📚 The Imperial Presidency: A Mathematical Analysis of Presidential Power | 帝王总统：总统权力的数学分析
The term 'imperial presidency' was coined by historian Arthur M. Schlesinger Jr. to describe a presidency that has extended its powers far beyond what the framers of the US Constitution intended. In an A-Level Decision Mathematics context, we can use power indices to measure how much influence the president truly wields within a weighted voting system representing the federal government. This article explores the Shapley-Shubik and Banzhaf power indices to quantify whether the American president has become an imperial figure, effectively a dictator in the mathematical sense.
'帝王总统'一词由历史学家小阿瑟·M·施莱辛格提出，用来形容美国总统权力大大超出宪法制定者初衷的现象。在A-Level决策数学中，我们可以使用权力指数，来衡量在一个代表联邦政府的加权投票系统中，总统实际掌握了多少影响力。本文探讨夏普利-舒比克指数和班扎夫指数，通过量化方法判断美国总统是否已成为数学意义上的独裁者，即一个帝王般的人物。
1. The Imperial Presidency Concept | 帝王总统的概念
The imperial presidency refers to a situation where the executive branch dominates the legislative and judicial branches, making the president nearly unchallengeable. Schlesinger argued that this phenomenon emerged especially during the Cold War, when national security concerns allowed the president to assume war-making powers without congressional approval. In extreme cases, the president may act unilaterally, resembling an elected monarch.
帝王总统指的是行政部门主导立法和司法部门，使总统几乎不受制约的局面。施莱辛格认为，这一现象尤其在冷战时期出现，国家安全忧虑使总统得以未经国会批准就行使战争权。在极端情况下，总统可能单边行事，形同民选君主。
2. Power Indices in Decision Mathematics | 决策数学中的权力指数
In Edexcel A-Level Decision Mathematics 1, we study how to measure the power of voters in a weighted voting system. A weighted voting system is defined by a quota q and a set of weights w₁, w₂, ..., wₙ. A coalition is winning if the sum of its members' weights meets or exceeds q. Power indices such as the Shapley-Shubik index and the Banzhaf index help us determine the proportion of power held by each player, irrespective of their nominal weight.
在Edexcel A-Level决策数学1中，我们学习如何衡量加权投票系统中投票人的权力。一个加权投票系统由配额q和一组权重w₁, w₂, ..., wₙ定义。如果一个联盟的成员权重之和达到或超过q，它就是获胜联盟。夏普利-舒比克指数和班扎夫指数等权力指数，可以帮助我们确定每个参与者所掌握的权力比例，而不取决于其名义权重大小。
3. The Shapley-Shubik Index | 夏普利-舒比克指数
treats the order in which voters join a coalition as important. For n players, there are n! actorial) possible sequential coalitions. In each sequence, we identify the pivotal player – apley-Shubik power index for player i is φᵢ = (number of sequences where i is pivotal) / n!. This index sums to 1 over all players.
夏普利-舒比克指数将投票人加入联盟的顺序视为重要因素。对于n个参与者，共有n!（n的阶乘）种可能的序列联盟。在每种序列中，我们确定关键玩家——即其加入使联盟首次成为获胜联盟的那一位。参与者i的夏普利-舒比克权力指数为 φᵢ = ( i 作为关键玩家的序列数 ) / n!。该指数在所有参与者之间总和为1。
4. The Banzhaf Index | 班扎夫指数
The Banzhaf index focuses on coalitions rather than sequences. A player is critical in a coalition if removing them causes the coalition to lose its winning status. The Banzhaf power index is the number of coalitions in which a player is critical divided by the total number of critical defections across all players. This index also sums to 1. Both indices can reveal discrepancies between voting weight and actual power.
班扎夫指数关注联盟而非序列。若从某一联盟中移除一个玩家导致该联盟失去获胜地位，则该玩家在该联盟中为关键玩家。班扎夫权力指数等于某玩家是关键玩家的联盟个数，除以所有玩家的总关键次数。该指数总和也为1。这两个指数都能揭示投票权重与实际权力之间的差异。
5. Modelling the US Government as a Weighted Voting System | 将美国政府建模为加权投票系统
f the president vetoes. A simplified model might assign weights of P = 1, H = 1, S = 1, with a quota q =
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