📚 The impact of different electoral systems on party representation and on voter choice | 不同选举制度对政党代表性和选民选择的影响
From the arithmetic of seat allocation to the strategic calculus of voting, mathematics provides a powerful lens for analysing how electoral systems shape party landscapes and individual behaviour. Understanding these mechanisms reveals why some nations fragment into multiparty coalitions while others stabilise into two‑party dominance, and how voter decisions are mathematically constrained by the rules of the game. This article explores the quantitative impact of major electoral systems on party representation and voter choice, using concepts from discrete mathematics, probability, and game theory that align with the Edexcel A‑Level Mathematics and Further Mathematics curricula.
从席位分配的算术到投票策略的博弈计算，数学为分析选举制度如何塑造政党格局与个人行为提供了一面强大的透镜。理解这些机制可以揭示为何有些国家分化为多党联合政府，而另一些国家则稳定在两个主要政党的主导之下，也揭示了选民的决策如何因规则而受到数学上的约束。本文运用离散数学、概率论和博弈论等契合 Edexcel A‑Level 数学与进阶数学大纲的概念，探讨几种主要选举制度对政党代表性和选民选择的数量化影响。
1. Electoral Systems as Mathematical Mappings | 选举制度作为数学映射
An electoral system can be modelled as a function that transforms a vector of vote shares into a vector of seat shares. The properties of this function – proportionality, monotonicity, susceptibility to strategic voting – can be rigorously defined and tested. In A‑Level terms, we are concerned with algorithms, inequalities, and optimisation problems that determine who holds power.
选举制度可以被建模为一个函数，它将得票率向量转换为席位占有率向量。该函数的性质——比例性、单调性、易受策略投票影响的程度——均可严格定义并加以检验。用 A‑Level 的语言来说，我们关注的是决定执政者的算法、不等式与优化问题。
2. Plurality/Majority Systems and Duverger’s Law | 多数制与迪韦尔热定律
The First‑Past‑the‑Post (FPTP) system awards a seat to the candidate with the most votes in each single‑member constituency, even if they fail to secure an absolute majority. Mathematically, FPTP can be expressed as a winner‑takes‑all function: for constituency i, sᵢ = 1 if vᵢ > vⱼ for all j ≠ i, else 0. Aggregated nationally, this often leads to significant disproportionality. The effective number of parties tends to reduce because voters abandon smaller parties that cannot win locally, a phenomenon known as Duverger’s Law. The tactical voter’s decision can be modelled by expected utility: vote for the preferred party only if its probability of winning exceeds a threshold derived from the difference in utility between the top two contenders.
简单多数制（FPTP）在每个单席选区中，不论是否获得绝对多数，都将席位赋予得票最多的候选人。从数学上看，FPTP 可表达为一个赢家通吃函数：对选区 i，若 vᵢ > vⱼ 对所有 j ≠ i 成立，则 sᵢ = 1，否则为 0。从全国汇总后，这常常导致严重的非比例性。有效政党数目趋于减少，因为选民会放弃无法在地方胜出的小党，这就是迪韦尔热定律。策略选民的决策可用期望效用建模：只有在该党获胜概率超过由前两名候选人效用差推导出的阈值时，才会投票给最偏好的政党。
3. Proportional Representation and Seat Allocation Algorithms | 比例代表制与席位分配算法
Proportional representation (PR) systems aim to minimise the disparity between vote shares and seat shares. The mathematical core lies in the allocation of a fixed number of seats among parties according to their vote totals. Two main families of algorithms exist: largest remainder methods (using quotas) and highest averages methods (using divisors). In Edexcel Decision Mathematics, the concept of an iterative algorithm that compares quotients is directly applicable. The Hare quota is defined as Q = total valid votes / total seats. Parties receive floor(votes / Q) seats initially, and remaining seats go to the largest remainders. This can lead to paradoxes: the Alabama paradox, where an increase in total seats can cause a party to lose a seat, illustrates the non‑monotonic property of some quota‑based methods.
