On the Cryptographic Hardness of Finding a Nash Equilibrium

Abstract

We prove that finding a Nash equilibrium of a game is hard, assuming the existence of indistinguishability obfuscation and one-way functions with sub-exponential hardness. We do so by showing how these cryptographic primitives give rise to a hard computational problem that lies in the complexity class PPAD, for which finding Nash equilibrium is complete.Previous proposals for basing PPAD-hardness on program obfuscation considered a strong "virtual black-box" notion that is subject to severe limitations and is unlikely to be realizable for the programs in question. In contrast, for indistinguishability obfuscation no such limitations are known, and recently, several candidate constructions of indistinguishability obfuscation were suggested based on different hardness assumptions on multilinear maps.Our result provides further evidence of the intractability of finding a Nash equilibrium, one that is extrinsic to the evidence presented so far. * MIT.

Hardness of Continuous Local Search: Query Complexity and Cryptographic Lower Bounds

“…Our results extend the recent result of Bitanski et al [12] and Garg et al [26], who constructed distributions of hard instances of End-of-Line based on indistinguishability obfuscation. We establish that similarly to the End-of-Line, under these cryptographic assumptions, there is a distribution of hard instances for End-of-Metered-Line.…”

“…In particular, they showed hardness of Sink-of-Verifiable-Line under the assumption of existence of one-way permutations and indistinguishability obfuscation both with polynomial security, and furthermore they showed an alternative construction of hard instances of Sink-ofVerifiable-Line assuming compact functional encryption (a special kind of public-key encryption which supports restricted secret keys that enable a key holder to learn a specific function of the encrypted data, but learn nothing else), which is a plausibly weaker cryptographic primitive than indistinguishability obfuscation. Using the reduction to End-of-Line of [2,12] their result implies cryptographic hardness for PPAD.…”

“…However, given that virtual black-box obfuscation is not achievable in general, it remained an obvious important open problem to show hardness for PPAD under weaker cryptographic assumptions. Bitanski et al [12] were the first to solve this open problem. They showed that, assuming one-way functions and indistinguishability obfuscation (both with subexponential security), there exist hard instances of the End-of-Line problem.…”

“…Bitanski, Paneth and Rosen [12] showed hardness of End-of-Line under the assumption of existence of injective one-way functions and indistinguishability obfuscation (both with sub-exponential security).…”

“…Recently, Bitanski, Paneth and Rosen [12] constructed hard instances of End-of-Line under cryptographic assumptions and showed the first white-box hardness result for PPAD (the cryptographic assumptions were improved in [26]). An overview of the above subclasses of TFNP and the known hardness results is depicted in Figure 1 (see Section 3 for the formal definitions of the various subclasses of TFNP).…”