Comprehensive Guide to Elections, Electoral Integrity, and Electoral Systems

Elections, Democratic Governance, and Global Trends

• Core Concepts and Definitions:

  • An electoral system is defined as a set of laws that regulate electoral competition between candidates, parties, or both.

  • Democracies are frequently classified and analyzed according to the specific properties and mechanics of their electoral systems.

  • An electoral formula determines the precise mathematical algorithm or rule through which cast votes are translated into legislative or executive seats.

• Global Proliferation of Direct Elections:

  • Elections are increasingly utilized to fill legislative and executive offices worldwide.

  • Almost all contemporary independent sovereign states utilize direct popular elections to elect members to the lower house of their national parliaments.

  • Over the post-World War II era, the decennial frequency of both legislative and presidential elections has grown substantially, accelerating sharply through the 1990s, 2000s, and 2010s.

Electoral Integrity, Malpractice, and Election Forensics

• Normative Standards of Electoral Integrity:

  • Electoral integrity refers to the extent to which the conduct of an election meets international standards and global norms concerning "good" elections.

  • These standards and norms are formally codified in international treaties, conventions, covenants, and guidelines issued by multilateral bodies and regional organizations.

  • Violations of international standards across any stage of the electoral cycle are designated as electoral malpractice.

• Cross-National Variation and Structural Determinants:

  • While democratic regimes generally demonstrate significantly higher levels of electoral integrity than dictatorships, substantial variation exists within both democratic and non-democratic categories.

  • Electoral integrity is shaped by four primary domestic and systemic factors:

    • Domestic structural constraints: Socioeconomic conditions, level of economic development, and historical state capacity.

    • The role of the international community: Diplomatic pressure, foreign aid conditionality, and treaty oversight.

    • Institutional design: The constitutional framework, separation of powers, and statutory rules governing contestation.

    • Electoral management bodies (EMBs): The degree of formal independence, administrative competence, and financial autonomy of election commissions.

  • Evaluative indices such as the Perception of Electoral Integrity (PEI) index assess integrity across multiple distinct sub-dimensions: electoral laws, administrative procedures, district boundaries, voter registration, party and candidate registration, media coverage, campaign finance, voting processes, the vote count, post-election results handling, and the autonomy of electoral authorities.

• Strategies for Identifying Electoral Fraud:

  • Two primary empirical methodologies are utilized to uncover malpractice:

    • Election monitoring: On-site qualitative and quantitative observation conducted by domestic civil society observer groups or international missions.

    • Election forensics: Statistical and mathematical analysis of official voting returns to detect irregularities, anomalous distributions, or manufactured vote counts.

• Benford's Law in Election Forensics:

  • Mathematical Basis: Benford's Law posits that across naturally occurring multi-digit numerical data, the distribution of leading digits is non-uniform. Lower digits appear as the first or second digit far more frequently than higher digits.

  • Forensic Application: Human agents fabricating data tend to assign digits uniformly or exhibit specific cognitive biases, violating the logarithmic distributions predicted by Benford's Law (Cho and Gaines 2007).

  • Digit Probabilities and Expectations:

    • The theoretical probability distribution of first and second digits under Benford's Law is formalized as follows:

• In forensic electoral studies, scholars generally evaluate the distribution of the second digit rather than the first digit, as second-digit tests are less vulnerable to district magnitude and baseline population scale constraints (Mebane 2006, 2008; Pericchi and Torres 2011; Cantu and Saiegh 2011).

• For instance, forensic analysis applied to official vote tallies from $45{,}692$ ballot boxes in the 2009 presidential election in Iran identified statistically significant deviations from Benford's Law, providing empirical evidence of systemic manipulation (Mebane 2013).

