Before revision:  This is in stark contrast with reachability games, which are known to be much more complicated under{\em   all-pay bidding than first-price bidding.
After revision:  This is in stark contrast with reachability games, which are known to be much more complicated under{\em in which in each turn, the player who moves is selected according to a (biased) coin toss. The equivalences for all-pay bidding than first-price bidding.