Fragility and Robustness in Mean-payoff Adversarial Stackelberg Games

Two-player mean-payoff Stackelberg games are nonzero-sum infinite duration games played on a bi-weighted graph by a leader (Player~0) and a follower (Player~1). Such games are played sequentially: first, the leader announces her strategy, second, the follower chooses his strategy. This pair of strategies defines a unique infinite path in the graph and both players receive their respective payoff computed as the mean of the rewards that they receive when traversing edges along the infinite path. As a consequence, if we assume that the follower is rational then we can deduce that the follower's response to the leader strategy is a strategy that maximizes his payoff against the strategy proposed by the leader; it is thus a best-response to this strategy. Knowing that, the leader should choose a strategy that maximizes the payoff that she receives when the follower chooses a best-response to her strategy. If we cannot impose which best-response is chosen by the follower, we say that the follower, though strategic, is \emph{adversarial} towards the leader. The maximal value that the leader can get in this nonzero-sum game is called the {\em adversarial Stackelberg value} of the game. First, we show that the nonzero-sum nature of the mean-payoff Stackelberg game makes it fragile against modelling imprecisions. This is in contrast with mean-payoff games in the zero-sum setting which are robust. Second, we show how robustness is recovered when considering $\epsilon$-best responses of the follower instead of best-responses only. This lead to the notion of $\epsilon$-adversarial Stackelberg value. Third, we provide algorithms to decide the threshold problem for this robust value as well as ways to compute it effectively. Finally, we characterize the memory needed by the strategies of the leader and the follower in these games.