Taxonomy of Electoral System Families

• Primary System Classifications:

  • Electoral systems are grouped into three overarching families according to their electoral formula:

    • Majoritarian systems

    • Proportional systems

    • Mixed systems

• Majoritarian Family Variants:

  • Plurality Rules:

    • Single-Member District Plurality (SMDP)

    • Single Nontransferable Vote (SNTV)

    • Block Vote (BV)

    • Party Block Vote (PBV)

    • Limited Vote (LV)

    • Borda Count (BC) and Modified Borda Count

  • Absolute Majority Rules:

    • Alternative Vote (AV)

    • Two-Round System: Majority-Runoff (TRS)

    • Two-Round System: Majority-Plurality (TRS)

• Proportional Family Variants:

  • Quota-Based Systems: Hare, Hagenbach-Bischoff, Droop, Imperiali, and Reinforced Imperiali quotas, paired with remainder allocation rules.

  • Divisor (Highest Average) Systems: d'Hondt, Sainte-Laguë, and Modified Sainte-Laguë divisors.

  • Ballot Architecture: List Proportional Representation (List PR) versus Single Transferable Vote (STV).

• Mixed Family Variants:

  • Independent Mixed Systems: Coexistence and Superposition.

  • Dependent Mixed Systems: Fusion, Correction (Compensatory), and Conditional.

Majoritarian Electoral Systems: Plurality and Absolute Majority

• Core Logic of Majoritarianism:

  • A majoritarian electoral system awards legislative or executive seats to the candidates or political parties that secure either a plurality or an absolute majority of cast votes.

• Single-Member District Plurality (SMDP):

  • Mechanism: Voters cast a single ballot for a individual candidate within a single-member electoral district. The candidate receiving the highest number of votes wins the seat, regardless of whether they achieve an absolute majority.

  • Theoretical Bounds: A candidate can mathematically win an SMDP district with an extremely low percentage or count (hypothetically, two votes if all other candidates receive one vote each).

  • Case Study (Bedford Constituency, UK Legislative Elections, 2019):

-   In this election, Mohammad Yasin won the seat with a plurality of $43.3\%$ of the total vote, defeating Ryan Henson by a margin of $145$ votes, despite $56.7\%$ of the electorate voting for other candidates.
-   **Institutional Strengths:**
    -   *Simplicity:* Understandable rules for electors and straightforward, inexpensive ballot tabulation.
    -   *Identifiable accountability:* With exactly one representative per geographic constituency, voters can directly locate political responsibility for district outcomes.
-   **Institutional Weaknesses:**
    -   High levels of unrepresented or "wasted" votes.
    -   Significant disproportionality between national vote share and total seats won.

• Single Nontransferable Vote (SNTV):

  • Mechanism: Voters cast a single, non-transferable, candidate-centered vote in a multimember electoral district. Candidates receiving the highest overall vote totals are elected until all district seats are filled.

  • Introduces intra-party competition, as multiple candidates from the same party contend against each other for district seats.

• The Alternative Vote (AV):

  • Mechanism: A preferential, candidate-centered system deployed in single-member districts. Voters rank candidates ordinally ($1, 2, 3, \dots$).

  • If a candidate secures an absolute majority ($> 50\%$) of first-preference votes, they are elected immediately.

  • If no candidate obtains an absolute majority, the candidate with the lowest number of first-preference votes is eliminated. Their ballots are redistributed among the remaining candidates according to designated second preferences.

  • This elimination and reallocation cycle recurs iteratively until a candidate achieves an absolute majority of total votes cast (in full preferential voting) or of valid ballots remaining (in optional preferential voting).

  • Case Study (Richmond Constituency, New South Wales, Australian Federal Election, 1990):

• Charles Blunt (National Party leader) initially led on the first count with $40.9\%$ against Neville Newell (Labor) at $26.7\%$.

• Through subsequent preference redistributions following the elimination of minor candidates and third-place independent candidate Helen Caldicott, Newell amassed sufficient transferred preferences to reach $34{,}664$ votes ($50.5\%$), overcoming Blunt's $33{,}980$ votes ($49.5\%$).

• Political parties utilize standardized instructional leaflets ("How-to-Vote cards") distributed at polling sites to coordinate voter rankings and direct preference flows toward preferred coalition partners.

• Majority-Runoff Two-Round Systems (TRS):

  • Mechanism: Electors cast a single candidate vote in a single-member district. If any candidate achieves an absolute majority ($> 50\%$) in the first round, they are elected immediately. If no candidate reaches this threshold, the two highest vote-earners advance to a decisive second-round runoff.

