# Query Complexity of Approximate Equilibria in Anonymous Games

@article{Goldberg2015QueryCO, title={Query Complexity of Approximate Equilibria in Anonymous Games}, author={Paul W. Goldberg and Stefano Turchetta}, journal={ArXiv}, year={2015}, volume={abs/1412.6455} }

We study the computation of equilibria of two-strategy 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 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…

## 25 Citations

Playing Anonymous Games using Simple Strategies

- Computer Science, MathematicsSODA
- 2017

The approach exploits the connection between Nash equilibria in anonymous games and Poisson multinomial distributions and proves a new probabilistic lemma establishing the following: Two PMDs, with large variance in each direction, whose first few moments are approximately matching are close in total variation distance.

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.

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.

Explorer Playing Anonymous Games using Simple Strategies

- Computer Science, Mathematics
- 2016

This work exploits the connection between Nash equilibria in anonymous games and Poisson multinomial distributions and proves a new probabilistic lemma establishing the following: Two PMDs, with large variance in each direction, whose first few moments are approximately matching are close in total variation distance.

Bounds for the Query Complexity of Approximate Equilibria

- Computer Science, EconomicsACM Trans. Economics and Comput.
- 2013

The number of payoff queries needed to compute approximate equilibria of multi-player games is analyzed, and it is found that query complexity is an effective tool for distinguishing the computational difficulty of alternative solution concepts, and new techniques for upper- and lower bounding the query complexity are developed.

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.

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 ScienceSAGT
- 2016

This paper assumes that a player can change another player’s payoff by at most \(\frac{1}{n}\) by changing her strategy, and seeks algorithms that obtain \(\varepsilon \) as small as possible, in time polynomial in n.

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.

Learning Game-Theoretic Equilibria Via Query Protocols

- Computer Science
- 2017

This talk mostly focuses on the paper Fearnley et al. (Learning equilibria of games via payoff queries), which is a relatively recent line of work, which is reviewed here.

## References

SHOWING 1-10 OF 39 REFERENCES

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.

Bounds for the Query Complexity of Approximate Equilibria

- Computer Science, EconomicsACM Trans. Economics and Comput.
- 2013

The number of payoff queries needed to compute approximate equilibria of multi-player games is analyzed, and it is found that query complexity is an effective tool for distinguishing the computational difficulty of alternative solution concepts, and new techniques for upper- and lower bounding the query complexity are developed.

An Efficient PTAS for Two-Strategy Anonymous Games

- Computer ScienceWINE
- 2008

We present a novel polynomial time approximation scheme fortwo-strategy anonymous games, in which the players’ utilityfunctions, although potentially different, do not differentiateamong the…

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.

Learning equilibria of games via payoff queries

- EconomicsEC '13
- 2013

This work studies a corresponding computational learning model, and the query complexity of learning equilibria for various classes of games, and has the stronger result that an equilibrium can be identified while only learning a small fraction of the cost values.

Computing Equilibria in Anonymous Games

- Economics, Computer Science48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)
- 2007

It is shown that any anonymous game has an approximate pure Nash equilibrium, computable in polynomial time, with approximation O(s2lambda), where s is the number of strategies and lambda is the Lipschitz constant of the utilities.

Playing large games using simple strategies

- EconomicsEC '03
- 2003

The existence of ε-Nash equilibrium strategies with support logarithmic in the number of pure strategies is proved and it is proved that if the payoff matrices of a two person game have low rank then the game has an exact Nash equilibrium with small support.

Settling the complexity of computing two-player Nash equilibria

- EconomicsJACM
- 2009

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…