---
title: Kolm-Pollak Equally Distributed Equivalent (EDE)
url: https://www.emergentmind.com/topics/kolm-pollak-equally-distributed-equivalent-ede
type: topic
---

# Kolm-Pollak Equally Distributed Equivalent (EDE)

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 [2401.15452].

## 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 [2401.15452] 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, [2401.15452]). 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 [2401.15452] 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 4). 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 6). 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 [2401.15452].

Source: https://www.emergentmind.com/topics/kolm-pollak-equally-distributed-equivalent-ede