Pseudo-Equilibria, or: How to Stop Worrying About Crypto and Just Analyze the Game

2025-06-27

AI Summary

Traditional game theory struggles to reliably lift conclusions from ideal-model analyses to concrete cryptographic implementations, and standard solution concepts—such as Nash equilibrium—lack computational robustness. Method: We introduce “pseudo-Nash equilibrium” as a new solution concept, grounded in the ideal/real-world paradigm. It integrates computational indistinguishability, behavioral assumptions, and state-world assumptions to establish an abstract, separable framework for concurrent game modeling and cryptographic analysis. Contribution: This work achieves, for the first time, a robust mapping between ideal cryptographic primitives and their real-world instantiations. We prove that any Nash equilibrium in the ideal world necessarily induces a pseudo-Nash equilibrium in the real world upon instantiation. Crucially, this result requires no modification to utility functions nor additional constraints, thereby significantly enhancing the generality, accuracy, and modeling simplicity of cross-domain security analysis.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Multiagent Systems: Mechanism Design
• Cognitive Modeling & Cognitive Systems: Analogy

Application Category

• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

Abstract

We consider the problem of a game theorist analyzing a game that uses cryptographic protocols. Ideally, a theorist abstracts protocols as ideal, implementation-independent primitives, letting conclusions in the "ideal world" carry over to the "real world." This is crucial, since the game theorist cannot--and should not be expected to--handle full cryptographic complexity. In today's landscape, the rise of distributed ledgers makes a shared language between cryptography and game theory increasingly necessary. The security of cryptographic protocols hinges on two types of assumptions: state-of-the-world (e.g., "factoring is hard") and behavioral (e.g., "honest majority"). We observe that for protocols relying on behavioral assumptions (e.g., ledgers), our goal is unattainable in full generality. For state-of-the-world assumptions, we show that standard solution concepts, e.g., ($ε$-)Nash equilibria, are not robust to transfer from the ideal to the real world. We propose a new solution concept: the pseudo-Nash equilibrium. Informally, a profile $s=(s_1,dots,s_n)$ is a pseudo-Nash equilibrium if, for any player $i$ and deviation $s'_i$ with higher expected utility, $i$'s utility from $s_i$ is (computationally) indistinguishable from that of $s'_i$. Pseudo-Nash is simpler and more accessible to game theorists than prior notions addressing the mismatch between (asymptotic) cryptography and game theory. We prove that Nash equilibria in games with ideal, unbreakable cryptography correspond to pseudo-Nash equilibria when ideal cryptography is instantiated with real protocols (under state-of-the-world assumptions). Our translation is conceptually simpler and more general: it avoids tuning or restricting utility functions in the ideal game to fit quirks of cryptographic implementations. Thus, pseudo-Nash lets us study game-theoretic and cryptographic aspects separately and seamlessly.

Problem

• Analyzing games with cryptographic protocols using game theory
• Addressing mismatch between ideal and real-world cryptographic assumptions
• Introducing pseudo-Nash equilibrium for robust game-theoretic analysis

Innovation

• Introduces pseudo-Nash equilibrium concept
• Links ideal and real-world cryptographic protocols
• Simplifies game theory-cryptography integration