📚 First Past the Post | 领先者当选的数学剖析
First Past the Post (FPTP) is an electoral system where the candidate with the most votes in a constituency wins, even without an absolute majority. While commonly discussed in politics, its mathematical implications reveal profound consequences for representation, proportionality, and strategic behaviour. This article explores FPTP through the lens of A‑Level Mathematics, applying probability, algebra, and statistical reasoning to dissect how this single‑member plurality method shapes outcomes.
“领先者当选”（FPTP）是一种选举制度，在某一选区中获得最高票数的候选人即告胜出，无需获得绝对多数票。该系统在政治领域广受讨论，但其数学内涵深刻影响着代表性与比例性，并引发了策略性投票行为。本文借助A‑Level数学的视角，运用概率、代数与统计推理，剖析这一简单多数制如何塑造选举结果。
1. The Counting Rule and Basic Algebraic Representation | 计票规则与基本代数表示
In an FPTP constituency with n candidates, let the vote tally for candidate i be vi. The winner is the argument that maximises vi, that is, winner = argmaxi vi. No threshold T is required; the maximum alone suffices. If m voters turn out, then the sum of all vi equals m, assuming no blank ballots. The winning fraction is simply max(vi)/m, which can be as low as just over 1/n in a perfectly split field.
在FPTP选区中有n名候选人，令候选人i的得票数为vi。胜出者即为使vi取最大值的参选人，即winner = argmaxi vi。无需任何阈值T，单一最大值即足矣。假设有m位选民投票且无空白票，则所有vi之和等于m。获胜得票比例仅为max(vi)/m，在完全均分的态势下，该值可低至略大于1/n。
2. Wasted Votes and the Efficiency Gap | 废票与效率差距
Mathematically, a vote is wasted if it does not contribute to electing a candidate. In FPTP, all votes cast for losing candidates and all surplus votes beyond the runner‑up’s tally for the winner are wasted. If the winner’s vote is W, and the runner‑up gets R, the wasted votes for that constituency are (sum of all losing votes) + (W − R − 1). The efficiency gap EG is defined as the difference in wasted votes between two major parties divided by total votes. EG = (wastedA − wastedB) / total votes. A large |EG| signals gerrymandering.
从数学上讲，若一张选票未促成候选人当选，即为废票。在FPTP中，投给落选者的所有选票，以及胜者超出亚军的得票数部分，均属废票。若胜者得票为W，亚军得票为R，则该选区的废票数为（所有落选者得票之和）+（W − R − 1）。效率差距EG定义为两大政党废票数之差除以总票数：EG = (wastedA − wastedB) / total votes。若|EG|值大，即暗示选区划分不公。
3. Probability of Winning with Plurality vs Majority | 简单多数与绝对多数获胜的概率
Consider three candidates A, B, C whose support in the electorate is independent and uniformly distributed. The probability that A wins under FPTP with a plurality is analytically derived by the volume of the region where vA > vB and vA > vC within the simplex vA+vB+vC=1. By symmetry, this is 1/3. However, the probability that any candidate secures an absolute majority (>½) is given by integrating the area where one coordinate exceeds 0.5, yielding 3 × (1/2)2 = 3/4? Wait, we must recalculate: the region where vA > 0.5 is a smaller triangle with vertices (0.5,0.5,0), etc. The area is 1/4 for each, so total probability of a majority is 3/4. Thus in a 3‑candidate contest, an absolute majority is more likely than not under uniform random voting, but a plurality winner might still lack a majority 25% of the time.
假设三位候选人A、B、C，其选民支持度相互独立且均匀分布。在FPTP下，A以简单多数胜出的概率，可通过在单纯形vA+vB+vC=1上满足vA > vB且vA > vC的区域的体积求得。由对称性可知该概率为1/3。然而，任一候选人获得绝对多数（>½）的概率，可以通过积分某个坐标超过0.5的区域来计算，每个候选人对应概率为1/4，故合计为3/4。因此，在均匀随机投票下，三人竞选中出现绝对多数的可能性超过50%，但仍有25%的概率简单多数获胜者并未得到过半票数。
4. The Spoiler Effect and Strategic Voting | 搅局效应与策略投票
FPTP encourages strategic voting: supporters of a minor party may vote for a major candidate to prevent the least preferred from winning. Mathematically, if a voter’s utilities for candidates A, B, C are U(A)=10, U(B)=5, U(C)=0, and polls suggest A and C are front‑runners, the voter should vote for A instead of sincere favourite B, because the pivot probability of a tie between A and C dwarfs that of a three‑way tie. The expected utility of voting for X is ΣY,Z pXYZ × ΔU, where pXYZ is the probability of a tie between X, Y, Z and ΔU the utility gain from breaking the tie. Under plurality, pABC is negligible, so the sincere vote for B is wasted.
