📚 Origins and development of international law and institutions | 国际法与制度的起源与发展：数学视角
International law and institutions have evolved over centuries as states sought predictable, peaceful mechanisms to manage their relations. By applying mathematical models—ranging from game theory and network analysis to optimisation—we can rigorously examine why sovereign states bind themselves to legal rules, how institutions emerge, and what makes them durable. This article explores the mathematical underpinnings of these historical processes, demonstrating that many core features of international law can be understood through the lens of strategic interaction, repeated games, and dynamic systems.
几个世纪以来，随着国家寻求可预测的和平机制来管理相互关系，国际法与制度逐步演进。通过应用博弈论、网络分析和最优化等数学模型，我们可以严谨地考察主权国家为何自愿受法律规则约束、制度如何产生以及什么使其持久。本文探讨这些历史过程的数学基础，表明国际法的许多核心特征可以通过战略互动、重复博弈和动态系统的视角加以理解。
1. Strategic interaction and the birth of international law | 战略互动与国际法的诞生
The Peace of Westphalia (1648) is often cited as the origin of modern international law, establishing the principle of state sovereignty. From a mathematical standpoint, this can be modelled as a shift from a Hobbesian state of nature toward a coordination equilibrium. Before 1648, the lack of mutually recognised boundaries made cooperation unstable; states were trapped in a one-shot Prisoner's Dilemma where defection dominated. The treaty created focal points, reducing transaction costs and allowing repeated interactions to generate cooperative norms.
1648年《威斯特伐利亚和约》常被视为现代国际法的起源，确立了国家主权原则。从数学角度看，这可以建模为由霍布斯自然状态向协调均衡的转变。此前，缺乏彼此承认的边界使合作不稳定，国家陷入单次囚徒困境，背信占优。条约创造了聚焦点，降低了交易成本，使重复互动能够生成合作规范。
- State interactions as non‑zero‑sum games
- 从零和博弈转向非零和博弈框架
- Focal points reduce strategic uncertainty
- 聚焦点降低战略不确定性
Payoff matrix for a generic coordination game: two symmetric equilibria at (C,C) and (D,D), but (C,C) is payoff‑dominant once institutions exist.
一般协调博弈的支付矩阵：存在两个对称均衡 (C,C) 和 (D,D)，但制度出现后 (C,C) 成为支付占优均衡。
| C | D | |
| C | 3, 3 | 0, 2 |
| D | 2, 0 | 1, 1 |
2. The Prisoner's Dilemma and early treaty compliance | 囚徒困境与早期条约遵守
In a one‑shot Prisoner's Dilemma, rational self‑interest leads to mutual defection. Yet history shows that even without a world government, states often comply with treaties. Mathematically, the resolution lies in iterating the game. If the discount factor δ is sufficiently high, the threat of future punishment can sustain cooperation in a subgame‑perfect equilibrium, as shown by the Folk Theorem. Early bilateral treaties—such as trade agreements—can be understood as infinite‑horizon repeated games where reputation becomes a strategic asset.
在单次囚徒困境中，理性自利导致相互背叛。然而历史表明，即使没有世界政府，国家也经常遵守条约。数学上，答案在于将博弈重复进行。如果贴现因子 δ 足够高，未来惩罚的威胁可以在子博弈完美均衡中维持合作，这由民间定理所证明。早期双边条约（如贸易协议）可理解为无限期重复博弈，声誉成为战略资产。
ondition: (T - R) / (T - P) ≤ δ, where T = temptation, R = reward, P = punishment payoffs.
合作条件：(T - R) / (T - P) ≤ δ，其中 T = 背信诱惑，R = 合作报酬，P = 惩罚收益。
n (5-3)/(5-1)=0.5 ≤ 0.8, so cooperation can be sustained. This threshold explains why long‑lived states with frequent interactions are more likely to develop binding customs.
若 δ = 0.8，且收益 T=5, R=3, P=1，则 (5-3)/(5-1)=0.5 ≤ 0.8，合作可维持。这一门槛解释了为何具有频繁互动的长寿国家更可能发展出有约束力的习惯。
3. Institutionalisation as a commitment device | 制度化作为承诺机制
International institutions such as the League of Nations or the United Nations can be viewed as commitment devices that alter the payoff structure. By creating monitoring mechanisms and dispute‑resolution procedures, they transform a Prisoner's Dilemma into an Assurance game, lowering the risk of unilateral defection. Mathematically, this is equivalent to adding a state‑contingent penalty function: the expected cost of violation includes not only retaliation but also institutional sanctions such as exclusion from decision‑making or loss of voting rights.
