Negreanu's tournament theory regarding big pots.

[JNash]

You make some very good points, such as the fact that a short stack is far more likely to be able to value-bet his good hands and get called. My counter to this is that the short stack gets into positions where he wants to value-bet so infrequently that this is not worth as much as the big stack's better chances at winning with bluffs.

I have purposely called this a "hypothesis" because I can't prove it, yet believe it to be true. The crux of the discussion is whether a short stack, beyond the obvious point that his stack is worth less than a bigger stack, is at a tactical advantage or disadvantage compared to a bigger stack.

I claim that because "big stacks get more respect" the big stack is in a better tactical position--i.e. a priori (before the hand is dealt) he has a positive chip-EV, while the short stack has a negative chip-EV.

You have listed some good reasons why a short stack might have an advantage, so the question is "net-net" which of these factors is more important.

I'll give you two things to ponder--one empirical, the other theoretical.

The empirical point is simpy based on what I've observed happen at various stages in a tournament. For example, at bubble time, there can be little question that short stacks play very conservatively (whether this is optimal or not is beside the point, it's simply a fact). At this point in a tournament, big stacks can (and often do) mop up lots of chips. This is consistent with saying that their a priori chip-EV is positive at this stage of the tourney.

Empirically, I would also say that even post-bubble, short stacks can be "forced" to play hands that they don't want to simply because the rising blinds are putting pressure on them. The "don't want to" means that they play even if they face negative chip-EVs.

My second argument is purely theoretical. It has been stated in many books (including TPFAP and Gambling Theory) that the payoff function is concave -- i.e. the chips you win are worth more than the chips you lose. Since the tournament is a zero-sum game, when you compute the fair value of the chips across all players, they have to add up to the total prize pool. So if player A has a concave function, there must be some player B who has a convex function. My contention is that the big stacks all have concave functions, and the short stacks convex ones. (This assumes equal skill -- with unequal skills this would be much messier, but I believe still directionally correct.)

[JNash]

It's definitely not based on the idea of two players being dealt the same cards in the same hand. It is simply a mental experiment that given any given set of two cards, a big stack can make more of it (has a higher chip-EV).

I believe that the ability to survive a run of "unplayable" cards is indeed part of the argument. A big stack can actually "play" some cards that a short stack wouldn't want to play. If you add the big stack's chances of winning the pot "fair and square" (i.e. play it to showdown) to his ability to force the opponent away from the pot, his probability of doing the latter allows him to play far more pots and thus do better even when the cards are unplayable.

At this point I can't see that position has much of an influence on the shape of the payoff function--except in extreme cases like when your stack is down to one big blind. In that case, the fair value of your chips is higher if you still have 8 hands to get lucky or whether you're forced to play the next hand.

[JNash]

It somewhat depends on exactly what you mean by "conservative" and "aggressive." You may want to read my answer to eastbay's post above.

My short answer is that the fact that the payoff function is convex for the short stack means that he is CORRECT (positive fair value-EV) to take 50/50 chances, or even some negative chip-EV situations. In other words, he "should" gamble it up.

However, the short stack can't afford to play as aggressively as the big stack (in the sense of trying to bully people out of a pot with sub-par cards), because he'll get called, so he needs to be far more selective as to when he picks his spots. His only saving grace (see pzhon's post) is that when he does enter a pot with premium cards, he's quite likely to get full value.

[Pensive Gerbil]

When I read Negreanu's article, I was surprised by what seemed to me to be his misunderstanding of basic tournament theory. I suppose it shows that excellent math/logic skills are not that important.

Sorry...I couldn't resist! [img]/images/graemlins/wink.gif[/img] I believe Daniel N. is a very talented and courageous player who may have a couple of leaks in the area of logical/theoretical thinking. Perhaps he's been lucky that his particular logical leaks have not led to more costly errors.

-PG

[Ezcheeze]

Whether or not having a larger stack helps you win the next hand is irrelevant. If your chances of doubling up are greater than .5 then, as many poeple have shown previously in the thread, your chances of winning the tourny do not double after you have doubled your chips.

It seems like what you might be saying is that when a skilled player has a small stack then his chances of doubling up decrease. I doubt that this is true, and it would have to decrease to less than .5 in order for him to want to take -EV situations in the hopes of doubling up. If this was true then the stack size it is true for is surely so small that the whole concept is useless anyway.

-Ezcheeze

[REL18]

I have a quick question for you probably not gonna answer but only takes a yes or no answer. Do you know Barry Carpenter