FPTP催生策略投票：小党派的支持者可能投票给主要候选人，以防止最讨厌的一方胜出。数学上，若选民对候选人A、B、C的效用分别为U(A)=10、U(B)=5、U(C)=0，且民调显示A与C领先，该选民就应投票给A而非真心支持的B，因为A与C平局的 pivot 概率远大于三方平局的概率。投票给X的期望效用为ΣY,Z pXYZ × ΔU，其中pXYZ是X、Y、Z之间平局的概率，ΔU是打破平局带来的效用增益。在简单多数制下，pABC微乎其微，因此对B的诚实投票便被浪费了。
5. Duverger’s Law and the Equilibrium Number of Parties | 杜瓦杰定律与均衡政党数量
Duverger’s law is a statistical regularity that FPTP tends to produce two‑party systems. We can model this using a spatial competition framework: suppose voters are uniformly distributed on [0,1] and two established parties position at a and b. A third party entering at c can only succeed if the share of voters closer to c than to a or b exceeds the existing shares. The equilibrium analysis shows that with single‑member districts and plurality rule, the entry of a third party is often unviable because it fragments its own ideological wing, leading to the opponent’s victory. The Nash equilibrium positions converge to the median voter, and only two parties are sustainable.
杜瓦杰定律是一条统计规律，即FPTP倾向产生两党制。我们可以用空间竞争模型加以刻画：假设选民在[0,1]上均匀分布，两个既有政党分别定位在a与b。第三党在c处进入，仅当其获得的选民份额超过既有政党份额时才能成功，这部分选民即离c比a或b更近的群体。均衡分析表明，在单席选区和简单多数规则下，第三党进入通常无法存活，因为它会分散自身意识形态阵营的选票，反令对手胜出。纳什均衡下的定位将收敛至中位选民，只有两个政党能够维持。
6. Disproportionality Measures: Loosemore–Hanby Index | 比例偏差度：卢斯莫尔–汉比指数
FPTP often yields a seat distribution far from the national vote share. The Loosemore–Hanby index D quantifies disproportionality: D = ½ Σ |vi − si|, where vi is the vote share and si the seat share of party i. A value of 0 indicates perfect proportionality, while 1 means maximum disproportionality. For real FPTP elections, D can range from 0.15 to 0.30, compared to below 0.10 for proportional systems. Algebraically, we can calculate D from election data using absolute deviations, demonstrating the magnification effect of winner‑takes‑all.
FPTP往往使议席分布严重偏离全国得票率。卢斯莫尔–汉比指数D量化了比例偏差度：D = ½ Σ |vi − si|，其中vi为政党i的得票份额，si为其议席份额。0表示完全比例代表，1表示最大偏差。实际FPTP选举中，D值可达0.15至0.30，而比例代表制下则低于0.10。从代数角度看，我们可以利用绝对偏差从选举数据中算出D，直观展现赢者通吃的放大效应。
7. Marginal Seats and the Cube Law | 边际席位与立方定律
A historical empirical observation is the cube law: in two‑party FPTP systems, the ratio of seats won by the two parties scales roughly as the cube of the ratio of their votes. That is, if votes are VA and VB, then SA/SB ≈ (VA/VB)3. This non‑linear relationship arises from the distribution of constituencies’ partisan lean. If underlying constituency means follow a normal distribution with a certain variance, integration shows that seat swings are much more sensitive to vote swings near the 50‑50 point. The cube law implies that a small lead in votes can command a landslide in seats.
一项历史经验规律是所谓立方定律：在两党制FPTP下，两党赢得的席次比大致与其得票比的立方成正比。即若得票数为VA与VB，则SA/SB ≈ (VA/VB)3。这一非线性关系源于各选区党派倾向的分布。若选区平均支持度服从特定方差的正态分布，积分结果表明，在50‑50附近，席次摆动对票数摆动的敏感度极高。立方定律意味着微弱的得票领先可转化为席次上的压倒性胜利。
8. Algebraic Spreadsheets: Simulating a National Election | 代数模拟：一次全国大选的电子表格推演
We can construct a simplified FPTP model using a spreadsheet. Suppose there are 10 constituencies, each with 1000 voters, and three parties X, Y, Z. We assign random vote totals for each party per constituency using a uniform distribution. We then apply the FPTP rule: the party with the highest number in a constituency gets 1 seat. Summing seats across constituencies shows that the national seat share often deviates significantly from the national vote share. For example, X might get 40% of the vote but 60% of the seats if its support is efficiently distributed. This simulation reinforces the algebraic principle of the winner’s bonus.
