[George Rice]

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I wasn't sure Snyder had actually made this claim, but after a careful re-reading of all 3 articles I can see that he did (at least some of that contained in the paragraph I quoted above). I'll need to go back and review your prior comments.


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I posted the cite from his second article in my second post in the thread in the magazine forum in response to your question then.

To save everyone the legwork:

His statement:

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An equal-skill tournament analysis would lead us to believe that if only two players remained at a final table, with one player holding 90% of the chips and the other holding 10%, the player with 10% would still have a 10% chance of winning the event. In fact, if these players were flipping coins to determine the winner, that would be true. But it’s not true if the player with the bigger chip stack is a skilled tournament player who understands how to use his chips. In this case, the player with only 10% of the chips has almost no chance of winning, even if that player with 10% of the chips matches the skill level of his opponent.

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My proof:

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Snyder has asserted that even when heads up the more chips you have the more each chip is worth. This fact is easily disproved. And unlike multi-table and multi-player situations, is not complicated to show dollar values for the chips.

For example, in David's 32 player tournament, it gets down to two players. And we'll assume it's winner take all for this example. Hero has 3199 chips and Goat has 1 chip. How much is this 1 chip worth? How much is the 3199 stack worth? We assign a value of X for one chip, and 3200-X for the value Hero's stack (because the total value must equal 3200). In order for Arnold's claim that chips gain value to be correct, the last chip won by Hero must be worth the most. Therefore it must be worth more than $1. If it was worth exactly $1, then Hero's stack would be worth 3199 (3200-1), the same as the number of chips in his stack, meaning his chips were worth $1 to begin with (in violation of Snyder's assertion). If the last chip won is worth less than $1, then Hero's stack is worth more than 3199, and the average value of each chip in that stack was worth more than $1 (again, in violation of Snyder's assertion). So to keep with Snyder's assertion, the last chip must be worth more than $1 (and hence, each chip in Hero's stack is worth less than $1)

But that means X>1. So Goat's 1 lone chip must also be worth more than $1. So Goat's lone chip is worth more than each chip in Hero's stack! And if each chip he gains is worth more than the previous chip, all Goats individual chips are worth more than Hero's individual chips (they can never be worth less than $1). You see where this is going. This is also in violation of Snyder's assertion. Therefore, Snyder's assertion that chips gain value in a head-up situation in not valid.

And this isn't a phenomenon that occurs only at the extreme ranges. Because when heads-up any EV or "SV" gained by a large stack has to be lost by the small stack. Any EV or "SV" lost by the large stack has to be gained by the small stack. EV doesn't appear from or vanish into thin air.

The conclusion you can draw from this is that when heads-up, individual chips neither gain nor lose value. Snyder proved this himself in his second article "Chip Value in Poker Tournaments."

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And, of course, I was talking about an equal skilled situation.