📚 Mathematical Analysis of Electoral Systems in the USA | 美国选举制度的数学分析
Electoral systems in the USA are often studied in politics, but underneath the rules are mathematical ideas: weighted averages, proportional allocation, sampling error, and probability. For Edexcel A-Level Mathematics students, these examples connect statistical and algebraic methods to real-world decision making.
美国的选举制度通常在政治课上学习，但这些规则背后是数学思想：加权平均、比例分配、抽样误差和概率。对 Edexcel A-Level 数学学生而言，这些例子将统计与代数方法同现实决策联系起来。
1. Why Electoral Systems Are a Mathematical Problem | 为什么选举制度是一个数学问题
An electoral system converts individual votes into seats or electoral votes. This conversion is a function: the input is a set of vote counts, and the output is a distribution of power. Different rules can produce different outputs from the same inputs, so we can compare systems quantitatively using ratios, shares, and efficiency measures.
选举制度将个人选票转换为席位或选举人票。这种转换是一个函数：输入是一组得票数，输出是权力分配。不同规则会从相同输入产生不同输出，因此我们可以用比率、份额和效率指标来定量比较不同制度。
For example, a candidate may win the presidency with fewer total votes than an opponent because the Electoral College uses a weighted sum rather than a simple national total. A-Level skills such as weighted averages and proportional reasoning help explain why this happens.
例如，一位候选人可能以较少的总票数赢得总统职位，因为选举人团使用的是加权和，而不是简单的全国总票数。加权平均和比例推理等 A-Level 技能有助于解释为什么会发生这种情况。
2. Types of Voting Rules Used in the USA | 美国使用的投票规则类型
The main systems in the USA include plurality voting, majority run-off, and the Electoral College. Plurality, also called first-past-the-post, is used in most congressional elections: the candidate with the largest number of votes wins, even if that number is below 50%.
美国使用的主要制度包括简单多数制、多数决选制和选举人团。简单多数制也称领先者当选，用于大多数国会选举：得票最多的候选人获胜，即使该票数低于 50%。
Majority run-off requires the winner to obtain more than 50% of the valid votes. If no candidate reaches this threshold in the first round, a second round is held between the top two candidates. The Electoral College is different: voters choose electors in each state, and those electors then select the president.
多数决选制要求获胜者获得超过 50% 的有效选票。如果第一轮没有候选人达到这一门槛，则在得票前两名之间进行第二轮投票。选举人团则不同：选民在每个州选择选举人，再由这些选举人选出总统。
Plurality winner: candidate with the highest vote count
Majority requirement: votes > 0.5 × total valid votes
3. The Electoral College as a Weighted Vote | 作为加权投票的选举人团
In the US presidential election, each state has a number of electoral votes roughly proportional to its population. The total number of electoral votes is 538, and a candidate needs at least 270 to win. Mathematically, this is a weighted sum: national support is measured by adding state-level electoral votes, not by adding all individual votes nationwide.
在美国总统选举中，每个州拥有大致与其人口成比例的选举人票。选举人票总数为 538 张，候选人至少需要 270 张才能获胜。数学上，这是一个加权和：全国支持度通过州的选举人票相加来衡量，而不是将所有个人选票简单相加。
Smaller states have a higher weight per person than larger states. For example, Wyoming has about 580,000 people and 3 electoral votes, so each electoral vote represents roughly 193,000 people. California has about 39,000,000 people and 54 electoral votes, so each electoral vote represents roughly 722,000 people. This shows that the system is not purely proportional.
较小的州人均权重高于较大的州。例如，怀俄明州约有 58 万人和 3 张选举人票，因此每张选举人票代表约 19.3 万人。加州约有 3900 万人和 54 张选举人票，因此每张选举人票代表约 72.2 万人。这表明该制度并非纯粹按人口比例分配。
Weighted result = Σ (state electoral votes × state winner indicator)
4. Apportionment: Allocating Seats by Population | 席位分配：按人口分配席位
House seats are allocated among the 50 states every ten years after the census. The method uses a divisor to produce quotas. A state’s quota is its population divided by the national average district size. Because seats must be whole numbers, rounding creates an apportionment problem.
