Title: A taxonomy of learning dynamics in 2 x 2 games,
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 x 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.