Modeling Access Differences to Reduce Disparity in Resource Allocation

TL;DR

This work formalizes resource allocation when access disparities align with social disadvantage, introducing a concrete access model and an access-aware allocation method that can reduce resource disparity while preserving geographic proportionality. The core method relies on a Poisson-based acquisition model with a gap parameter $η$, yielding naïve and exact acquisition functions; a tractable approximation enables practical optimization and an iterative heuristic to handle saturation effects. Empirical validation using county-level COVID-19 vaccination data, vulnerability metrics, and global data supports the linear relationship between vulnerability and access, and demonstrates substantial disparity reductions under various resource levels and gap assumptions. The framework offers a actionable, justice-oriented approach to resource allocation with broad applicability across scales, while highlighting the importance of addressing broader access barriers such as information, transportation, and infrastructure.

Abstract

Motivated by COVID-19 vaccine allocation, where vulnerable subpopulations are simultaneously more impacted in terms of health and more disadvantaged in terms of access to the vaccine, we formalize and study the problem of resource allocation when there are inherent access differences that correlate with advantage and disadvantage. We identify reducing resource disparity as a key goal in this context and show its role as a proxy to more nuanced downstream impacts. We develop a concrete access model that helps quantify how a given allocation translates to resource flow for the advantaged vs. the disadvantaged, based on the access gap between them. We then provide a methodology for access-aware allocation. Intuitively, the resulting allocation leverages more vaccines in locations with higher vulnerable populations to mitigate the access gap and reduce overall disparity. Surprisingly, knowledge of the access gap is often not needed to perform access-aware allocation. To support this formalism, we provide empirical evidence for our access model and show that access-aware allocation can significantly reduce resource disparity and thus improve downstream outcomes. We demonstrate this at various scales, including at county, state, national, and global levels.

Figures (5)

• Figure 1: Visualizations --- (\ref{['fig:exact-vs-naive']}) Acquisition behavior versus time (with $\eta=0.5$, $\beta=0.5$). Blue and red plots are resources acquired by all and by the advantaged, respectively for $P=10, 100, 1000, 10000$ (faintest to darkest), for the maximal expected time. For a specific $N$, resources are exhausted sooner (the orange example). For each case, the average resource is the solid line. The error bars represent the $5$ to $95$ percentile range. The limits (dotted red and blue) correspond to the approximation model in Section \ref{['sec:approx-rho']}. These are indistinguishable from the average $P=1000$ case and from the entire confidence range for $P=10000$. The acquisition function is always the expected fraction of resources acquired by the disadvantaged upon exhaustion. This can have two distinct behaviors, based on whether only exhaustion occurs or both saturation and exhaustion occur. (\ref{['fig:robustness']}) Constraint set of the example in Sec. \ref{['sec:robustness']} (interior of the blue region). Direction of the vector $c_j$ of $\mathsf{RD}$ in Eq. \ref{['eq:disparity-alt']}, as $\eta$ varies (in pink). The narrow fan implies that there exists a single discrepancy-optimizing allocation that is optimal for all $\eta$, and thus robust to lack of specification of $\eta$.
• Figure 2: Vaccination rate vs. vulnerability --- (\ref{['fig:access-evidence']}) Each bubble represents a county, with area proportional to its population. The $x$-axis is the estimated percentage of vulnerable in the county ($\beta$). The $y$-axis is the percentage of the overall population with their first COVID-19 vaccination dose as of 12/30/2021. The orange curve is a soft nearest-neighbor interpolation $\hat{y}(\beta)$ of the relationship between $\beta$ and $y$. The dotted red is a visual guide illustrating the common dominant behavior across states. (\ref{['fig:wholeworld']}) Global manifestation of the same phenomenon as in Fig. \ref{['fig:access-evidence']} --- A roughly linear inverse relationship between vulnerability numbers and vaccination rates across countries.
• Figure 3: Behavior of Allocations --- Each point is a location $j$. The $x$-axis is the percentage vulnerable $\beta$ at location $j$ and the $y$-axis is the ratio of allocation to its population $\frac{n}{p}$. The allocation for $\alpha=0.1$, $\alpha=0.5$, and $\alpha=0.9$ are also indicated with dotted lines. Main observations --- (\ref{['fig:ellone-behavior']}) $n^\star_1$ specifies two thresholds, does not allocate below the lower, fully allocates above the higher, and maintains $\alpha$ in between. (\ref{['fig:ellrelinf-behavior']}) $n^\star_\infty$ fixes a threshold and alters the proportional allocation slightly lower below it and slightly higher above it.
• Figure 4: Outcomes of Allocations --- (Left) Resource (vaccination) disparity ($y$-axis, $\mathsf{RD}$) of both the access-aware allocation $n^\star_\infty$ offers and the proportional allocation (faint plots), for $\alpha=0.1, 0.5, 0.9$. (Right) Difference between the two, which captures the improvement in disparity by using the access-aware allocation. The $x$-axis is the access gap, $\eta$. Main observations --- (\ref{['fig:ellone-outcome']}) For $n^\star_1$, improvements are more marked when the access gap is larger, i.e. when $\eta$ is smaller. Counterintuitively, more availability (larger $\alpha$) offers relatively more opportunity to mitigate disparity, as explained in the text. (\ref{['fig:ellrelinf-outcome']}) For $n^\star_\infty$, the general behavior is comparable to that of $n^\star_1$. However, in this case we are more constrained not to deviate from proportionality, and the gains are thus less.
• Figure 5: Sorted $\mathsf{RD}$ values at every vertex of the constraint polytope for allocation across counties for $4$ states, CT, MA, ME, and VT. The orange $\times$ indicates the $\mathsf{RD}$ of proportional allocation. The dots represent the $\mathsf{RD}$ of the access-aware allocation, either in the heuristic form of the paper, or augmented via noisy restarts.

Theorems & Definitions (3)

• Proposition 4.1
• Proposition 4.2
• Proposition 5.1