  • Single-Round Decisive Outcome (Burundi Presidential Election, 2020):

• Because Évariste Ndayishimiye obtained an absolute majority ($71.45\%$), no runoff election was triggered.

• Two-Round Runoff Reversal (Uruguay Presidential Election, 2019):

• In Round 1, Daniel Martinez led Luis Alberto Lacalle Pou by $10.79$ percentage points ($40.49\%$ vs. $29.70\%$).

• Prior to Round 2, Lacalle Pou consolidated backing from eliminated candidates (notably Talvi and Manini Ríos), securing $50.79\%$ in the runoff to win the presidency.

• Compared to SMDP, majority-runoff systems expand voter choice in the primary contest while ensuring the eventual winner commands majority backing in the head-to-head runoff.

Proportional Representation: Quota and Divisor Formulas

• Foundational Principles of Proportional Representation (PR):

  • PR systems operate within multimember districts using either mathematical quotas or divisor algorithms.

  • The core theoretical objective is the proportional reflection of electoral support: a party winning $x\%$ of the votes should secure approximately $x\%$ of the legislative seats.

  • The two primary sub-types are List PR systems (allocating seats to party slates) and the Single Transferable Vote (STV) (allocating seats directly to candidates via preferential ballots).

• Mathematical Quotas:

  • A quota defines the exact quantity of votes required to guarantee a single seat in a given district.

  • The general formula for calculating an electoral quota is:

$Q(n) = \frac{V_d}{M_d + n}$

• Variable definitions:

  • $V_d$ = total number of valid votes cast in district $d$.

  • $M_d$ = district magnitude (total seats available in district $d$).

  • $n$ = integer modifier determining the quota variant.

• Standard quota variants include:

  • Hare Quota: $n = 0$

  • Hagenbach-Bischoff Quota: $n = 1$

  • Imperiali Quota: $n = 2$

  • Reinforced Imperiali Quota: $n = 3$

  • Droop Quota: The Hagenbach-Bischoff quota rounded down to the nearest integer plus one:

$Q_{\text{Droop}} = \left\lfloor \frac{V_d}{M_d + 1} \right\rfloor + 1$

• Seat Allocation via Quotas and the Largest Remainder Method:

  • Stage 1 (Automatic Seats): A party's total district vote is divided by the quota. The integer quotient represents the number of "automatic" seats allocated immediately to that party.

  • Stage 2 (Remainder Allocation): Because integer seat allocations rarely equal the district magnitude ($M_d$), the remaining seats are assigned sequentially to parties possessing the largest fractional remainders of unused votes.

  • Worked Example (Hare Quota with Largest Remainders):

    • District parameters: $M_d = 10$ seats; $V_d = 100{,}000$ valid votes.

    • Hare Quota calculation:

$Q = \frac{100{,}000}{10 + 0} = 10{,}000$

-   Automatic allocation distributes $7$ seats. The remaining $3$ seats are awarded to the three largest fractional remainders: Party A ($0.70$), Party E ($0.61$), and Party B ($0.60$).

• Divisor (Highest Average) Systems:

  • Instead of establishing an absolute quota, a divisor system iteratively divides each party's raw vote total by a predetermined sequence of divisors to produce a matrix of quotients. District seats are assigned one by one to the largest quotients across the matrix.

  • Common divisor sequences:

    • d'Hondt: $1, 2, 3, 4, 5, \dots$

    • Sainte-Laguë: $1, 3, 5, 7, 9, \dots$

    • Modified Sainte-Laguë: $1.4, 3, 5, 7, 9, \dots$

  • Worked Example (d'Hondt Seat Allocation):

    • Same baseline data ($M_d = 10$ seats, $100{,}000$ votes):

• Comparative Observation: Under Hare Largest Remainders, Party E ($6{,}100$ votes) gained a seat through its remainder, leaving Party C with one seat. Under d'Hondt, the final seat (Seat 10) fell to Party C's quotient of $7{,}900$, denying Party E a seat. The d'Hondt formula systematically favors larger parties relative to pure quota-remainder systems.

