Score vs. winrate in score-based games: which reward for reinforcement learning?

In the last years, DeepMind algorithm AlphaZero has become the state of the art to efficiently tackle perfect information two-player zero-sum games with a win/lose outcome. However, when the win/lose outcome is decided by a final score difference, AlphaZero may play score-suboptimal moves, because all winning final positions are equivalent from the win/lose outcome perspective. This can be an issue, for instance when used for teaching, or when trying to understand whether there is a better move. Moreover, there is the theoretical quest of the perfect game. A naive approach would be training a AlphaZero-like agent to predict score differences instead of win/lose outcomes. Since the game of Go is deterministic, this should as well produce outcome-optimal play. However, it is a folklore belief that "this does not work".In this paper we first provide empirical evidence to this belief. We then give a theoretical interpretation of this suboptimality in a general perfect information two-player zero-sum game where the complexity of a game like Go is replaced by randomness of the environment. We show that an outcome-optimal policy has a different preference for uncertainty when it is winning or losing. In particular, when in a losing state, an outcome-optimal agent chooses actions leading to a higher variance of the score. We then posit that when approximation is involved, a deterministic game behaves like a nondeterministic game, where the score variance is modeled by how uncertain the position is. We validate this hypothesis in a AlphaZero-like software with a human expert.