📚 Electoral Reform: The Mathematics of Voting Systems | 选举改革：投票系统中的数学
Every election depends on a set of rules that translate millions of individual preferences into a collective decision. When these rules are debated – Should we keep first-past-the-post? Should we adopt proportional representation? – we are talking about electoral reform. Hidden behind that political debate is a rich layer of mathematics, from apportionment algorithms and fairness criteria to paradoxes and impossibility theorems. This article unpacks the mathematical foundations of electoral reform for A‑Level students, showing how numbers shape democracy.
每一次选举都依赖一套规则，将数百万个人的偏好转化为集体决定。当我们争论是否保留简单多数制、是否采用比例代表制时，我们谈论的正是选举改革。在这场政治争论背后，隐藏着一层丰富的数学知识——从席位分配算法和公平性准则，到悖论与不可能定理。本文为 A‑Level 学生剖析选举改革的数学根基，展示数字如何塑造民主。
1. What Is Electoral Reform? | 什么是选举改革？
Electoral reform means changing the system by which votes are cast, counted, and converted into seats. Reforms often aim to correct disproportionality, reduce wasted votes, or guarantee that outcomes reflect the will of the people more accurately.
选举改革意味着改变投票、计票和将选票转化为席位的方式。改革通常旨在纠正比例失调、减少浪费选票或确保结果更准确地反映民意。
rent function f: votes → seats, and analysing its properties forms the core of this topic.
常见的动机包括希望票数份额与席位份额更加紧密地挂钩、保障少数群体代表权以及消除策略性投票。从数学角度看，每一项改革提案都可以被视为不同的函数 f：选票 → 席位，而分析该函数的性质正是本主题的核心。
2. First-Past-the-Post: Plurality Voting | 简单多数制：相对多数投票
In a first-past-the-post (FPTP) system, each voter selects one candidate, and the candidate with the most votes in a constituency wins that single seat. The winner does not need an absolute majority; a simple plurality is enough.
在简单多数制中，每位选民选择一位候选人，在选区中获得最多票数的候选人即赢得该单一席位。获胜者无需获得绝对多数，简单相对多数即足矣。
If we denote the number of votes for candidate i as vᵢ, the winner is argmax(v₁, v₂, …, vₖ). This rule can produce highly disproportional national outcomes: a party with 35% of the vote nationally can win 60% of the seats if its support is efficiently distributed, while a party with 20% of the vote but spread evenly may win very few seats.
若将候选人 i 的得票数记为 vᵢ，则胜选者为 argmax(v₁, v₂, …, vₖ)。这一规则可能产生高度不成比例的国家结果：一个获得全国 35% 选票的政党，如果其支持者分布高效，可能赢得 60% 的席位；而一个获得全国 20% 选票但均匀分布的政党，则可能仅获极少席位。
FPTP’s mathematical simplicity masks complex consequences, such as ‘wasted votes’ – all votes cast for losing candidates and any votes above the minimum needed to win. Reformers often seek to minimise these wasted votes.
简单多数制的数学简单性掩盖了复杂的后果，例如“废票”——所有投给落选候选人的票，以及任何超出获胜所需最低票数的票。改革者常常试图减少这些废票。
3. Ranked Choice and Alternative Vote (IRV) | 排序投票与可转移投票（即时决选投票）
Ranked‑choice voting (RCV), also known as instant‑runoff voting (IRV), asks voters to rank candidates in order of preference. If no candidate receives over 50% of first‑preference votes, the candidate with the fewest first preferences is eliminated and their votes are transferred to the next available preference on each ballot. This process repeats until a candidate crosses the 50% threshold.
排序投票要求选民对候选人按偏好排序。如果没有候选人获得超过 50% 的第一偏好票，则第一偏好票最少的候选人被淘汰，其选票按每张选票上的下一可用偏好进行转移。重复此过程，直至有候选人跨越 50% 的门槛。
Mathematically, the elimination algorithm can be described as: while no candidate’s first‑preference count cᵢ > ½Σcᵢ, eliminate the candidate with min(cₖ), and redistribute ballots to the next preference. This method is designed to produce a majority winner and reduce tactical voting, but it does not guarantee the election of a Condorcet winner (a candidate who would beat every other candidate in head‑to‑head comparisons).
