A Taxonomy of Learning Dynamics in 2 × 2 Games

35 Pages Posted: 8 Feb 2017 Last revised: 9 Feb 2017

See all articles by Marco Pangallo

Marco Pangallo

University of Oxford

James Sanders

University of Manchester

Tobias Galla

University of Manchester - School of Physics and Astronomy

J. Doyne Farmer

University of Oxford - Institute for New Economic Thinking at the Oxford Martin School; Santa Fe Institute

Date Written: February 7, 2017

Abstract

Learning would be a convincing method to achieve coordination on an equilibrium. But does learning converge, and to what? We answer this question in generic 2-player, 2-strategy games, using Experience-Weighted Attraction (EWA), which encompasses many extensively studied learning algorithms. We exhaustively characterize the parameter space of EWA learning, for any payoff matrix, and we understand the generic properties that imply convergent or non-convergent behaviour in 2 × 2 games.

Irrational choice and lack of incentives imply convergence to a mixed strategy in the centre of the strategy simplex, possibly far from the Nash Equilibrium (NE). In the opposite limit, in which the players quickly modify their strategies, the behaviour depends on the payoff matrix: (i) a strong discrepancy between the pure strategies describes dominance-solvable games, which show convergence to a unique fixed point close to the NE; (ii) a preference towards profiles of strategies along the main diagonal describes coordination games, with multiple stable fixed points corresponding to the NE; (iii) a cycle of best responses defines discoordination games, which commonly yield limit cycles or low-dimensional chaos.

While it is well known that mixed strategy equilibria may be unstable, our approach is novel from several perspectives: we fully analyse EWA and provide explicit thresholds that define the onset of instability; we find an emerging taxonomy of the learning dynamics, without focusing on specific classes of games ex-ante; we show that chaos can occur even in the simplest games; we make a precise theoretical prediction that can be tested against data on experimental learning of discoordination games.

Keywords: Behavioural Game Theory, EWA Learning, Convergence, Equilibrium, Chaos

JEL Classification: C62, C73, D83

Suggested Citation

Pangallo, Marco and Sanders, James and Galla, Tobias and Farmer, J. Doyne, A Taxonomy of Learning Dynamics in 2 × 2 Games (February 7, 2017). Available at SSRN: https://ssrn.com/abstract=2913183 or http://dx.doi.org/10.2139/ssrn.2913183

Marco Pangallo (Contact Author)

University of Oxford ( email )

Mansfield Road
Oxford, Oxfordshire OX1 4AU
United Kingdom

James Sanders

University of Manchester ( email )

Oxford Road
Manchester, M13 9PL
United Kingdom

Tobias Galla

University of Manchester - School of Physics and Astronomy ( email )

United Kingdom

J. Doyne Farmer

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

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

HOME PAGE: http://www.inet.ox.ac.uk/people/view/4

Santa Fe Institute ( email )

1399 Hyde Park Road
Santa Fe, NM 87501
United States
505-984-8800 (Phone)
505-982-0565 (Fax)

HOME PAGE: http://www.santafe.edu/~jdf/

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