  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.
