📚 Coalition of Supporters: Analysing Power in Weighted Voting Systems | 支持者联盟：加权投票系统中的权力分析
Behind every political decision, corporate board vote or international resolution, coalitions of supporters determine the outcome. In A‑Level Decision Mathematics, we move beyond simple majority counting and enter the world of weighted voting systems, where not all votes are equal. This article explains how coalitions form, how to measure the influence of each supporter using the Banzhaf and Shapley‑Shubik power indices, and how to approach typical exam questions with confidence.
在每一次政治决策、公司董事会投票或国际决议背后，支持者联盟决定着最终结果。在A‑Level决策数学中，我们不再局限于简单的多数计票，而是进入加权投票系统的世界——在这里，并非所有票的效力都相同。本文将解释联盟如何形成，如何使用班扎夫指数和沙普利‑舒比克权力指数衡量每位支持者的影响力，以及如何自信地应对典型的考试题目。
1. Introduction to Weighted Voting Systems | 加权投票系统简介
A weighted voting system is a decision‑making framework in which each voter (called a player) is assigned a positive integer weight. The system is defined by a quota q and a list of weights w₁, w₂, …, wₙ. A coalition of supporters is any subset of players, and it is winning precisely when the sum of weights of its members meets or exceeds the quota. The notation [q: w₁, w₂, …, wₙ] fully captures the voting rule.
加权投票系统是一种决策框架，每位投票者（称为参与者）被赋予一个正整数权重。该系统由一个定额 q 以及一组权重 w₁、w₂、…、wₙ 定义。支持者联盟就是参与者的任意子集，当且仅当其成员权重之和达到或超过定额时，该联盟即为获胜联盟。记法 [q: w₁, w₂, …, wₙ] 完整刻画了这一投票规则。
2. Basic Terminology: Players, Quota, Weight | 基本术语：参与者、定额、权重
eadlock can occur; if too high, a single player may effectively block everything. The word “coalition” simply means a group of supporters who agree to vote together.
在任何加权投票分析中，我们首先要确定参与者及其权重。定额 q 设定了获胜的门槛；通常，½（总权重） < q ≤ 总权重。如果 q 过低，可能出现僵局；如果过高，单个参与者就可能有效地否决一切。“联盟”一词指的就是一群同意共同投票的支持者。
3. Coalitions: Simple Coalitions and Winning Coalitions | 联盟：简单联盟与获胜联盟
alitions. All others are losing coalitions. For the system [6: 4, 3, 2, 1], the winning coalitions (with total in brackets) are: {A,B} (7), {A,C} (6), {A,B,C} (9), {A,B,D} (8), {A,C,D} (7), {B,C,D} (6), and {A,B,C,D} (10). Notice that {A} alone (weight 4) is losing.
对于 n 个参与者，存在 2ⁿ 种可能的联盟，包括空联盟。只有总权重 ≥ q 的联盟才被称为获胜联盟，其余都是失败联盟。对于系统 [6: 4, 3, 2, 1]，获胜联盟（括号内为总权重）有：{A,B}(7)、{A,C}(6)、{A,B,C}(9)、{A,B,D}(8)、{A,C,D}(7)、{B,C,D}(6) 和 {A,B,C,D}(10)。注意 {A} 单独（权重4）是失败的。
4. Critical Players in a Winning Coalition | 获胜联盟中的关键参与者
ritical. Remove C: {A,B} has 7 ≥ 6, so C is not critical. Recognising critical players is the foundation of power indices.
在一个获胜联盟中，如果移除某参与者会导致联盟总权重降至定额以下，则该参与者就是关键的（或称摇摆投票人）。在系统 [6:4,3,2,1] 的联盟 {A,B,C} 中，总权重为9。移除 A：剩余 {B,C} 权重 3+2=5 < 6，因此 A 是关键。移除 B：{A,C} 权重 4+2=6 ≥ 6，B 不是关键。移除 C：{A,B} 权重 7 ≥ 6，C 不是关键。识别关键参与者是权力指数计算的基础。
5. The Banzhaf Index of Power | 班扎夫权力指数
The Banzhaf index measures each player’s ability to change a coalition from winning to losing. For every player, count the total number of swings (critical appearances) across all winning coalitions. The Banzhaf power for player i is then the number of swings for i divided by the total number of swings for all players. The resulting ratios indicate the proportion of power each supporter holds.
班扎夫指数衡量每位参与者将联盟从获胜变为失败的能力。对每位参与者，统计其在所有获胜联盟中出现的摇摆次数（关键次数）。参与者 i 的班扎夫权力值等于 i 的摇摆次数除以全体参与者的总摇摆次数。所得比值显示了每位支持者所拥有的权力比例。
6. Calculating the Banzhaf Index: Step-by-Step | 计算班扎夫指数：分步指南
Let us calculate the Banzhaf indices for the system [6:4,3,2,1]. We list all winning coalitions and underline the critical players in each.
我们以系统 [6:4,3,2,1] 为例计算班扎夫指数。列出所有获胜联盟，并在每个联盟中标注关键参与者。
| Winning Coalition | Weight | Critical Players |
|---|---|---|
| {A,B} | 7 | A, B |
| {A,C} | 6 | A, C |
| {A,B,C} | 9 | A only |
| {A,B,D} | 8 | A, B, D |
| {A,C,D} | 7 | A, C, D |
| {B,C,D} | 6 | B, C, D |
| {A,B,C,D} | 10 | A only |
β(B)=3/13, β(C)=3/13, β(D)=2/13. This shows supporter A holds the largest share of power.
