Regret Minimization in Stackelberg Games with Side Information

In its most basic form, a Stackelberg game is a two-player game in which a leader commits to a (mixed) strategy, and a follower best-responds. Stackelberg games are perhaps one of the biggest success stories of algorithmic game theory over the last decade, as algorithms for playing in Stackelberg games have been deployed in many real-world domains including airport security, anti-poaching efforts, and cyber-crime prevention. However, these algorithms often fail to take into consideration the additional information available to each player (e.g. traffic patterns, weather conditions, network congestion), a salient feature of reality which may significantly affect both players' optimal strategies. We formalize such settings as Stackelberg games with side information, in which both players observe an external context before playing. The leader then commits to a (possibly context-dependent) strategy, and the follower best-responds to both the leader's strategy and the context. We focus on the online setting in which a sequence of followers arrive over time, and the context may change from round-to-round. In sharp contrast to the non-contextual version, we show that it is impossible for the leader to achieve good performance (measured by regret) in the full adversarial setting (i.e., when both the context and the follower are chosen by an adversary). However, it turns out that a little bit of randomness goes a long way. Motivated by our impossibility result, we show that no-regret learning is possible in two natural relaxations: the setting in which the sequence of followers is chosen stochastically and the sequence of contexts is adversarial, and the setting in which the sequence of contexts is stochastic and the sequence of followers is chosen by an adversary.