Question about Tony Guerrera article

[helter skelter]

I've been trying to learn about ICM and have been reading threads referred to in FAQ's, their associated links, etc. In one of them, I came across this. Since I am interested in MTT's I read the article by Tony Guerrera referred to in this post:



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Ultimately, the proper way to do tournament analysis is to take into account everyone's pushing and calling distributions as a function of stack size and to calculate explicitly the optimal moves to make as a function of those distributions and the current relative stack sizes.

However, once the average stack is less than 10BB, there's not too much of a skill edge that you can exploit, especially at the higher buy-in STTs.

In general, the amount of risk you should shy away from in tournaments is grossly over estimated. I know this is an STT forum, but here's a link to an article I wrote about this topic, as it applies to early round play in MTTs:

http://www.pokerhelper.com/when-to-r...tournament.php

As people gravitate more towards playing an optimal endgame in STTs, it's going to become tough to have any sort of significant edge in the endgame, meaning that your edge actually needs to come from outplaying your opponents early on and getting yourself to the endgame crapshoot with a chip advantage more consistently than your opposition.

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(This is the link to the thread, for anyone interested.)


http://forumserver.twoplustwo.com/sh...age=0&vc=1


Anyway, I understood the article (mostly) until I got to this point:


"P(2nd) equals the probability of doubling up to half the chips in the tournament minus the probability of winning the tournament. P(3rd) is the probability of doubling up to a third of the chips in the tournament minus the probabilities of finishing in second and first. In general, the probability of finishing in ith place is the probability of doubling up to a stack of minus the sums of the probabilities of finishing in a higher place."



I can conceptualize P(2nd). When you get to 2nd, its only 1 double up to 1st, but I don't understand how P(3rd) would be the probabilty of getting 1/3 the chips.

It's hard to conceptualize, but the only way I can conceptualize P(3rd) would be taking half the chips from the tourney and putting them in 1 stack, the other 2 players each having 1/4 chips and then one of the quarter stacks doubling through the other 1/4 chips. So the player knocked out would have only reached 1/4 chips.

Preceding that, you'd have 1/2 chips in one stack, 1/4 in one stack and two 1/8 chip stacks, etc.

I'm sure I must be missing something, but is it even possible to double to 1/3 the chips? If you doubled up again, then you'd have 2/3 the chips.


He goes on to say:


"In general, the probability of finishing in ith place is the probability of doubling up to a stack of N/i minus the sums of the probabilities of finishing in a higher place"


Am I right in the way I am looking at it? Is he right in the way he is looking at it? Or are we both wrong?

[helter skelter]

I should note that he defined N as the number of players in the tournament.

[pzhon]

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I'm sure I must be missing something, but is it even possible to double to 1/3 the chips? If you doubled up again, then you'd have 2/3 the chips.

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That's not the point. With equal skill, the probability of reaching n times your stack before busting out is about 1/n. This doesn't require n to be a power of 2. He has a generalization to to situations where you assume that a player has a skill advantage.

His estimate of the probability of finishing in nth place is the probability of winning a tournament with 1/n times as many players minus the probability of finishing in higher places. This is a model which is only suitable for the start of a tournament. It's not clear whether this is a good model or a bad model.

I think he comes up with really bad conclusions using this model. His results are still much more conservative than from other models. I'm not sure why. I think he made a serious miscalculation.

[helter skelter]

OK, I see now how he presumes that part of the equation. I had to think of it in terms other than doubling.

As each player busts out, he evenly distributes his chips to the other players, so using the 16 player tourney:

Player 16 goes out and now each of the other players has 1/15 the chips

Player 15 goes out and now each of the other players has 1/14 the chips

. . . .

Player 4 goes out and now each of the other players has 1/3 the chips

So rather than saying doubling, he could say:

"In general, the probability of finishing in ith place is the probability of aquiring up to a stack of N/i minus the sums of the probabilities of finishing in a higher place"

Doubling is just the chip aquisition method he is discussing


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I think he comes up with really bad conclusions using this model. His results are still much more conservative than from other models. I'm not sure why. I think he made a serious miscalculation.

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The general conclusion is something I've seen alluded to in other posts; that even a superior player can't afford to pass up high EV+ situations.

His example uses a player that has an ROI of 100% and he concludes that D (representing the probability that you would win an all-in confrontation) would only need to be .58 in the smaller tourneys and .56 in the larger tourneys (given typical payout schedules).

Are you saying D is higher using other models?

[pzhon]

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I think he comes up with really bad conclusions using this model. His results are still much more conservative than from other models. I'm not sure why. I think he made a serious miscalculation.

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The general conclusion is something I've seen alluded to in other posts; that even a superior player can't afford to pass up high EV+ situations.

His example uses a player that has an ROI of 100% and he concludes that D (representing the probability that you would win an all-in confrontation) would only need to be .58 in the smaller tourneys and .56 in the larger tourneys (given typical payout schedules).

Are you saying D is higher using other models?

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No, it is much lower in every other model I've seen.

Even with this one, I don't see how he goes from the prize distribution to a value of D other than the probability of doubling up. In his model, if the tournament pays less than half of the players, you have to double up to get any prize. Either there is a critical part of his model which he didn't include, or else he miscalculated.

His model of chip accumulation is missing a few things, including time. If I double up on the first hand, this is very different to me than doubling up at the end of the 4th level. It means I can still apply my skill advantage to accumulate more chips (on average) for 4 extra levels. This means doubling up on the first hand is better than doubling up at the typical time, and I should accept a lower probability of doubling up in order to do so quickly.

For example, suppose I start with 1000 chips, and expect to accumulate 100 chips on average by doubling up 55% of the time at the end of level 4, and busting out 45% of the time. If I have 2000 chips at the start of the tournament, how many should I expect to have at the end of level 4? I think the answer should be between 2100 and 2200. If 2000 chips at time 0 is worth 2200 at time 4, then D~50%. If it is worth 2100 at time 1, then D~52.5%. Both of these are lower than my 55% probability of doubling up.

That points out how implausible it is to require a value like 56%, higher than the probability of doubling up normally in Guerrera's examples. It also shows that this model is not so great, and isn't worth studying that deeply.

[pzhon]

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His example uses a player that has an ROI of 100%


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Ah, now I think I see what happened. I think the above is the missing (and highly implausible) assumption. He assumes that the player being analyzed has an expected return of 2 buy-ins no matter what. He assumes that you know the player's ROI is 100% ROI regardless of the size of the field or the payout structure. No one has a ROI of 100% in SNGs played for at least $1, but many players may have a ROI greater than 400% in large tournaments like the WSOP main event.

I believe the implausibly high requirements come from assuming a high ROI in smaller tournaments than were considered by others, combined with ignoring the value of time to a winning player.

[helter skelter]

I don't think finding an exact number would be worthwhile, anyway. It looks like the numbers are close enough to 0.5 in most situations that even a great player would be wrong to pass up coinflips. By the time you know it's a coinflip, there is probably already enough money in the pot to push it well over 0.5. Add to that the fact that you rarely can be sure if it's a coinflip.

Thanks for the explanations.