Negreanu's tournament theory regarding big pots.

[fnurt]

Before the double up, you were 1.5% to win the tournament. After the double up, you were 2.5% to win. That sure sounds like less than twice to me.

[JNash]

Hello David

I understand your point, but would like to pose a slightly more complex and perhaps more realistic question.

The question I am after is the relationship between stack size and fair value of the stack AT A PARTICULAR POINT IN TIME, i.e assuming same # of players left, same distribution of chips, etc. If an outside player of a given skill level could buy a 20000 chip stack or a 40000 chip stack, should he pay more, less or exactly 2x for the latter?

You point out that if he has above average skill, the answer is that 40K is worth less than 2x the 20K stack. Fine so far.

But suppose that your probability of winning the next hand for any given 2 cards you are dealt CHANGES depending on your stack size relative to field???

If you are short stacked and enter a pot, you are more likely to get picked on--you have an adverse selection problem. Your chances of winning a bluff are less than if your stack were bigger, your chances of winning blinds are less, etc. For a big stack, the reverse is true--you are more likely to win bluffs, steal blinds, etc.

This leads me to what I'll call the "S-curve hypothesis."

Assume everyone has equal inherent skill. By the middle of a tournament, some players have bigger stacks than others simply as a result of luck/random walk.

Now plot a curve that has the "fair value" (FV) of a stack on the y-axis and the stack size on the x-axis. What is the "fair value" FV that an outsider should pay for different stack sizes? My hypothesis is that initially that relationship would be convex, and later on concave. It would be an S-curve, with the inflection point somewhere near the average stack size.

My heuristic proof is that if your chip count is quite low relative to the average stack size, not only is your chance of winning the tournament low (i.e. the FV is low), but your chances of winning the next pot with any given hand are lower than average (it's as if your skill were suddenly lower than average). From these low levels, doubling your chip count will MORE than double the FV of your stack (i.e. the function is convex here).

Conversely, if you have a big stack, your chances of winning the next pot are BETTER than those of the average stack. Even though you have the same inherent skill as all the others, your "effective" skill is now greater than average. Doubling your chips now less than doubles your FV (per your point).

If the S-curve hypothesis is true, a number of things follow logically:
1) If you are a short stack, you would be mathematcially correct to take some "gambles" where your CEV is negative--because the chips you win have a higher $EV than those you lose.
2) If you are a big stack, you would be mathematically correct to avoid some positive chip-EV situations because the chips you lose are worth more $EV than those you lose.

Please note that this does not necessarily imply that a large stack should play fewer hands than a small stack. Remember, the probabilities a big stack would use to calculate the chip EV are better than those of a short stack--even if you have been dealt identical cards. While the shape of the FV-chip count relationship would lead a big stack to be more conservative, the fact that his chip-EV is much better than average may overcome this and lead him to correctly play more hands than a short stack.

Carrying this thinking further, one can speculate how the shape of this S-curve changes as the tournament progresses, and as the size of the blinds changes relative to the average stack size. Obviously, as fewer players are left, the y-axis intercept moves up (e.g. if you're in the money even a small chip count is guaranteed something). As the blinds get big relative to the average stack size, I would think that the curve would get "more bowed"--i.e. it becomes more like a step-function--low chip levels are virtually guaranteed not to advance much further, while big stacks are virtually guaranteed to do well. At the low end of the curve, it has become more convex, and for above-average chip counts it becomes more concave.

Sorry to ramble on for so long, but this has been bouncing around my head for quite some time...

[David Sklansky]

My point is not about tournaments per se. It is pure irrefutable logic and it is unacceptable if anyone on this forum doesn't understand it. I'm hoping I don't have to explain it.

[fnurt]

You are in the wrong forum to be so smart [img]/images/graemlins/wink.gif[/img] but what you say sounds correct to me.

[JNash]

Hi Aces.

Thanks, your explanation is clear and compelling.

Now, take a peak at my othe post regarding the S-Curve Hypothesis in this thread. What if the probability of winning a hand (your "effective skill level") changes depending on your stack size? I believe the answer then becomes that for low stacks, doubling up is worth more than 2x, while for large stacks it is less than 2x.

[Rushmore]

I would have posited that your point was that the more chips one has, the less value each chip has, thusly diminishing the overall value of one's stack relative to a smaller stack, and therefore diminishing the odds of winning the tournament relative to the percentage increase in chips achieved.

But then you made that disclaimer about tournaments per se, and now I'm in the weeds again.

You are rolling this out like it's the easiest thing in the world.

I seem to recall a similar tone in TPFAP ("...if you do not know how I got this number, you are not ready for this book...").

I read the book anyway.

[eastbay]

[ QUOTE ]
Before the double up, you were 1.5% to win the tournament. After the double up, you were 2.5% to win. That sure sounds like less than twice to me.

[/ QUOTE ]

Huh?

You said: "You are in a 100 player tournament and have a 1% chance of winning."

Ok, 1%.

"But if you double, you are so good at playing a big stack that even though you have 1/50 of the chips in play, you have a 1/40 (2.5%) chance of winning."

Last time I checked 2.5% is more than 2x 1%.

eastbay

[fnurt]

You shouldn't have stopped reading. I went on to demonstrate why the initial 1% chance of winning, while it might be what most people would assume, is inaccurate if you have a 60% chance of doubling up.

[eastbay]

[ QUOTE ]
You shouldn't have stopped reading. I went on to demonstrate why the initial 1% chance of winning, while it might be what most people would assume, is inaccurate if you have a 60% chance of doubling up.

[/ QUOTE ]

So let me get this straight. You started from a false premise, and then from that, logically derived a conclusion which was, in fact, the correct premise? QED?

Uh, ok.

eastbay

[Eldog605]

yeah, I would have to say that was one of the smartest, yet best explained "theoroms" I've read in some time. Its nice to hear a mix of math and the english language.