Degressive proportionality
Degressive proportionality is an apportionment principle in representative assemblies comprising entities of varying sizes, whereby smaller entities receive a disproportionately higher number of seats relative to their population compared to larger entities, resulting in an increasing population-to-seat ratio as entity size grows.[1] This method ensures that no single large population dominates proceedings, while still allocating more total seats to populous entities.[1] In the European Parliament, degressive proportionality is mandated by Article 14(2) of the Treaty on European Union, requiring representation to be degressively proportional with a minimum of six and a maximum of ninety-six seats per Member State.[2] Allocation methods, such as the Cambridge Compromise—which assigns a base of five seats plus additional seats proportional to population—or the Power Compromise using adjusted population units, operationalize this principle to meet fixed total seat numbers like 705 or 751.[1] The approach balances demographic equity with the need for smaller states to maintain influence, as evidenced by disparities where an MEP from Malta represents about 78,000 citizens compared to over 850,000 from Germany.[1] However, prospective EU enlargement to 30-40 members could intensify representational inequalities, prompting debates over formulas like square-root scaling or supplementary EU-wide lists to preserve legitimacy without fully equalizing per-capita representation.[3] Analogous structures appear in the U.S. Electoral College, where smaller states gain amplified votes per capita due to a minimum of three electors each, though not formally termed degressive.[4]Definition and Principles
Core Concept and Distinction from Other Systems
![Seats in the European Parliament versus population per country, 2014-2019][float-right] Degressive proportionality refers to an apportionment method in which seats in a legislative body are allocated to jurisdictions such that the ratio of seats to population decreases as the population of the jurisdiction increases, while still granting larger jurisdictions more total seats than smaller ones.[5] This results in smaller jurisdictions receiving greater per capita representation to ensure their influence is not overwhelmed by more populous entities, yet the allocation remains tied to population size rather than being uniform.[1] Mathematically, it implies a concave allocation function where the marginal seat gain per additional inhabitant diminishes for larger populations, often formalized with a minimum seat threshold and a cap on maximum seats.[6] The principle serves as a compromise in multi-level governance systems, balancing the need for equitable voice among disparate-sized units against the democratic ideal of one-person-one-vote representation.[7] In the European Union's Treaty of Lisbon, effective December 1, 2009, it mandates that European Parliament seats be distributed degressively proportionally, with each member state receiving at least six seats and no state exceeding 96, ensuring ratios of citizens per seat range from about 1:500,000 in smaller states like Malta to 1:850,000 in larger ones like Germany as of the 2019-2024 term.[1] Distinct from pure proportional representation, which allocates seats linearly with population or vote shares to achieve exact equivalence (e.g., seats = constant × population), degressive proportionality deliberately deviates by over-representing smaller units, preventing dominance by large populations at the expense of minority interests.[8] [7] It contrasts with equal representation systems, such as the U.S. Senate's fixed two seats per state since 1789, where per capita disparities can exceed 60:1 between Wyoming and California, as degressive systems preserve increasing seat totals for larger units without the full rigidity of equality.[7] This hybrid approach mitigates the vulnerabilities of strict proportionality to majority tyranny while avoiding the inefficiencies of absolute equality in diverse unions.[8]Mathematical and Apportionment Methods
Degressive proportionality in apportionment systems allocates seats to jurisdictional units such that the allocation function s(p) for population p is strictly increasing but concave, ensuring that the representation ratio p/s(p) rises with p, thereby granting smaller units a higher per capita share relative to larger ones.[9][10] This contrasts with strict proportionality, where s(p) \propto p, and equal representation, where s(p) is constant; degressiveness bridges the two by applying a sublinear scaling, often formalized as s(p) \propto p^\alpha with $0 < \alpha < 1, though practical implementations incorporate bounds like minimum and maximum seats per unit. One parametric approach is the fixed proportionality scheme (FPS), which decomposes total seats h into equal, proportional, and subproportional components:A(x) = F \cdot \frac{h}{100 \cdot n} + P \cdot \frac{h \cdot x}{100 \cdot \sum p_i} + S \cdot \frac{h \cdot \sqrt{x}}{100 \cdot \sum \sqrt{p_i}},
where n is the number of units, x is a unit's population, F + P + S = 100 (percentages), and the square-root term (S) induces degressiveness via a power of 0.5.[9] Parameters are tuned to meet treaty constraints, such as F=10, P=50, S=40 for the European Parliament's 720 seats across 27 states, yielding allocations verified as degressive prior to integer rounding via methods like Webster's.[9] Alternative methods adapt divisor-based apportionment for degressiveness. The Cambridge Compromise assigns a base of 5 seats per unit, then distributes the remainder using the Adams method (upward rounding with divisors), ensuring a minimum of 6 seats while maintaining s(p_i) > s(p_j) for p_i > p_j and rising p/s(p). More general base-plus-power schemes use
A(t) = \frac{M(t^d - p^d) + m(P^d - t^d)}{P^d - p^d},
with base m=6, cap M=96, total population P, minimum population p, and exponent d \approx 0.9 tuned to fixed h, producing concave allocations that satisfy monotonicity and bounds.[10] These approaches, often combined with largest-remainder adjustments post-initial allocation, operationalize degressiveness without a unique formula, prioritizing empirical fit to population data over pure theory.[10] In practice, degressive methods require iterative numerical solution or optimization to balance total seats, monotonicity, and the degression condition that no unit's marginal seat gain exceeds another's for equivalent population increments.[9] For instance, under EU Treaty rules (minimum 6, maximum 96 seats), power-parameter tuning minimizes deviations from negotiated outcomes while enforcing s(p)/p non-decreasing.[10] Such systems extend classical divisor methods (e.g., Jefferson, Webster) by incorporating subproportional elements, though they may introduce super-proportionality for smallest units to amplify degressiveness.[10]