从数学上讲，该淘汰算法可描述为：当没有任何候选人的第一偏好票数 cᵢ > ½Σcᵢ 时，淘汰 min(cₖ) 的候选人，并将选票重新分配给下一偏好。该方法旨在产生多数获胜者并减少策略性投票，但不能保证选出孔多塞胜者（即在两两对决中击败所有其他候选人的候选人）。
4. Condorcet Methods and Pairwise Comparisons | 孔多塞方法与两两比较
A Condorcet method declares the winner to be the candidate who, when compared in turn with each other candidate, is preferred by a majority of voters. If such a candidate exists, they are the Condorcet winner. The matrix of pairwise victories can be built from ranked ballots: for each pair (X, Y), count the number of voters who rank X above Y, denoted N(X>Y).
孔多塞方法宣布的胜者是，在与每一位其他候选人逐一比较时，获得多数选民偏好的候选人。如果存在这样的候选人，他就是孔多塞胜者。两两对决的胜负矩阵可根据排序选票构建：对每一对候选人 (X, Y)，统计将 X 排在 Y 之前的选民人数，记为 N(X>Y)。
Condorcet’s paradox occurs when collective preferences become cyclic, e.g. N(A>B) > N(B>A), N(B>C) > N(C>B), but N(C>A) > N(A>C). In such cases, no Condorcet winner exists, and different methods (e.g. Copeland, Schulze, minimax) break the cycle using distinct scoring rules. The Schulze method, for instance, uses a beatpath algorithm based on the strongest paths in the pairwise comparison graph.
当集体偏好出现循环时，便产生孔多塞悖论，例如 N(A>B) > N(B>A)，N(B>C) > N(C>B)，但 N(C>A) > N(A>C)。此时不存在孔多塞胜者，不同的方法（如科普兰法、舒尔茨法、最大最小法）利用不同的计分规则打破循环。例如，舒尔茨法使用基于两两比较图中最强路径的胜路径算法。
5. Proportional Representation: The Need for Apportionment | 比例代表制：席位分配的必要性
ere are S seats, the ideal quota for party i is (Vᵢ/V) × S, but this is rarely an integer.
比例代表制旨在按政党得票比例分配席位。这需要一种席位分配方法：将分数席位定额转化为整数席位分配的规则。若政党 i 的得票数为 Vᵢ，总票数为 V = ΣVᵢ，共有 S 个席位，则政党 i 的理想定额为 (Vᵢ/V) × S，但很少为整数。
Apportionment methods differ in their divisor sequences or quota rules. Two famous families are largest‑remainder methods (e.g. Hare quota, Droop quota) and highest‑average (divisor) methods (e.g. D’Hondt, Sainte‑Laguë). Understanding these requires careful arithmetic and a sense of fairness.
分配方法在除数序列或定额规则上存在差异。两大类著名方法是最大余数法（例如黑尔定额、特罗普定额）和最高平均数（除数）法（例如德洪特法、圣拉古法）。理解它们需要仔细的算术运算和公平感。
6. The D’Hondt (Jefferson) Method | 德洪特（杰斐逊）分配法
The D’Hondt method (known as Jefferson’s method in the US) is a highest‑average method that favours larger parties. For each party, the votes Vᵢ are divided successively by the divisor sequence 1, 2, 3, 4, …, producing a table of quotients. The S highest quotients across all parties are identified, and each quotient gives a seat to its party.
德洪特法（在美国称为杰斐逊法）是一种最有利于大党的最高平均数方法。对每个政党，将得票 Vᵢ 依次除以除数序列 1, 2, 3, 4, …，生成商数表。找出所有政党中最大的 S 个商数，每个商数为其政党赢取一席。
The method can be written mathematically as: allocate a seat to the party j for which Vⱼ/(sⱼ + 1) is maximal, where sⱼ is the number of seats already allocated to party j. Repeat S times. A worked example with 5 seats and votes A:5400, B:2400, C:1600, D:600 illustrates the quotient matrix.
该方法可数学表述为：将席位分配给使 Vⱼ/(sⱼ + 1) 最大的政党 j，其中 sⱼ 是政党 j 已获席位数。重复 S 次。以 5 席位、得票 A:5400、B:2400、C:1600、D:600 为例，可得商数矩阵。
Example Table – D’Hondt Quotients
| Party | Votes | ÷1 | ÷2 | ÷3 | ÷4 |
|---|---|---|---|---|---|
| A | 5400 | 5400 | 2700 | 1800 | 1350 |
| B | 2400 | 2400 | 1200 | 800 | 600 |
| C | 1600 | 1600 | 800 | 533.3 | 400 |
| D | 600 | 600 | 300 | 200 | 150 |
The five highest quotients (bold) secure seats: A gets 3 seats, B gets 1, C gets 1, and D gets 0. D’Hondt systematically advantages the party with the largest vote share, raising the effective threshold for smaller parties.
