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The behaviour of multi-agent learning in competitive settings is oftenconsidered under the restrictive assumption of a zero-sum game. Only under thisstrict requirement is the behaviour of learning well understood; beyond this,learning dynamics can often display non-convergent behaviours which preventfixed-point analysis. Nonetheless, many relevant competitive games do notsatisfy the zero-sum assumption.  Motivated by this, we study a smooth variant of Q-Learning, a popularreinforcement learning dynamics which balances the agents' tendency to maximisetheir payoffs with their propensity to explore the state space. We examine thisdynamic in games which are `close' to network zero-sum games and find thatQ-Learning converges to a neighbourhood around a unique equilibrium. The sizeof the neighbourhood is determined by the `distance' to the zero-sum game, aswell as the exploration rates of the agents. We complement these results byproviding a method whereby, given an arbitrary network game, the `nearest'network zero-sum game can be found efficiently. As our experiments show, theseguarantees are independent of whether the dynamics ultimately reach anequilibrium, or remain non-convergent.

Beyond Strict Competition: Approximate Convergence of Multi Agent Q-Learning Dynamics