📚 Informal Cooperation Among States: A Game Theory Perspective | 国家间的非正式合作：博弈论视角
In the complex arena of international relations, states frequently manage their interactions through informal understandings rather than legally binding treaties. These tacit agreements can govern everything from tariff reductions to arms control, often succeeding where formal contracts might fail. Game theory, a rigorous mathematical tool for analysing strategic interactions, offers a compelling explanation for how such informal cooperation emerges, why it persists, and when it is likely to break down. This article explores the key mathematical models that illuminate the delicate balance between cooperation and competition among sovereign states.
在复杂的国际关系舞台上，各国常常通过非正式的理解而非具有法律约束力的条约来管理彼此间的互动。这些默契的协议可以管辖从关税削减到军备控制的一切事务，并且往往在正式合约可能失败的地方取得成功。博弈论作为分析战略互动的严谨数学工具，为这种非正式合作为何出现、为何持续以及何时可能破裂提供了令人信服的解释。本文将探讨那些揭示主权国家间合作与竞争微妙平衡的关键数学模型。
1. The Prisoner's Dilemma Framework | 囚徒困境框架
The most fundamental model for understanding informal cooperation between states is the Prisoner's Dilemma. In this two-player game, each state can choose to Cooperate (C) or Defect (D). The payoff structure creates a tension: mutual cooperation yields a good outcome for both, but each player has a private incentive to defect regardless of what the other does, leading to a suboptimal Nash equilibrium.
理解国家间非正式合作的最基本模型是囚徒困境。在这个双人博弈中，每个国家都可以选择合作（C）或背叛（D）。收益结构制造了一种张力：相互合作为双方带来良好结局，但每个玩家都有独自背叛的动机，不论对方如何选择，这导致一个次优的纳什均衡。
Consider a typical payoff matrix for two states, A and B, where the first number in each cell is A's payoff and the second is B's. We assume T > R > P > S, using the canonical notations Temptation, Reward, Punishment, Sucker's payoff.
考虑两个状态A和B的典型收益矩阵，每个单元格中第一个数字是A的收益，第二个是B的收益。我们假设T > R > P > S，使用经典记号：诱惑、奖励、惩罚和傻瓜收益。
| B: Cooperate | B: Defect | |
| A: Cooperate | (R, R) | (S, T) |
| A: Defect | (T, S) | (P, P) |
nt strategy for each state is to defect, resulting in the equilibrium (D, D) with payoff (1, 1), even though (C, C) with (3, 3) would be better for both. This mirrors the difficulty of achieving informal cooperation.
为了具体起见，设T = 5，R = 3，P = 1，S = 0。该博弈表明每个国家的主导策略是背叛，导致均衡为（D, D）以及收益（1, 1），尽管（C, C）的（3, 3）对双方都更好。这映射了实现非正式合作的困难。
2. Nash Equilibrium and Rational Self-Interest | 纳什均衡与理性自利
In a one-shot Prisoner's Dilemma, the unique Nash equilibrium is mutual defection. Formally, a strategy profile (D, D) is a Nash equilibrium because neither player can unilaterally improve their payoff by switching to Cooperate. If A switches to C while B plays D, A's payoff drops from 1 to 0; if B does the same, B's payoff also falls. This mathematical fact seems to doom any informal cooperation: rational, self-interested states should always defect when interaction ends after a single move.
在单次囚徒困境中，唯一的纳什均衡是相互背叛。形式上，策略组合（D, D）是一个纳什均衡，因为任何一方单方面改为合作都无法提高自己的收益。如果A改为C而B坚持D，A的收益从1降到0；若B做同样的事，B的收益也会下降。这一数学事实似乎宣告了任何非正式合作的死亡：当互动仅在一次行动后就结束时，理性自利的国家应当总是选择背叛。
This reasoning explains why formal treaties often include monitoring and enforcement mechanisms to change the payoff structure; without them, the incentives to cheat are too strong. However, international reality shows much informal cooperation, suggesting that the one-shot model is insufficient.
