用博弈论解释为何理性对手会维持系统低效稳定状态
Rational Adversaries and the Maintenance of Fragility: A Game-Theoretic Theory of Rational Stagnation
- 构建基于潜在损失的理性对手模型，解释系统为何陷入低效均衡
- 发现合作脆弱带：在特定互认比率下，合作与背叛均可能为均衡
- 揭示三种策略态，适用于分析社交算法与政治信任等现实场景
协作系统常长期处于次优但稳定的状况。本文将此现象称为‘理性停滞’，并将其解释为由理性对手维持的均衡，其效用遵循潜在损失原则 $u_{D} = U_{ideal} - U_{actual}$。从囚徒困境出发，我们证明变换 $u_{i}' = a hinspace u_{i} + b hinspace u_{j}$ 与互认比值 $w = b/a$ 可生成一个脆弱合作带 $[w_{ ext{min}}, hinspace w_{ ext{max}}]$，在此区间内 (C,C) 与 (D,D) 均为均衡。扩展至具有随机合作收益 $R_t$ 与干预成本 $(C_c, hinspace C_m)$ 的动态模型，通过贝尔曼方程分析得出三种战略状态：即时破坏、理性停滞、干预放弃。附录进一步将效用推广至参考依赖非线性形式，并证明其在参考点变动下的稳定性，确保框架鲁棒性。应用示例涵盖社交媒体算法与政治信任，说明敌对理性如何刻意维持脆弱性。
原文摘要 · Abstract (English)
Cooperative systems often remain in persistently suboptimal yet stable states. This paper explains such "rational stagnation" as an equilibrium sustained by a rational adversary whose utility follows the principle of potential loss, $u_{D} = U_{ideal} - U_{actual}$. Starting from the Prisoner's Dilemma, we show that the transformation $u_{i}' = a\,u_{i} + b\,u_{j}$ and the ratio of mutual recognition $w = b/a$ generate a fragile cooperation band $[w_{\min},\,w_{\max}]$ where both (C,C) and (D,D) are equilibria. Extending to a dynamic model with stochastic cooperative payoffs $R_{t}$ and intervention costs $(C_{c},\,C_{m})$, a Bellman-style analysis yields three strategic regimes: immediate destruction, rational stagnation, and intervention abandonment. The appendix further generalizes the utility to a reference-dependent nonlinear form and proves its stability under reference shifts, ensuring robustness of the framework. Applications to social-media algorithms and political trust illustrate how adversarial rationality can deliberately preserve fragility.
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