[eastbay]

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1) In TPFAP, there is a very elegant proof "by symmetry" of the proportionality argument. It basically relies on the reasoning that if I have, say, 20% of the chips, and you have 80% of the chips, and we have equal skill, then I have a 50/50 chance of doubling up. The critical element of the proof is that, with equal skill, I always have exactly a 50/50 chance of doubling up. I started to question whether this was indeed true. Might it be possible that the optimal game-theoretic strategy actually depends on the stack sizes? In that case, all bets are off and the proof doesn't work.


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Let's not confuse issues. Equal skill by no means implies both players are playing optimally (game theoretic sense). The so-called "proof" of S&M is obviously wrong by counterexample.

Both players play the following (pathological, but equally "skilled") strategy: push if you have more than half the chips, fold if you have less than half. Flip a coin if stacks are even.

You need (at least) some kind of stack independence of strategy condition before this symmetry argument has any hope of being sufficient to guarantee a linear relation. Otherwise, you're left with a family of curves which are all admissible by symmetry, from linear to step function, with the only symmetry requirement being f(x)+f(1-x)=1.

Now, I grant that S&M may have been trying to simplify the discussion for a general audience, but I would prefer they not use the word "proof" in that context.

As an aside, buried somewhere, I think in this thread, is someone's offering of a proof of what you conclude in your discussion: that the "extra" chips are strategically useless for optimal play. I never took the time to try to convince myself that it is correct, but I suspect that it is.

eastbay