这一推理解释了为什么正式条约通常包含监督和强制执行机制以改变收益结构；没有这些，欺骗的激励就太强了。然而，国际现实显示出大量的非正式合作，这表明单次模型是不充分的。
3. Repeated Games and the Shadow of the Future | 重复博弈与未来的阴影
The key to understanding informal cooperation lies in repeated interactions. Most state relationships are ongoing, and so the Prisoner's Dilemma is played repeatedly. In a repeated game, a player's strategy can depend on past actions, and the threat of future retaliation can deter defection. The mathematical framework shifts from a single payoff to the sum of discounted future payoffs.
理解非正式合作的关键在于重复互动。大多数国家间关系是持续性的，因此囚徒困境被反复进行。在重复博弈中，玩家的策略可以依赖于过去的行为，未来报复的威胁可以威慑背叛。数学框架从单次收益转向了贴现未来收益的总和。
ayoffs relative to present ones. The total payoff for an infinite sequence of actions is ∑t=0∞ δt ut, where ut is the payoff in round t. A higher δ indicates a greater weight on the future; long-term relationships with patient states favour cooperation.
设贴现因子为δ（0 < δ < 1），它衡量玩家相对于当前收益对未来收益的重视程度。无限序列行动的总收益为∑t=0∞ δt ut，其中ut是第t轮的收益。较高的δ表示对未来赋予更大权重；耐心的国家之间长期关系有利于合作。
4. The Folk Theorem and Equilibria in Repeated Games | 大众定理与重复博弈中的均衡
The Folk Theorem states that in an infinitely repeated game, any payoff profile that gives each player at least their minimax payoff can be sustained as a Nash equilibrium, provided the discount factor is sufficiently close to 1. This means that nearly any pattern of behaviour, including sustained mutual cooperation, can be rationalised as an equilibrium outcome.
大众定理指出，在无限重复博弈中，任何至少给予每个玩家其最小最大收益的收益组合，只要贴现因子足够接近1，就可以作为纳什均衡被支撑。这意味着几乎任何行为模式，包括持续的相互合作，都可以作为均衡结果合理化。
For our Prisoner' minimax payoff for each player is 1 (they can always guarantee 1 by defecting). Thus, (3,3) is feasible and strictly better than the minimax, so by the Folk Theorem, it can be an equilibrium if players are patient enough. This provides a mathematical foundation for informal cooperation.
对于我们的囚徒困境，T=5，R=3，P=1，S=0，每个玩家的最小最大收益是1（他们总可以通过背叛保证1）。因此，（3,3）是可行的且严格优于最小最大值，所以根据大众定理，如果玩家有足够的耐心，它可以成为一个均衡。这为非正式合作提供了数学基础。
5. Trigger Strategies and Condition for Cooperation | 触发策略与合作条件
The most famous strategy that sustains cooperation is the Grim Trigger: cooperate in the first round and continue cooperating as long as the opponent cooperates; if the opponent ever defects, defect forever thereafter. To check whether this is an equilibrium, compare the payoff from adhering to the strategy with the payoff from a one-time deviation.
支撑合作最著名的策略是严厉触发策略：第一轮合作，此后只要对手合作就继续合作；一旦对手背叛，就永远背叛。要检验这是否是均衡，需比较坚持该策略的收益与一次性偏离的收益。
eration is sustained if R/(1 − δ) ≥ T + δP/(1 − δ). Rearranging gives δ ≥ (T − R)/(T − P).
如果双方都使用严厉触发策略，他们每期得到R，总收益 = R + δR + δ²R + ... = R/(1 − δ)。如果玩家A在第一期背叛而B合作，A立即得到T，但此后双方都背叛，所以A从第二期起得到P：总和 = T + δP + δ²P + ... = T + δP/(1 − δ)。合作被维持的条件是R/(1 − δ) ≥ T + δP/(1 − δ)。整理得 δ ≥ (T − R)/(T − P)。
1, this requires δ ≥ (5 − 3)/(5 − 1) = 2/4 = 0.5. So as long as a state values next period's payoff at least half as much as today' relationships encourage informal understandings: the shadow of the future is heavy enough.
