[Paper Review] Inapproximability Results for Approximate Nash Equilibria

Argyrios Deligkas, John Fearnley+1 more|arXiv (Cornell University)|Aug 11, 2016

Game Theory and Applications20 references4 citations

TL;DR

This paper establishes conditional quasi-polynomial time lower bounds for computing approximate Nash equilibria with constrained social welfare in bimatrix games, under the Exponential Time Hypothesis (ETH). It proves that solving the (1/8 − O(δ))-NE O(δ)-SW problem requires n~Ω(log n) time, extending hardness results to multiple decision problems involving payoff maximization, support constraints, and welfare approximation for ε-NE with ε < 1/8.

ABSTRACT

We study the problem of finding approximate Nash equilibria that satisfy certain conditions, such as providing good social welfare. In particular, we study the problem $ε$-NE $δ$-SW: find an $ε$-approximate Nash equilibrium ($ε$-NE) that is within $δ$ of the best social welfare achievable by an $ε$-NE. Our main result is that, if the exponential-time hypothesis (ETH) is true, then solving $\left(\frac{1}{8} - \mathrm{O}(δ) ight)$-NE $\mathrm{O}(δ)$-SW for an $n imes n$ bimatrix game requires $n^{\mathrm{\widetilde Ω}(\log n)}$ time. Building on this result, we show similar conditional running time lower bounds on a number of decision problems for approximate Nash equilibria that do not involve social welfare, including maximizing or minimizing a certain player's payoff, or finding approximate equilibria contained in a given pair of supports. We show quasi-polynomial lower bounds for these problems assuming that ETH holds, where these lower bounds apply to $ε$-Nash equilibria for all $ε&lt; \frac{1}{8}$. The hardness of these other decision problems has so far only been studied in the context of exact equilibria.

Motivation & Objective

• To investigate the computational hardness of finding approximate Nash equilibria that satisfy specific constraints, such as high social welfare.
• To extend inapproximability results from exact equilibria to approximate equilibria, particularly for ε-NE with bounded social welfare deviation.
• To establish conditional time lower bounds for decision problems involving approximate equilibria under the Exponential Time Hypothesis (ETH).
• To show that even approximate equilibria with desirable properties are computationally hard to compute, beyond the known PPAD-completeness of exact equilibria.
• To unify and strengthen prior inapproximability results by linking them to ETH and the hidden clique problem.

Proposed method

• Reduces the problem of finding a (1/8 − O(δ))-NE within δ of optimal social welfare to the hidden clique problem in random graphs.
• Constructs a bimatrix game G'' where the existence of a specific strategy in the row player's support depends on the maximum independent set size in a hypergraph F.
• Uses the Exponential Time Hypothesis (ETH) to argue that solving the hidden clique problem in G(n,1/2) requires n~Ω(log n) time.
• Proves that if ω(F) = 1 (maximum independent set size is full), then G'' admits an ε*-WSNE with the target strategy in support; otherwise, it does not.
• Applies the Lipton-Markakis-Mehta (LMM) algorithm framework to analyze support size and payoff bounds in ε-NE and ε-WSNE.
• Employs a reduction from the hidden clique problem to show that solving ε-NE δ-SW requires quasi-polynomial time under ETH.

Experimental results

Research questions

• RQ1 Can we establish conditional time lower bounds for computing approximate Nash equilibria with bounded social welfare deviation?
• RQ2 Is the problem of finding an ε-NE within δ of the best achievable social welfare for ε < 1/8 hard under ETH?
• RQ3 Do decision problems involving payoff maximization or support constraints in approximate equilibria also exhibit quasi-polynomial time hardness?
• RQ4 Can the hardness of finding constrained approximate equilibria be linked to the hidden clique problem under ETH?
• RQ5 How does the hardness of approximate equilibria compare to that of exact equilibria in terms of computational complexity?

Key findings

• Under the Exponential Time Hypothesis (ETH), solving the (1/8 − O(δ))-NE O(δ)-SW problem for n×n bimatrix games requires n~Ω(log n) time.
• The hardness result extends to decision problems such as maximizing or minimizing a player’s payoff in an ε-NE, or finding an ε-NE within a given pair of supports.
• For all ε < 1/8, these decision problems require quasi-polynomial time under ETH, establishing a conditional lower bound.
• The reduction shows that solving ε-NE δ-SW is at least as hard as finding a hidden clique of size O(log n) in G(n,1/2), a problem assumed to require quasi-polynomial time.
• The result strengthens prior inapproximability results by linking them to ETH and showing that even approximate equilibria with good welfare are hard to compute.
• The paper establishes that the LMM algorithm’s quasi-polynomial time complexity is essentially optimal for ε-NE under ETH, as no faster algorithm exists unless ETH fails.