📚 Party Systems | 政党系统：选举投票的数学分析
In decision mathematics, the study of party systems and voting methods provides a powerful lens through which to analyse how collective choices are made. By treating votes as data inputs and electoral rules as algorithms, we can explore the mathematical properties that determine which party or candidate emerges victorious. This article examines the most common voting systems used around the world, from simple plurality to sophisticated proportional methods, and reveals their underlying mathematical logic, paradoxes, and implications for party representation.
在决策数学中，政党系统和投票方法的研究为分析集体选择提供了一个有力视角。将选票视为数据输入，将选举规则视为算法，我们可以探索决定哪一个政党或候选人胜出的数学性质。本文考察了世界各地最常用的投票系统，从简单的相对多数制到复杂的比例代表制，揭示了它们背后的数学逻辑、悖论及其对政党代表性的影响。
1. Introduction to Voting and Party Systems | 投票与政党系统简介
A voting system is a mathematical rule that translates individual preferences into a collective outcome. In the context of party systems, elections determine the composition of a legislative body, influencing which party or coalition forms a government. The system chosen has profound consequences on the number of viable political parties and the fairness of representation.
投票系统是将个体偏好转化为集体结果的数学规则。在政党体系中，选举决定了立法机构的组成，影响哪一个政党或联盟组建政府。所选择的系统对可行的政党数量和代表的公平性有着深远的影响。
2. The Plurality System (First-Past-The-Post) | 相对多数制（简单多数制）
In a plurality election, each voter casts a single vote for one candidate, and the candidate with the most votes wins – even without an absolute majority. This system is used in the UK House of Commons and the US Congress. Mathematically, if there are n candidates and Vᵢ is the number of votes for candidate i, the winner is the candidate for whom Vᵢ > Vⱼ for all j ≠ i. No transfer of preferences occurs, meaning a candidate can win with far less than 50% of the vote when the opposition is fragmented.
在相对多数制选举中，每位选民投一张票给一位候选人，得票最多的候选人获胜——即使没有获得绝对多数。英国下议院和美国国会都使用这一制度。从数学上看，如果有 n 位候选人，第 i 位候选人获得 Vᵢ 张票，则获胜者是满足对所有 j ≠ i 有 Vᵢ > Vⱼ 的候选人。不会发生选票偏好的转移，这意味着当反对党选票分散时，候选人可能以远低于 50% 的得票率获胜。
This system tends to produce a two-party system due to Duverger's law: the wasted-vote effect discourages votes for smaller parties. From a decision mathematics perspective, it is a simple maximisation problem but often fails to capture the true collective preference.
由于杜瓦杰法则，这一制度倾向于产生两党制：废票效应使得选民不愿投票给小党。从决策数学的角度看，这是一个简单的最大化问题，但往往无法捕捉真实的集体偏好。
3. Majority and Runoff Systems | 绝对多数与两轮决选制
re a winner has broad support, majority systems require a candidate to secure more than 50% he top two candidates proceed to a second round. Mathematically, let V₁ be the total first‑ 0.5 × V₁, a second ballot is held with the set {i, j} that corresponds to the two highest‑placed candidates.
为确保获胜者得到广泛支持，绝对多数制要求候选人获得超过 50% 的选票。常见方式是两轮决选制：如果第一轮没有候选人获得绝对多数票，得票前两位的候选人进入第二轮。从数学上看，设 V₁ 为第一轮总票数；若 max(Vᵢ) ≤ 0.5 × V₁，则对得票最高的两位候选人进行第二轮投票。
west votes is eliminated and their votes are redistributed to the next preference. This iterative allocation continues until one candidate passes the 50% tion f(P, R) → W, where P is a profile of preference orders and R is the elimination rule.
一种变体是排序复选制（AV），选民按偏好给候选人排序。如果没有候选人在第一选择中获得绝对多数票，得票最少的候选人被淘汰，其选票被重新分配给下一偏好。这一迭代分配过程一直持续到一位候选人跨越 50% 的门槛。排序复选制可以建模为一个阶梯函数 f(P, R) → W，其中 P 是偏好顺序组合，R 是淘汰规则。
4. Proportional Representation and Party Lists | 比例代表制与政党名单
r example, the Hare quota is Q = V / S, where V is the total valid votes and S is the number of seats. Each party receives ⌊Vₚ / Q⌋ seats initially, with remaining seats allocated via largest remainders.
