Kolm-Pollak Equally Distributed Equivalent (EDE)

• Kolm–Pollak EDE is a welfare and equity metric that summarizes burden distributions by equating heterogeneous outcomes to a uniform welfare measure using an exponential inequality-aversion utility.
• Its derivation includes a linearized proxy that converts nonlinear exponential and logarithmic components into a MILP-friendly form, enabling efficient integration into large-scale facility location models.
• Empirical validations show that the approach improves equity by reducing worst-case access burdens while maintaining computational scalability across urban planning applications.

The Kolm–Pollak Equally Distributed Equivalent (EDE) is a welfare- and equity-based metric for summarizing distributions of burdens such as access distances to essential services. It defines a single value, the EDE, such that if every individual experienced this value, social welfare (measured with an exponential inequality-aversion utility) would match observed welfare under the true, heterogeneous distribution. Recent advances have produced a linearized proxy of the Kolm–Pollak EDE for efficient integration into large-scale facility location models, improving equity of access without sacrificing computational scalability (Horton et al., 27 Jan 2024).

1. Formal Definition

Let a population be partitioned into units $i=1,\dots,N$, each with outcome $z_i$ (e.g., distance to nearest facility). The Kolm–Pollak EDE, denoted $\mathcal K$, is defined as the solution to the equation

$$\mathcal K(\mathbf z) = -\frac{1}{\kappa}\ln\left[\frac{1}{N}\sum_{i=1}^N \exp(-\kappa z_i)\right]$$

where $\epsilon < 0$ is an inequality-aversion parameter and $\kappa = \alpha \epsilon$ with scaling factor $$\alpha = \frac{\sum_{i=1}^N z_i}{\sum_{i=1}^N z_i^2}$$. In population-weighted settings, where origin $r$ with population $p_r$ has outcome $z_r$, and total population $T = \sum_r p_r$,

$$\mathcal K(\mathbf z) = -\frac{1}{\kappa}\ln\left[\frac{1}{T}\sum_{r} p_r \exp(-\kappa z_r)\right].$$

Because $\epsilon < 0$ for burdens such as travel distances, $\mathcal K$ is always greater than or equal to the mean; greater aversion to inequality (more negative $\epsilon$) raises $\mathcal K$ toward the maximum.

2. Linearized Proxy Derivation

Direct optimization over the Kolm–Pollak EDE is infeasible for large-scale discrete facility location, due to exponential and logarithmic nonlinearities. However, by exploiting assignment structure, the objective can be transformed and linearized. Let $y_{r,s} \in \{0, 1\}$ indicate assignment of population node $r$ to facility $s$, with $d_{r,s}$ the associated burden (e.g., travel distance), so

$z_r = \sum_{s} y_{r,s} d_{r,s}.$

The Kolm–Pollak EDE objective becomes

$$\mathcal K(\mathbf y) = -\frac{1}{\kappa}\ln\left[\frac{1}{T}\sum_{r} p_r \exp\left(-\kappa \sum_s y_{r,s}d_{r,s}\right)\right].$$

Monotonic transformations allow dropping $-1/\kappa$, the logarithm, and $1/T$, yielding a nonlinear sum

$$\breve{\mathcal K}(\mathbf y) = \sum_r p_r \exp\left(-\kappa \sum_s y_{r,s} d_{r,s}\right).$$

Given assignment constraints $\sum_s y_{r,s}=1$, $y_{r,s} \in\{0,1\}$, Proposition 2.1 (Horton et al., 27 Jan 2024) demonstrates that

$$\breve{\mathcal K}(\mathbf y) = \sum_{r,s} p_r y_{r,s} \exp(-\kappa d_{r,s}),$$

a function linear in $y_{r,s}$ for fixed $\kappa$. This linear proxy is denoted

$$\overline{\mathcal K}(\mathbf y) = \sum_{r,s} p_r y_{r,s} e^{-\kappa d_{r,s}}.$$

In practice, $\alpha$ (and thus $\kappa$) is set via data estimation or warm-start, with post-hoc updates ensuring negligible error in the realized inequity-aversion.

