Debates about the relative powers of the two houses | 两院相对权力的辩论

📚 Debates about the relative powers of the two houses | 两院相对权力的辩论

In any bicameral legislature, one of the most enduring political debates centres on the relative powers of the two chambers. Using mathematical models from decision mathematics, we can move beyond rhetoric and quantify how much actual voting power each house possesses under different constitutional rules. This article applies power indices—specifically the Banzhaf and Shapley-Shubik indices—to the UK Parliament, showing how the Commons and Lords compare and how formal rules like the Parliament Acts dramatically shift the balance.

在任何两院制立法机构中，关于两院相对权力的争论始终是经久不衰的政治议题。借助决策数学中的数学模型，我们可以超越空泛的言辞，量化在不同宪制规则下每个议院实际拥有的投票权。本文将权力指数——具体是班扎夫指数和沙普利-舒比克指数——运用于英国议会，展示下院和上院如何比较，以及《议会法案》等正式规则如何彻底改变权力平衡。

1. Introduction to Power Indices | 权力指数引论

Power indices are tools from cooperative game theory, frequently studied in Edexcel Further Mathematics Decision 2. They measure the influence of a voter in a weighted voting system, based not merely on the number of votes held but on the number of times that voter can change a losing coalition into a winning one. This concept is essential for understanding debates about institutional design.

权力指数是合作博弈论中的工具，经常在爱德思高等数学D2中学习。它们衡量的是加权投票系统中投票者的影响力，所依据的不仅仅是所持票数，而是该投票者能将失败联盟转变为获胜联盟的次数。这一概念对于理解制度设计的辩论至关重要。

2. Weighted Voting Systems | 加权投票系统

A weighted voting system is denoted by [q; w₁, w₂, …, wₙ], where q is the quota required to pass a measure and wᵢ are the weights of the n voters. In a two-house legislature, we can simplify the system to two voters: the lower house with weight wₗ and the upper house with weight wᵤ, along with a quota q. Different constitutional arrangements alter these parameters and, therefore, the distribution of power.

加权投票系统记作 [q; w₁, w₂, …, wₙ]，其中 q 是通过议案所需的票数定额，wᵢ 是 n 个投票者的权重。在两院制立法机构中，我们可以将系统简化为两个投票者：下院权重为 wₗ，上院权重为 wᵤ，配额为 q。不同的宪制安排会改变这些参数，从而改变权力的分配。

3. The UK Parliament as a Two-Chamber System | 英国议会作为两院制系统

Historically, the House of Commons and the House of Lords had equal legislative standing, but the Parliament Acts of 1911 and 1949 transformed the relationship. Today, the Commons can pass certain bills without the Lords’ consent after a delay. From a decision mathematics perspective, this creates multiple possible weighted voting models depending on the type of legislation, giving us a rich context for power index analysis.

历史上，下院和上院在立法上地位平等，但1911年和1949年的《议会法案》改变了这一关系。如今，下院可以在延迟一段时间后、不经上院同意便通过某些法案。从决策数学的视角看，这根据立法类型创造了多种可能的加权投票模型，为权力指数分析提供了丰富的背景。

4. Quantifying Power: Banzhaf Index | 量化权力：班扎夫指数

The absolute Banzhaf index for voter i, denoted b(i), counts the number of winning coalitions in which i is critical—meaning that removing i would turn the coalition from winning to losing. The normalised Banzhaf index β(i) = b(i) / Σ b(j) gives the voter’s share of total power. In a two-player game, b(i) is simply the number of winning coalitions where i’s defection would cause the coalition to fail.

