Different Electoral Systems | 不同的选举制度

📚 Different Electoral Systems | 不同的选举制度

Electoral systems are not only the machinery of democracy but also a fascinating application of mathematical reasoning. From simple majority counts to complex preference‑based methods, each system can be analysed using axioms of fairness, preference aggregation, and algorithm design. In the Decision Mathematics component of A‑Level, we explore how mathematical models expose the strengths and hidden flaws of different voting rules — preparing us to think critically about the algorithms that shape political outcomes.

选举制度不仅是民主政治的运行机制，也是数学推理的一个迷人应用。从简单的多数计数到复杂的基于偏好的方法，每种制度都可以用公平性公理、偏好汇聚和算法设计来进行分析。在 A‑Level 决策数学（Decision Mathematics）中，我们探索数学模型如何揭示不同投票规则的优点与隐藏缺陷，让我们能够批判性地思考那些塑造政治结果的算法。

1. The Plurality Voting System | 简单多数制

Under plurality voting (also known as first‑past‑the‑post), each voter casts a single vote for their favourite candidate, and the candidate with the highest number of votes wins — no absolute majority is required. The algorithm is trivial: tally first preferences and output the candidate with max count.

在简单多数制（亦称“领先者当选”）下，每位选民只投一票给自己最喜欢的候选人，得票最高的候选人获胜——不需要绝对多数。算法极其简单：统计第一偏好，输出票数最高的候选人。

While computationally efficient, plurality voting often violates the Condorcet criterion (a candidate who would beat every other in head‑to‑head contests may still lose) and is highly susceptible to vote splitting among similar candidates.

尽管计算高效，简单多数制常常违反孔多塞准则（即在两两对决中能击败所有其他候选人的候选人可能仍然落选），并且极易因相似候选人之间的选票分散而导致不公平结果。

Example: In an election with three candidates A, B, C, and preference profile 40% A ≻ B ≻ C, 35% C ≻ B ≻ A, 25% B ≻ A ≻ C, A wins with 40% despite B being preferred by a majority when compared to A alone.

示例：在一次有 A、B、C 三位候选人的选举中，偏好分布为 40% A ≻ B ≻ C, 35% C ≻ B ≻ A, 25% B ≻ A ≻ C，A 以 40% 的得票率获胜，但如果单独比较 A 与 B，多数人更偏好 B。

2. Majority Voting with Runoff | 两轮多数制

When no candidate secures an overall majority, a second round is held between the top two candidates. This runoff method ensures the eventual winner has majority support among those who vote in the second round, but it still discards lower‑ranked preferences from the first round.

当没有候选人获得半数以上选票时，在得票最高的两人之间举行第二轮投票。这种决选制确保最终的获胜者在第二轮投票中获得多数支持，但仍然丢弃了第一轮中排名偏好的信息。

From a mathematical viewpoint, the two‑round system can be modelled as a threshold algorithm: if max(votes) > 0.5 × total, then winner = argmax; else restrict to top two and re‑run the preference elicitation.

从数学角度看，两轮制可以建模为一种阈值算法：如果最高票数 > 总票数一半，则赢家为得票最高者；否则限制在得票前两名，并重新采集偏好。

This system still fails the Condorcet criterion and can incentivise tactical voting in the first round to influence which two candidates proceed.

该制度仍然不能满足孔多塞准则，而且可能激励选民在第一轮进行策略性投票，以影响进入决选的两位候选人。

3. Alternative Vote (Instant Runoff) | 替代投票制（即时决选）

Alternative Vote (AV) simulates a series of runoffs using a single ranked ballot. Voters rank candidates in order of preference. If no candidate has a majority of first preferences, the candidate with the fewest votes is eliminated, and their votes are redistributed to the next available preference on each ballot. This process repeats until one candidate achieves a majority.

替代投票制（AV）通过一次排序投票模拟多轮淘汰。选民按偏好顺序排列候选人。如果没有候选人在第一偏好中获得多数票，则票数最少的候选人被淘汰，其选票按每张选票上的下一顺位重新分配，直到有候选人获得多数票为止。

AV can be described as a recursive elimination algorithm: while no candidate has > 50% of active votes, eliminate min‑vote candidate and transfer ballots. Its mathematical elegance lies in gradually narrowing the field while respecting individual preference orderings.

AV 可描述为一种递归淘汰算法：当没有候选人获得超过 50% 的活跃票数时，淘汰得票最少的候选人并转移选票。其数学上的优雅之处在于，在逐步缩小候选人范围的同时尊重了个体的偏好顺序。

AV satisfies the later‑no‑harm criterion but can still fail monotonicity: gaining more support can sometimes cause a winning candidate to lose under certain preference distributions.

