Bayesian Approach to Cooperative Security in Distributed Signature

Distributed threshold signature schemes provide the foundation for decentralized security, enabling consensus and asset custody without a single point of failure. However, the transition from centralized control to voluntary participation introduces a critical vulnerability: strategic non-cooperation. While cryptographic primitives guarantee security against malicious adversaries, they often fail to account for rational agents who may withhold participation to minimize private costs. Furthermore, existing literature predominantly models these interactions under the assumption of complete information, neglecting the population uncertainty inherent to open, permissionless networks. This thesis addresses this gap by developing a Bayesian game-theoretic framework to analyze cooperative incentives in distributed signatures. We model the system as a game with incomplete information, where agents must determine their strategy based on a probabilistic estimate of the network size. The analysis contrasts two distinct network environments: a fragile regime modeled by the Geometric distribution and a resilient regime modeled by the Poisson distribution. Our findings reveal a fundamental tension between network stability and incentive compatibility. We demonstrate that in fragile networks, the inherent risk of node insufficiency acts as a natural coordination mechanism, compelling high participation to prevent system failure. In contrast, in resilient networks, statistical confidence in the population size fosters a free-rider problem, leading to rational apathy and coordination failure. Through the construction of dynamic force maps, we quantify a critical design trade-off. Fragile networks exhibit extreme sensitivity to cost increases, whereas resilient networks demonstrate strategic instability, requiring strong initial beliefs to sustain cooperation. We conclude that physical robustness does not imply strategic security, necessitating the design of external incentive mechanisms that balance the robustness of large networks with the participatory urgency of small ones.