比例代表制（PR）的目标是按照各政党获得的总票数比例分配议席。在封闭名单比例代表制中，每个政党提交一份候选人排名清单，选民投票给政党。接着使用当选基数或除数法计算席位分配。例如，黑尔基数定义为 Q = V / S，其中 V 为有效票总数，S 为议席数。每个政党首先获得 ⌊Vₚ / Q⌋ 个席位，剩余席位按最大余数分配。
Alternatively, divisor methods such as the D'Hondt (Jefferson) method use a sequence of divisors d₁, d₂, d₃, … to allocate seats. For each party, the votes‑per‑seat ratio Vₚ/k is calculated, and seats are awarded sequentially to the highest such quotients. Mathematically, this minimises disproportionality according to a specific criterion. PR systems tend to encourage multi‑party systems and coalition governments.
此外，除数法（如东特法/杰斐逊法）使用一系列除数 d₁, d₂, d₃ … 来分配席位。对每个政党，计算每席平均得票数 Vₚ/k，座位依次分配给具有最高比值的政党。从数学上讲，这根据特定准则最小化了比例偏差。比例代表制往往鼓励多党制和联合政府。
5. Single Transferable Vote (STV) and Preferential Voting | 单记移让投票制与偏好投票
Single Transferable Vote (STV) is a proportional system used in multi‑member constituencies. Voters rank candidates in order of preference. A candidate reaching the quota (usually the Droop quota: ⌊V / (S+1)⌋ + 1) is elected immediately. Surplus votes above the quota are transferred to the next available preference at a fractional value, preserving the proportional will of the electorate. Eliminations of low‑polling candidates proceed similarly to AV.
单记移让投票制（STV）是多议席选区中使用的一种比例代表制。选民按偏好顺序给候选人排名。达到当选基数（通常为德罗普基数：⌊V / (S+1)⌋ + 1）的候选人立即当选。超过基数的剩余选票以分数值转移到下一个有效偏好，以保持选民的等比例意愿。低票淘汰过程则与排序复选制类似。
s / total received). These fractional votes are then redistributed to remaining candidates according to the next preference. This complex algorithm ensures both local representation and overall proportionality.
STV 的数学涉及迭代剩余选票转移和数值缩放。设权值 w 表示选票的转移值；初始 w = 1。如果某候选人获得超出基数的选票，转移值变为 w × (剩余票数 / 总获得票数)。这些分数选票随后按下一偏好重新分配给其余候选人。这一复杂算法保障了地方代表性和整体比例性。
6. Condorcet Criteria and Pairwise Comparisons | 孔多塞准则与两两比较
pairwise comparisons. To find such a winner, we construct an n × n matrix M where entry Mᵢⱼ er of voters who prefer candidate i over candidate j. If there exists a row i such that Mᵢⱼ > Mⱼ cept that a truly representative candidate should be preferred by a majority in every head‑to‑head match.
孔多塞赢家是指在所有两两比较中都能击败其他每一位候选人的候选人。要找到这样的赢家，我们构建一个 n × n 矩阵 M，其中 Mᵢⱼ 表示偏好候选人 i 胜于候选人 j 的选民数量。如果存在某一行 i 使得对所有 j ≠ i 有 Mᵢⱼ > Mⱼᵢ，那么候选人 i 就是孔多塞赢家。这种方法聚焦于一个理念：一个真正具有代表性的候选人应当在每一场一对一较量中都获得多数选民的偏好。
However, Condorcet cycles can occur: A beats B, B beats C, but C beats A. This reveals a cyclic majority preference, which is a profound impossibility in collective decision‑making. The existence of such cycles shows that aggregated preferences may not be transitive, challenging the notion of a ‘best’ party system.
然而，可能出现孔多塞循环：A 胜 B，B 胜 C，而 C 胜 A。这就揭示了循环多数偏好，是集体决策中的一个深刻的不可能性。这种循环的存在表明，聚合偏好可能不具有传递性，对“最佳”政党系统的概念提出了挑战。
7. Arrow's Impossibility Theorem | 阿罗不可能定理
Kenneth Arrow's theorem states that no ranked voting system can simultaneously satisfy a small set of seemingly reasonable conditions: unrestricted domain (all possible preference orders are allowed), non‑dictatorship, Pareto efficiency, and independence of irrelevant alternatives (IIA). In other words, there is no perfect voting system that translates individual rankings into a single collective ranking without encountering a paradox or violating one of these axioms.
肯尼斯·阿罗的定理指出，没有任何排序投票制能同时满足一组看似合理的条件：无限制域（允许所有可能的偏好顺序）、非独裁性、帕累托效率以及无关选项独立性（IIA）。换句话说，不存在这样一种完美的投票系统，能够将个人排序转化为单一的集体排序而不会遇到矛盾或违反其中一条公理。
This result has profound implications for party systems: every electoral rule is a compromise, and the choice of system inherently shapes political outcomes. In decision mathematics, Arrow's theorem is often proved by constructing a decisive voter and showing that they become an effective dictator under IIA; the proof uses combinatorics and binary relations.
