Best-Response Dynamics, Playing Sequences, and Convergence to Equilibrium in Random Games

33 Pages Posted: 18 Feb 2021 Last revised: 18 Nov 2022

See all articles by Torsten Heinrich

Torsten Heinrich

Faculty for Economics and Business Administration, Chemnitz University of Technology, 09111 Chemnitz, Germany; University of Oxford - Institute for New Economic Thinking at the Oxford Martin School; Oxford Martin Programme on Technological and Economic Change, Oxford Martin School, University of Oxford

Yoojin Jang

University of Oxford

Luca Mungo

University of Oxford - Mathematical Institute

Marco Pangallo

CENTAI Institute

Alex Scott

University of Oxford - Mathematical Institute

Bassel Tarbush

University of Oxford - Merton College

Samuel Wiese

University of Oxford - Mathematical Institute

Date Written: January 11, 2021

Abstract

We analyze the performance of the best-response dynamic across all normal-form games using a random games approach. The playing sequence---the order in which players update their actions---is essentially irrelevant in determining whether the dynamic converges to a Nash equilibrium in certain classes of games (e.g. in potential games) but, when evaluated across all possible games, convergence to equilibrium depends on the playing sequence in an extreme way. Our main asymptotic result shows that the best-response dynamic converges to a pure Nash equilibrium in a vanishingly small fraction of all (large) games when players take turns according to a fixed cyclic order. By contrast, when the playing sequence is random, the dynamic converges to a pure Nash equilibrium if one exists in almost all (large) games.

Keywords: Best-response dynamics, equilibrium convergence, random games, learning models in games

JEL Classification: C62, C72, C73, D83

Suggested Citation

Heinrich, Torsten and Jang, Yoojin and Mungo, Luca and Pangallo, Marco and Scott, Alex and Tarbush, Bassel and Wiese, Samuel, Best-Response Dynamics, Playing Sequences, and Convergence to Equilibrium in Random Games (January 11, 2021). Available at SSRN: https://ssrn.com/abstract=3764151 or http://dx.doi.org/10.2139/ssrn.3764151

Torsten Heinrich

Faculty for Economics and Business Administration, Chemnitz University of Technology, 09111 Chemnitz, Germany ( email )

D-09107 Chemnitz
Germany

HOME PAGE: http://https://www.tu-chemnitz.de/wirtschaft/vwl2/personal/heinrich.php.en

University of Oxford - Institute for New Economic Thinking at the Oxford Martin School ( email )

Eagle House
Walton Well Road
Oxford, OX2 6ED
United Kingdom

Oxford Martin Programme on Technological and Economic Change, Oxford Martin School, University of Oxford ( email )

University of Oxford
34 Broad Street
Oxford, OX1 3BD
United Kingdom

Yoojin Jang

University of Oxford ( email )

Mansfield Road
Oxford, Oxfordshire OX1 4AU
United Kingdom

Luca Mungo

University of Oxford - Mathematical Institute ( email )

Radcliffe Observatory, Andrew Wiles Building
Woodstock Rd
Oxford, Oxfordshire OX2 6GG
United Kingdom

Marco Pangallo (Contact Author)

CENTAI Institute ( email )

Corso Inghilterra 3
10138
Torino
Italy

Alex Scott

University of Oxford - Mathematical Institute ( email )

Radcliffe Observatory, Andrew Wiles Building
Woodstock Rd
Oxford, Oxfordshire OX2 6GG
United Kingdom

Bassel Tarbush

University of Oxford - Merton College ( email )

Merton College
Oxford, OX1 4JD
United Kingdom

Samuel Wiese

University of Oxford - Mathematical Institute ( email )

Radcliffe Observatory, Andrew Wiles Building
Woodstock Rd
Oxford, Oxfordshire OX2 6GG
United Kingdom

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