📚 AS WJEC Politics: Formula & Theorem Quick Reference Handbook | AS WJEC 政治：公式定理速查手册
This handbook brings together the essential formulas, quantitative models, and key theorems you will encounter in the AS WJEC Politics specification. It covers electoral mathematics, party system dynamics, voting behaviour models, game theory tools, and classic political science laws – all presented as ready-to-use references. Use these pages to consolidate your understanding of the analytical side of politics alongside your case studies.
本手册汇集了 AS WJEC 政治课程中你将遇到的核心公式、量化模型和重要定理。内容涵盖选举数学、政党制度动态、投票行为模型、博弈论工具以及经典政治学定律，全部以便捷查阅的形式呈现。用这些页面结合你的案例分析，夯实政治学的分析性知识。
1. Effective Number of Parties (Laakso–Taagepera Index) | 有效政党数（拉索拉–塔格佩拉指数）
The most widely used measure of party system fragmentation is the effective number of parties (ENP). It weights parties by their size, revealing how many sizable parties shape competition. For vote shares vᵢ or seat shares sᵢ, the index is calculated as:
衡量政党制度碎片化最常用的指标是有效政党数。它根据政党规模加权，显示有多少个具备影响力的政党在塑造竞争。基于得票份额 vᵢ 或席次份额 sᵢ，计算公式为：
ENP = 1 / ∑ pᵢ²
where pᵢ is the fractional share of party i (votes or seats). For example, if three parties win 50%, 30% and 20% of the vote, ENP = 1 / (0.5² + 0.3² + 0.2²) = 1 / 0.38 ≈ 2.63, suggesting that competition resembles a two-and-a-half party system.
其中 pᵢ 是政党 i 的得票份额或席次份额。例如，若三党分别获得 50%、30%和 20% 的选票，ENP = 1 / (0.5² + 0.3² + 0.2²) = 1 / 0.38 ≈ 2.63，表明竞争格局近似于两个半政党体制。
2. Electoral Quotas: Hare Quota and Droop Quota | 选举配额：黑尔配额与德鲁普配额
In proportional representation systems that use the single transferable vote (STV) or largest remainder methods, quotas determine how many votes a candidate needs to win a seat. The Hare quota is the simplest:
在采用单记可转让投票（STV）或最大余数法的比例代表制中，配额决定了候选人赢得一个席位所需的票数。黑尔配额是最简单的形式：
QH = V / S
where V is the total number of valid votes and S is the number of seats to be filled. The Droop quota, used in many STV elections, lowers the threshold slightly to avoid wasting votes:
其中 V 是总有效票数，S 是待填补的席位数。德鲁普配额在许多 STV 选举中使用，略微降低门槛以避免选票浪费：
QD = ⌊ V / (S + 1) ⌋ + 1
For instance, with 100,000 votes and 4 seats, the Hare quota is 25,000 votes; the Droop quota is ⌊100,000/5⌋ + 1 = 20,001 votes. Droop makes it mathematically impossible for more candidates to reach the quota than there are seats.
例如，有 100,000 票和 4 个席位，黑尔配额是 25,000 票；德鲁普配额为 ⌊100,000/5⌋ + 1 = 20,001 票。德鲁普配额使得达到配额的人数在数学上不可能超过席位总数。
3. The Cube Law | 立方定律
The cube law describes the translation of votes into seats in first-past-the-post (FPTP) systems. It predicts that the ratio of seats won by two major parties approximates the cube of their vote ratio:
立方定律描述了简单多数制下选票转化为席位的方式。它预测两大政党赢得的席位比约等于其得票比的立方：
SA / SB ≈ (VA / VB)³
If Party A wins 55% of the two-party vote and Party B wins 45%, the expected seat ratio is roughly (0.55/0.45)³ ≈ 1.824, giving Party A a significantly inflated seat bonus. This mechanical effect helps explain why FPTP tends to produce single-party majorities in the UK House of Commons.
如果 A 党赢得两党选票的 55%，B 党赢得 45%，预计的席位比约为 (0.55/0.45)³ ≈ 1.824，这使得 A 党获得明显放大的席次红利。这种机械效应有助于解释为何简单多数制在英国下议院通常产生单一政党的多数席位。
4. Duverger’s Law and Duverger’s Hypothesis | 杜瓦杰定律与杜瓦杰假设
Duverger’s Law states that single-member plurality electoral systems tend to favour two-party systems. The mechanical effect (small parties are under-represented) and the psychological effect (voters avoid wasting their votes on likely losers) reinforce bipartism. Duverger’s Hypothesis extends this reasoning: proportional representation systems tend to foster multiparty systems, although the link is weaker than the law.
杜瓦杰定律指出，单一选区相对多数制倾向于造就两党制。机械效应（小党代表性不足）和心理效应（选民不愿将选票浪费在可能落败者身上）共同强化了两党格局。杜瓦杰假设将此逻辑延伸：比例代表制倾向于促成多党制，但这一关联不如定律牢固。
5. Median Voter Theorem | 中位选民定理
In a one-dimensional policy space with single-peaked preferences, two vote-maximising parties will converge on the policy preferred by the median voter. Formally, if voters’ ideal points are arrayed along a left–right line, the median position cannot be defeated by any alternative in a pairwise majority vote (Black’s theorem). Under strict conditions, the theorem predicts centrist convergence and policy stability.