我们可以借助电子表格构建简化的FPTP模型。设有10个选区，每区1000名选民，三个政党X、Y、Z。利用均匀分布为每党在各区的得票随机赋值，再应用FPTP规则：每区最高票数党获1席。汇总各区席次后会发现，全国议席比例经常严重偏离全国得票比例。例如，若X的选票集中分布于关键选区，它可能以40%的普选票赢取60%的席位。这一模拟印证了胜者红利的代数原理。
9. Mathematical Critiques: Arrow’s Impossibility Theorem and FPTP | 数学批判：阿罗不可能定理与FPTP
Arrow’s theorem states that no rank‑order voting system can simultaneously satisfy unrestricted domain, non‑dictatorship, Pareto efficiency, and independence of irrelevant alternatives (IIA). FPTP blatantly violates IIA: the presence of a third candidate can alter the outcome between the first two. Mathematically, if societal preferences are such that A beats B beats C in pairwise contests, but the introduction of C splits the vote and causes B to win under FPTP, the outcome is not independent of C. This is the classic spoiler effect formalised.
阿罗定理指出，没有任何一种排序投票制能同时满足无限制定义域、非独裁、帕累托效率与无关选项独立性（IIA）。FPTP公然违反IIA：第三位候选人的存在足以改变前两者之间的胜负。数学上，若社会偏好呈A击败B、B击败C的序对关系，但由于C的加入分走了选票，使得B在FPTP下获胜，该结果就依赖于无关选项C。这正是经典搅局效应的形式化描述。
10. Proportionality and the Deviation from Ideal: Gallagher Index | 比例性与理想偏离：加拉格尔指数
The Gallagher index (least squares index) is another disproportionality metric: LSq = √(½ Σ (vi − si)2). It gives more weight to large deviations than the Loosemore–Hanby index. For FPTP, typical values are between 10 and 20, while PR systems lie below 5. By applying this formula to UK general election results, students can compute LSq and discuss how district magnitude and geographic clustering influence the index. This demonstrates the power of quadratic penalties in measuring fairness.
加拉格尔指数（最小二乘指数）是另一种比例偏差度指标：LSq = √(½ Σ (vi − si)2)。相比卢斯莫尔–汉比指数，它对大偏差赋予更高权重。FPTP的典型取值在10至20之间，而比例代表制则低于5。将此公式应用于英国大选结果，学生可自行算出LSq，并探讨选区规模与地理集聚如何影响该指数。这展示了二次惩罚在衡量公平性方面的力量。
11. Mathematical Redistricting: Cracking, Packing and the Polsby–Popper Ratio | 选区重划的数学：裂化、堆叠与珀尔斯比–波普比比率
Gerrymandering in FPTP is mathematically framed by techniques of cracking (dispersing opposition voters) and packing (concentrating them). The Polsby–Popper ratio for a district is a compactness measure: PP = 4πA / P2, where A is area and P perimeter. Highly irregular districts with low PP scores are often indicators of gerrymandering. By exploring geometric properties, students see how FPTP invites manipulation of boundary shapes to engineer wasted votes across districts, altering the efficiency gap.
FPTP下的选举地理操纵，数学上可通过裂化（分散反对选民）与堆叠（集中反对选民）手法实现。珀尔斯比–波普比比率是衡量选区紧凑度的指标：PP = 4πA / P2，其中A为面积，P为周长。高度不规则的选区其PP值极低，通常是操纵选区的信号。通过探究几何属性，学生可以看到FPTP如何诱使人们操纵边界形态，在各选区之间人为制造废票，从而改变效率差距。
12. From Plurality to Majority: Runoff and Mathematical Comparison | 从简单多数到绝对多数：两轮决选与数学比较
FPTP’s single‑round simplicity contrasts with two‑round runoff systems. Algebraically, the transfer of votes in a runoff can be modelled by assuming preferences are single‑peaked. If the top two candidates advance, the median voter theorem guarantees the Condorcet winner is elected under certain conditions. By comparing FPTP and runoff in a 3‑candidate model using preference orderings, students can compute the probability that a non‑Condorcet winner wins under each system, illustrating FPTP’s vulnerability to minority victories when the field is fragmented.
FPTP的单轮简单性与两轮决选制形成对照。代数上，决选中的票数转移可在单峰偏好假设下建模。若前两名候选人进入第二轮，中位选民定理保证在特定条件下选出孔多塞胜者。通过用偏好排序比较三人竞选下的FPTP与决选制，学生可以计算各制度下非孔多塞胜者当选的概率，从而揭示当候选人领域分散时FPTP易产生少数胜利的脆弱性。
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