As a further note, though, if by

doesn't seem likely that you could actually end up doing worse that those with fewer slips

you mean I should be able even without tearing up a slip or otherwise limiting my own options, then no--I was in a weaker bargaining position at the beginning, and game-theoretically I should have ended up worse than my opponent. That was a key finding of Thomas Schelling's, though he applied it to nuclear warfare (see The Strategy of Conflict and also the link at the beginning of this post for more info on his bare-knuckle game theory).