AV 满足“后来无伤害”准则，但仍可能违反单调性：在某些偏好分布下，获得更多支持有时反而会使获胜的候选人落选。

4. Borda Count | 博尔达计数

The Borda count is a position‑based method: in an election with n candidates, each ballot awards n−1 points to the top preference, n−2 points to the second, down to 0 points for the last. The candidate with the highest total points wins.

博尔达计数是一种基于排名位置的方法：在有 n 名候选人的选举中，每张选票给第一偏好 n−1 分，第二偏好 n−2 分，依次递减，最后一名得 0 分。总分最高的候选人获胜。

This system can be expressed as a weighted sum over all ballots: Borda score of candidate X = Σ (n − rank_X). It uses more preference information than plurality or runoff methods and often selects a broadly acceptable winner, but it is vulnerable to strategic nomination (cloning) and can violate the majority criterion.

该系统可表示为所有选票的加权和：候选人 X 的博尔达得分 = Σ (n − 排名_X)。它比简单多数制或决选制利用了更多的偏好信息，通常会选出广泛可接受的人选，但容易受到策略性提名（克隆人）的影响，并可能违反多数准则。

Example: With three candidates A, B, C and the same 40‑35‑25 profile as before, Borda scores are: A = 40×2 + 35×0 + 25×1 = 105, B = 40×1 + 35×1 + 25×2 = 125, C = 40×0 + 35×2 + 25×0 = 70; hence B wins, illustrating how the Condorcet‑preferred candidate can triumph.

示例：用之前的 40‑35‑25 偏好分布，三位候选人 A、B、C 的博尔达得分分别为：A = 40×2 + 35×0 + 25×1 = 105，B = 40×1 + 35×1 + 25×2 = 125，C = 40×0 + 35×2 + 25×0 = 70；因此 B 获胜，这体现了孔多塞偏好的候选人如何胜出。

5. Condorcet Methods | 孔多塞方法

A candidate is the Condorcet winner if they defeat every other candidate in pairwise comparisons. Condorcet methods aim to elect such a candidate when one exists. The Condorcet criterion is a desirable property that plurality, runoff, and AV all fail.

如果一位候选人在两两比较中击败了所有其他候选人，则该候选人为孔多塞胜者。孔多塞方法的目标是当选存在这种候选人时就选出他/她。孔多塞准则是简单多数制、决选制和 AV 都未能满足的一项优良属性。

Methods such as the Copeland rule or the Schulze method construct a pairwise preference matrix M where M_ij = number of voters preferring i to j. The Condorcet winner is the row with all entries > total voters/2. In absence of a Condorcet winner, different Condorcet‑compatible rules use different tie‑breaking algorithms based on the magnitudes of victories.

诸如科普兰法（Copeland rule）或舒尔茨法（Schulze method）等方法构建两两比较矩阵 M，其中 M_ij = 偏好 i 胜于 j 的选民数。孔多塞胜者所在的行所有元素都大于总选民数的一半。在没有孔多塞胜者的情况下，不同的孔多塞兼容规则会根据胜利幅度采用不同的平局处理算法。

Condorcet methods highlight the mathematical depth of electoral design: they aim to find the “closest” median or strongest path in a directed preference graph, often requiring graph‑theoretic algorithms.

孔多塞方法突显了选举设计的数学深度：它们旨在有向偏好图中找到“最接近”的中位数或最强路径，通常需要图论算法。

6. Arrow’s Impossibility Theorem | 阿罗不可能定理

Kenneth Arrow proved that no ranked‑voting electoral system can simultaneously satisfy a set of seemingly reasonable axioms: unrestricted domain, non‑imposition, non‑dictatorship, independence of irrelevant alternatives (IIA), and the Pareto condition. This theorem is a cornerstone of social choice theory and shows that every voting system involves trade‑offs.

肯尼斯·阿罗证明，没有任何排序投票选举制度能同时满足一组看似合理的公理：无限制定义域、非强加性、非独裁性、无关备选方案独立性（IIA）和帕累托条件。该定理是社会选择理论的核心，并表明每种投票制度都包含权衡。

From a decision‑mathematics perspective, Arrow’s theorem is a proof that a perfect algorithm for aggregating preferences does not exist; every system must sacrifice at least one desired property. This encourages quantitative analysis of which axioms to relax for a given context.

从决策数学的角度看，阿罗定理证明了完美汇聚偏好的算法是不存在的；每种制度都必须牺牲至少一项理想性质。这鼓励我们对在给定情境下放松哪些公理进行定量分析。

7. Proportional Representation Systems | 比例代表制

Proportional representation (PR) systems allocate seats to parties in proportion to votes received. Unlike winner‑takes‑all methods, PR uses multi‑member constituencies and algorithms like the D’Hondt method or Sainte‑Laguë method to convert vote shares into seat numbers.

