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A taxonomy of learning dynamics in 2 x 2 games

Marco Pangallo, James Sanders, Tobias Galla, Doyne Farmer

posted on 08 February 2017

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Learning would be a convincing method to achieve coordination on a Nash Equilibrium (NE). 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 most 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 2x2 games.
Irrational choice and lack of incentives imply convergence to a mixed strategy in the centre of the strategy simplex, possibly far from the 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.

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