国际联盟或联合国等国际机构可视为承诺装置，改变支付结构。通过创设监督机制和争端解决程序，它们将囚徒困境转化为保证博弈，降低了单方面背信的风险。数学上，这等价于添加一个状态相依的惩罚函数：违约的预期成本不仅包括报复，还包括制度性制裁，如被排除在决策之外或失去投票权。
β·S(compliance) ; πᵢ(defect) = T - γ·F(defection), where β,γ are institutional parameters.
新支付：πᵢ(合作) = R + β·S(合规) ；πᵢ(背信) = T - γ·F(违约)，其中 β,γ 是制度参数。
The effectiveness of an institution depends on the size of γ and the credibility of the penalty. Through statistical analysis, we can measure how the presence of a permanent secretariat (γ↑) correlates with higher treaty compliance rates.
制度的有效性取决于 γ 的大小和惩罚的可信度。通过统计分析，我们可以衡量常设秘书处的存在（γ↑）如何与更高的条约遵守率相关。
4. Network theory and the spread of legal norms | 网络理论与法律规范的扩散
Legal norms do not spread randomly; they diffuse through diplomatic, trade, and cultural networks. Using graph theory, we can represent states as nodes and treaties or diplomatic ties as edges. The degree distribution and clustering coefficient of the treaty network reveal how quickly a new norm—such as the prohibition of chemical weapons—can cascade. Small‑world properties often observed in these networks imply that a few highly connected states can accelerate global acceptance.
法律规范并非随机传播，而是通过外交、贸易和文化网络扩散。利用图论，我们可以将国家表示为节点，条约或外交关系表示为边。条约网络的度分布和聚类系数揭示了新规范——如禁止化学武器——可以多快发生级联。这些网络中常见的小世界特性意味着少数高度连通的国家可以加速全球接受。
Basic epidemic threshold for norm adoption: if the spectral radius ρ of the adjacency matrix exceeds 1/λ (adoption rate), a global cascade occurs.
规范采纳的基本流行阈值：若邻接矩阵的谱半径 ρ 超过 1/λ（采纳率），则发生全局级联。
For example, the Geneva Conventions of 1949 were ratified quickly because major powers acted as hubs, effectively lowering the threshold for other states through diplomatic pressure and aid conditionality.
例如，1949年日内瓦公约得以迅速批准，是因为大国充当了枢纽节点，通过外交压力和援助条件有效降低了其他国家的采纳门槛。
5. Evolutionary game theory and customary international law | 演化博弈论与习惯国际法
Customary international law emerges from consistent state practice accompanied by a sense of legal obligation (opinio juris). This process mirrors an evolutionary game where successful strategies replicate. Consider a population of states playing either 'obey custom' (O) or 'violate' (V). If the payoff to O rises with the proportion of others obeying, the system can evolve toward a stable equilibrium where custom becomes self‑enforcing, even without central enforcement.
习惯国际法源于一贯的国家实践和法律确念（opinio juris）。这一过程类似于演化博弈，其中成功策略得以复制。设想一组国家采用“遵守习惯”(O) 或“违反”(V)。如果 O 的收益随其他国家的遵守比例上升，则系统可以朝着一个稳定均衡演化，习惯变得自我实施，无需中央执法。
Replicator dynamics: dx/dt = x(1 - x)[E(O) - E(V)], where x = fraction of O‑strategists.
复制者动态：dx/dt = x(1 - x)[E(O) - E(V)]，其中 x = O 策略者的比例。
d such rules, but positive mutual benefits and retaliatory costs drove the population toward near‑universal compliance, stabilising at x = 1 if the incentives are correctly aligned.
历史上的例子包括禁止海盗行为及外交豁免的发展。起初只有少数国家尊重这些规则，但正和互惠与报复成本驱使整体向近乎普遍遵守演化，若激励匹配正确，将稳定在 x = 1。
6. Public goods and multilateral institutions | 公共品与多边机构
Many international institutions aim to provide global public goods—such as peace, clean air, or financial stability. Their provision suffers from a free‑rider problem, classically modelled as a prisoners’ dilemma with n players. The optimal contribution of each state depends on others’ efforts, leading to under‑provision unless a mechanism of selective incentives or matching commitments is introduced. Large institutions like the World Health Organization (WHO) design funding rules to partially internalise the externality.
许多国际机构旨在提供全球公共品——如和平、清洁空气或金融稳定。其供给面临搭便车问题，经典模型为 n 人囚徒困境。每个国家的最优贡献依赖于他人努力，除非引入选择性激励或匹配承诺机制，否则会导致供给不足。世界卫生组织等大机构设计资金规则以部分内化外部性。
gate supply of public good G = Σ gᵢ, with individual utility Uᵢ = B(Σ gᵢ) - C(gᵢ), where B' > 0, C' > 0. Nash equilibrium gives gᵢ such that B'(G) = C'(gᵢ), which is less than Pareto optimal.
