Markets for Public Decision-making

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.

💡 Research Summary

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The paper investigates how market mechanisms can be adapted to collective decision‑making problems where a set of binary issues must be resolved and each participant (agent) has a preferred alternative for every issue. Agents receive an equal endowment of artificial currency that can only be spent on purchasing probability mass for their favored alternatives; randomized outcomes are allowed, making the decision analogous to allocating divisible private goods.

The authors first examine the simplest “per‑issue pricing” scheme, where a single price is posted for each issue and all agents face the same price vector. They prove (Section 3) that this approach can lead to equilibria whose Nash welfare, utilitarian welfare, and egalitarian welfare are each a factor of Θ(n) worse than the optimal, where n is the number of agents. This negative result shows that a uniform price across agents is insufficient when agents have opposing preferences on the same issue.

Motivated by this, the core contribution is the pairwise issue expansion reduction. For every issue and every unordered pair of agents that disagree on that issue, a distinct fictitious good is created. Buying a unit of this good corresponds to a “pairwise negotiation” in which the two agents split the probability of the issue’s outcomes according to their preferences. The whole public‑decision instance is thus transformed into a standard Fisher market: a set of divisible goods (the pairwise goods) and a set of agents each endowed with a budget. Crucially, the reduction guarantees equivalence at equilibrium: any market‑clearing price vector in the constructed Fisher market induces a “pairwise pricing equilibrium” in the original public‑decision problem.

The authors show that this pairwise pricing equilibrium maximizes Nash welfare in the public‑decision setting. Because the reduction yields a Fisher market with nested utility structures (e.g., nested Leontief or nested CES‑Leontief), all known polynomial‑time algorithms for Fisher markets can be imported directly. Specifically they obtain:

1. A strongly polynomial algorithm for two‑agent instances with arbitrary utilities (via the algorithm of Cole and Fleischer).
2. A strongly polynomial algorithm for Leontief utilities with weights in {0,1} (based on the work of Devanur et al.).
3. A polynomial‑time O(log n)‑approximation algorithm that simultaneously approximates Nash, utilitarian, and egalitarian welfare for general Leontief utilities (building on the algorithm of Cole, Gkatzelis, and Goel).
4. A discrete‑time tâtonnement process that converges in polynomial time for markets with nested CES‑Leontief utilities (following the method of Barman, Gkatzelis, and Kannan).

Through the reduction, each of these results translates into the corresponding public‑decision guarantees: efficient computation of Nash‑optimal outcomes, simultaneous approximation of multiple welfare criteria, and a convergent price‑adjustment dynamics that can be implemented without ever exposing the underlying Fisher market to the participants.

The paper also discusses the relationship to Lindahl equilibria (personalized prices) and shows that while Lindahl’s framework allows arbitrary individualized prices, the pairwise pricing scheme achieves comparable efficiency with far fewer price variables—only one price per disagreeing pair per issue. Moreover, the authors argue that traditional market‑based mechanisms such as Quadratic Voting or Trading‑Post pricing suffer the same O(n) inefficiency demonstrated for per‑issue pricing.

A substantial portion of the work is devoted to tâtonnement for public decisions. The proposed dynamics present agents with current pairwise prices, collect their demanded probability shares, and update prices based on excess demand. The process respects the public‑decision constraint that all agents must share the same realized outcome distribution, yet it retains the classic market‑clearing update rule. The authors prove convergence to the Nash‑optimal equilibrium, thereby providing a practical iterative voting protocol that could be deployed in participatory budgeting or multi‑issue referenda.

Strategic behavior is not addressed; the authors assume truthful reporting of preferences, noting that any mechanism that is both strategy‑proof and Pareto‑optimal in this setting would be dictatorial (citing existing impossibility results). They leave the design of incentive‑compatible extensions as future work.

In summary, the paper makes three major contributions: (i) it identifies the severe inefficiency of uniform per‑issue pricing in public decision markets, (ii) it introduces a novel pairwise issue expansion that reduces any public‑decision instance to a Fisher market while preserving equilibrium properties, and (iii) it leverages the rich algorithmic literature on Fisher markets to obtain polynomial‑time Nash‑optimal solutions, simultaneous welfare approximations, and convergent tâtonnement dynamics for public decision‑making. This bridges the gap between private‑goods market theory and collective decision processes, opening avenues for scalable, fair, and efficient voting mechanisms in large‑scale democratic settings.