📚 Electoral Systems: Mathematical Models and Algorithms | 选举制度：数学模型与算法
Electoral systems are often seen as political rules, but at their core they are mathematical procedures for converting votes into seats. Understanding the underlying arithmetic, algorithms, and fairness criteria is essential for any student applying mathematics to real-world decision making.
选举制度常被视为政治规则，但其核心是将选票转化为席位的数学程序。理解其背后的算术、算法和公平性准则，对于任何将数学应用于现实决策的学生来说都至关重要。
1. Why Mathematics Underpins Electoral Systems | 为什么数学是选举制度的基础
An electoral system takes a set of vote counts and produces a set of seat allocations. This is a deterministic mapping from non-negative integers to non-negative integers, constrained by the total number of seats.
选举制度接收一组选票数并产生一组席位分配。这是从非负整数到非负整数的确定性映射，受席位总数的约束。
The key mathematical challenge is proportionality: if a party wins 40% of the votes, it should ideally receive about 40% of the seats, but exact equality is rarely possible because seats are indivisible.
关键的数学挑战是比例性：如果一个政党赢得了40%的选票，理想情况下应获得约40%的席位，但由于席位不可分割，精确相等几乎不可能。
For Edexcel A-Level Maths students, these ideas connect to algorithms, rounding procedures, and discrete optimisation. You may encounter similar logic in Decision Mathematics, where systematic procedures are used to solve allocation and scheduling problems.
对于Edexcel A-Level数学学生而言，这些思想与算法、取整程序和离散优化相关。在决策数学中，你会遇到类似的逻辑，即使用系统化程序解决分配和调度问题。
2. Vote Counting and Proportionality | 计票与比例性
Let V be the total number of valid votes and S be the total number of seats to be filled. The simple proportion of seats for party i with votes vᵢ is given by:
设 V 为有效选票总数，S 为待分配席位总数。政党 i 获得票数 vᵢ 的简单席位比例为：
Seat share = (vᵢ / V) × S
This value is rarely an integer, so an allocation rule must round it up or down while preserving the total number of seats. The discrepancy between the exact proportional share and the integer seat allocation is called the apportionment error.
该值很少是整数，因此分配规则必须对其进行向上或向下取整，同时保持席位总数不变。精确比例份额与整数席位分配之间的差异称为分摊误差。
Two main families of methods exist: quota methods and divisor methods. Quota methods start from a fixed quota, while divisor methods divide vote totals by a sequence of divisors. Each family has distinct mathematical properties and potential paradoxes.
存在两大类方法：配额法和除数法。配额法从固定配额出发，而除数法将选票总数除以一系列除数。每个家族都有不同的数学性质和潜在的悖论。
3. Quota Formulas: Hare and Droop | 配额公式：黑尔配额与德鲁普配额
The Hare quota (also called the simple quota) is the total votes divided by the total seats:
黑尔配额（也称简单配额）是总票数除以总席位数：
Q_H = V / S
The Droop quota is used in single transferable vote systems and is slightly smaller:
德鲁普配额用于单一可转移投票制，其值略小：
Q_D = floor[ V / (S + 1) ] + 1
The floor function rounds down to the nearest integer. Droop’s formula ensures that no more candidates can reach the quota than there are seats. This is an elegant use of the integer-part operation to avoid over-allocation.
向下取整函数舍入到最近的整数。德鲁普公式确保达到配额的人数不会超过席位数。这是利用取整运算避免超额分配的优雅方法。
For example, if V = 1000 and S = 4, then Q_H = 250 and Q_D = floor(1000/5)+1 = 201. A candidate needs 201 votes under Droop, not 250, to be elected.
例如，若 V = 1000，S = 4，则 Q_H = 250，Q_D = floor(1000/5)+1 = 201。在德鲁普配额下，候选人需要201票即可当选，而非250票。
4. Divisor Methods: D’Hondt and Sainte-Laguë | 除数法：东特法与圣拉格法
Divisor methods allocate seats iteratively. In each round, the party with the largest quotient vᵢ / dᵢ receives the next seat, where dᵢ depends on the number of seats already won.
除数法迭代分配席位。在每一轮中，商 vᵢ / dᵢ 最大的政党获得下一个席位，其中 dᵢ 取决于该党已赢得的席位数。
The D’Hondt method uses divisors 1, 2, 3, 4, … for each party. The Sainte-Laguë method uses divisors 1, 3, 5, 7, … which gives smaller parties a better chance because the divisors grow more slowly.
东特法对各政党使用除数 1、2、3、4…… 圣拉格法使用除数 1、3、5、7…… 由于除数增长更慢，这给较小的政党提供了更好的机会。
Algorithmically, D’Hondt can be described as: initialise all seat counts to zero; repeat S times: select the party j that maximises vⱼ / (sⱼ + 1); increment sⱼ by one. This is a greedy algorithm.