对于投票者 i，绝对班扎夫指数记作 b(i)，计算的是 i 在其中起关键作用的获胜联盟数量——所谓关键，是指移除 i 后联盟将从获胜变为失败。标准化班扎夫指数 β(i) = b(i) / Σ b(j) 给出了该投票者占总权力的份额。在两人博弈中，b(i) 仅仅是 i 的背叛会导致联盟失败的获胜联盟数。

5. Calculating Banzhaf for the Commons and Lords | 计算下院和上院的班扎夫指数

Consider a classical bicameral model where a bill requires approval from both houses. We can represent this as [2; 1, 1], with a quota of 2 and both houses having weight 1. The winning coalitions are {Commons, Lords}. In this coalition, both members are critical because removing either reduces the total weight below 2. Hence, b(Commons) = b(Lords) = 1, and β(Commons) = β(Lords) = 1/2. Under this equal-rights model, mathematical power is perfectly balanced, reflecting the pre-1911 constitution.

考虑一个经典的两院制模型，即法案需要两院都批准。我们可以将其表示为 [2; 1, 1]，配额为2，两院权重均为1。获胜联盟是{下院, 上院}。在此联盟中，两名成员都具备关键性，因为移去任何一方都会使总权重低于2。因此，b(下院)=b(上院)=1，且β(下院)=β(上院)=1/2。在此平等权利模型下，数学权力完全平衡，反映了1911年之前的宪制。

6. Shapley-Shubik Index Approach | 沙普利-舒比克指数方法

The Shapley-Shubik index considers all possible sequential orders in which a coalition can be formed. A voter is pivotal if their addition makes the cumulative weight reach or exceed the quota for the first time. For [2; 1, 1], there are 2! = 2 orders: (Commons → Lords) and (Lords → Commons). In both, the second voter is pivotal. Thus the Shapley-Shubik index gives each house 1/2, identical to the normalised Banzhaf index in this symmetric case.

沙普利-舒比克指数考虑所有可能的联盟形成顺序。如果某位投票者的加入首次使得累积权重达到或超过配额，则该投票者即为关键枢轴。对于 [2; 1, 1]，共有 2! = 2 种顺序：(下院→上院) 和 (上院→下院)。两种顺序中，第二位投票者是枢轴。因此沙普利-舒比克指数赋予每个议院1/2，与这个对称情形下的标准化班扎夫指数相同。

7. Sequential Coalitions and Pivotal Players | 顺序联盟与关键参与者

When we move to more complex models, the Shapley-Shubik index can differ sharply from Banzhaf because it accounts for the ordering of commitments. For instance, if we introduce a ‘financial privilege’ rule where the Commons can ultimately pass a money bill alone, the system becomes [1; 1, 0] effectively for such bills, giving the Commons a Shapley-Shubik index of 1 and the Lords 0. This illustrates how constitutional conventions can be rigorously captured by the choice of voting system parameters.

当我们转向更复杂的模型时，沙普利-舒比克指数可能与班扎夫指数产生显著差异，因为它考虑了承诺的顺序。例如，如果引入“财政特权”规则，即下院可以最终单独通过财政法案，那么对于这类法案，系统实际上变为 [1; 1, 0]，下院的沙普利-舒比克指数为1，上院为0。这表明宪法惯例可以通过投票系统参数的选择被严谨地刻画。

8. Comparing Indices: Which House Holds More Power? | 指数比较：哪个院权力更大？

Mathematically, the answer depends entirely on the chosen q and weights. A wide range of plausible constitutional models can be constructed. For ordinary public bills, the current system after the Parliament Acts can be approximated by a model where the Commons has a weight of 2 and the Lords a weight of 1, but the quota remains 2, giving the Commons a greater Banzhaf power. The table below summarises several scenarios.