比例代表制（PR）旨在最小化得票率与席位占有率之间的差距。其数学核心在于根据各党得票总额分配固定数量的席位。存在两大类算法：最大余数法（使用基数）和最高平均数法（使用除数）。在 Edexcel 决策数学中，比较商值的迭代算法概念可直接适用。黑尔基数的定义为 Q = 有效选票总数 / 总席位数。各党初始获得 ⌊得票数 / Q⌋ 个席位，剩余席位按最大余数分配。这可能导致悖论：阿拉巴马悖论——增加总席位反而导致某党失去一个席位——就展示了一些以基数为基础的方法的非单调性。
4. The D’Hondt and Sainte‑Laguë Divisors: Mathematical Bias | 汉狄法与圣拉格法：数学偏差
Highest averages methods work by repeatedly awarding seats to the party with the highest quotient v / (s + d), where v is the party’s votes, s is seats already won, and d is a modifier. For D’Hondt, d = 1; for Sainte‑Laguë, d = 0.5. These formulas can be analysed using inequalities. D’Hondt favours larger parties because the divisor sequence (1, 2, 3, …) reduces the quotient more slowly than Sainte‑Laguë’s (0.5, 1.5, 2.5, …). The bias B can be quantified: under D’Hondt, a party with vote share p can expect a seat share approximately p / (1 + p), creating a systematic advantage for the largest party. This directly impacts voter choice, as supporters of small parties may defect to larger allies to ensure their vote is not wasted – a rational decision justified by the mathematics of seat thresholds.
最高平均数法通过反复将席位分配给商值 v / (s + d) 最高的政党来运作，其中 v 为政党得票，s 为已获席位数，d 为修正参数。汉狄法取 d = 1；圣拉格法则取 d = 0.5。这些公式可用不等式加以分析。汉狄法偏向大党，因为其除数序列 (1, 2, 3, …) 削减商值的速度比圣拉格法的序列 (0.5, 1.5, 2.5, …) 更慢。偏差 B 可被量化：在汉狄法下，得票率为 p 的政党预计可获得约 p / (1 + p) 的席位率，这就为最大党创造了系统性优势。这直接影响了选民选择，因为小党的支持者可能会转向较大的同盟以确保选票不被浪费——基于席位阈值的数学推导证明这一决策是理性的。
5. Effective Number of Parties: Laakso‑Taagepera Index | 有效政党数目：拉克斯托—塔格培拉指数
To measure the impact of an electoral system on party representation, we can compute the effective number of electoral parties (ENEP) and parliamentary parties (ENPP). The index is N = 1 / Σ pᵢ², where pᵢ is the vote or seat share of party i. This is the reciprocal of the Herfindahl‑Hirschman concentration index. For a pure two‑party system, N ≈ 2. In FPTP systems, ENEP is often reduced relative to the actual number of parties entering the election, because voters desert weak parties. By comparing ENEP and ENPP, the mechanical and psychological effects of the electoral system can be disentangled. In Edexcel Statistics, the sum of squared shares resembles the calculation of variance; it provides a concise measure of fragmentation that can be tracked over time.
为了衡量选举制度对政党代表性的影响，我们可以计算有效选举政党数（ENEP）和有效议会政党数（ENPP）。该指数为 N = 1 / Σ pᵢ² ，其中 pᵢ 为政党 i 的得票率或席位率。这是赫芬达尔—赫希曼集中度指数的倒数。对于纯粹的二元政党体制，N ≈ 2。在 FPTP 体系中，ENEP 通常会相对于实际参选政党数有所缩小，因为选民会放弃弱势政党。通过比较 ENEP 与 ENPP，选举制度的机械效应和心理效应可以被区分开来。在 Edexcel 统计学中，份额平方和类似于方差的计算；它提供了一种简洁的碎片化度量，可随时间追踪。
6. Electoral Thresholds and Strategic Fragmentation | 选举阈值与策略性政党分裂
Many PR systems impose a legal threshold T% (e.g., 5%) that a party must surpass to win any seats. The mathematical condition for representation is v/V > T/100, where v is the party’s votes and V is the national total. This creates a step function: seat share = 0 if vote share < T, while above T the allocation follows the proportional formula. From a voter’s perspective, the expected utility of voting for a minor party diminishes sharply near the threshold, leading to an equilibrium where rational voters coalesce around parties safely above T. This can be modelled using discontinuous expected value functions, which in A‑Level Further Mathematics relate to piecewise definitions and decision analysis under uncertainty.