这一结论对政党系统具有深远影响：每一种选举规则都是一种妥协，而制度的选择天然地塑造了政治结果。在决策数学中，阿罗定理的证明通常通过构造一位决定性选民，并表明在 IIA 条件下该选民实质上成为了独裁者；其证明过程运用了组合数学和二元关系。
8. Apportionment Methods: Hamilton, Jefferson, and Webster | 议席分配方法：汉密尔顿法、杰斐逊法与韦伯斯特法
es)/(total seats) and grants each party the integer part of its quotient. Leftover seats go to the parties with the largest fractional remainders. This method, however, is susceptible to the Alabama paradox: an increase in total seats can cause a party to lose a seat.
议席分配是指将固定数量的议席按照政党或州的人口比例进行分配的数学问题。汉密尔顿法使用基数 Q = (总票数)/(总席位数)，给予每个政党其商数的整数部分，剩余席位分配给余数最大的政党。然而，这一方法容易受到阿拉巴马悖论的影响：总席位数增加反而可能导致某个政党失去一个席位。
e divisor d until the rounded numbers of seats sum to the required total. The Jefferson (D'Hondt) method rounds down each quotient Vₚ / d, while the Webster (Sainte‑Laguë unding rule has genuine political impact, favouring larger parties (Jefferson) or smaller parties (Webster).
除数法则通过调整除数 d 直到四舍五入后的席位总数等于所需总数来避免这一悖论。杰斐逊法（东特法）将每个商数 Vₚ / d 向下取整，而韦伯斯特法（圣拉盖法）使用标准四舍五入。这些方法可以表示为：选择 d 使得 Σ round(Vₚ / d) = S。舍入规则的选择具有真实的政治影响，杰斐逊法倾向于大党，韦伯斯特法则对小党更有利。
9. Power Indices in Coalitions | 联盟中的权力指数
In a party system where no single party commands a majority, coalitions form. Mathematical power indices such as Shapley‑Shubik and Banzhaf measure the actual influence of a party within a voting bloc. The Banzhaf index counts how many coalitions become losing if a given party withdraws, weighted by the number of winning coalitions. The Shapley‑Shubik index considers all possible orders of joining a coalition and assigns power based on the proportion of orderings in which the party is pivotal.
在没有任何单一政党占据多数的政党体系中，联盟就会形成。数学权力指数，如夏普利-舒比克指数和班扎夫指数，用于衡量一个政党在投票集团中的实际影响力。班扎夫指数计算的是如果某一政党退出，有多少个联盟会变成失败联盟，并根据获胜联盟的数量进行加权。夏普利-舒比克指数则考虑所有可能的加入联盟的顺序，并根据该政党作为关键加入者的顺序比例来分配权力。
For a weighted voting game {q; w₁, w₂, …, wₙ}, where q is the quota and wᵢ are the seat weights, a coalition is winning if the sum of its weights meets or exceeds q. The computation of power indices helps explain why small parties in a close parliament can hold disproportionate influence. Such analysis is rooted in cooperative game theory, a standard topic in decision mathematics.
对于一个加权投票博弈 {q; w₁, w₂, …, wₙ}，其中 q 是法定门槛，wᵢ 是席位数，当一个联盟的席位总数达到或超过 q 时即为获胜联盟。权力指数的计算有助于解释为什么势均力敌的议会中小党能拥有不成比例的影响力。这类分析植根于合作博弈论，是决策数学中的标准课题。
10. Conclusion: The Mathematics of Party Systems | 结论：政党系统的数学
The study of party systems through the lens of decision mathematics reveals that every voting method encodes a particular definition of fairness. Whether using simple plurality, proportional allocation, or sophisticated preference transfers, each system has its own mathematical strengths and weaknesses. Paradoxes such as Condorcet cycles, the Alabama paradox, and Arrow's impossibility remind us that no system is perfect. As students of mathematics, understanding these algorithms deepens our appreciation of why democracies choose the electoral rules they do – and how small changes in the formula can reshape an entire political landscape.
通过决策数学的镜头审视政党系统，我们发现每一种投票方法都编码了一种特定的公平定义。无论是采用简单多数制、比例分配还是复杂的偏好转移，每种制度都有其独特的数学优势和弱点。孔多塞循环、阿拉巴马悖论和阿罗不可能定理等悖论提醒我们，没有任何一个系统是完美的。作为数学学习者，理解这些算法将加深我们对民主社会为何选择如此选举规则的认识——以及公式的微小变动如何重塑整个政治版图。
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