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

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 Erdős-Rényi 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 − ǫ -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.

On the Tractability of Public Persuasion with No Externalities

“…[24,4] There exists a constant c ∈ (0, 1) and constant a ∈ (0, 1) such that it is NP-hard to approximate f + (a) to within c (additively) for any monotone submodular function f . 6 In the remainder of the reduction, we show that any (1 − δ)-approximate ǫ-persuasive private persuasion in the constructed instances can be converted to an additively (2ǫ/a + 2δ)-optimal algorithm for computing f + (x). Invoking Lemma B.1, this implies the NP-hardness of designing a (1 − ǫ)-approximate ǫ-persuasive private signaling scheme in poly(n 1/ǫ , |Θ|) time.…”

Computing Equilibria of Prediction Markets via Persuasion

“…the logarithm of the probability assigned to e. Its "expected score function" is G(w) = e w e log w e = −H(w), the negative of Shannon entropy. The quadratic scoring rule is R(w, e) = 2w e − w 2 2 . Its expected score function is G(w) = w 2 2 .…”

“…The quadratic scoring rule is R(w, e) = 2w e − w 2 2 . Its expected score function is G(w) = w 2 2 . Both are strictly proper.…”

“…When the total number of grid points are polynomially bounded, we obtain efficient algorithms. This idea has been employed in algorithmic persuasion (e.g., [6,2]).…”

“…. For any δ > 0, let ǫ be as defined in Equation (10) and K = log(2d/ǫ)d 2 2ǫ 2 . There always exists a (4Lǫ + δ)-optimal signaling scheme whose posterior beliefs are all K-uniform (i.e., in ∆ d (K)).…”