许多比例代表制设定了政党必须超出的法定门槛 T%（例如 5%）才能获得任何席位。获得代表权的数学条件为 v/V > T/100，其中 v 为该党得票，V 为全国总票数。这就产生了一个阶梯函数：若得票率 < T，则席位率为 0；一旦超过 T，席位分配即按比例公式执行。从选民角度而言，投票给小党的期望效用会在门槛附近急剧下降，从而形成一个理性选民围绕安全高于 T 的政党聚拢的均衡。这可以用不连续的期望值函数来建模，在 A‑Level 进阶数学中涉及分段定义与不确定条件下的决策分析。
7. Tactical Voting and Game‑Theoretic Equilibria | 策略投票与博弈均衡
Voter choice under any electoral system can be analysed as a game where the payoff depends on the election outcome. Define utility uᵢ(c) that voter i derives from candidate c winning. In a plurality election, voting sincerely yields the highest payoff only if the preferred candidate is viable. Otherwise, the rational strategy is to vote for the most preferred among the front‑runners to prevent the least preferred from winning. This can be formalised using a payoff matrix and solved for Nash equilibria. At A‑Level, students encounter similar payoff structures in zero‑sum and coordination games. The prevalence of tactical voting is directly measurable through the “wasted vote” metric: vote share minus seat share. The greater the gap, the stronger the incentive for voters to defect.
在任何选举制度下，选民选择都可以作为一个博弈来分析，博弈的支付取决于选举结果。定义选民 i 从候选人 c 获胜中所获得的效用 uᵢ(c)。在相对多数选举中，只有当首选候选人有胜选可能时，真诚投票才能带来最高支付。否则，理性策略是将票投给领先者中最偏好的那位，以阻挠最不偏好的候选人当选。这可以用一个支付矩阵加以形式化，并求解纳什均衡。在 A‑Level 阶段，学生会遇到零和博弈和协调博弈中类似的支付结构。策略投票的普遍程度可通过“废票”指标直接度量：得票率减去席位率。这一差距越大，选民倒戈的激励就越强。
8. Mixed‑Member Systems and the Compensation Mechanism | 混合成员制与补偿机制
Mixed‑member proportional (MMP) systems, used in Germany and New Zealand, combine single‑member districts with PR top‑up seats. The math involves solving for the number of list seats needed to bring each party’s total seats into proportionality. Let dᵢ be district seats won, and let sᵢ* be the ideal seat share from votes. The top‑up tᵢ = max(0, round(sᵢ* × S – dᵢ)), where S is total seats. Constraints such as a 5% threshold or three direct mandates override this pure formula, producing a piecewise linear compensation function. This hybrid model gives voters two choices: one for a local candidate (possibly tactical) and one for a party list (usually sincere). The separation of these choices allows researchers to isolate the psychological effect of electoral systems on voter sincerity.
混合成员比例制（MMP）在德国和新西兰使用，将单席选区与比例代表制的增补席位相结合。其数学涉及求解为使各党总席位达到比例性所需的政党名单席位数。设 dᵢ 为赢得的选区席位，sᵢ* 为由得票率得出的理想席位率。增补席位 tᵢ = max(0, round(sᵢ* × S – dᵢ))，其中 S 为总席位。5% 门槛或三个直接当选名额等约束条件会凌驾于这一纯公式之上，从而产生一个分段线性的补偿函数。这种混合模式为选民提供了两个选择：一是对地方候选人的选择（可能是策略性的），一是对政党名单的选择（通常是真诚的）。这两个选择的分离使得研究者可以分离出选举制度对选民真诚度的心理效应。
9. Gerrymandering and the Geometry of District Drawing | 杰利蝾螈与选区划分的几何学
In systems with single‑member districts, the drawing of boundaries can radically alter the translation of votes to seats. The efficiency gap, a mathematical measure of gerrymandering, compares wasted votes for each party. A vote is wasted if it is cast for a losing candidate or for a winning candidate beyond the margin needed. The formula is EG = (Wₐ – W_b) / V, where Wₐ and W_b are wasted votes for the two main parties. An absolute value above 0.08 is often considered indicative of gerrymandering. This connects to A‑Level topics on optimisation, area partitioning, and inequalities. Voter choice is affected because safe seats reduce the incentive to vote, while competitive districts increase it – a phenomenon that can be modelled with turnout functions dependent on the margin of victory predicted by polls.