3. Theoretical Properties and Approximation Guarantees

The linear proxy $\overline{\mathcal K}$ retains the same optimizers as the original nonlinear Kolm–Pollak objective, provided fixed $\kappa$ (Corollary 2.3, (Horton et al., 27 Jan 2024)). Its linearity in assignment variables $y_{r,s}$ enables direct embedding in mixed-integer linear programming (MILP) frameworks. When $\alpha$ is estimated (rather than computed with final optimal assignments), approximation guarantees are provided (Theorem 2.10); a single update step—recomputing $\alpha$ from an initial solution—brings the output inequity parameter $\epsilon_{\rm out}$ within $\ll1$ of the intended $\epsilon_{\rm in}$.

Extensions to the basic model, including capacity constraints, fractional assignment, and facility penalties, can be handled by incorporating additional linear or piecewise-linear constraints. Theorems 2.16 and 2.18 give provable error bounds for penalty linearization and for capacity variants, ensuring the overall integrity of the proxy’s equity-optimization properties.

4. Integration into Facility Location Models

The Kolm–Pollak Linear proxy (KPL) facility location model is formulated as: $$\begin{aligned} \min_{x,y}\quad &\sum_{r,s} p_r y_{r,s} e^{-\kappa d_{r,s}} \ \text{subject to}\quad &\sum_{s} x_s = k, \ &y_{r,s} \leq x_s\quad \forall r,s, \ &\sum_{s} y_{r,s} = 1 \quad \forall r, \ &x_s \in \{0,1\},\quad y_{r,s}\in\{0,1\}\quad \forall r,s. \end{aligned}$$ Here $x_s$ are facility location binaries indicating which sites are open. The model supports variant constraints (capacities, penalties, splits) with the addition of further linear or piecewise-linear conditions, without losing MILP tractability. The model thus bridges equity theory and practical optimization by embedding distributionally sensitive objectives into a scalable workflow.

5. Computational and Scaling Strategies

Efficient large-scale solution of Kolm–Pollak-based models is achieved through several computational approaches:

• Fixing $\alpha$ before optimization (optionally updating once) enables precomputation of MILP objective coefficients $e^{-\kappa d_{r,s}}$.
• Assignment variables $y_{r,s}$ can be pruned by excluding combinations exceeding a threshold $d_{\max}$, simplifying the search space.
• Standard high-performance MILP solvers (Gurobi, SCIP) on large-memory HPC nodes reliably scale to problems with $>200$ million binaries; runs may use up to 2 TB RAM for the largest instances.
• Relaxing optimality conditions with Gurobi's default MIP gap (0.01%) or looser criteria (e.g., 0.5% gap with time limits) allows controlled trade-offs between run-time and solution quality.
• Piecewise-linear penalty approximations are constructed using exactly spaced or tightly clustered points to ensure at-worst machine-epsilon errors (Theorem 2.18).

6. Empirical Validation and Applications

Empirical testing (Horton et al., 27 Jan 2024) validates the Kolm–Pollak linear proxy in real-world problem instances:

Food-desert application (500 U.S. cities):

Opening $k=1,5,10$ new supermarkets (with preexisting ones fixed) shows that the KPL model achieves solve times similar to $p$-median and is one order of magnitude faster than $p$-center (Figure 1). Across all cities, average distance increases by $\approx$ 8 m (for $k=1$) compared to $p$-median, while maximum distance decreases by $\approx$ 432 m, demonstrating significant improvement for worst-off residents (Table 7).

Polling-location application (five mid-sized cities):

When relocating all $k$ sites under capacity constraints, the KPL approach matches or outperforms $p$-center in both maximum distance and Kolm–Pollak scores, maintaining mean distance within $0.5\%$ of that achieved by $p$-median (Table 6, Figure 2). Reestimation of $\alpha$ after a single update keeps output inequity aversion within $\sim5\%$ of the target parameter (Tables 2–3).

These results confirm that the proxy enables optimization of equity-driven public facility planning at realistic national and metropolitan scales, with transparent theoretical guarantees and robust numerical performance.

7. Extensions, Limitations, and Significance

The linear proxy approach for the Kolm–Pollak EDE generalizes to facility location models with heterogeneous site penalties, hard/soft capacity constraints, and split demand assignments. Each extension inherits the proxy’s scalability and tight provable error bounds, retaining direct MILP compatibility. The method offers a principled means to balance average access and worst-off outcomes, addressing policy concerns around equity in public planning.

A plausible implication is that further improvements in computational infrastructure or solver technology will enhance scalability and applicability in domains beyond urban access planning.

For detailed proofs, implementation, and case data, see (Horton et al., 27 Jan 2024).