Dynamic Allocation of Public Goods with Approximate Core Equilibria

TL;DR

This work studies online allocation of multiple shareable public goods without money, where agent values are additive and drawn per period from a joint distribution that may exhibit correlations. It introduces a black-box reduction that uses an artificial currency to simulate monetary mechanisms in a repeated, non-monetary setting, and analyzes value-scaling pacing strategies that yield equilibria where the outcome closely approximates the core. The authors show that the core-approximation achieved in the online setting is tightly linked to the Dead-Weight Loss of the underlying one-shot monetary mechanism, with bounds of the form $(\mathcal{H}_n-1, o(1))$ in the common IC cases, and a robust extension to multi-item, submodular-cost settings via an ex-ante core relaxation. The results demonstrate a practical, distribution-agnostic approach to achieving fair and efficient allocation of public goods online, even under complex valuation correlations and global constraints.

Abstract

We consider the problem of repeatedly allocating multiple shareable public goods that have limited availability in an online setting without the use of money. In our setting, agents have additive values, and the value each agent receives from getting access to the goods in each period is drawn i.i.d. from some joint distribution $\mathcal{D}$ (that can be arbitrarily correlated between agents). The principal also has global constraints on the set of goods they can select over the horizon, which is represented via a submodular allocation-cost function. Our goal is to select the periods to allocate the good to ensure high value for each group of agents. We develop mechanisms for this problem using an artificial currency, where we give each agent a budget proportional to their (exogenous) fair share. The correlated value distribution makes this an especially challenging problem, as agents may attempt to free-ride by declaring low valuations for the good when they know other agents have high values-hoping those agents will bear a larger share of the cost of the resource. We offer a black-box reduction from monetary mechanisms for the allocation of a costly excludable public good. We focus on pacing strategies, the natural strategies when using AI agents, where agents report a scaled version of their value to the mechanism. Our main results show that when using a truthful monetary mechanism as our building block, the resulting online mechanism has a focal equilibrium in which each agent plays a pacing strategy whose outcome results in an allocation that is a $(\mathcal{H}_n-1)$-approximation of the core, where $\mathcal{H}_n$ is the Harmonic number, and $n$ is the number of agents. Remarkably, we are able to achieve an approximate core solution as a Nash outcome without explicit collaboration or coordination between the agents.

Figures (1)

• Figure 1: This diagram illustrates the fact that half-spaces are Pareto optimal, which is the key idea to the proof of \ref{['lem: core-characterization']}. Assume the region $H$ above the dashed black line (corresponding to $\{x \in [0, 1]^2: \sum_{i\in S}\frac{x_i}{\beta_i} = 1\}$ for some $\beta$), and the region $R$ above the solid blue curve (corresponding to some ex-ante allocation policy), have equal measure. Then, the measure of the green region is equal to the measure of the red region. Furthermore, for all points in the green region $\sum_{i\in S}\frac{x_i}{\beta_i} \geq 1$ and for all points in the red region $\sum_{i\in S}\frac{x_i}{\beta_i} < 1$. Hence, $\mathop{\mathrm{\mathbb{E}}}\limits_{V \sim \mathcal{D}}\left[\sum_{i\in S}\frac{x_i}{\beta_i} \cdot \mathbbm{1}\left[V \in H\right]\right] \geq \mathop{\mathrm{\mathbb{E}}}\limits_{V \sim \mathcal{D}}\left[\sum_{i\in S}\frac{V_i}{\beta_i} \cdot \mathbbm{1}\left[x \in R\right]\right]$. Thus, $R$ cannot Pareto dominate $H$. This argument generalizes to any number of dimensions.

Theorems & Definitions (88)

• Definition 2.1: $(\gamma, \delta)$-Approximate Core
• Definition 2.1
• Proposition 3.1
• proof
• Definition 3.2: Average Expected Utility
• Definition 3.3: Expected Cost Before Depletion
• Definition 3.4: Approximate Best Response and Approximate Nash Equilibrium
• Theorem 3.5
• Theorem 3.6: Pacing Equilibrium
• Theorem 3.7