Re: S-Curve Hypothesis
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Of course it is going to be possible for a player to have a chance of doubling up with the small stack of under .5 and with the large stack of over .5, and in this case obviously what you say will be correct. I'm just saying that this player might think there is nothing he can do about this fact but as long as the blinds are still reasonable in comparison to his stack and we arn't in a bubble situation with work his game should be able to improve to get this probability of doubling up with a smaller stack to above .5.
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That is the essence of the S-curve hypothesis. Even if all players have equal skill, their chip count influences their probability of winning. You state it in terms of the probability of doubling up, but I believe the point is even more general: a priori, before the cards are dealt, the short stack has a negative chip-EV and the big stack a positive chip-EV. Incidentally "by symmetry" (since this is after all a zero-sum game) if there were only two players, the amount of positive chip-EV for the big-stack is exactly equal to the amount of negative chip-EV of the small stack.
The practical question is when in a tourney this S-curve effect actually becomes pronounced, versus when the payoff function is simply linear. I believe that when the blinds are low compared to the average chip count, the curve is pretty much linear, and you ring-game EV calculations are correct. But as the blinds increase relative to the average stack size, I believe the curve becomes more S-shaped--certainly around bubble-time, but even beyond that.
Further, I believe that my hypothesis holds true no matter how steep the payout structure--i.e. it would include winner-take all situations. Mind you, the exact shape of the payoff function will certainly depend on the particulars of the payout structure, but the general convex-concave feture should not be affected by this.
I even believe (gasp--dare I say it?) that this applies to heads-up situations in no-limit winner-take-all settings, as long as the blinds continue to increase regularly. [I say this knowing that this sounds heretical and contrary to all accepted gospel, but I'll just need to go to the theory forum and take my licking...]
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