{j,0}\) negative.

Those works take great care to show conditions analogous to strong convexity or Lipschitz continuity of the gradient in the cases of interest.

It is well known that the primal objective function is strictly concave, and so \(p^*\), the optimal dual solution, is unique.

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Title: Markets for public decision-making
Authors: Nikhil Garg
Ashish Goel
Benjamin Plaut
Publication date: 18-11-2020
Publisher: Springer Berlin Heidelberg
Published in: Social Choice and Welfare / Issue 4/2021
Print ISSN: 0176-1714
Electronic ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-020-01298-4

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