公共品总供给 G = Σ gᵢ，个体效用 Uᵢ = B(Σ gᵢ) - C(gᵢ)，B' > 0, C' > 0。纳什均衡给出 gᵢ 满足 B'(G) = C'(gᵢ)，低于帕累托最优。
The Kyoto Protocol and Paris Agreement incorporate mechanisms like emissions trading (a market‑based matching scheme) to shift the equilibrium closer to the social optimum—a clever institutional design rooted in mechanism design theory.
《京都议定书》和《巴黎协定》纳入了排放交易等机制（一种基于市场的匹配方案），使均衡向社会最优靠近——这是扎根于机制设计理论的巧妙制度设计。
7. Power indices and voting in international organisations | 权力指数与国际组织的表决
Weighted voting systems in the IMF, World Bank, and UN Security Council reflect power asymmetries. The Banzhaf and Shapley‑Shubik power indices quantify the a priori influence of each member state on outcomes. For instance, in the UN Security Council, the five permanent members hold veto power, giving them a Shapley‑Shubik index far above their share of seats. This mathematical structure explains why institutional reform is so contested: small changes in voting weights can drastically redistribute power.
国际货币基金组织、世界银行和联合国安理会的加权表决制反映了权力不对称。班扎夫权力指数和沙普利‑舒比克权力指数量化了每个成员国对结果的先验影响力。例如，在联合国安理会，五个常任理事国拥有否决权，其沙普利‑舒比克指数远高于其席位比例。这一数学结构解释了为何制度改革争议如此之大：投票权重微调可能急剧再分配权力。
Shapley value for player i: φᵢ = Σ (|S|! (n - |S| - 1)! / n!) [v(S ∪ {i}) - v(S)] where S is a coalition without i.
局中人 i 的沙普利值：φᵢ = Σ (|S|! (n - |S| - 1)! / n!) [v(S ∪ {i}) - v(S)]，其中 S 不含 i。
In the IMF, quota reforms often involve intense bargaining because a shift of even 0.5% of total votes can alter whether a country holds a blocking minority. Mathematical modelling of voting weights helps negotiators understand the precise levers of influence.
在 IMF，份额改革常伴随激烈讨价还价，因为总投票权 0.5% 的变动就可能改变一国是否拥有少数否决权。投票权重的数学建模帮助谈判者理解影响力的精确杠杆。
8. Conflict bargaining and the role of law of war | 冲突谈判与战争法的作用
War can be modelled as costly bargaining under asymmetric information. Fearonʼs rationalist explanation shows that war occurs when states cannot credibly communicate their resolve. The laws of war (jus in bello) serve as a signalling mechanism and a constraint set. Mathematically, they alter the payoff space by imposing costs on prohibited actions, shaping the set of feasible enforceable agreements during crises.
战争可建模为不对称信息下的昂贵谈判。费伦的理性主义解释表明，战争发生于国家无法可信地传达决心之时。战争法（jus in bello）充当信号机制和约束集。数学上，它们通过为禁止行为施加成本来改变支付空间，塑造危机期间可行可执行协议集。
: EUₐ = p·πₐ(win) + (1-p)·πₐ(lose) - cₐ, where cₐ includes legal sanctions for violations.
国家 A 的战争期望效用：EUₐ = p·πₐ(胜) + (1-p)·πₐ(败) - cₐ，其中 cₐ 包含对违法的法律制裁。
The 1899 and 1907 Hague Conventions codified many customary rules and can be seen as raising the cost parameter for atrocity, thereby reducing the likelihood of extreme violence even in the midst of war—a classic legal engineering of the conflict game.
1899 年和 1907 年海牙公约编纂了许多习惯规则，可视为提高了暴行的成本参数，从而在战争中降低了极端暴力的可能性——这是对冲突博弈的典型法律工程。
9. Trade law and optimal tariff games | 贸易法与最优关税博弈
The General Agreement on Tariffs and Trade (GATT) and later the World Trade Organization (WTO) address a fundamental prisoner’s dilemma in trade: each country benefits from its own tariffs but loses from others’ retaliation. The Nash equilibrium tariff is higher than optimal, so a binding agreement to reduce tariffs can make everyone better off. The WTO dispute settlement mechanism adds a repeated‑game punishment clause that supports the cooperative equilibrium.
