[Paper Review] ETH-Hardness for Symmetric Signaling in Zero-Sum Games

Aviad Rubinstein|arXiv (Cornell University)|Oct 16, 2015

Complexity and Algorithms in Graphs11 references3 citations

TL;DR

This paper establishes that, under the exponential time hypothesis, computing an ϵ-approximately optimal symmetric signaling scheme in two-player zero-sum games requires quasi-polynomial time, resolving an open problem from Dughmi (2014). It further proves that achieving a multiplicative approximation is NP-hard, establishing strong complexity bounds for symmetric signaling in zero-sum games.

ABSTRACT

We prove that, assuming the exponential time hypothesis, finding an \epsilon-approximately optimal symmetric signaling scheme in a two-player zero-sum game requires quasi-polynomial time. This is tight by [CCDEHT'15] and resolves an open question of [Dughmi'14]. We also prove that finding a multiplicative approximation is NP-hard.

Motivation & Objective

• To resolve the computational complexity of finding approximately optimal symmetric signaling schemes in two-player zero-sum games.
• To determine whether symmetric signaling schemes can be computed efficiently under standard complexity assumptions.
• To close a longstanding open question posed by Dughmi (2014) regarding the hardness of symmetric signaling in zero-sum games.
• To establish tight complexity bounds for both additive and multiplicative approximation in symmetric signaling.

Proposed method

• The authors use the exponential time hypothesis (ETH) as a foundational complexity assumption to derive lower bounds on the time required to compute ϵ-approximately optimal symmetric signaling schemes.
• They construct reductions from known hard problems to the symmetric signaling problem, demonstrating that solving it in sub-quasi-polynomial time would violate ETH.
• The proof leverages structural properties of zero-sum games and the duality between signaling schemes and equilibrium outcomes.
• The authors extend their analysis to multiplicative approximation, proving NP-hardness via a reduction from a known NP-complete problem.
• They rely on results from [CCDEHT'15] to show that their quasi-polynomial lower bound is tight.
• The method combines complexity-theoretic reasoning with game-theoretic modeling of signaling in zero-sum settings.

Experimental results

Research questions

• RQ1 Is computing an ϵ-approximately optimal symmetric signaling scheme in two-player zero-sum games solvable in polynomial time under the exponential time hypothesis?
• RQ2 What is the exact complexity threshold for symmetric signaling when approximating the optimal scheme within additive error ϵ?
• RQ3 Can a multiplicative approximation to the optimal symmetric signaling scheme be computed efficiently, or is it NP-hard?
• RQ4 Does the quasi-polynomial lower bound established by [CCDEHT'15] represent a tight limit for symmetric signaling in zero-sum games?
• RQ5 Is there a gap between the complexity of symmetric and asymmetric signaling in zero-sum games?

Key findings

• Finding an ϵ-approximately optimal symmetric signaling scheme in a two-player zero-sum game requires quasi-polynomial time under the exponential time hypothesis.
• The quasi-polynomial time lower bound is tight, as confirmed by prior work [CCDEHT'15].
• Computing a multiplicative approximation to the optimal symmetric signaling scheme is NP-hard.
• The result resolves an open question posed by Dughmi (2014) regarding the complexity of symmetric signaling in zero-sum games.
• The paper establishes strong intractability results for both additive and multiplicative approximations in symmetric signaling.