Structural Determinants of Proportionality: Magnitude, Thresholds, and List Types

• District Magnitude ($M$):

  • District magnitude refers to the total number of legislative representatives elected from a single district.

  • Magnitude is the single most critical institutional variable dictating the proportionality of PR systems: as district magnitude increases, proportionality increases, and the barrier to entry for smaller political parties decreases.

  • Empirical variation across states:

    • Single national district ($M = \text{total legislature}$): Ukraine utilized a nationwide single district of $M = 450$ in the 2006 and 2007 parliamentary elections. Serbia currently elects its parliament from a single nationwide district where $M = 250$.

    • Low-magnitude systems: Chile historically employed a binomial system with $M = 2$, creating strong majoritarian incentives within a nominally proportional framework.

• Electoral Thresholds:

  • An electoral threshold specifies the minimum percentage or absolute number of votes a party must attain to be eligible for any legislative seat allocation.

  • Natural (Effective) Threshold: An implicit mathematical threshold inherent to the mechanics of the system, defined primarily by district magnitude ($M$) and formula:

$T_{\text{natural}} \approx \frac{100\%}{M + 1}$

-   **Formal (Legal) Threshold:** A statutory minimum vote share explicitly prescribed in the electoral code (e.g., $5\%$ at the national level).
-   *Negative Consequences of High Thresholds:* Restricting entry via high formal thresholds can disenfranchise large shares of the electorate and distort outcomes:
    -   *Turkey (2002):* A national threshold of $10\%$ resulted in $46\%$ of total votes cast being wasted on parties that failed to pass the threshold.
    -   *Poland (1993):* Strict thresholds led to $34\%$ wasted votes, allowing former Communist successor parties to win a substantial governing majority despite receiving a minority of the popular vote.

• Party List Types:

  • Closed Party List: The party hierarchy strictly predetermines the internal ranking of its candidates. Electors vote solely for the political party. If a party wins $k$ seats, the first $k$ candidates listed on the slate are elected. Voters cannot indicate preference for individual candidates.

• Open Party List: Electors can vote either for the party as a whole or indicate a personal preference for a specific candidate within that party list. Individual candidates amass personal preference votes, which determine their final rank order for seat acquisition.

• Free Party List (Panachage): Voters possess multiple votes (often equal to district magnitude) that they can distribute flexibly among individual candidates on a single party list, across different party lists (panachage), or accumulate multiple votes on a single favored candidate (cumulation).

The Single Transferable Vote (STV)

• Core Mechanics and Definitions:

  • The Single Transferable Vote (STV) is the only proportional representation system that does not utilize party lists.

  • STV is a candidate-centered preferential voting system deployed in multimember electoral districts ($M \ge 2$).

  • Voters ordinally rank candidates ($1, 2, 3, \dots$) across parties or independent candidacies.

  • Any candidate who surpasses a specified quota (typically the Droop quota) of first-preference votes is elected immediately.

  • In subsequent counting rounds:

    • Surplus votes from candidates who exceed the quota are transferred to remaining active candidates in proportion to lower preferences designated on their ballots.

    • If no candidate attains the quota, the lowest-ranking candidate is eliminated, and all their ballots are transferred to designated next preferences.

    • This procedure continues iteratively until all district seats are filled.

• Comprehensive Worked Example of STV:

  • District Parameters: District Magnitude $M = 3$; Total Electorate $V = 20$ voters.

  • Candidates: Bruce, Shane, Sheila, Glen, Ella.

  • Droop Quota Calculation:

$Q = \left\lfloor \frac{20}{3 + 1} \right\rfloor + 1 = 5 + 1 = 6$

-   Each candidate requires exactly $6$ votes to win election.
-   *Distribution of Cast Ballots (20 Total):*
    -   $4$ ballots (Group AAAA): [1] Bruce, [2] Shane, [3] Ella
    -   $2$ ballots (Group BB): [1] Shane, [2] Bruce, [3] Glen
    -   $8$ ballots (Group CCCCCCCC): [1] Sheila, [2] Glen
    -   $4$ ballots (Group DDDD): [1] Sheila, [2] Ella
    -   $1$ ballot (Group E): [1] Glen
    -   $1$ ballot (Group F): [1] Ella

• Step-by-Step Round Progression:

• Count 1: Sheila receives $12$ first-preference votes ($8$ from C, $4$ from D), surpassing the quota of $6$. Sheila is elected with a surplus of $6$ votes ($12 - 6 = 6$).

