## AbstractOne issue in multi-agent co-adaptive learning concerns convergence. When two (or more) agents play a game with different information and different payoffs, the general behaviour tends to be oscillation around a Nash equilibrium. Several algorithms have been proposed to force convergence to mixed-strategy Nash equilibria in imperfect-information games when the agents are aware of their opponent's strategy. We consider the effect on one such algorithm, the lagging anchor algorithm, when each agent must also infer the gradient information from observations, in the infinitesimal time-step limit. Use of an estimated gradient, either by opponent modelling or stochastic gradient ascent, destabilises the algorithm in a region of parameter space. There are two phases of behaviour. If the rate of estimation is low, the Nash equilibrium becomes unstable in the mean. If the rate is high, the Nash equilibrium is an attractive fixed point in the mean, but the uncertainty acts as arrow-band coloured noise, which causes dampened oscillations.
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