Coalition of Supporters: Voting Power and Mathematical Modelling | 支持者联盟：投票权力与数学建模

📚 Coalition of Supporters: Voting Power and Mathematical Modelling | 支持者联盟：投票权力与数学建模

In A-Level Mathematics, especially in Decision Mathematics and Further Mathematics options, the idea of a ‘coalition of supporters’ appears when we model weighted voting, board decisions, and negotiation games. A coalition is simply a set of supporters who join forces to reach a required threshold, and mathematical tools such as combinatorics, probability and power indices help us analyse how influential each supporter really is. This article builds the key ideas step by step, with worked examples and exam-focused guidance.

在 A-Level 数学中，尤其是在决策数学和进阶数学选修模块里，“支持者联盟”的概念出现在加权投票、董事会决策和谈判博弈的建模中。联盟就是一组支持者联合起来以达到某个规定门槛的集合，而组合数学、概率和权力指数等数学工具能帮助我们分析每个支持者真正的影响力。本文逐步构建关键概念，并给出实例和备考指导。

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1. What is a Coalition of Supporters? | 什么是支持者联盟？

A coalition is any group of players, voters or supporters who agree to act together to achieve a common goal, such as passing a motion or winning a vote. In mathematics, we formalise this by treating each supporter as having a certain weight, and a coalition as a subset of all supporters.

联盟是指同意共同行动以实现共同目标（例如通过一项动议或赢得投票）的任何一组参与者、选民或支持者。在数学中，我们通过把每个支持者赋予一定权重，并将联盟视为所有支持者的子集来形式化这一概念。

For example, in a small committee with supporters A, B, C and D, the possible coalitions include {A}, {A, C}, {B, D}, {A, B, C} and even the full set {A, B, C, D}. The empty coalition, containing no supporters, is not usually counted as a meaningful coalition because it cannot influence the outcome.

例如，在一个由支持者 A、B、C 和 D 组成的小型委员会中，可能的联盟包括 {A}、{A, C}、{B, D}、{A, B, C}，甚至全集 {A, B, C, D}。空联盟（不含任何支持者）通常不算作有意义的联盟，因为它无法影响结果。

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2. Weighted Voting Systems | 加权投票系统

A weighted voting system is normally written in the form [q; w₁, w₂, …, wₙ], where q is the quota needed to win, and w₁, w₂, …, wₙ are the weights of the n supporters. A coalition wins if the sum of its members’ weights is at least q.

加权投票系统通常写成 [q; w₁, w₂, …, wₙ] 的形式，其中 q 是获胜所需的配额，w₁, w₂, …, wₙ 是 n 个支持者的权重。如果一个联盟成员的权重总和至少为 q，该联盟就获胜。

Total weight of a coalition = Σ wᵢ for each i in the coalition

Quotas are often set above half of the total weight to ensure that no two disjoint coalitions can both win. For example, if the total weight is 10, a quota of 6 is common: any winning coalition must control more than half of the total voting weight.

配额通常设定为高于总权重的一半，以确保两个互不相交的联盟不可能同时获胜。例如，如果总权重为 10，常用配额为 6：任何获胜联盟必须控制超过一半的总投票权重。

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3. Counting Possible Coalitions | 计算可能的联盟数

For n supporters, each supporter can either be in a coalition or not, so the total number of subsets is 2ⁿ. Excluding the empty set gives the number of non-empty coalitions:

对于 n 个支持者，每个支持者可以加入或不加入一个联盟，因此子集总数为 2ⁿ。排除空集后，非空联盟数为：

Number of non-empty coalitions = 2ⁿ – 1

For n = 3, there are 2³ – 1 = 7 possible non-empty coalitions. For n = 4, there are 15, for n = 5 there are 31, and so on. This rapid growth is why a systematic listing method is essential in exam questions.

当 n = 3 时，有 2³ – 1 = 7 个非空联盟。当 n = 4 时，有 15 个；n = 5 时有 31 个，依此类推。这种快速增长正是考试中需要系统枚举方法的原因。

One useful way to list all coalitions is to use binary numbers. Write each supporter as a digit: 1 means the supporter is included, 0 means not included. For four supporters A, B, C, D, the binary string 1010 represents the coalition {A, C}.

一种列出所有联盟的实用方法是使用二进制数。将每个支持者写成一个数字：1 表示该支持者在联盟中，0 表示不在。例如四个支持者 A、B、C、D，二进制串 1010 表示联盟 {A, C}。

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4. Winning Coalitions and Quotas | 获胜联盟与配额

A coalition is called a winning coalition if the sum of its members’ weights is greater than or equal to the quota q. If the sum is strictly less than q, the coalition is losing. Blocking coalitions are those that cannot pass a motion but may be able to prevent others from reaching the quota.

如果一个联盟成员的权重总和大于或等于配额 q，则称该联盟为获胜联盟。如果总和严格小于 q，则该联盟为失败联盟。阻挡联盟是指自身不能通过动议，但可能阻止其他联盟达到配额的联盟。

Consider the weighted voting system [6; 4, 3, 2, 1]. The coalition {A, B} has weight 4 + 3 = 7, which reaches the quota 6, so it is winning. The coalition {B, C} has weight 3 + 2 = 5, so it is losing.

考虑加权投票系统 [6; 4, 3, 2, 1]。联盟 {A, B} 的权重为 4 + 3 = 7，达到配额 6，因此它是获胜联盟。联盟 {B, C} 的权重为 3 + 2 = 5，因此它是失败联盟。

In exam problems, always write down the quota clearly and add weights carefully. A common error is to compare the number of supporters rather than the sum of their weights; in a weighted system, not every vote carries the same power.

在考试题中，务必清楚地写下配额，并仔细相加权重。一个常见错误是比较支持者的人数而不是权重总和；在加权系统中，并非每一票都具有相同的权力。

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5. Minimal Winning Coalitions | 最小获胜联盟

A minimal winning coalition is a winning coalition in which every member is essential: if any member leaves, the coalition becomes losing. Minimal winning coalitions are important because they identify the smallest groups that can actually pass a measure.

最小获胜联盟是指每个成员都不可或缺的获胜联盟：任何成员离开，联盟就会变成失败联盟。最小获胜联盟很重要，因为它们确定了实际上能够通过议案的最小群体。

Using the system [6; 4, 3, 2, 1], the coalition {A, B} is minimal winning: removing A gives {B} with weight 3, which loses; removing B gives {A} with weight 4, which also loses. However, {A, B, C} is winning but not minimal, because removing C still leaves {A, B} with weight 7, which wins.

在系统 [6; 4, 3, 2, 1] 中，联盟 {A, B} 是最小获胜联盟：移除 A 得到权重为 3 的 {B}，失败；移除 B 得到权重为 4 的 {A}，也失败。然而，{A, B, C} 是获胜联盟但不是最小获胜联盟，因为移除 C 后仍留下权重为 7 的 {A, B}，依然获胜。

To test whether a winning coalition is minimal, check each member in turn: if any member can be removed without losing, the coalition is not minimal.

要检验一个获胜联盟是否最小，逐一检查每个成员：如果移除某个成员后联盟仍未失败，则该联盟不是最小获胜联盟。

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6. Critical Players in a Coalition | 联盟中的关键支持者

A supporter is critical in a winning coalition if removing that supporter causes the coalition to become losing. Critical players are the ones without whom the coalition would not have enough weight to meet the quota.

如果在获胜联盟中移除某个支持者会导致该联盟变成失败联盟，则该支持者就是关键支持者。关键支持者是那些没有他们

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