比例代表制（PR）根据各党得票比例分配议席。与“赢者通吃”的方法不同，PR 采用多席选区，并使用如顿特法（D’Hondt）或圣拉格法（Sainte‑Laguë）等算法将票数比例转化为议席数。

For example, the D’Hondt method sequentially allocates seats by dividing each party’s vote total by 1, 2, 3, … after each seat award and assigning the next seat to the party with the highest resulting quotient. This can be seen as a greedy algorithm that favours larger parties slightly.

例如，顿特法按顺序分配议席：每次分配一个议席后，将各党票数除以 1、2、3……，然后将下一个议席分配给商最大的政党。这可以看作一个稍微偏向较大政党的贪心算法。

Mathematically, PR systems raise questions about the measure of disproportionality and how thresholds (e.g., a 5% hurdle) alter proportionality, encouraging simulation and modelling to compare electoral outcomes under different systems.

从数学上讲，比例代表制引发了关于非比例性的度量以及门槛（例如 5% 门槛）如何改变比例性的问题，鼓励通过模拟和建模来比较不同制度下的选举结果。

8. Fairness Criteria and System Evaluation | 公平性准则与制度评估

To compare electoral systems mathematically, we define fairness criteria such as the majority criterion, Condorcet criterion, monotonicity, participation criterion, and independence of clones. Each system can be checked against these criteria using counterexamples or logical proofs.

为了在数学上比较选举制度，我们定义了诸如多数准则、孔多塞准则、单调性、参与准则和克隆独立性等公平性准则。可以利用反例或逻辑证明检验每种制度是否符合这些准则。

Criterion / 准则	Plurality / 简单多数	AV / 替代投票	Borda / 博尔达	Condorcet / 孔多塞方法
Majority / 多数	Yes	Yes	No	Yes
Condorcet / 孔多塞	No	No	No	Yes
Monotonicity / 单调性	Yes	No	Yes	Depends on variant
Participation / 参与	Yes	No	No	No

Such a comparison table is a common tool in decision mathematics to visualise trade‑offs and understand why no system dominates all others.

这种比较表格是决策数学中常用的工具，用来直观地展示权衡关系，并理解为什么没有一种制度能全面优于其他制度。

9. Strategic Voting and Manipulation | 策略投票与操纵

When voters misrepresent their true preferences to achieve a more favourable outcome, it is called strategic or tactical voting. The Gibbard–Satterthwaite theorem states that any non‑dictatorial voting system with at least three possible outcomes is susceptible to manipulation by some voter.

当选民为了得到更有利的结果而歪曲自己的真实偏好，这就是策略投票或战术投票。吉巴德–萨特斯韦特定理指出，任何具有至少三种可能结果且非独裁的投票制度，都容易被某些选民操纵。

Mathematical models of rational voting treat an election as a game and seek Nash equilibria. For example, in plurality voting, supporters of a minor candidate may vote for a major candidate they dislike less to prevent their least‑preferred choice from winning.

理性投票的数学模型将选举视为一场博弈，并寻求纳什均衡。例如，在简单多数制下，小党候选人的支持者可能会投票给一个不那么讨厌的主要政党候选人，以防止他们最不喜欢的候选人获胜。

Decision mathematics helps quantify the extent of distortion introduced by tactical behaviour and can simulate how different systems either encourage or suppress such manipulation.

决策数学有助于量化战术行为引入的扭曲程度，并可以模拟不同制度是如何鼓励或抑制这种操纵的。

10. Algorithms for Determining Winners | 确定获胜者的算法

Each voting system is an algorithm, and implementing them correctly is a core part of decision mathematics. For instance, an AV count requires iterative elimination and transfer, while a Condorcet method involves computing a matrix and testing for cycles in preferences.

每种投票制度都是一个算法，正确实现它们是决策数学的核心部分。例如，AV 计票需要进行迭代淘汰和转移选票，而孔多塞方法则需要计算矩阵并检测偏好循环。

The computational complexity differs: plurality is O(n); Borda is O(n); AV may require multiple rounds but is still polynomial; finding a Schulze winner requires computing the strongest path in a directed graph, which can be done by using a variant of the Floyd–Warshall algorithm.

计算复杂度各有不同：简单多数制为 O(n)；博尔达计数为 O(n)；AV 可能需要多轮但仍是多项式的；寻找舒尔茨获胜者则需计算有向图中的最强路径，这可以通过弗洛伊德–沃肖尔算法的一个变种来完成。

Understanding these algorithmic aspects ensures that electronic voting systems and simulations are built on rigorous, verifiable mathematical foundations.

