One Vote, Several Parliaments: An Empirical Analysis of the Algorithmic Ambiguity of the Italian Electoral Law on the 2022 General Election Data

Technical Summary: One Vote, Several Parliaments

Problem Statement
The paper investigates the algorithmic ambiguity inherent in the Italian electoral law for the Chamber of Deputies (D.P.R. 361/1957, as amended by Law 165/2017, the "Rosatellum"). While the statutory text describes the procedure for distributing proportional seats among electoral constituencies (Article 83(1)(h)), it fails to specify the operational schedule for executing the exclusion clause that prevents units from exceeding their national seat entitlement. As demonstrated in prior theoretical work by Crafa, this textual underspecification admits at least three distinct algorithmic interpretations (A, B, and C). The central empirical question addressed here is whether these admissible interpretations, when executed on real-world election data, produce measurably different parliaments, specifically altering the identity of elected deputies and their territorial assignment.

Methodology
The authors developed a complete, open-source implementation of the seat-allocation pipeline (Articles 77, 83, 83-bis, 84, and 85) using Python and pandas. The implementation processes raw municipal-level data from the 2022 Italian general election (September 25, 2022).

1. Data Processing: The system reconstructs list tallies by redistributing "candidate-only" votes (votes cast for a district candidate without a list) pro-rata among coalition lists, as mandated by Article 77(1)(c). It handles data inconsistencies, such as duplicate candidate entries and encoding errors, to ensure exact alignment with official records.
2. Validation: The implementation was validated against official outcomes at four levels:
   • Level 1: Exact reproduction of the national apportionment (245 proportional seats) from raw data.
   • Level 2: Agreement of national list totals within ±0.06% (attributed to the difference between near-definitive open data and certified counts).
   • Level 3: Name-by-name comparison with official proclamations, matching 389 of 391 modeled seats (99.5%). The two mismatches correspond to seats under formal investigation by the Chamber's Committee on Elections due to internal inconsistencies in official records or margins within the uncertainty of vote counts.
   • Level 4: Step-by-step comparison with the National Central Electoral Office's minutes (transcribed via OCR), confirming agreement on intermediate decimal parts, percentage tallies, and recovery chains.
3. Experimental Design: The study runs the full pipeline under three interpretations:
   • Algorithm A (Sequential): Processes constituencies one by one, dynamically updating the exclusion set.
   • Algorithm B (Mattarella-style): Assigns all integer parts first, then assigns residuals with dynamic exclusion.
   • Algorithm C (Operational Practice): Assigns all integer parts, computes the exclusion set statically, then assigns all residuals.
     The study also tests the sensitivity of Algorithm A to the processing order of the 27 constituencies by running 1,000 uniformly random permutations.

Key Contributions

1. Faithful Implementation: The paper provides the first complete, executable model of the Rosatellum's proportional allocation pipeline that reproduces the official 2022 outcome exactly (save for the two contested seats).
2. Empirical Quantification of Ambiguity: The study moves beyond abstract examples to quantify the impact of statutory ambiguity on a real national election, identifying specific deputies who gain or lose seats based solely on the interpretation of the text.
3. Order-Dependence Analysis: The paper demonstrates that the sequential interpretation (Algorithm A) is highly sensitive to the order of constituency processing. Reversing the official order replaces 6 deputies and displaces 1. A sample of 1,000 random orders yields 560 distinct outcomes, with up to 13 deputies entering or leaving the Chamber relative to the practice outcome.
4. Detection of Structural Defects: The analysis reveals that under 29% of random orders, Algorithm A fails to assign one or two seats entirely because no eligible unit has a usable decimal part in the remaining constituencies. In another 9% of cases, the Chamber is filled but with a different party composition.
5. Robustness Verification: A Monte Carlo analysis confirms that the differences between interpretations are robust to input noise well beyond the residual uncertainty of the data (±0.06%), as the divergences are anchored to integer-valued saturation events or decimal margins exceeding the noise by one to two orders of magnitude.

Results

• Algorithm A vs. C: Under the official processing order, Algorithm A differs from the operational practice (Algorithm C) by 8 deputies (4 replaced by 4).
• Algorithm B: Under the Mattarella-style interpretation, 2 seats cannot be assigned by the rules stated in the text under any order, a defect that historically occurred in 1994, 1996, and 2001.
• Party Composition: While the statutory compensation procedure preserves national party totals under the executions documented in practice (C), arbitrary processing orders in Algorithm A can alter party composition.
• Individual Impact: Across 1,000 sampled orders, 458 distinct individuals are elected. 339 are elected under every order, while 119 are elected only under specific orders.

Significance and Claims
The paper claims to provide concrete empirical form to the conceptual point that formalizing a statute does not merely implement it but can reveal that the enacted text fails to specify a unique decision function. The ambiguity documented is operational rather than lexical; the text prescribes elementary operations without fixing their schedule.

The authors assert that this ambiguity is largely invisible to doctrinal reading because individual clauses are clear in isolation. However, when executed, different admissible schedules of the same words elect different parliaments. The study confirms Crafa's central claim: on real data, the admissible readings of the Italian electoral law elect measurably different parliaments, down to named individuals, with margins that survive the residual uncertainty of the data. The paper concludes that the selection of the currently applied algorithm (C) is performed by materials external to the statute (manuals, practice, tradition) rather than by the statute itself.

The work is situated within the "law as code" tradition, inverting the usual direction by taking the enacted text as-is, executing all admissible readings, and measuring the residual freedom's impact, rather than proposing a new formal notation for the law.