Allocating Public Goods via Dynamic Max-Min Fairness: Long-Run Behavior and Competitive Equilibria

TL;DR

This paper studies dynamic max-min fair allocation (DMMF) for repeated single-resource sharing among $n$ agents with i.i.d. round values. It proves that no pure Nash equilibrium exists under fixed-threshold strategies, and introduces a data-driven Win-Rate Matching (WRM) policy that induces an approximate equilibrium with significantly improved welfare: at least $1-1/e$ of each agent’s ideal utility for general distributions and $1-O(rac{ ext{log} n}{n})$ for uniforms. The technical core develops a subgroup state-space collapse framework, a decomposition into stable agent groups, and drift-based analysis to show convergence of win rates under WRM to welfare-optimal levels. The results illuminate how simple, decentralized policies can achieve near-optimal welfare in a robust, distribution-agnostic dynamic allocation setting, and provide a blueprint for inducing equilibria via data-driven strategy guidance beyond DMMF.

Abstract

Dynamic max-min fair allocation (DMMF) is a simple and popular mechanism for the repeated allocation of a shared resource among competing agents: in each round, each agent can choose to request or not for the resource, which is then allocated to the requesting agent with the least number of allocations received till then. Recent work has shown that under DMMF, a simple threshold-based request policy enjoys surprisingly strong robustness properties, wherein each agent can realize a significant fraction of her optimal utility irrespective of how other agents' behave. While this goes some way in mitigating the possibility of a 'tragedy of the commons' outcome, the robust policies require that an agent defend against arbitrary (possibly adversarial) behavior by other agents. This however may be far from optimal compared to real world settings, where other agents are selfish optimizers rather than adversaries. Therefore, robust guarantees give no insight on how agents behave in an equilibrium, and whether outcomes are improved under one. Our work aims to bridge this gap by studying the existence and properties of equilibria under DMMF. To this end, we first show that despite the strong robustness guarantees of the threshold based strategies, no Nash equilibrium exists when agents participate in DMMF, each using some fixed threshold-based policy. On the positive side, however, we show that for the symmetric case, a simple data-driven request policy guarantees that no agent benefits from deviating to a different fixed threshold policy. In our proposed policy agents aim to match the historical allocation rate with a vanishing drift towards the rate optimizing overall welfare for all users. Furthermore, the resulting equilibrium outcome can be significantly better compared to what follows from the robustness guarantees.

Figures (2)

• Figure 1: A sample path of allocations under DMMF with $4$ agents in the symmetric setting (i.e., with $\alpha_i = 1/4$ for all $i$). The agents all use threshold strategies, with strategy vector $(\texttt{S}^{\texttt{Thr}}_{0.1},\texttt{S}^{\texttt{Thr}}_{0.2},\texttt{S}^{\texttt{Thr}}_{0.25},\texttt{S}^{\texttt{Thr}}_{0.5})$. Note that while for all agents $W_i[t]$ grows linearly, they split up into $3$ stable subgroups, each with a different average win rate. Note also that DMMF gives agent $1$ the highest priority and agent $4$ the lowest priority in every round after round $70$; in fact, we formally prove that the trajectories of agents in different subgroups almost surely never intersect after some finite time.
• Figure 2: Utility of an agent who follows Win-Rate Matching or the robust threshold of fikioris2023online, when every other agent follows Win-Rate Matching. The utility is normalized using the total ideal utility up to that round. We assume there are $5$ agents with value distributions uniform in $[0, 1]$. In addition, to get faster empirical convergence, we use $\zeta(t) = 1$ and $\eta(t)$ that quickly decreases linearly until $0.05$.

Theorems & Definitions (48)

• proposition 1
• proposition 2
• definition 1: Subgroup Stability Condition
• theorem 1: Global State-Space Collapse for the Allocation Process
• lemma 1
• lemma 2
• theorem 2: Non-Existence of Equilibria in Fixed Thresholds
• proposition 3
• lemma 3
• lemma 4