理解这些算法方面的知识，能确保电子投票系统和模拟建立在严格、可验证的数学基础之上。

11. Extended Worked Example | 扩展示例解析

Consider an election with 100 voters and four candidates P, Q, R, S. The preference schedule is:
30 voters: P ≻ R ≻ S ≻ Q
25 voters: Q ≻ R ≻ S ≻ P
20 voters: R ≻ S ≻ P ≻ Q
15 voters: S ≻ R ≻ P ≻ Q
10 voters: P ≻ Q ≻ R ≻ S

考虑一个有 100 名选民和 P、Q、R、S 四位候选人的选举。偏好排程如下：
30 名选民：P ≻ R ≻ S ≻ Q
25 名选民：Q ≻ R ≻ S ≻ P
20 名选民：R ≻ S ≻ P ≻ Q
15 名选民：S ≻ R ≻ P ≻ Q
10 名选民：P ≻ Q ≻ R ≻ S

Under plurality: P gets 30+10=40, Q gets 25, R gets 20, S gets 15. P wins with 40%. But pairwise comparisons show R beats P 60‑40, beats Q 65‑35, beats S 65‑35 — so R is the Condorcet winner, highlighting a failure of plurality.

简单多数制下：P 得 30+10=40，Q 得 25，R 得 20，S 得 15。P 以 40% 胜出。但两两比较显示 R 以 60‑40 击败 P，以 65‑35 击败 Q，以 65‑35 击败 S——因此 R 是孔多塞胜者，这突显了简单多数制的缺陷。

Using AV: first round S is eliminated (15 votes) → transferred to R (next preference on S ballots is R, so R gets 20+15=35). Now tallies: P=40, R=35, Q=25. Q is eliminated (25 votes) → 25 Q votes have 2nd preference R, so R becomes 35+25=60, beating P 60‑40. Thus AV elects the Condorcet winner here.

采用 AV：首轮 S 被淘汰（15 票）→ 转移给 R （S 的选票上下一偏好是 R，因此 R 获 20+15=35）。此时票数为：P=40，R=35，Q=25。Q 被淘汰（25 票）→ 25 张 Q 选票的第二偏好是 R，于是 R 变为 35+25=60，以 60‑40 击败 P。因此 AV 在此例中选出了孔多塞胜者。

The Borda count with 4 candidates gives 3‑2‑1‑0 points. Compute scores: P: 40×3 + 0 + 20×1 + 15×1 + 10×3 = 120+20+15+30=185; Q: 0 + 25×3 + 0 + 0 + 10×2 = 75+20=95; R: 30×2 + 25×2 + 20×3 + 15×2 + 10×1 = 60+50+60+30+10=210; S: 30×1 + 25×1 + 20×2 + 15×3 + 0 = 30+25+40+45=140. R wins with 210, again selecting the Condorcet winner. Such worked examples build intuition about how preference aggregation behaves.

博尔达计数，4 位候选人，分值为 3‑2‑1‑0。计算得分：P: 40×3 + 0 + 20×1 + 15×1 + 10×3 = 120+20+15+30=185；Q: 0 + 25×3 + 0 + 0 + 10×2 = 75+20=95；R: 30×2 + 25×2 + 20×3 + 15×2 + 10×1 = 60+50+60+30+10=210；S: 30×1 + 25×1 + 20×2 + 15×3 + 0 = 30+25+40+45=140。R 以 210 分获胜，再次选出孔多塞胜者。这样的算例能直观培养对偏好汇聚机制的直觉。

12. Real‑world Relevance and Revision Tips | 现实意义与复习提示

Understanding electoral systems through decision mathematics not only prepares you for exam questions on algorithms and criteria but also equips you to evaluate real‑world democratic processes — from parliamentary elections to award voting. Always practise writing clear algorithms and testing systems with small preference profiles to spot failures of criteria.

通过决策数学理解选举制度，不仅能为考试中的算法和准则题型做好准备，也能让你评估现实世界的民主程序——从议会选举到奖项投票。务必练习写出清晰的算法，并用小规模偏好分布测试各制度，以发现违反准则的情况。

When revising, memorise the main criteria and their violations for each system. Be ready to carry out AV, Borda, and Condorcet calculations step‑by‑step, and to interpret output in the context of social choice theory.

复习时，记住主要准则以及每种制度违反哪些准则。准备好逐步进行 AV、博尔达和孔多塞方法的手动计算，并能在社会选择理论的背景下解释结果。

Focus on the link between algorithms and fairness — many exam marks come from explaining why a particular result is unfair according to a given criterion, rather than just computing a winner.

重点学习算法与公平性之间的联系——许多考试分数来自于解释为什么根据某一准则某个结果是不公平的，而不仅仅是算出一个胜者。

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Different Electoral Systems | 不同的选举制度