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DOI: 10.4230/LIPIcs.ITCS.2024.37

Smooth Nash Equilibria: Algorithms and Complexity

Authors: Constantinos Daskalakis, Noah Golowich, Nika Haghtalab, and Abhishek Shetty

Published in: LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

Abstract

A fundamental shortcoming of the concept of Nash equilibrium is its computational intractability: approximating Nash equilibria in normal-form games is PPAD-hard. In this paper, inspired by the ideas of smoothed analysis, we introduce a relaxed variant of Nash equilibrium called σ-smooth Nash equilibrium, for a {smoothness parameter} σ. In a σ-smooth Nash equilibrium, players only need to achieve utility at least as high as their best deviation to a σ-smooth strategy, which is a distribution that does not put too much mass (as parametrized by σ) on any fixed action. We distinguish two variants of σ-smooth Nash equilibria: strong σ-smooth Nash equilibria, in which players are required to play σ-smooth strategies under equilibrium play, and weak σ-smooth Nash equilibria, where there is no such requirement. We show that both weak and strong σ-smooth Nash equilibria have superior computational properties to Nash equilibria: when σ as well as an approximation parameter ϵ and the number of players are all constants, there is a {constant-time} randomized algorithm to find a weak ϵ-approximate σ-smooth Nash equilibrium in normal-form games. In the same parameter regime, there is a polynomial-time deterministic algorithm to find a strong ϵ-approximate σ-smooth Nash equilibrium in a normal-form game. These results stand in contrast to the optimal algorithm for computing ϵ-approximate Nash equilibria, which cannot run in faster than quasipolynomial-time, subject to complexity-theoretic assumptions. We complement our upper bounds by showing that when either σ or ϵ is an inverse polynomial, finding a weak ϵ-approximate σ-smooth Nash equilibria becomes computationally intractable. Our results are the first to propose a variant of Nash equilibrium which is computationally tractable, allows players to act independently, and which, as we discuss, is justified by an extensive line of work on individual choice behavior in the economics literature.

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Constantinos Daskalakis, Noah Golowich, Nika Haghtalab, and Abhishek Shetty. Smooth Nash Equilibria: Algorithms and Complexity. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 37:1-37:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{daskalakis_et_al:LIPIcs.ITCS.2024.37,
  author =	{Daskalakis, Constantinos and Golowich, Noah and Haghtalab, Nika and Shetty, Abhishek},
  title =	{{Smooth Nash Equilibria: Algorithms and Complexity}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{37:1--37:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-309-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{287},
  editor =	{Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.37},
  URN =		{urn:nbn:de:0030-drops-195657},
  doi =		{10.4230/LIPIcs.ITCS.2024.37},
  annote =	{Keywords: Nash equilibrium, smoothed analysis, PPAD}
}

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DOI: 10.4230/LIPIcs.ITC.2020.15

Pure Differentially Private Summation from Anonymous Messages

Authors: Badih Ghazi, Noah Golowich, Ravi Kumar, Pasin Manurangsi, Rasmus Pagh, and Ameya Velingker

Published in: LIPIcs, Volume 163, 1st Conference on Information-Theoretic Cryptography (ITC 2020)

Abstract

The shuffled (aka anonymous) model has recently generated significant interest as a candidate distributed privacy framework with trust assumptions better than the central model but with achievable error rates smaller than the local model. In this paper, we study pure differentially private protocols in the shuffled model for summation, a very basic and widely used primitive. Specifically: - For the binary summation problem where each of n users holds a bit as an input, we give a pure ε-differentially private protocol for estimating the number of ones held by the users up to an absolute error of O_{ε}(1), and where each user sends O_{ε}(log n) one-bit messages. This is the first pure protocol in the shuffled model with error o(√n) for constant values of ε. Using our binary summation protocol as a building block, we give a pure ε-differentially private protocol that performs summation of real numbers in [0, 1] up to an absolute error of O_{ε}(1), and where each user sends O_{ε}(log³ n) messages each consisting of O(log log n) bits. - In contrast, we show that for any pure ε-differentially private protocol for binary summation in the shuffled model having absolute error n^{0.5-Ω(1)}, the per user communication has to be at least Ω_{ε}(√{log n}) bits. This implies (i) the first separation between the (bounded-communication) multi-message shuffled model and the central model, and (ii) the first separation between pure and approximate differentially private protocols in the shuffled model. Interestingly, over the course of proving our lower bound, we have to consider (a generalization of) the following question that might be of independent interest: given γ ∈ (0, 1), what is the smallest positive integer m for which there exist two random variables X⁰ and X^1 supported on {0, … , m} such that (i) the total variation distance between X⁰ and X^1 is at least 1 - γ, and (ii) the moment generating functions of X⁰ and X^1 are within a constant factor of each other everywhere? We show that the answer to this question is m = Θ(√{log(1/γ)}).

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Badih Ghazi, Noah Golowich, Ravi Kumar, Pasin Manurangsi, Rasmus Pagh, and Ameya Velingker. Pure Differentially Private Summation from Anonymous Messages. In 1st Conference on Information-Theoretic Cryptography (ITC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 163, pp. 15:1-15:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{ghazi_et_al:LIPIcs.ITC.2020.15,
  author =	{Ghazi, Badih and Golowich, Noah and Kumar, Ravi and Manurangsi, Pasin and Pagh, Rasmus and Velingker, Ameya},
  title =	{{Pure Differentially Private Summation from Anonymous Messages}},
  booktitle =	{1st Conference on Information-Theoretic Cryptography (ITC 2020)},
  pages =	{15:1--15:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-151-1},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{163},
  editor =	{Tauman Kalai, Yael and Smith, Adam D. and Wichs, Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITC.2020.15},
  URN =		{urn:nbn:de:0030-drops-121208},
  doi =		{10.4230/LIPIcs.ITC.2020.15},
  annote =	{Keywords: Pure differential privacy, Shuffled model, Anonymous messages, Summation, Communication bounds}
}

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Smooth Nash Equilibria: Algorithms and Complexity

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Pure Differentially Private Summation from Anonymous Messages

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