从算法上讲，东特法可描述为：将所有政党的席位数初始化为零；重复 S 次：选择使 vⱼ / (sⱼ + 1) 最大化的政党 j；将 sⱼ 增加一。这是一个贪心算法。
These methods are widely used because they avoid the paradoxes that can occur with simple rounding of quotas, such as a party receiving fewer seats after the total number of seats increases.
这些方法被广泛使用，因为它们避免了简单取整配额时可能出现的悖论，例如在总席位数增加后某个政党反而获得更少席位。
5. Worked Example: Allocating 8 Seats | 示例：分配 8 个席位
Suppose three parties A, B, and C receive 10000, 8000, and 5000 votes respectively, and 8 seats must be allocated using the D’Hondt method.
假设三个政党 A、B、C 分别获得 10000、8000 和 5000 票，需要用东特法分配 8 个席位。
We compute quotients vᵢ / 1, vᵢ / 2, vᵢ / 3, … and select the eight largest values. The table below shows the first few quotients, with the selected ones labelled in order.
我们计算商 vᵢ / 1、vᵢ / 2、vᵢ / 3…… 并选择最大的八个值。下表显示了前几个商，被选中的值按顺序标注。
| Divisor | A (10000) | B (8000) | C (5000) |
|---|---|---|---|
| 1 | 10000 (1st) | 8000 (2nd) | 5000 (4th) |
| 2 | 5000 (3rd) | 4000 (5th) | 2500 (8th) |
| 3 | 3333 (6th) | 2667 (7th) | 1667 |
| 4 | 2500 | 2000 | 1250 |
Reading the eight highest quotients in order gives A four seats, B three seats, and C one seat. The exact proportional shares were 3.48, 2.78, and 1.74, so D’Hondt slightly favours the larger parties.
按顺序读取最大的八个商，得出 A 获得四席，B 获得三席，C 获得一席。精确比例份额分别为 3.48、2.78 和 1.74，因此东特法略微有利于较大的政党。
If Sainte-Laguë were used with divisors 1, 3, 5, the first few quotients would be different, and the allocation would shift slightly, often giving smaller parties an extra seat at the expense of the largest party.
如果使用圣拉格法，除数为 1、3、5，前几个商会有所不同，分配也会略有变化，通常会让较小的政党从最大政党手中多获得一个席位。
6. Paradoxes in Apportionment | 分摊中的悖论
The Alabama paradox occurs when increasing the total number of seats causes a party to lose a seat. This can happen with some quota methods, but divisor methods such as D’Hondt avoid it.
阿拉巴马悖论是指增加总席位数反而导致某个政党失去席位。某些配额法可能发生这种情况，但东特法等除数法则避免了这一悖论。
Another issue is the population paradox: a party can lose a seat to another party even though its vote total grows faster. These paradoxes show why a seemingly fair rounding rule can produce counter-intuitive results.
另一个问题是人口悖论：即使某个政党的选票增长更快，它也可能将席位输给另一个政党。这些悖论表明，看似公平的取整规则也可能产生违反直觉的结果。
Mathematicians use axiomatic approaches to classify apportionment methods. A method is called house-monotone if increasing S never reduces any party’s seat count. Divisor methods are house-monotone.
数学家使用公理化方法对分摊方法进行分类。如果增加 S 不会减少任何政党的席位数，则称该方法具有众议院单调性。除数法具有众议院单调性。
7. Fairness Criteria and Arrow’s Theorem | 公平准则与阿罗不可能定理
A fair electoral system should satisfy several mathematical properties: monotonicity (more votes never lead to fewer seats), quota satisfaction, and absence of the Alabama paradox.
公平的选举制度应满足若干数学性质：单调性（选票增加不会导致席位减少）、配额满足性，以及不存在阿拉巴马悖论。
Arrow’s impossibility theorem states that no ranked voting system can simultaneously satisfy all reasonable fairness criteria when there are at least three candidates. The criteria include unrestricted domain, non-dictatorship, Pareto efficiency, and independence of irrelevant alternatives.
阿罗不可能定理指出，当至少有三名候选人时，没有任何排序投票制能同时满足所有合理的公平准则。这些准则包括无限制域、非独裁、帕累托效率和无关备选方案独立性。
This deep result shows that all electoral systems involve trade-offs; mathematics helps quantify and compare these trade-offs rather than eliminate them entirely.
这一深刻结果表明，所有选举制度都涉及权衡；数学帮助量化和比较这些权衡，而不是完全消除它们。
8. Voting Power Indices | 投票权力指数
In committee voting or coalitions, the number of votes a member has is not the same as their actual
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