使用T=5，R=3，P=1，这要求δ ≥ (5 − 3)/(5 − 1) = 2/4 = 0.5。因此只要一个国家将下一期的收益至少视为今天收益的一半，合作就可以成为均衡。这解释了为什么稳定持久的关系促进非正式理解：未来的阴影足够沉重。
6. Tit-for-Tat and Other Reciprocal Strategies | 以牙还牙及其他互惠策略
In practice, states rarely employ the extreme Grim Trigger; instead they use more forgiving but still reciprocal strategies. The Tit-for-Tat (TFT) strategy simply repeats the opponent's previous move: start with cooperation, then copy whatever the opponent did in the last period. Mathematically, it has been shown in Axelrod's tournaments that TFT is a robustly successful strategy, promoting cooperation while allowing for recovery after accidental defections.
在实践中，国家很少采用极端的严厉触发策略；相反，它们使用更宽容但仍具互惠性的策略。以牙还牙（TFT）策略仅仅是重复对手上一轮的行动：以合作开始，然后照搬对手上期所做的一切。从数学上讲，阿克塞尔罗德锦标赛已表明TFT是一个强健成功的策略，它促进合作同时允许在意外背叛后恢复。
If two TFT players meet, they cooperate forever. If a TFT player meets a constant defector, it defects from period 2 onward, limiting losses. This balance of strict reciprocity and forgiveness makes TFT a natural model for informal state interactions, where accidental policy missteps can be corrected without total collapse of the relationship.
如果两个TFT玩家相遇，他们会永远合作。如果一个TFT玩家遇到一个总是背叛的玩家，它从第二期起背叛，从而限制损失。这种严格互惠与宽容的平衡使TFT成为国家间非正式互动的自然模型，在意外政策失误后关系不会完全崩溃。
7. The Role of Reputation and Signalling | 声誉与信号传递的作用
Informal cooperation often hinges on reputation: a state known for honouring informal deals attracts more cooperation, while a state with a history of defection is shunned. In repeated games with incomplete information, the beliefs about another player's type (e.g., "cooperative" or "rational defector") update via Bayes' rule, linking current actions to future payoffs through their reputational effect.
非正式合作往往取决于声誉：以遵守非正式协议闻名的国家能吸引更多合作，而有背叛历史的国家则被回避。在不完全信息的重复博弈中，对另一玩家类型（例如“合作型”或“理性背叛型”）的信念通过贝叶斯法则更新，将当前行动通过声誉效应与未来收益联系起来。
Mathematically, even a small probability that the opponent is a "committed cooperator" can significantly raise the value of cooperation, because defecting destroys that belief and yields lower long-term payoffs. This formalises why states go to great lengths to build a trustworthy image, and why informal understandings can be so stable: reputation becomes a form of collateral.
从数学上讲，即使对方是“坚定合作者”的概率很小，也能显著提升合作的价值，因为背叛会摧毁这一信念并带来更低的长期收益。这正式说明了为什么国家不遗余力地建立可信形象，以及为什么非正式理解可以如此稳定：声誉成为了一种抵押品。
8. Discount Factor Variability Across States | 各国贴现因子的异质性
Not all states have the same δ. A state facing domestic political pressure for quick results, or one expecting a sudden regime change, effectively has a low δ, making informal cooperation harder. Conversely, a stable democratic state with long-term economic interests typically has a high δ, making it a reliable partner.
并非所有国家都有相同的δ。一个面临国内政治压力要求快速见效的国家，或预期发生突然政权更迭的国家，实际上具有较低的δ，使得非正式合作更困难。相反，一个具有长期经济利益的稳定民主国家通常有较高的δ，使其成为可靠的合作伙伴。
We can model this heterogeneity by assuming each player i has a private δi. In a bilateral relationship, cooperation requires both δi the threshold (T − R)/(T − P) for their respective grim trigger calculations. If either δi is too low, informal understanding collapses, which mirrors real-world failures when short-sighted actors enter negotiations.
我们可以通过假设每个玩家i有一个私人δi来模拟这种异质性。在双边关系中，合作要求在各自的严厉触发计算中两个δi都≥阈值(T − R)/(T − P)。如果任一δi过低，非正式理解就会崩溃，这正好反映了现实世界中目光短浅的行为体参与谈判时失败的场景。
9. Coordination and the Stag Hunt Game | 协调与猎鹿博弈
Not all informal cooperation is a Prisoner's Dilemma. Some situations resemble the Stag Hunt, where there are two equilibria: both cooperate (hunt stag) and both defect (hunt hare). Here, the risk of unilateral cooperation is high, but mutual cooperation yields the best payoff. The mathematical payoff structure features R > T and P > S, with a mixed-strategy equilibrium that determines the likelihood of coordination failure.