在单席选区制度中，边界划分可以根本性地改变票数与席位之间的转换关系。作为衡量杰利蝾螈的一种数学工具，效率差距比较了各党派的废票数量。凡是被投给落选候选人或超出获胜所需裕度投给胜选候选人的选票，都算作废票。其公式为 EG = (Wₐ – W_b) / V，其中 Wₐ 与 W_b 是两大政党的废票。绝对值若大于 0.08，通常被视为存在杰利蝾螈行为。这与 A‑Level 中优化、区域划分和不等式等主题密切相关。选民选择受到影响，因为安全席会降低投票激励，而竞争性选区则会增强激励——这一现象可以通过投票率函数建立模型，该函数取决于民调预测的胜选裕度。
10. Power Indices and Coalition Dynamics | 权力指数与联盟动态
Under PR, no single party often commands a majority, so coalitions form. Power indices like Shapley–Shubik and Banzhaf quantify a party’s influence on the outcome, which may be disproportionate to its seat count. In a weighted voting game with quota q (e.g., 50%+1), the Shapley–Shubik index counts the number of orderings in which a party is the pivotal voter. The mathematical process involves factorials and permutations, directly linking to the permutations and combinations topic in Edexcel Statistics. For instance, with three parties holding 45, 35, and 20 seats and a quota of 51, the smallest party’s power index may be zero if it can never be pivotal. Voters, aware of post‑election negotiations, may therefore vote for a party that is a likely kingmaker rather than their sincere preference – a second‑order strategic choice.
在比例代表制下，往往没有单一政党占据多数，因此需要组建联盟。沙普利—舒比克指数和班扎夫指数等权力指数可以量化一个政党对结果的影响力，这种影响力可能与席位数量不成正比。在一个有配额 q（例如 50%+1）的加权投票博弈中，沙普利—舒比克指数计算的是某党成为关键投票者的顺序数目。这一数学过程涉及阶乘和排列，直接关联到 Edexcel 统计学中的排列与组合主题。例如，有 A、B、C 三党分别拥有 45、35 和 20 个席位，配额为 51，如果最小政党在任何排列中都绝不可能成为关键投票者，则其权力指数为零。因此，预见到选后谈判的选民可能会投票给一个可能成为“造王者”的政党，而不是自己的真诚偏好——这是一种二阶的策略性选择。
11. Measuring Representativeness: Gallagher and Loosemore‑Hanby Indices | 代表性的衡量：加拉赫指数与鲁斯莫尔—汉比指数
Disproportionality between votes and seats can be measured by indices that are essentially statistical distance metrics. The Gallagher least‑squares index is G = √(½ Σ (vᵢ – sᵢ)²), while the Loosemore‑Hanby index is D = ½ Σ |vᵢ – sᵢ|. These are directly analogous to variance and absolute deviation used in A‑Level descriptive statistics. By computing these values for different elections, one can quantify the representational distortion introduced by FPTP versus PR. A higher index indicates weaker party representation, which typically depresses voter turnout among supporters of disadvantaged parties and encourages strategic entry or exit decisions by party elites.
选票与席位之间的非比例性可以通过若干指数加以衡量，这些指数本质上都是统计距离量度。加拉赫最小二乘指数为 G = √(½ Σ (vᵢ – sᵢ)²)，而鲁斯莫尔—汉比指数为 D = ½ Σ |vᵢ – sᵢ|。它们与 A‑Level 描述性统计中使用的方差和绝对离差直接对应。通过计算不同选举中的这些数值，可以量化 FPTP 相对于比例代表制所引入的代表性扭曲。指数越高，表明政党的代表性越弱，这通常会抑制弱势政党支持者的投票率，并鼓励政党精英做出策略性参选或退出的决策。
12. Conclusion: Mathematical Insights into Electoral Design | 结语：从数学洞察选举设计
Mathematics not only describes but also predicts the consequences of electoral system choices. The interplay of algorithms, threshold functions, and game‑theoretic incentives shapes party competition and voter behaviour in measurable ways. For students of Edexcel A‑Level Mathematics, electoral systems offer a rich applied context for exploring topics from discrete decision algorithms to statistical modelling and combinatorial power analysis. By quantifying party representation and voter choice, mathematics empowers us to design fairer democratic institutions and to understand the rational choices of voters who must navigate the rules of the game.
数学不仅可以描述选举制度选择所带来的后果，还能加以预测。算法、阈值函数与博弈激励的相互作用，以可度量的方式塑造着政党竞争和选民行为。对于 Edexcel A‑Level 数学课程的学生而言，选举制度为探索从离散决策算法到统计建模以及组合权力分析等多个主题提供了丰富的应用场景。通过量化政党代表性和选民选择，数学赋予我们设计更加公平的民主制度的力量，并帮助我们理解选民在博弈规则下必须做出的理性抉择。
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