[JNash]

Sorry for posting so long after the fact-- I have not been to the site in a long time. Since I did the original S-Curve post, I thought I should comment.

Two comments: first, I now disavow the S-curve hypothesis. I believe that I was wrong. (See below.) Second, doing an empirical study of the ACTUAL chipEV-->tourneyEV function would be very interesting.

Starting with the idea of emprical testing first...In that case, rather than focusing on heads-up, you might as well test how stack size at certain points in a tourney affects tournament EV. For example, in large MTT tourneys, after about an hour the field is usually cut by 50% and there is a pretty wide dispersion between chip leaders and small stacks. With enough tournament samples, you could estimate pretty easily what the emprical tournament EV is for someone in the top 10% of stack size, with an average stack, etc. If you had enough data and the patience of a PhD student, you could even run some regressions to try to separate the effect of skill (observed win rate) and stack size.

Now, as to the S-Curve hypothesis. As my nick indicates, I am interested in game theory, so my original post was in the theory forum. So, I take as a given all the standard theoretical assumptions, i.e. equal skill, optimal game-theoretic play, etc. Under these conditions, I now believe that the following Sklansky/Malmuth assertions are absolutely correct (in theory, that is):
1) EV of winning a heads-up freezeout is proportional to the fraction of total chips you have (i.e. linear)
2) In any winner-take-all tournament (heads-up is just a special case of this), the probability of winning is proportional to the chips you have.
3)) Tournament EV is a concave fuction of chip EV--i.e. NOT the S-curve I had hypothesized.

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.

I have since then convinced myself that the fact that the big stack has some "extra" chips in reserve does not affect the optimal strategy for the two players at all. The only thing that matters in determining the optimal play is the size of the blinds relative to the size of the stacks. If I have 200 chips out of 2000 in play, and you have 1800, we are currently effectively playing a game where we each have 200, since the most we can bet is allin. I don't have a "proof" proof, (of the fact that the optimal strategy for the two players is independent of any "extra" chips one of them may have), but I believe this to be true.

2) If optimal optimal play depends only on the size of the smallest stack involved in a confrontation, and the availability of extra chips for some players does not matter, then the TPFAP proof for the multi-player winner-take-all tourney also goes through without a hitch.

3) Finally, the concavity question, which is the same thing as the TPFAP assertion that "the chips you win are always worth less than the chips you lose."

I now believe that this is always true in the case of multiple payouts and more than 3 players.

I'll give a heuristic argument for this effect. First, suppose you have 80% of the chips in play, 5 places pay, and there are 5 players left. You are very, very likely to win 1st place. If you win a confrontation that busts out one of the players, you gain some EV, but some of it "leaks" to all the other players who are now assured of finishing one place higher. So, your gain in chips does not give you a linear pickup in tournament EV.

Extending this further, suppose you are an above-average stack, and you win a confrontation with a small stack. You've pushed the small stack closer to elimination, which benefits not only you, but also other players--i.e. you get concavity again.

My original S-curve argument assumed that big stacks have an advantage over small stacks because they can "bully" the short stack. While this may still be empirically true in actual play, I now believe that the game-theorically correct play for the small stack does NOT depend on the presence of extra chips for the big stack. The small stack plays strictly based on pot-odds, the size of the blinds, and the maximum amount that can be bet--i.e. his own stack size.

Anyway, sorry for sending y'all on a blind goose chase, but I now no longer believe in the S-curve...

May your flops be disguised and your rivers kind!