统计摇摆次数：A 关键出现 5 次，B 出现 3 次，C 出现 3 次，D 出现 2 次。总摇摆次数 = 5+3+3+2=13。因此班扎夫指数：β(A)=5/13，β(B)=3/13，β(C)=3/13，β(D)=2/13。由此可知支持者 A 掌握最大的权力份额。
7. The Shapley‑Shubik Index: A Different Perspective | 沙普利‑舒比克指数：另一个视角
While the Banzhaf index treats all winning coalitions equally, the Shapley‑Shubik index considers the order in which supporters join a coalition one by one. A player is pivotal if, at the moment they join, the accumulated weight reaches or exceeds the quota for the first time. The Shapley‑Shubik power for a player is the number of sequential coalitions (permutations) in which they are pivotal, divided by n!.
班扎夫指数平等地看待所有获胜联盟，而沙普利‑舒比克指数则考虑支持者依次加入联盟的顺序。如果某参与者在加入的那一刻使累计权重首次达到或超过定额，则该参与者就是关键加入者。某参与者的沙普利‑舒比克权力值等于其作为关键加入者的顺序总数除以 n!。
8. Shapley‑Shubik Index Calculation via Pivotal Players | 通过关键加入者计算沙普利‑舒比克指数
Using the same system [6:4,3,2,1], we list all 4! ng all 24 permutations, we find the pivotal counts: A pivotal in 12, B in 4, C in 4, D in 4. So the Shapley‑ 1/6. Again A dominates, but the power distribution differs slightly from the Banzhaf view.
以相同的系统 [6:4,3,2,1] 为例，列出参与者 A、B、C、D 的全部 4!=24 种排列。对每种排列，确定首先使累计权重 ≥ 6 的那个参与者。例如，顺序 (B, D, A, C) 中：B 权重 3，D 权重 1，累计 4，然后 A 加入，累计 4+4=8 ≥ 6，A 即为关键加入者。在 (C, B, A, D) 中：C=2，B=3 累计 5，A 使累计达 9，A 关键。检查全部 24 种排列后，关键加入次数为：A 关键 12 次，B 关键 4 次，C 关键 4 次，D 关键 4 次。因此沙普利‑舒比克指数为 φ(A)=12/24=1/2，φ(B)=4/24=1/6，φ(C)=1/6，φ(D)=1/6。仍然是 A 占主导，但权力分布与班扎夫视角略有不同。
9. Comparing Banzhaf and Shapley‑Shubik Indices | 班扎夫与沙普利‑舒比克指数对比
Both indices measure voting power, but their philosophies differ. The Banzhaf index focuses on coalitions where a defection can break a win; it is simpler to compute and often used in legal settings. The Shapley‑Shubik index captures the sequential logic of coalition building, making it valuable when the order of persuasion matters. In many straightforward weighted systems, the two indices give similar rankings, but discrepancies arise when weights are uneven or when dummy players (zero power) exist.
两种指数都衡量投票权力，但其哲学理念不同。班扎夫指数侧重于只要一票叛离就会破坏获胜的联盟；它计算更简单，常用于法律领域。沙普利‑舒比克指数抓住了联盟建立的顺序逻辑，因此在说服顺序重要的场合很有价值。在许多直接的加权系统中，两种指数的排名相似，但当权重不均衡或存在傀儡参与者（权力为零）时，两者就会出现差异。
10. Applications in Real-World Coalitions | 现实世界中的联盟应用
Weighted voting systems appear in shareholders’ meetings, the EU Council of Ministers, and the International Monetary Fund. For an A‑Level student, the key is to recognise that a coalition of supporters can often achieve more than the raw vote count suggests. By computing power indices, we uncover hidden imbalances: a minority player may hold zero power if they are never critical, despite having a positive weight.
加权投票系统出现在股东大会、欧盟部长理事会和国际货币基金组织等场合。对于 A‑Level 学生而言，关键在于认识到支持者联盟往往能达成比原始票数所显示的要多的结果。通过计算权力指数，我们可以发现隐藏的不平衡：一个拥有正权重的少数参与者，如果从未成为关键者，其实际权力可能为零。
11. Limitations and Assumptions | 局限性与假设
The models assume that all coalitions are equally likely (Banzhaf) or that all permutations are equally likely (Shapley‑Shubik). In practice, ideological or political constraints make some coalitions impossible. Additionally, the indices ignore strategic behaviour, such as threats or side payments. However, for the exam, you must work within the pure mathematical models, listing coalitions exhaustively and counting swings or pivotal positions without second‑guessing real‑world alliances.
这些模型假设所有联盟等可能（班扎夫）或所有排列等可能（沙普利‑舒比克）。在实际中，意识形态或政治限制会使某些联盟不可能成立。此外，权力指数忽略了策略行为，如威胁或私下交易。但在考试中，你必须在纯数学模型内工作，彻底列出联盟，并计算摇摆次数或关键位置，而不去猜测现实世界的联盟关系。
12. Summary and Exam Tips | 总结与应试技巧
To master coalition and power index questions, always start by writing down the weighted voting system in the [q: w₁, w₂, …] notation. Systematically list all winning coalitions; a logical approach is to work in order of coalition size. For Banzhaf, underline critical players in each winning coalition, then tally swings. For Shapley‑Shubik, use a factorial counting method or list permutations carefully. Check your total power adds to 1. With practice, this topic becomes a reliable source of marks on the Decision Mathematics paper.
要掌握联盟和权力指数题目，务必先将加权投票系统写成 [q: w₁, w₂, …] 的形式。系统地列出所有获胜联盟；一种合理的方法是按照联盟规模顺序进行。对于班扎夫指数，在每个获胜联盟中标注关键参与者，然后统计摇摆次数。对于沙普利‑舒比克指数，可使用阶乘计数法或仔细列出所有排列。检查权力总和是否为1。通过练习，该主题将成为决策数学试卷中可靠的得分点。
Published by TutorHao | Decision Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图