关税与贸易总协定（GATT）及后来的世界贸易组织（WTO）处理贸易中的基本囚徒困境：每个国家从本国关税获益，但因他国报复而受损。纳什均衡关税高于最优水平，因此有约束力的降税协议可改善所有各方。WTO 争端解决机制添加了重复博弈惩罚条款，支撑合作均衡。
ormula: t* = 1 / (ε*), where ε* is the foreign export supply elasticity. However, in a non‑cooperative Nash: tN > t*.
最优关税公式：t* = 1 / (ε*)，其中 ε* 为外国出口供给弹性。然而在非合作纳什中：tN > t*。
The most‑favoured‑nation principle can be modelled as a multilateral equal‑treatment constraint that prevents bilateral opportunism, thereby reducing the discount factor needed to sustain cooperation across many trading partners.
最惠国待遇原则可建模为多边平等对待约束，防止双边机会主义，从而降低在众多贸易伙伴间维持合作所需的贴现因子。
10. Dynamic systems and the evolution of treaty regimes | 动态系统与条约制度的演化
tion or withdrawal rate. Institutional design can raise r (by providing technical assistance) or lower m (by imposing withdrawal penalties), thus shifting the steady state from N* = 0 to N* = K(1 - m/r), as long as r > m.
条约制度常呈现出参与滚雪球式的临界点。逻辑斯蒂微分方程可以捕捉采纳动态。若 N 为批准国数量，K 为最大值，则 dN/dt = rN(1 - N/K) - mN，其中 r 为规范接受的内禀增长率，m 为退出率。制度设计可提高 r（通过提供技术援助）或降低 m（通过施加退出惩罚），从而将稳态从 N* = 0 移动至 N* = K(1 - m/r)，只要 r > m。
* = K(1 - m/r) is stable and positive if r > m; otherwise, N* = 0 is the global attractor.
稳态分析：若 r > m，N* = K(1 - m/r) 稳定且为正；否则 N* = 0 是全局吸引子。
The 1997 Mine Ban Treaty is a prime example: a low initial r was boosted by civil society campaigns (increasing r dramatically) and stigmatisation of non‑ratifiers (increasing m for defectors) until the regime reached near‑universality.
1997 年《禁雷公约》是一个典型例子：起初较低的 r 经由公民社会运动（大幅提高 r）和对非批准国的污名化（提高背离者的 m）得以推升，直至制度接近普遍参与。
11. Agent‑based modelling and institutional resilience | 基于主体的建模与制度韧性
Agent‑based models (ABMs) simulate heterogeneous states with bounded rationality, which better captures the development of international law than equilibrium‑focused models. By programming states with different preferences, learning rules, and network positions, researchers can study how institutions emerge spontaneously. An ABM of international environmental law might show that even when collective action is rational, institutions require a minimum critical mass of intrinsically motivated states to ignite cooperation, consistent with historical patterns of leadership by “green” states.
基于主体的模型（ABM）模拟具有有限理性的异质性国家，比聚焦均衡的模型更能捕捉国际法的发展。通过为国设定不同偏好、学习规则和网络位置，研究者可研究制度如何自发涌现。一个国际环境法的 ABM 可能表明，即使集体行动是理性的，制度也需要最低临界质量的内生激励国家来点燃合作，这与历史上“绿色”国家发挥引领作用的模式一致。
(u_cooperate - u_average), where pᵢ is the propensity to cooperate, α the adaptation rate.
简单学习规则：pᵢ(t+1) = pᵢ(t) + α(u_合作 - u_平均)，其中 pᵢ 为合作倾向，α 为适应速率。
These simulations reveal that small institutional innovations, such as a rapid‑response fund, can reduce the variance of payoffs and tip the system toward a cooperative basin of attraction.
这些模拟表明，小型制度创新，如快速反应基金，可以减少收益方差，使系统向合作的吸引盆倾斜。
12. Conclusion: mathematical literacy and the future of international law | 结论：数学素养与国际法的未来
The co‑evolution of international law and institutions with mathematical modelling is more than an academic exercise. As global challenges become increasingly complex—climate change, cyber security, pandemics—policymakers equipped with the analytical tools of game theory, network science, and dynamical systems can design more effective treaties and organisations. Understanding the mathematical logic behind compliance, power distribution, and norm diffusion enables a deeper appreciation of why legal frameworks succeed or fail, and how we can engineer them for a more cooperative world.
国际法与制度与数学建模的共同演进不仅是学术练习。随着全球挑战日益复杂——气候变化、网络安全、大流行——具备博弈论、网络科学和动力系统分析工具的决策者可以设计更有效的条约和组织。理解合规、权力分配和规范扩散背后的数学逻辑，使我们能更深刻地认识法律框架何以成败，以及如何设计它们以构建一个更具合作性的世界。
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