  In this paper, we study the notion of adversarial Stackelberg value for
two-player non-zero sum games played on bi-weighted graphs with the mean-payoff
and the discounted sum functions. The adversarial Stackelberg value of Player 0
is the largest value that Player 0 can obtain when announcing her strategy to
Player 1 which in turn responds with any of his best response. For the
mean-payoff function, we show that the adversarial Stackelberg value is not
always achievable but epsilon-optimal strategies exist. We show how to compute
this value and prove that the associated threshold problem is in NP. For the
discounted sum payoff function, we draw a link with the target discounted sum
problem which explains why the problem is difficult to solve for this payoff
function. We also provide solutions to related gap problems.
