Stackelberg equilibria have become increasingly important as a solution concept in computational game theory, largely inspired by practical problems such as security settings. In practice, however, there is typically uncertainty regarding the model about the opponent. This paper is, to our knowledge, the first to investigate Stackelberg equilibria under uncertainty in extensive-form games, one of the broadest classes of game. We introduce robust Stackelberg equilibria, where the uncertainty is about the opponent's payoffs, as well as ones where the opponent has limited lookahead and the uncertainty is about the opponent's node evaluation function. We develop a new mixed-integer program for the deterministic limited-lookahead setting. We then extend the program to the robust setting for Stackelberg equilibrium under unlimited and under limited lookahead by the opponent. We show that for the specific case of interval uncertainty about the opponent's payoffs (or about the opponent's node evaluations in the case of limited lookahead), robust Stackelberg equilibria can be computed with a mixed-integer program that is of the same asymptotic size as that for the deterministic setting. In a Stackelberg equilibrium, a leader commits to a strategy first, and then a follower chooses a strategy for herself. The model was first introduced in the context of competition between firms where the leader picks a quantity to supply, and then the follower picks a quantity to supply (von Stackelberg 1934). Stackelberg equilibria have become important as a solution concept in computational game theory, largely inspired by practical problems such as security settings, where the leader is a defender who picks a mixed (i.e., potentially randomized) strategy first, and then the follower who is the attacker decides where to attack, if at all. Most work on Stackelberg equilibria has focused on normal-form (aka. matrix-form) games. Conitzer and Sandholm (2006) studied the problem of computing an optimal strategy to commit to in normal-form games. That line of work has been extended to many security-game applications. In practice, there is typically uncertainty about the opponent's payoffs. In normal-form games this has been studied as Bayesian Stackelberg games where the players have private information about their own payoffs, and there is common knowledge of the prior distribution over the pay- Extensive-form games (EFGs)-i.e., tree-form gamesare a very general game representation language. EFGs are exponentially more compact and also more expressive than normal-form games. Letchford and Conitzer (2010) study how to compute an optimal strategy to commit to in EFGs and prove hardness results under several assumptions about the game structure. Bošanskỳ et al. (2015) provide further results specifically for perfect-information EFGs. Bosansky and Cermak (2015) develop a mixed-integer program (MIP) for computing a Stackelberg strategy, and Cermak et al. (2016) develop an iterative approach based on upperbounding solutions from extensive-form correlated Stackelberg equilibria. To our knowledge, we are the first to consider uncertainty about the opponent in Stackelberg strategies for EFGs. This is important because EFGs are a powerful representation language and because in practice there is typically uncertainty about the opponent. We take a robust approach to modeling this uncertainty. We introduce robust Stackelberg equilibria for EFGs, where the uncertainty is about the opponent's payoffs, as well as ones where the opponent has limited lookahead and the uncertainty is about the opponent's node evaluation function. We develop a new MIP for the deterministic limitedlookahead setting. We then extend the MIP to the robust setting for Stackelberg equilibrium under unlimited and under limited lookahead by the opponent. We show that for the specific case of interval uncertainty about the opponent's payoffs (or about the opponent's node evaluations in the case of limited lookahead), robust Stackelberg equilibria can be computed with a MIP that is of the same asymptotic size as that for the deterministic setting. Our results for robust Stackelberg equilibria in EFGs are relevant to security-game settings with sequential interactions, where EFG models can more compactly represent certain games, as compared to a normal-form representation (Bosansky and Cermak 2015) . Robust models are important in security games, where opponent models often have uncertainty, both in standard security games (KiekarXiv:1711.08080v1 [cs.GT]
Also, whereas learning in the context of security games has appeared in the literature (Balcan et al. 2015; REF , this paper introduces an RL-based approach to threat screening for the first time.
Robust Stackelberg Equilibria in Extensive-Form Games and Extension to Limited Lookahead
ArXiv
ArXiv
Mathematics Computer Science