五个最高商数（粗体）决定席位归属：A 获 3 席，B 获 1 席，C 获 1 席，D 获 0 席。德洪特法系统性地偏袒得票最多的政党，提高了小党的实际门槛。
7. The Sainte‑Laguë (Webster) Method | 圣拉古（韦伯斯特）分配法
The Sainte‑Laguë method, also called Webster’s method, uses the odd divisor sequence 1, 3, 5, 7, … instead of 1, 2, 3, 4, … This makes it more favourable to smaller parties. The iterative rule is: allocate a seat to the party maximising Vⱼ/(2sⱼ + 1).
圣拉古法又称韦伯斯特法，使用奇除数序列 1, 3, 5, 7, …，而非 1, 2, 3, 4, …。这使得它对小党更有利。迭代规则为：将席位分配给使 Vⱼ/(2sⱼ + 1) 最大的政党。
A modified version (used in some Nordic countries) replaces the first divisor with 1.4 instead of 1, to reduce the advantage for very small parties. Mathematically, the choice of divisor sequence dramatically changes the proportionality profile.
某些北欧国家使用的修正版本将第一个除数由 1 替换为 1.4，以削弱极小型政党的优势。数学上，除数序列的选择极大地改变了比例性特征。
Applying pure Sainte‑Laguë to the same votes Vₐ = 5400, Vb = 2400, Vc = 1600, Vd top five are 5400 (A), 2400 (B), 1800 (A), 1600 (C), 1080 (A), so A gets 3, B 1, C 1, D 0 – identical to D’Hondt in this case, but with different vote distributions Sainte‑Laguë often awards an extra seat to a smaller party.
将纯圣拉古法应用于相同票数 Vₐ=5400、Vb=2400、Vc=1600、Vd=600 和 5 个席位，所得商数为：A/1=5400, A/3=1800, A/5=1080, B/1=2400, B/3=800, C/1=1600, C/3≈533.3, D/1=600。前五大商数为 5400 (A), 2400 (B), 1800 (A), 1600 (C), 1080 (A)，因此 A 获 3 席，B 1 席，C 1 席，D 0 席——在此例中与德洪特法结果相同，但在其他票数分布下，圣拉古法常会将一个额外席位授予小党。
8. Paradoxes and Fairness Criteria | 悖论与公平性准则
Apportionment mathematics is haunted by paradoxes. The Alabama paradox occurs when increasing the total number of seats causes a party to lose a seat. This famously happened to Alabama after the 1880 US census when using Hamilton’s method. The population paradox happens when a party with a faster‑growing population loses a seat to a party with a slower‑growing population.
席位分配的数学被各种悖论所萦绕。阿拉巴马悖论指增加总席位数反而导致某政党失去席位，这在 1880 年美国人口普查后使用哈密尔顿法时发生在阿拉巴马州。人口悖论则指人口增长较快的政党反而将席位输给增长较慢的政党。
Fairness criteria attempt to capture what we want from an electoral system. Quota rule: each party should receive either the floor or the ceiling of its exact quota. House monotonicity: increasing the total number of seats should not reduce any party’s allocation. Balinski and Young proved that no apportionment method can simultaneously satisfy quota and avoid all paradoxes.
公平性准则试图捕捉我们对选举制度的期望。定额规则：各政党获得的席位应是其准确配额的下取整或上取整。议院单调性：增加总席位数不应减少任何政党的分配。巴林斯基和杨证明了没有任何分配方法能同时满足定额并避免所有悖论。
9. Arrow’s Impossibility Theorem | 阿罗不可能定理
Kenneth Arrow’s celebrated theorem states that no ranking‑based voting system can convert individual preference orders into a collective order while simultaneously satisfying a small set of seemingly reasonable conditions: unrestricted domain, non‑dictatorship, Pareto efficiency (if every voter prefers X over Y, then society prefers X over Y), and independence of irrelevant alternatives (the relative ranking of X and Y depends only on individual preferences between X and Y).
肯尼思·阿罗的著名定理指出，没有一种基于排序的投票系统能将个体偏好排序转化为集体排序，同时满足一小套看似合理的条件：无限制域、非独裁、帕累托效率（若每位选民都偏好 X 甚于 Y，则社会偏好 X 甚于 Y）以及无关选项独立性（X 与 Y 的相对排序仅取决于个体对 X 与 Y 的偏好）。
The theorem is a profound mathematical result proved using concepts similar to those in Boolean algebra and set theory. It implies that every democratic voting rule involves some compromise or susceptibility to paradox. For A‑Level students, the logical structure of the proof – using decisive coalitions and contradiction – showcases how formal mathematics applies to social science.
