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Technologos comments on Playing the Meta-game - Less Wrong
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You're right--and as my professor said, since I had a better initial situation, I should have been able to do at least as well as my opponent.
Tearing up two slips would have been unlikely to beat tearing up one--the latter creates the necessary scarcity (and causes the auction) while not diminishing the total number of deals that much. In fact, since my opponent beat five dollars, I couldn't have won with two torn slips, but I should at least have been even with my opponent, and might (as my reply to CronoDAS notes) have been able to win via a trick.
I could refuse a bad deal, but my partners knew that I was trying to win $20, where they stood to lose at most a dollar relative to agreeing to the deal; they had less to lose, and thereby had a stronger bargaining position (the literature on Nash bargaining is relevant).
As a further note, though, if by
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).