Hardness Results for Signaling in Bayesian Zero-Sum and Network Routing Games

TL;DR

This paper establishes new computational hardness results for signaling problems in Bayesian zero-sum and network routing games, showing that optimal signaling strategies are generally NP-hard to compute or approximate.

Contribution

It proves NP-hardness of approximating signaling in Bayesian zero-sum and network routing games, and identifies cases where PTAS are possible under certain conditions.

Findings

NP-hard to obtain an additive FPTAS for maximizing equilibrium utility in Bayesian zero-sum games

NP-hard to approximate the minimal average latency within a factor of 4/3 in Bayesian network routing games

A PTAS exists for structured zero-sum games with Lipschitz payoff matrices

Abstract

We study the optimization problem faced by a perfectly informed principal in a Bayesian game, who reveals information to the players about the state of nature to obtain a desirable equilibrium. This signaling problem is the natural design question motivated by uncertainty in games and has attracted much recent attention. We present new hardness results for signaling problems in (a) Bayesian two-player zero-sum games, and (b) Bayesian network routing games. For Bayesian zero-sum games, when the principal seeks to maximize the equilibrium utility of a player, we show that it is NP-hard to obtain an additive FPTAS. Our hardness proof exploits duality and the equivalence of separation and optimization in a novel way. Further, we rule out an additive PTAS assuming planted clique hardness, which states that no polynomial time algorithm can recover a planted clique from an Erd\H{o}s-R\'enyi random graph. Complementing these, we obtain a PTAS for a structured class of zero-sum games (where obtaining an FPTAS is still NP-hard) when the payoff matrices obey a Lipschitz condition. Previous results ruled out an FPTAS assuming planted-clique hardness, and a PTAS only for implicit games with quasi-polynomial-size strategy sets. For Bayesian network routing games, wherein the principal seeks to minimize the average latency of the Nash flow, we show that it is NP-hard to obtain a (multiplicative) $(4/3 - \epsilon)$-approximation, even for linear latency functions. This is the optimal inapproximability result for linear latencies, since we show that full revelation achieves a $(4/3)$-approximation for linear latencies.