在一维政策空间且偏好为单峰的条件下，两个追求选票最大化的政党将会聚到中位选民偏好的政策上。形式上，若选民的最优点沿左右光谱排列，中位点在两两多数决中无法被任何替代方案击败（布莱克定理）。在严格条件下，该定理预测政党将向中间靠拢，政策趋于稳定。
6. Iron Law of Oligarchy | 寡头铁律
Proposed by Robert Michels, the iron law of oligarchy asserts that all large organisations, including democratic parties and trade unions, inevitably evolve into oligarchies. The technical demands of organisation create a professionalised leadership that consolidates power, making genuine internal democracy very difficult to sustain. This ‘law’ is a foundational concept in elite theories of politics.
米歇尔斯提出的寡头铁律主张，包括民主政党和工会在内的一切大型组织，都不可避免地演变为寡头统治。组织的技术性需求缔造出一个集权的职业化领导层，使得真正的内部民主难以持久。这一“定律”是精英政治理论的基础概念。
7. Logic of Collective Action | 集体行动的逻辑
Mancur Olson’s logic of collective action challenges the assumption that groups automatically form to pursue common interests. Large groups face a free-rider problem: rational individuals will not voluntarily contribute to a public good they can enjoy regardless. Selective incentives (benefits reserved for participants) or compulsion are therefore needed to organise latent groups. This theorem explains why producer lobbies are often more potent than consumer or taxpayer groups.
奥尔森的集体行动逻辑挑战了团体会自然形成以追求共同利益的假设。大型集团面临搭便车问题：理性个体不会自愿为一种无论是否付出都能享受的公共物品做贡献。因此，需要选择性激励（仅限参与者获得的利益）或强制手段才能把潜在团体组织起来。这一定理解释了为何生产者游说集团往往比消费者或纳税人团体更具影响力。
8. Prisoner’s Dilemma: A Game Theory Framework | 囚徒困境：博弈论框架
The prisoner’s dilemma models situations where rational individuals choose strategies leading to a collectively suboptimal outcome. Two players, each choosing to Cooperate or Defect, face a payoff structure where T > R > P > S and (T + S) / 2 < R. A common payoff matrix (Player A, Player B) is:
囚徒困境模拟了理性个体选择策略却导致集体次优结果的局面。两个参与者各自选择合作或背叛，面对收益结构 T > R > P > S 且 (T + S) / 2 < R。常见的收益矩阵（参与方 A，参与方 B）如下：
| B cooperates | B defects | |
|---|---|---|
| A cooperates | R=3, R=3 | S=0, T=5 |
| A defects | T=5, S=0 | P=1, P=1 |
Defection is a dominant strategy, yet mutual cooperation yields a higher joint payoff. The model illuminates problems such as arms races, climate change negotiations, and candidate attack strategies in elections.
背叛是占优策略，但相互合作却能带来更高的共同收益。这一模型揭示了军备竞赛、气候变化谈判以及选举中的攻击性策略等问题。
9. Banzhaf Power Index | 班扎夫权力指数
In weighted voting bodies (e.g. the EU Council, US Electoral College via state blocs), the Banzhaf index measures a voter’s power as the proportion of winning coalitions in which the voter is critical – where their defection would turn the coalition into a losing one. For a player i:
在加权投票机构（如欧盟理事会、经由州集团运作的美国选举人团）中，班扎夫指数将一个投票者的权力衡量为其作为关键者的获胜联盟占总关键投票次数的比例——此处“关键”是指该投票者的退出会使联盟从获胜转为失败。对于参与者 i：
Banzhaf(i) = swings by i / total swings of all players
A higher index indicates disproportionate influence. Calculating it requires enumerating all possible coalitions and identifying critical players, a fundamental tool for analysing constitutional design and legislative leverage.
指数值越高代表影响力越不成比例。计算它需要列举所有可能的联盟并识别关键玩家，这是分析宪制设计和立法影响力的基本工具。
10. Political Business Cycle and the Phillips Curve | 政治商业周期与菲利普斯曲线
The political business cycle theory suggests that incumbent governments manipulate fiscal or monetary policy before elections to stimulate growth and lower unemployment, boosting their re-election chances. The short-run trade-off is captured by the expectations-augmented Phillips curve:
政治商业周期理论认为，现任政府会在选举前操纵财政或货币政策以刺激增长、降低失业，从而提升连任机会。这种短期权衡可通过附加预期的菲利普斯曲线加以刻画：
π = πᵉ − β(u − un)
where π is actual inflation, πᵉ is expected inflation, u is unemployment, un is the natural rate, and β > 0. Post-election austerity often follows, creating a boom–bust economic rhythm linked to the electoral calendar.
其中 π 为实际通胀率，πᵉ 为预期通胀率，u 为失业率，un 为自然失业率，β > 0。选举后常常跟随紧缩，形成与选举日程同步的繁荣–萧条经济节律。
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