众议院席位每十年人口普查后在 50 个州之间重新分配。该方法使用除数产生配额。一个州的配额等于其人口除以全国平均选区人口。由于席位必须是整数，取整会带来分配问题。
A common approach is Hamilton’s method. First, compute each state’s quota. Then give each state the integer part of its quota. Finally, assign the remaining seats to the states with the largest fractional parts. This ensures the total number of seats is exact, but it can lead to paradoxes when the number of seats changes.
一种常见方法是汉密尔顿法。首先计算每个州的配额。然后给每个州配额的整数部分。最后将剩余席位分配给小数部分最大的州。这确保了席位总数精确，但当席位数量变化时可能产生悖论。
Quota for a state = state population ÷ (total population ÷ total seats)
5. Proportional Representation and Quota Methods | 比例代表制与配额方法
Although the USA mostly uses winner-take-all rules, some city councils and primary elections use proportional methods. Proportional representation allocates seats to parties or groups according to their vote share. If a party wins p% of the votes, it should receive roughly p% of the seats.
尽管美国大多采用赢家通吃规则，但一些市议会和初选使用比例代表制。比例代表制按票数份额向政党或团体分配席位。如果一个政党获得 p% 的选票，它应获得约 p% 的席位。
Quota methods determine how many votes are needed for one seat. The Hare quota is the simplest: total valid votes divided by total seats. The Droop quota is slightly smaller, which tends to favour larger parties. These quotas are used in different proportional systems to decide which candidates are elected.
配额方法确定获得一个席位需要多少选票。黑尔配额最简单：有效票总数除以席位数。德鲁普配额略小，这往往有利于较大的政党。这些配额用于不同的比例代表制，以决定哪些候选人当选。
Hare quota = total valid votes ÷ total seats
Droop quota = (total valid votes ÷ (total seats + 1)) + 1
6. Efficiency Gap and Gerrymandering | 效率差距与选区划分不公
Gerrymandering can be measured mathematically. The efficiency gap counts wasted votes: votes for a losing candidate and votes for a winning candidate beyond the 50% needed to win. It compares the two parties’ wasted votes as a share of all votes cast. A large gap suggests one party has an advantage due to district boundaries.
选区划分不公可以用数学衡量。效率差距统计浪费的选票：投给落选候选人的选票，以及投给当选候选人中超过 50% 获胜所需部分的选票。它将两党的浪费选票占总票数的份额进行比较。差距较大表明一个政党因选区边界而获得优势。
If Party A wastes many fewer votes than Party B, then Party A can win more seats with the same total vote share. The formula below gives a signed measure: a positive value indicates an advantage for Party A, while a negative value indicates an advantage for Party B.
如果 A 党浪费的选票远少于 B 党，那么 A 党可以在相同总票数份额下赢得更多席位。下面的公式给出了一个有符号的度量：正值表示 A 党占优势，负值表示 B 党占优势。
Efficiency Gap = (wasted votes for B − wasted votes for A) ÷ total votes
7. Polling, Sampling and Margin of Error | 民调、抽样与误差幅度
Before elections, opinion polls use random sampling. A-Level Statistics requires understanding sample size n, sample proportion p̂, and the margin of error for a 95% confidence interval. For large samples, the margin of error is approximately 1.96 × √(p̂(1 − p̂) ÷ n).
选举前的民调使用随机抽样。A-Level 统计学要求理解样本量 n、样本比例 p̂ 以及 95% 置信区间的误差幅度。对于大样本，误差幅度约为 1.96 × √(p̂(1 − p̂) ÷ n)。
This is why poll results often report ±3%. If a poll of 1,000 voters finds support at 52%, the true support could plausibly lie between 49% and 55%. Only if the gap between candidates exceeds the margin of error can we say the lead is statistically significant at the 5% level.
这就是为什么民调结果常报告 ±3%。如果对 1000 名选民的民调发现支持率为 52%，真实支持率可能在 49% 到 55% 之间。只有当候选人之间的差距超过误差幅度时，我们才能在 5% 显著性水平上说领先是统计显著的。
MoE ≈ 1.96 × √(p̂(1 − p̂) ÷ n)
8. Probability Models for Election Outcomes | 选举结果的概率模型
We can model the chance that a candidate wins using the binomial or normal distribution. If a candidate’s true support is p and n voters are sampled, the number of supporters X follows the binomial distribution B(n, p). For large n, X is approximately normal with mean np and variance np(1 − p).
我们可以用二项分布或正态分布对候选人获胜的概率建模。如果候选人的真实支持率为 p，且抽样了 n 位选民，则支持者
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