该定理是一个深刻的数学结论，其证明运用了类似布尔代数和集合论的概念。它表明每种民主投票规则都涉及某种妥协或易受悖论影响。对 A‑Level 学生而言，证明的逻辑结构——运用决定性联盟和矛盾法——展示了形式数学如何应用于社会科学。
10. Gerrymandering and the Efficiency Gap | 杰利蝾螈与效率差距
Gerrymandering is the manipulation of electoral district boundaries to advantage one party. Mathematics provides tools to detect and measure it. A widely cited metric is the efficiency gap, EG, which counts ‘wasted’ votes. A wasted vote for a party is any vote cast for their candidate beyond the 50%+1 needed to win in a district, plus all votes cast for losing candidates.
杰利蝾螈是指操纵选区边界以偏袒某一政党。数学提供了检测和衡量它的工具。一个广泛引用的指标是效率差距 EG，它统计“浪费的”票数。某政党在选区中的废票是指超出获胜所需 50%+1 票数的任何投票，加上投给落选候选人的所有票。
For a two‑party election across n districts, let WA, WB be total wasted votes for parties A and B. The efficiency gap is EG = (WA − WB) / V, where V is the total votes cast. A magnitude |EG| > 0.07 (or 7%) is often considered indicative of potential gerrymandering. This is just the beginning; more sophisticated techniques use Markov chain Monte Carlo to compare a given plan with ensembles of neutral maps.
对于 n 个选区中的两党选举，设 WA、WB 分别为政党 A 和 B 的废票总数。效率差距为 EG = (WA − WB) / V，其中 V 为总投票数。当 |EG| > 0.07（即 7%）时，常常被视为可能存在杰利蝾螈的迹象。这还只是开始；更复杂的技术使用马尔可夫链蒙特卡洛方法，将特定的选区划分与大量中立地图的集合进行比较。
Courts increasingly rely on such mathematical measures, making geometry and statistics powerful allies in the quest for fair representation.
法院越来越依赖这类数学指标，使几何与统计成为追求公平代表制的强大盟友。
11. Referendums and Supermajorities | 全民公投与绝对多数门槛
Electoral reform often involves referendums, where a binary choice is offered. The mathematics of referendums includes setting thresholds for validity. For example, some countries require a participation quorum (e.g. at least 50% turnout) or an approval quorum (e.g. the proposal must be approved by at least 25% of registered voters). These thresholds interact in mathematically subtle ways.
选举改革往往涉及全民公投，提供二选一的选择。公投的数学包括设定有效门槛。例如，有些国家要求参与法定人数（如至少 50% 投票率）或批准法定人数（如提案须获得至少 25% 登记选民的支持）。这些门槛以数学上微妙的方式相互作用。
Consider a supermajority rule: instead of 50%+1, a 60% threshold for constitutional changes. This can be modelled as requiring the proportion of yes‑votes p > t, where t is the threshold. The mathematical effect is to give a minority a blocking power, which alters the expected outcome under various voter distributions. Game‑theoretic models predict strategic voting in such settings.
考虑绝对多数规则：宪法修正需 60% 而非 50%+1 的门槛。这可以建模为要求赞成票比例 p > t，其中 t 为门槛。其数学效果是赋予少数派阻挠权，从而改变在不同选民分布下的预期结果。博弈论模型预测在此类情境中的策略性投票。
12. Conclusion: The Role of Mathematics in Reform | 结论：数学在改革中的作用
Electoral reform is not just a political slogan; it is a deeply mathematical field. From divisor methods and pairwise comparison matrices to impossibility theorems and gerrymandering metrics, mathematics provides the language for evaluating trade‑offs between proportionality, stability, and simplicity. Every voting system is a mathematical model of collective decision‑making, and understanding its properties empowers citizens and reformers alike.
选举改革不只是一句政治口号，它是一个深具数学性的领域。从除数法和两两比较矩阵到不可能定理和杰利蝾螈指标，数学为评估比例性、稳定性与简单性之间的权衡提供了语言。每种投票制度都是集体决策的数学模型，理解其性质能赋予公民和改革者力量。
For A‑Level students, these interconnected topics illustrate how pure arithmetic, algorithms, and logic intersect with real‑world governance. Whether you are studying decision mathematics, statistics, or core pure concepts, electoral mathematics offers a window into the machinery of democracy.
对 A‑Level 学生而言，这些相互关联的话题展示了纯粹的算术、算法和逻辑如何与现实世界的治理交织在一起。无论你在学习决策数学、统计学还是核心纯数，选举数学都为你打开一扇了解民主机制的窗口。
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