This is a good observation. While being "fair" and "irrational" may lose you the $1 you may have otherwise gained, it also forces the others to offer more than the $1 because they do need your participation (or at least the participation of a majority).
If you are the top guy and a majority of the others won't settle for $1, wouldn't you offer, say $100k each so that you can get perhaps $400k instead of nothing?
If you aren't playing "fair" and are being entirely rational, all you will get is the $1 if you are not the top guy. Why settle for that when by being irrational you can get more?
Merged post follows:
Consecutive posts merged
And then sometimes you have to play it out so that others know you aren't just bluffing. So what if you, as exec # 7, lose $1? Its not a big deal because on the next time everyone else knows they have to offer more, perhaps considerably more, to get your participation.