Provable Optimization of Quantal Response Leader-Follower Games with Exponentially Large Action Spaces

Leader-follower games involve a leader committing strategies before her followers. We consider quantal response leader-follower games, where the followers' response is probabilistic due to their bounded rationality. Moreover, both the leader's and followers' action spaces are exponentially large with respect to the problem size, hence rendering the overall complexity to solve these games beyond NP-complete. We propose the XOR-Game algorithm, which converges in linear speed towards the equilibrium of convex quantal response leader-follower games (#P-hard to find the equilibrium even though convex). XOR-Game combines stochastic gradient descent with XOR-sampling, a provable sampling approach which transforms highly intractable probabilistic inference into queries to NP oracles. We tested XOR-Game on zero-sum and distribution matching leader-follower games. Experiments show XOR-Game converges faster to a good leader's strategy compared to several baselines. In particular, XOR-Game helps to find the optimal reward allocations for the Avicaching game in the citizen science domain, which harnesses rewards to motivate bird watchers towards tasks of high scientific value.