Gradient-based learning in multi-agent systems is difficult because the gradient derives from a first-order model which does not account for the interaction between agents' learning processes. LOLA (arXiv:1709.04326) accounts for this by differentiating through one step of optimization. We propose to judge joint policies by their long-term prospects as measured by the meta-value, a discounted sum over the returns of future optimization iterates. We apply a form of Q-learning to the meta-game of optimization, in a way that avoids the need to explicitly represent the continuous action space of policy updates. The resulting method, MeVa, is consistent and far-sighted, and does not require REINFORCE estimators. We analyze the behavior of our method on a toy game and compare to prior work on repeated matrix games.

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