A general predictive account of Duverger's two effects of district magnitude: small-party seat-entry thresholds and the institutional conditions of the psychological effect

Shugert's (1985) 'crucial experiment' on Venezuela's fused legislative ballots separated Duverger's mechanical effect of district magnitude (the number of seats per electoral district), underrepresentation of small parties in seat allocation, from the psychological effect, voters' anticipatory desertion of small parties, and showed that the two interact compensatorily. What remains open in the research program that study seeded: (1) a threshold rule that predicts, from a district's party vote-share distribution and the electoral formula, the district magnitude at which further increases begin to admit the smallest parties to seats, and (2) a predictive account of the institutional conditions governing the strength of the psychological effect, so that the net consequence of changing district magnitude for disproportionality and party-system fragmentation can be anticipated outside the single-country configuration in which the two channels were first separated. Later literature has quantified the mechanical channel (distribution-free thresholds; logical seat-vote models) and has repeatedly found the psychological channel real but weaker, noisier, and conditional on social heterogeneity and ballot structure; neither the distribution-dependent seat-entry rule nor a general model of the psychological effect's moderators exists, so the problem remains open.

Background

Comparative research on electoral systems asks how the translation of votes into seats shapes party competition. The central institutional variable is district magnitude, $M$: the number of seats filled in an electoral district. Following Maurice Duverger's classic distinction, electoral rules operate through two channels. The mechanical effect is arithmetic: where few seats are contested, the seat-allocation mathematics of proportional representation systematically under-represents small parties. The psychological effect is behavioral: voters (and party elites) anticipate these losses and desert small parties at the ballot box. The two channels usually occur together and push in the same direction, so their separate strengths cannot normally be estimated from observed election results.

Matthew S. Shugart's 1985 study of Venezuela exploited an institutional configuration that separates them. Venezuelan voters cast a single party vote that was simultaneously used to allocate seats in three legislative tiers with different district magnitudes: the Senate ($M=2$ everywhere), the Chamber of Deputies (variable $M$ by state), and state legislative assemblies elected at substantially larger magnitudes. Because only one vote is cast, the psychological effect can attach to only one tier, leaving the other tiers to display the mechanical effect nearly in isolation. Shugert measured the mechanical channel by the deviation from proportionality $D=\frac{1}{2}\sum_i|s_i-v_i|$ (the Loosemore-Hanby index, where $v_i$ and $s_i$ are party $i$'s vote and seat shares) and the behavioral channel by the effective number of electoral parties $N=(\sum_i v_i^2)^{-1}$ (the Laakso-Taagepera index, a vote-share-weighted count of parties). For Chamber elections of 1958-1968 he fitted the power law $D=0.5M^{-0.60}$ at the district level, showed that $N$ rises with $M$ across all six elections studied (1958-1983), and demonstrated the interaction: in Senate seats paired with high-$M$ Chamber districts, where voter psychology is blocked from adjusting to the Senate's low $M$, disproportionality is exceptionally high, while tiers in which psychology is restrained by a low-$M$ principal tier show exceptionally low disproportionality. The psychological restraint partially compensates the mechanical tendency of low-$M$ districts to waste small-party votes.

Two limitations of this analysis define the open problem. First, everything rests on one country's unique fused ballot, and the paper itself flagged systems (reportedly Austria) in which increases in $M$ did not produce more small-party voting. Second, the mechanical $M$-$D$ relation is itself conditional on the distribution of party vote shares: in the 1973-1983 elections, when the party system had two large parties and a 'gap' (few or no middle-sized parties), the earlier log-linear decline of $D$ with $M$ largely disappeared, because the smallest parties gain nothing from moderate increases in $M$ until $M$ is large enough to admit them to seats. Classical results give distribution-free bounds for such entry: the threshold of exclusion (the maximum vote share that can still fail to win a seat, a worst-case bound) and the threshold of representation (the minimum vote share at which a party can win a seat, a best-case bound), derived by Rae-Hanby-Loosemore and by Lijphart-Gibberd as functions of $M$ and the electoral formula for list PR variants such as d'Hondt (highest-average) allocation. Later 'logical model' work by Taagepera and Shugart predicts the aggregate effective number of parties from $M$ and assembly size alone. None of these, however, yields a rule that takes a district's actual party-size distribution, including gap distributions, as input and returns the magnitude at which the smallest parties begin to win seats; and none predicts how strong the psychological compensation will be in a given institutional context.

Problem Statement

The problem is to give a general predictive account of Duverger's two effects of district magnitude, $M$ (the number of seats per electoral district), beyond the single-country fused-ballot configuration in which they were first cleanly separated. It comprises two linked components of one objective:

(1) A threshold rule, closed-form or estimable, that maps a district's party vote-share distribution, including 'gap' distributions with two large parties and only tiny parties, together with the seat-allocation formula, to the magnitude at which further increases in $M$ begin to admit the smallest parties to seats, and to the resulting deviation from proportionality $D$ (one half the sum of the absolute differences between each party's vote share and seat share). The rule is formulated at district level and takes the vote-share distribution as given; its output is the seat-entry point of the smallest parties, whether expressed as the entry magnitude of each such party or as the corresponding schedule of entry magnitudes, together with the resulting $D$ at each magnitude. Existing distribution-free thresholds of representation and exclusion and aggregate logical models of the effective number of parties do not provide this; the rule must be distribution-dependent, and so must cover gap-type party systems such as Venezuela 1973-1983, where the documented decline of $D$ with $M$ breaks down.

(2) A predictive account of the institutional conditions that govern the strength of the psychological effect (measured as the gap between the observed disproportionality and effective party numbers and the mechanical-only prediction computed from the vote-share distribution that would obtain absent anticipatory desertion), so that the compensatory interaction between the two channels, and hence the net effect of a change in $M$ on disproportionality and on the effective number of electoral and parliamentary parties (the vote-share-based and seat-share-based effective party counts), can be anticipated in systems other than Venezuela, including in reported cases (reportedly Austria) where increases in $M$ did not produce increased small-party voting. In this component the party vote-share distribution is itself treated as responding to $M$ through voter and elite expectations, so the two components compose rather than double-count the behavioral channel: (1) gives the seat allocation and the resulting district-level $D$ for a given distribution, (2) how that distribution shifts with the institutional setting. Since district-level allocations aggregate, the net effect sought in (2) concerns system-level disproportionality and effective party numbers as well as their district-level components. Conditions discussed in this literature include ballot structure (fused versus separate votes, and the presence of a more permissive tier on the same ballot, as in mixed-member contamination, where a proportional-representation vote on the same ballot boosts small-party support in the majoritarian tier), the relative salience of tiers, social heterogeneity, and the information available to voters for coordinating on viable parties. The object sought is an account that states, for the general class of systems rather than for any single condition in isolation, the relation between these conditions and that measured strength of the effect; a list of conditions, or an assertion of their directional tendency, with that relation left unstated, does not by itself answer this component.

The two channels are to be distinguished rather than conflated, and the account sought is general beyond the Venezuelan fused-ballot configuration: a re-description of that single case, or bounds that hold only in the worst or best case, does not by itself answer either component.

References

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