• Surplus Reallocation (Count 2): Sheila's surplus of $6$ votes represents exactly $50\%$ of her total vote pool ($\frac{6}{12} = 0.5$). The transfer value applied to all her ballots is $0.5$:

  • The $8$ C ballots transfer $8 \times 0.5 = 4$ votes to Glen (indicated as CCCC).

  • The $4$ D ballots transfer $4 \times 0.5 = 2$ votes to Ella (indicated as DD).

  • Glen now holds $1 + 4 = 5$ votes; Ella holds $1 + 2 = 3$ votes. Bruce has $4$; Shane has $2$.

  • No remaining candidate meets the quota of $6$. The candidate with the fewest votes, Shane ($2$ votes), is eliminated.

• Elimination Reallocation (Count 3): Shane's $2$ original ballots (Group BB) designate Bruce as second preference. Both ballots transfer to Bruce at full value ($+2$). Bruce reaches $4 + 2 = 6$ votes, matching the quota exactly. Bruce is elected.

• Final Allocation (Count 4): Bruce has no surplus. Between the remaining un-elected candidates, Glen holds $5$ votes and Ella holds $3$ votes. Ella is eliminated. Glen is the sole remaining candidate for the third and final seat and is declared elected.

Mixed Electoral Systems: Independent and Dependent Designs

• Institutional Framework and Electoral Tiers:

  • A mixed electoral system is one in which representatives are elected simultaneously via two distinct systems: one majoritarian and one proportional.

  • Most mixed systems operate across multiple electoral tiers (levels at which votes are aggregated and translated into seats):

    • Lower tier: Single-member district/constituency level (typically employing SMDP majoritarian formulas).

    • Upper tier: Regional or national level (employing List PR proportional formulas).

• Independent Mixed Systems (Parallel Systems / Majoritarian-Dominant Mixed):

  • Mechanism: The majoritarian and proportional components are executed completely independently of one another. The seat allocation in the proportional tier occurs without regard to seats won by parties in the district majoritarian tier.

  • Worked Model (Independent Mixed System):

    • Assume $5$ single-member districts with $5{,}000$ voters each ($25{,}000$ total votes). In every district, Party A wins $3{,}000$ votes ($60\%$) and Party B wins $2{,}000$ votes ($40\%$).

    • District tier: $5$ seats via SMDP. Party list tier: $5$ seats via PR allocated nationally.

-   Outcome: Party A converts $60\%$ of the popular vote into $80\%$ of the total seats ($8$ out of $10$), preserving substantial disproportionality.

• Dependent Mixed Systems (Mixed Member Proportional / Compensatory PR):

  • Mechanism: The application of the proportional formula directly depends on the distribution of seats or votes produced by the majoritarian tier. The upper-tier List PR seats are utilized as compensatory "top-up" seats to ensure that each party's total legislative seat share matches its nationwide party list vote share.

  • Dual-Ballot Architecture: Electors cast two separate votes on a split ballot:

    • Electorate / Constituency Vote: A vote for a specific local candidate running in a single-member district.

    • Party List Vote: A vote for a political party slate in the upper tier. The party vote strictly determines the total overall seat entitlement of each party in parliament.

• Worked Model (Dependent Mixed System):

  • Identical baseline: Party A wins $60\%$ ($15{,}000$ votes) and all $5$ SMDP districts; Party B wins $40\%$ ($10{,}000$ votes) and $0$ districts. Total legislature = $10$ seats.

  • Total entitlement: Party A is entitled to $60\%$ of $10 = 6$ total seats. Party B is entitled to $40\%$ of $10 = 4$ total seats.

• Outcome: In the dependent mixed framework, the List PR tier neutralizes the majoritarian bias created in the district contests, yielding a final seat distribution that is strictly proportional to overall popular support.