Re: ??
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You're playing games with terms trying to define the issue the way that suits your argument. I don't agree that the cash value of an individual chip is zero. If the tournament were stopped prematurely for some reason and payouts were made on remaining chips, they would have value. Any additional value based on skill would be subjective and would have to be ignored. So the chips have value independent of skill.
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I do not play my hands (nor does anyone else to my knowledge) with is consideration, so I do not see how it can ever apply to the strategy?
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And I agree that the "utility" value as you use the term is not linear based on stack size. And trying to plot a curve based on stack sizes, and skills of the remaining players would be hard, and such a curve would be complex. And no two curves would ever look the same.
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Absolutely agree, obv.
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But if you claim that chips added to a stack are worth more than chips already in that stack at any place on such a curve then the burden is on you to prove it. No one has done this. Some have demonstrated that they don't, at least on average.
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If you graph the utility value of a single chip based on the stack size in relation to eiter blinds or average stack - you will have a basic sideways hyperbole(SP?) pattern where the less chips you have - the less utility value they have (can't do MUCH with a single chip), as you get more chips their utility value will increase. At some point the line will level out to the point where doubling your stack will have ALMOST NONE additional utility value. At the early-middle of the curve (where it's steepest) is the place where typically chips added to your stack will increase it's utility value MORE than just the straight chip-count. Easy example - your M is 4, you are in push-fold mode, almost no FE. Once you double up - you are now in push/fold mode but with some FE - congrats, the UTILITY value of your stack has grown by MORE than just the straight chip count. If you double up again - you now have M=16 and now you can start re-stealing with FE and maybe even playing small pots in position and what-not.
Ok, another old beat-up and much discussed (and not mine) example. Everyone at your table has 100 chips, you have 150, one guy has 50. If you can get your hands on those 50 chips - you'll have 2x chips of anyone else at the table, and that extra 100 chips has significant utility value that is (for some players anyway) is more than double of utility value of the 50 extra chips we currently have. Gigabet states in his theory of stack sizes (if you haven't read it - you should, it's an interesting read) that he will KNOWINGLY take a -EV gamble for those 50 chips for a chance to get to 200, since if he loses the "race" he's down to 100 and no worse off, really, but if he wins he is getting advantage that is sufficient to overcome the long-term -EV gamble he had to take to get those chips.
The same, essentially, applies to taking a coinflip early in a tournamet - AS LONG AS YOU CAN EXTRACT MORE VALUE FROM LARGER STACK. Which, once again, brings us back to skill... By gambling to double-up you're giving up a certain edge (ability to outplay your opponents) in order to try and get a big stack because you feel that you have MORE of an edge over your opponents when you have a BIG stack.
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One place where a chip might be more than a previous chip is when such a chip gives you an additional hand, blind or round of hands.
Example: You are last to act in a heads-up hand on the river. The game is a no-limit holdem tournament and the blinds are 50/100. The next hand the blinds go to 100/200. There is 400 in the pot and you have 650 left. If you check and take the pot you'll have 1050, enough for three rounds plus part of the big blind. If you bet 100 you have 1150 if you are called and win. You'll have enough for three rounds plus the big blind and part of the small blind. But if you bet 200 and are called and win you'll have 1250, enough for four rounds plus. The second 100 you win by betting 200 is probably worth more than the 100 you'd win by betting 100.
But while the second 100 chip that you win may be worth more than the first 100 chip that you win, the second 100 chip you lose if you bet 200 and lose the hand is worth more than the second 100 chip should you win.
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I see your point (even though it's a bit contrived - I'm not entering the pot with 8 BB's if I do not intend to get ALL of my chips on the table, EVER - even if it's a limit game). However, once again - being able to survive through blinds has little enough value - you need a stack to work with, you need FE, and in CLOSE situations like these you are looking to double-up or go busto...
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