# Bounds for the Query Complexity of Approximate Equilibria

@article{Goldberg2013BoundsFT, title={Bounds for the Query Complexity of Approximate Equilibria}, author={Paul W. Goldberg and Aaron Roth}, journal={ACM Transactions on Economics and Computation (TEAC)}, year={2013}, volume={4}, pages={1 - 25} }

We analyze the number of payoff queries needed to compute approximate equilibria of multi-player games. We find that query complexity is an effective tool for distinguishing the computational difficulty of alternative solution concepts, and we develop new techniques for upper- and lower bounding the query complexity. For binary-choice games, we show logarithmic upper and lower bounds on the query complexity of approximate correlated equilibrium. For well-supported approximate correlated…

## 32 Citations

Query Complexity of Approximate Equilibria in Anonymous Games

- Computer Science, EconomicsWINE
- 2015

It is proved that $$\varOmega n \log {n}$$ payoffs must be queried in order to find any $$epsilon $$-well-supported Nash equilibrium, even by randomized algorithms, which is the first one to obtain an inverse polynomial approximation in poly-time.

Lower Bounds for the Query Complexity of Equilibria in Lipschitz Games

- Economics, MathematicsSAGT
- 2021

This work develops a query-efficient reduction from more general games to Lipschitz games, and provides an exponential lower bound on the deterministic query complexity of finding -approximate correlated equilibria of n-player, m-action, λ-Lipschitzer games for strong values of , motivating the consideration of explicitly randomized algorithms in the above results.

Finding Approximate Nash Equilibria of Bimatrix Games via Payoff Queries

- Economics, Computer ScienceACM Trans. Economics and Comput.
- 2016

It is shown that randomized algorithms require Ω(k2) payoff queries in order to find an ϵ-Nash equilibrium with ϵ < 1/4k, even in zero-one constant-sum games, which rules out query-efficient randomized algorithms for finding exact Nash equilibria.

Logarithmic Query Complexity for Approximate Nash Computation in Large Games

- Economics, Computer ScienceTheory of Computing Systems
- 2018

A randomised algorithm is presented that achieves ε approaching 18$\frac {1}{8}$ for 2-strategy games in a completely uncoupled setting, where each player observes her own payoff to a query, and adjusts her behaviour independently of other players’ payoffs/actions.

Finding approximate nash equilibria of bimatrix games via payoff queries

- Economics, Computer ScienceEC
- 2014

It is shown that randomized algorithms require Omega(k2) payoff queries in order to find a 1/6k-Nash equilibrium, even in zero-one constant-sum games, which rules out query-efficient randomized algorithms for finding exact Nash equilibria.

Settling the Complexity of Computing Approximate Two-Player Nash Equilibria

- Computer Science2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)
- 2016

We prove that there exists a constant ε > 0 such that, assuming the Exponential Time Hypothesis for PPAD, computing an ε-approximate Nash equilibrium in a two-player (n × n) game requires…

Simple Approximate Equilibria in Games with Many Players

- Economics, Computer ScienceEC
- 2017

This work considers ε-equilibria notions for a constant value of ε in n-player m-action games and proves its equivalence to the well-known Beck-Fiala conjecture from discrepancy theory, the first result that introduces a connection between game theory and discrepancy theory.

Well-Supported vs. Approximate Nash Equilibria: Query Complexity of Large Games

- Computer ScienceITCS
- 2017

It is proved that, for some constant $\epsilon>0$, any randomized oracle algorithm that computes an $\epSilon$-ANE in a binary-action, $n$-player game must make $2^{\Omega(n/\log n)}$ payoff queries.

Smoothed Efficient Algorithms and Reductions for Network Coordination Games

- Computer ScienceITCS
- 2020

The approach combines and generalizes the local-max-cut approaches of [ER14,ABPW17] to handle the multi-strategy case and defines a notion of smoothness-preserving reduction among search problems, which allows for the extension of smoothed efficient algorithms from one problem to another.

Learning Convex Partitions and Computing Game-theoretic Equilibria from Best-response Queries

- Computer ScienceACM Trans. Economics and Comput.
- 2021

Two algorithms for Constant-Dimension Generalised Binary Search provide bounds on the best-response query complexity of computing approximate well-supported equilibria of bimatrix games in which one of the players has a constant number of pure strategies.

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It is proved that $$\varOmega n \log {n}$$ payoffs must be queried in order to find any $$epsilon $$-well-supported Nash equilibrium, even by randomized algorithms, which is the first one to obtain an inverse polynomial approximation in poly-time.

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We prove that Bimatrix, the problem of finding a Nash equilibrium in a two-player game, is complete for the complexity class PPAD (Polynomial Parity Argument, Directed version) introduced by…