🤖 AI

Constituency Optimisation Through Hamiltonian Representation Of Mandates (COTHROM): Algorithmic Redistricting of Irish Election Boundaries

This paper introduces COTHROM, a novel computational framework that applies statistical physics and multi-criterion decision analysis to algorithmically optimize Irish electoral redistricting under the PR-STV system by balancing competing constitutional objectives through a Potts Hamiltonian model.

Original authors: Ruaidhrí Campion, Matthew Fenlon, Joshua Cooney Mercedal, Casey Farren-Colloty, Eliza Somerville, Michael A. J. Mitchell

Published 2026-06-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ruaidhrí Campion, Matthew Fenlon, Joshua Cooney Mercedal, Casey Farren-Colloty, Eliza Somerville, Michael A. J. Mitchell

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine Ireland's map as a giant jigsaw puzzle made of thousands of tiny pieces called "Electoral Divisions." Every few years, the government has to redraw the lines that group these pieces together into larger "constituencies" so people can vote for their representatives.

The rules for this puzzle are strict but also vague. The Constitution says:

Fairness: Every person should have roughly the same number of votes (Proportional Representation).
Shape: The pieces should be packed tightly together, not stretched out like a snake (Compactness).
Borders: Try not to cut through county lines (like not splitting a county in half).
Connectivity: All pieces in a constituency must touch each other.

The problem is that these rules often fight each other. If you make a constituency perfectly fair in population, you might have to slice through a county line. If you keep the county lines perfect, the population might become unbalanced. Currently, humans draw these maps by hand, making judgment calls on how to balance these fights.

The Paper's Solution: COTHROM
The authors created a computer program called COTHROM (Constituency Optimisation Through Hamiltonian Representation Of Mandates) to solve this puzzle automatically. Here is how they did it, using some fun analogies:

1. The "Magnetic" Map (The Physics Analogy)

Instead of thinking of the map as a political problem, the authors treated it like a physics problem. Imagine every tiny puzzle piece (Electoral Division) is a tiny magnet.

Each magnet wants to belong to a specific group (constituency).
The "rules" of the election (fairness, shape, borders) act like magnetic forces.
- If a group is too big or too small, the magnets feel a "push" to rearrange.
- If a group is stretched out, the magnets feel a "pull" to snap back into a tight ball.
- If a group crosses a county line, the magnets feel a "friction" penalty.

The computer calculates the total "energy" of the map. A map with bad boundaries has high energy (like a tense spring), and a good map has low energy (like a relaxed spring). The goal is to find the state where the map is most "relaxed."

2. The "Cooling" Process (Simulated Annealing)

How do you find the best arrangement among billions of possibilities? You can't just check them all.
The authors use a technique called Simulated Annealing, which is like cooling molten metal.

Hot: Imagine the magnets are vibrating wildly (high temperature). They jump around randomly, trying out crazy, messy map shapes. This helps the computer escape "bad" local solutions (like a map that looks okay but isn't the best).
Cooling Down: Slowly, the computer lowers the temperature. The magnets stop jumping so wildly and start settling into stable positions.
Frozen: By the time it's cold, the magnets have settled into the most efficient, lowest-energy arrangement possible.

3. The "Menu of Trade-offs" (Pareto Optimality)

Here is the tricky part: The law doesn't say how much to value fairness versus county borders. Should we care 50% about fairness and 50% about borders? Or 90% fairness?
Instead of guessing one "perfect" balance, the authors ran the computer many times with different "weights" (like turning a dial).

They didn't just give you one map.
They gave them a menu of the best possible maps (called a Pareto Front).
On this menu, every map is a "non-dominated" solution. This means: "If you want more fairness, you must accept less respect for county borders. If you want more respect for borders, you must accept less fairness."
This makes the trade-offs visible. It shows the decision-makers exactly what they are giving up to get something else.

4. What They Found (The Cork Test)

They tested this on County Cork.

They compared their computer-generated maps against the actual legal maps drawn by the Electoral Commission in 2023.
The Result: For almost every way of weighing the rules, the computer found maps that were better than the human-drawn ones. The computer maps had less population imbalance and were more compact, while still respecting the rules.
Note: The authors admit their computer didn't account for things like "community feelings" or specific rivers, which humans consider. But purely on the mathematical rules of the Constitution, the computer won.

Why This Matters

The paper argues that this method makes the process transparent and fair.

No Hidden Judgments: Currently, humans draw lines and say, "This is fair." No one can prove if a better map exists.
Auditability: With this code, anyone can see the math. You can say, "If we value county borders this much, here is the best possible map."
Reproducibility: Anyone can run the same code and get the same result, removing the guesswork from redistricting.

In short, the authors built a "magnetic puzzle solver" that shows us exactly how hard it is to balance the rules of Irish elections, proving that computers can find better, more mathematically perfect maps than the current manual process.

Technical Summary: Constituency Optimisation Through Hamiltonian Representation Of Mandates (COTHROM)

Problem Statement
The paper addresses the computational challenge of redistricting electoral boundaries in Ireland under the Proportional Representation Single Transferable Vote (PR-STV) system. The process is governed by the Irish Constitution and the Electoral Reform Act 2022, which mandate specific "hard" constraints (e.g., 171–181 total seats, constituencies of 3–5 seats, contiguity) and "soft" objectives (e.g., proportional representation, avoiding county boundary breaches, compactness, and geographic considerations).