并非所有非正式合作都是囚徒困境。有些情形类似于猎鹿博弈，其中有两个均衡：双方合作（猎鹿）和双方背叛（猎兔）。在此，单方面合作的风险很高，但相互合作产生最高收益。数学收益结构的特征是R > T和P > S，并存在一个混合策略均衡，决定了协调失败的可能性。
he coordination problem explains why states sometimes fail to achieve mutually beneficial informal agreements even when the incentives seem strong: they fear being left cooperating alone while the other cheats.
例如，设R=4，T=3，P=2，S=0。协调问题解释了为什么国家有时即使激励看似很强，也无法达成互利的非正式协议：它们害怕自己独自合作而对方欺骗。
10. Multiple States and the n-Player Dilemma | 多国博弈与n人困境
International informal cooperation often involves many states, as in climate change or trade norms. This can be modelled as an n-player Prisoner's Dilemma or a public goods game, where each state chooses a contribution level. The Nash equilibrium involves free-riding: each state contributes less than the socially optimal level because of the private cost, unless an enforcement mechanism or reputation spillovers create incentives.
国际非正式合作往往涉及众多国家，如在气候变化或贸易规范中。这可以建模为一个n人囚徒困境或公共品博弈，每个国家选择贡献水平。纳什均衡涉及搭便车：由于私人成本，每个国家的贡献低于社会最优水平，除非有执行机制或声誉溢出创造激励。
Mathematically, let the payoff for state i be ui = b(∑xj) − c xi, where b is the benefit from total contributions, c is the marginal cost, and xi is i' , but because aggregate benefit is shared, the equilibrium contribution is too low. This formalises the fragility of large-group informal cooperation.
从数学上看，设国家i的收益为ui = b(∑xj) − c xi，其中b是总贡献带来的收益，c是边际成本，xi是i的贡献量。在没有关联的情况下，每个i让边际收益=c，但因为总收益被共享，均衡贡献量过低。这正式说明了大群体非正式合作的脆弱性。
11. Empirical Evidence and Real-World Applications | 经验证据与现实世界应用
Game-theoretic predictions align well with observed state behaviour. For example, the General Agreement on Tariffs and Trade (GATT) functioned largely through informal reciprocity before being formalised. Compliance was sustained by the expectation of repeated trade interactions and the threat of retaliatory tariffs, exactly as a repeated Prisoner's Dilemma suggests. Data on diplomatic exchanges and arms races also show patterns predictable by these models.
博弈论的预测与观察到的国家行为高度一致。例如，关税及贸易总协定（GATT）在被正式化之前主要通过非正式互惠运作。对重复贸易互动的预期和报复性关税的威胁维持了履约，恰如重复囚徒困境所揭示的那样。外交交流与军备竞赛的数据也显示出这些模型可预测的模式。
Moreover, studies of international environmental agreements show that informal cooperation emerges most reliably when monitoring is easy and the shadow of the future is long, both conditions corresponding to high δ and transparency in the mathematical models.
此外，关于国际环境协议的研究表明，当监测容易且未来的阴影很长时，非正式合作最容易出现，这两项条件均对应于数学模型中的高δ和透明度。
12. Conclusion: The Mathematics of Trust Without Treaties | 结论：无条约信任的数学原理
Game theory demonstrates that informal cooperation among states is not a mere matter of goodwill but a rational outcome of repeated strategic interaction. Key parameters such as the discount factor, the payoff structure, and the availability of trigger strategies mathematically determine whether cooperation can be sustained. While formal treaties alter the payoff matrix through legal enforcement, informal understandings rely wholly on the self-enforcing logic of repeated games, making the mathematics of reciprocity essential to understanding why states so often cooperate without writing a single rule.
博弈论证明，国家间的非正式合作不仅仅是善意的结果，而是重复策略互动的理性产物。贴现因子、收益结构以及触发策略的可获得性等关键参数从数学上决定了合作能否被维持。尽管正式条约通过法律执行改变了收益矩阵，非正式理解完全依赖于重复博弈的自我执行逻辑，这使得互惠的数学原理对于理解为何国家经常在不写下任何规则的情况下合作至关重要。
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