[Paper Review] Query Complexity of Approximate Equilibria in Anonymous Games

Paul W. Goldberg, Stefano Turchetta|arXiv (Cornell University)|Dec 19, 2014

Game Theory and Applications30 references4 citations

TL;DR

This paper presents a randomized, query-efficient algorithm that computes an $O(n^{-1/4})$-approximate Nash equilibrium in anonymous games with two strategies using $\tilde{O}(n^{3/2})$ payoff queries and runtime, improving prior methods. It establishes a tight $\Omega(n\log n)$ query lower bound for finding $\epsilon$-well-supported equilibria, even with randomization, and shows exact equilibria cannot be computed query-efficiently.

ABSTRACT

We study the computation of equilibria of anonymous games, via algorithms that may proceed via a sequence of adaptive queries to the game's payoff function, assumed to be unknown initially. The general topic we consider is \emph{query complexity}, that is, how many queries are necessary or sufficient to compute an exact or approximate Nash equilibrium. We show that exact equilibria cannot be found via query-efficient algorithms. We also give an example of a 2-strategy, 3-player anonymous game that does not have any exact Nash equilibrium in rational numbers. However, more positive query-complexity bounds are attainable if either further symmetries of the utility functions are assumed or we focus on approximate equilibria. We investigate four sub-classes of anonymous games previously considered by \cite{bfh09, dp14}. Our main result is a new randomized query-efficient algorithm that finds a $O(n^{-1/4})$-approximate Nash equilibrium querying $ ilde{O}(n^{3/2})$ payoffs and runs in time $ ilde{O}(n^{3/2})$. This improves on the running time of pre-existing algorithms for approximate equilibria of anonymous games, and is the first one to obtain an inverse polynomial approximation in poly-time. We also show how this can be utilized as an efficient polynomial-time approximation scheme (PTAS). Furthermore, we prove that $Ω(n \log{n})$ payoffs must be queried in order to find any $ε$-well-supported Nash equilibrium, even by randomized algorithms.

Motivation & Objective

• To investigate the query complexity of computing approximate Nash equilibria in anonymous games where the payoff function is accessed via adaptive queries.
• To determine whether exact Nash equilibria can be computed efficiently under the query model, especially given that their representation may not be rational.
• To establish tight query complexity bounds for approximate and well-supported equilibria in two-strategy anonymous games.
• To develop a new randomized algorithm that achieves an inverse polynomial approximation in poly-time with improved query and runtime efficiency.

Proposed method

• The algorithm uses a randomized approach to sample payoff queries and applies binary search over a discretized strategy space to locate a fixed point of the smoothed game.
• It leverages the structure of Poisson Binomial Distributions to compute expected utilities efficiently from queried payoffs.
• The method assumes players randomize between two strategies with probabilities $\zeta/k$ and $1 - (k-1)\zeta/k$, focusing on $k=2$ for tractability.
• A key component is the use of a smooth game approximation where the payoff function is interpolated to ensure Lipschitz continuity, enabling efficient fixed-point search.
• The algorithm ensures that all players' incentives to deviate are bounded by $O(n^{-1/4})$, achieving an $\epsilon$-NE with $\epsilon = O(n^{-1/4})$.
• Lower bounds are derived using an adversarial distribution over games and the minimax principle, reducing the problem to learning a random $\log n$-bit string from query responses.

Experimental results

Research questions

• RQ1 Can exact Nash equilibria in anonymous games be computed via query-efficient algorithms, even approximately?
• RQ2 What is the minimum number of payoff queries required to compute an $\epsilon$-well-supported Nash equilibrium in two-strategy anonymous games?
• RQ3 Is it possible to design a query-efficient algorithm that computes an $\epsilon$-approximate Nash equilibrium with $\epsilon$ inverse polynomial in $n$?
• RQ4 Can the algorithmic approach be generalized to $k$-strategy anonymous games while maintaining query efficiency?
• RQ5 Does the query complexity of finding $\epsilon$-NE scale logarithmically with $n$, even for $\epsilon < 1/2$?

Key findings

• The paper presents a randomized algorithm that computes an $O(n^{-1/4})$-approximate Nash equilibrium in $\tilde{O}(n^{3/2})$ queries and runtime, improving on previous algorithms.
• The algorithm achieves an inverse polynomial approximation in polynomial time, which is the first such result for anonymous games.
• A lower bound of $\Omega(n\log n)$ queries is proven for any randomized algorithm to compute an $\epsilon$-well-supported Nash equilibrium, even for $\epsilon < 1$.
• The paper constructs a 2-strategy, 3-player anonymous game with no exact Nash equilibrium in rational numbers, showing that rational exact equilibria do not always exist.
• It is shown that exact equilibria cannot be computed via query-efficient algorithms, due to the possibility of irrational equilibria and the inherent complexity of the query model.
• The query complexity for $\epsilon$-WSNE is at least $\Omega(n\log n)$, indicating that logarithmic query overhead is unavoidable for well-supported equilibria.