从数学上说，答案完全取决于所选择的 q 和权重。我们可以构建一系列合理的宪制模型。对于普通公共法案，《议会法案》之后的现行体制可以近似为：下院权重为2，上院权重为1，但配额仍为2，使得下院拥有更大的班扎夫权力。下表总结了几种情形。

Scenario	System [q; wC, wL]	β(Commons)	β(Lords)
Pre-1911 equality	[2; 1, 1]	1/2	1/2
Post-1949 ordinary bill (Commons can override Lords after one year)	[2; 2, 1]	2/3	1/3
Money bill (Commons exclusive)	[1; 1, 0]	1	0
Salisbury convention (Lords do not block manifesto bills)	[2; 1.5, 0.5]	0.75	0.25

These numerical values crystallise the debate: the Commons’ dominance is not absolute in legislation but is heavily weighted in its favour, particularly on financial matters.

这些数值澄清明朗了这场辩论：下院在立法中的主导地位并非绝对，但权重严重偏向于它，尤其在财政事务上。

9. Criticisms of Power Index Models | 权力指数模型的批评

While decision mathematics provides valuable quantitative insight, critics rightly point out that formal rules capture only part of the picture. Power indices treat all winning coalitions as equally likely and ignore political parties, patronage, and the threat of reform. The Lords’ power to delay and amend, even without a veto, can exert significant real-world influence that a simple Banzhaf score may undervalue.

尽管决策数学提供了宝贵的量化洞见，批评者正确地指出，正式规则只能捕捉图景的一部分。权力指数将所有获胜联盟视作等概率，并忽略了政党、庇护关系以及改革的威胁。即使没有否决权，上院延迟和修改法案的权力也能在现实中施加重大影响，而简单的班扎夫得分可能低估了这一点。

10. Real-World Constraints: Parliament Acts | 现实限制：议会法案

The Parliament Act 1911 abolished the Lords’ veto over money bills and restricted its power over other public bills to a two-year delay, later reduced to one year by the 1949 Act. In decision mathematics terms, this delay can be modelled as a time-dependent payoff or as a dynamic game where the Commons has an “override” move after a certain number of stages, effectively altering the quota. This transforms a static power index into a multistage analysis.

1911年《议会法案》废除了上院对财政法案的否决权，并将其对其他公共法案的权力限制为延迟两年，1949年法案又将延迟缩短为一年。用决策数学的术语来说，这种延迟可以模型化为与时间相关的收益，或者作为一个动态博弈，其中下院在特定阶段后拥有“强行通过”的举措，从而实质上改变了配额。这就将静态的权力指数转变为多阶段分析。

11. Extensions: Multiple Parties and Coalitions | 扩展：多党与联盟

In a more granular model, the two houses can be disaggregated into individual MPs and peers, each with their own weighted votes in intra-chamber divisions. A combined power index across chambers then requires considering cross-chamber coalitions. This becomes a large-scale weighted voting game, often analysed computationally. Such simulations often show that the government’s majority in the Commons dramatically reduces the Lords’ relative power further.

在更细致的模型中，两院可以分解为一个个下院议员和上院贵族，各自在院内表决中拥有加权票。跨议院的综合权力指数就需要考虑跨院联盟。这就变成一个大规模的加权投票博弈，通常通过计算进行分析。这类模拟往往显示，政府在下院的多数席位会进一步大幅降低上院的相对权力。

12. Conclusion: The Maths Behind the Debate | 结论：辩论背后的数学

Debates about the relative powers of the two houses cease to be merely rhetorical when we introduce the rigorous framework of power indices. The House of Commons inevitably emerges with a larger mathematical share of power under nearly every post-1911 model, but the precise magnitude depends on the assumptions made. Decision mathematics thus equips us with a language to move from vague claims of “dominance” to testable, quantitative comparisons, while reminding us that no single index can capture the full texture of political influence.

当我们引入权力指数的严谨框架后，关于两院相对权力的辩论就不再仅仅是修辞之争。在1911年之后的几乎所有模型下，下院都不可避免地拥有更大的数学权力份额，但确切的大小取决于所做出的假设。因此，决策数学赋予我们一种语言，得以从模糊的“主导”断言转向可检验的量化比较，同时提醒我们，没有任何单一指数能够完全捕捉政治影响力的全貌。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）