The core difficulty lies in the combinatorial explosion of possible boundary configurations and the inherent tension between soft objectives. For instance, a configuration that strictly adheres to county boundaries may result in significant deviations from proportional representation, while a mathematically optimal proportional map may require numerous county breaches. Currently, the Electoral Commission (EC) draws boundaries manually, a process that lacks a systematic basis for comparing trade-offs or demonstrating that a proposed plan is near-optimal relative to the vast space of admissible alternatives.

Methodology
The authors propose COTHROM, a computational framework that models the redistricting problem using statistical physics, specifically the Potts model.

Hamiltonian Formulation:
The legal and constitutional requirements are encoded as terms in a Hamiltonian (energy function), $H(\sigma)$ , where $\sigma$ represents a specific configuration of electoral divisions (EDs) assigned to constituencies. The total energy is a weighted sum of specific objective functions:
- $H_P$ (Proportional Representation): Penalizes deviations of constituency population from the national average per seat.
- $H_C$ (Contiguity): Penalizes constituencies split into multiple disconnected components.
- $H_D$ (Compactness): A discrete cut-edge metric penalizing non-globular shapes (incentivizing compactness over "salamandering").
- $H_B$ (County Boundaries): Counts EDs assigned to a constituency that lie outside the constituency's primary county.
The objective function is $H = \sum J_\alpha (H_\alpha / Z_\alpha)$ , where $J_\alpha$ are coupling constants representing the relative weight of each objective, and $Z_\alpha$ are normalization factors.
Optimization via Simulated Annealing:
To find configurations that minimize $H$ , the authors employ Markov Chain Monte Carlo (MCMC) methods with a simulated annealing schedule.
- Algorithm: A Gibbs sampler is used to update the constituency assignment of individual EDs. Unlike the Metropolis algorithm which proposes random moves, the Gibbs sampler draws from the conditional distribution of all possible assignments for a given ED, improving efficiency in high-energy landscapes.
- Annealing: The system starts at a high "temperature" (allowing broad exploration of the configuration space) and is slowly cooled according to a geometric schedule ( $T_k = \alpha^k T_0$ ). This allows the system to escape local minima and converge toward a global minimum (ground state) representing an optimal boundary set for a given set of weights.
Multi-Criterion Decision Analysis (MCDA):
Recognizing that the relative weighting of objectives ( $J_\alpha$ ) is a normative choice not dictated by law, the framework does not commit to a single weighting. Instead, it utilizes MCDA to map the Pareto front. A configuration is Pareto optimal if no other configuration improves one objective without worsening another. This approach makes the trade-offs between competing constitutional goals explicit and quantifiable.

Key Results
The framework was validated using data from County Cork, a region with 20 seats across five constituencies.

Convergence: The system demonstrated clear phase transitions, where random initial configurations "crystallized" into structured, contiguous constituencies as the temperature decreased.
Optimality: When comparing the 2023 EC boundaries against the COTHROM-generated Pareto front, the existing legal boundaries were found to be dominated. COTHROM consistently produced configurations with significantly lower variance from proportional representation for the same level of compactness, or better compactness for the same level of proportional representation.
County Boundaries: In a test case involving midland counties, the framework demonstrated the ability to trade off proportional representation for strict adherence to county boundaries. By increasing the coupling constant for county boundaries ( $J_B$ ), the algorithm produced maps that nearly exactly reproduced 2023 boundaries but at the cost of higher population variance.
Pareto Analysis: The study revealed that the current EC boundaries do not lie on the Pareto front, suggesting that improvements in representation or compactness are possible without sacrificing other objectives, provided the specific trade-offs are accepted.

Significance and Claims
The paper claims to present the first computational framework specifically designed for Irish electoral redistricting that systematically optimizes across multiple constitutional requirements. Its primary contributions are:

Transparency and Auditability: By encoding legal mandates into an explicit mathematical model, the framework allows for the reproduction of results and the inspection of how specific weightings influence outcomes. This contrasts with the current manual process, where the optimality of a plan cannot be rigorously demonstrated.
Explicit Trade-off Quantification: The use of Pareto front analysis moves the debate from implicit deliberation to explicit quantification. It allows decision-makers to see exactly how much proportional representation must be sacrificed to gain a specific improvement in compactness or county boundary integrity.
Algorithmic Governance: The work suggests a pathway for "algorithmic governance" where the EC can generate mathematically optimized starting points for public consultation, ensuring that the final boundaries are not just compliant with the law, but are near-optimal within the defined legal landscape.
Limitations and Scope: The authors modestly note that their current model does not yet incorporate complex geographical features (mountains, rivers) or the nuanced insights from public consultations regarding community ties, though they propose these as future extensions. They also acknowledge that the definition of a "county boundary breach" is currently a simplified heuristic based on area fractions, requiring further refinement to match the EC's specific legal interpretations.

In summary, COTHROM provides a rigorous, physics-based tool to navigate the complex, multi-objective optimization landscape of Irish redistricting, offering a method to generate, compare, and justify electoral boundaries that better satisfy the constitutional mandate of proportional representation while respecting geographic constraints.