Negreanu's tournament theory regarding big pots.

[fnurt]

My post was clear to others, I apologize if it wasn't clear to you.

Let me try to explain it in mathematical terms.

At the beginning of the tournament, your chance of winning is X. If you successfully double up, your chance of winning will be Y.

We have arbitrarily assumed that your chance of doubling up before busting is 60%. What David Sklansky was pointing out was that this assumption, in and of itself, defines the mathematical relationship between X and Y. Since you have a 60% chance of increasing your winning chances from X to Y, X must be 60% of Y. Thus, you have not doubled your winning chances by doubling your stack; you have increased them from 0.6Y to Y.

By the same token, if you have less than a 50% chance of doubling up before you bust, then doubling up will always more than double your chances of winning the tournament. As Sklansky said, this is simply a mathematical fact.

[eastbay]

[ QUOTE ]
My post was clear to others, I apologize if it wasn't clear to you.

Let me try to explain it in mathematical terms.

At the beginning of the tournament, your chance of winning is X. If you successfully double up, your chance of winning will be Y.

We have arbitrarily assumed that your chance of doubling up before busting is 60%. What David Sklansky was pointing out was that this assumption, in and of itself, defines the mathematical relationship between X and Y. Since you have a 60% chance of increasing your winning chances from X to Y, X must be 60% of Y. Thus, you have not doubled your winning chances by doubling your stack; you have increased them from 0.6Y to Y.

By the same token, if you have less than a 50% chance of doubling up before you bust, then doubling up will always more than double your chances of winning the tournament. As Sklansky said, this is simply a mathematical fact.

[/ QUOTE ]

Ok, I do see what you meant. I do, however, think it was written in a misleading way.

I apologize for jumping on you.

eastbay

[Slo Pok]

J Nash commentary:

Interesting combination of left brain and right brain application. Beyond the linear and sequencial thought patterns of left brain logic in my opinion.

Thanks [img]/images/graemlins/smile.gif[/img]

[eastbay]

Nice post.

There does appear to be a self-balancing effect in what you describe, doesn't there?

The advantages you describe of the big stack I think can be called aggressive tactics - more bluffing opportunities, more blind stealing opportunities.

But the consequence is that the big stack should play more conservatively - pass up small +chipEV situations, as you say.

It seems there is a balancing effect - if the bigger stack has an advantage in being aggressive, the conclusion is that he should play more conservatively?

Can you give any more detail about how you reconcile this?

eastbay

[TStoneMBD]

I fully understand DS's last post, but I agree with JNash's standpoint here and hope that DS has the time to argue if JNash's logic is flawed. JNash's standpoint was what I was trying to define earlier, but am not as sophisticated with the terminology as he is.

[aces961]

[ QUOTE ]


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).


[/ QUOTE ]

The initial comment David made was for winner take all tourneys, and we were commenting on the probabilities of winning the tourney not about ev in a tourney with a very complicated pay schedule. When this pay schedule becomes very complicated as say a party 1000 person tourney with 100 different payouts then things can change. I would ask you to note the flatness of most payout schedules after they have jumped to zero to right above the buy in. Most of them stay pretty flat until you get to the final table, online at least. The winner take all tourney calculations are a pretty good approximation of how you should play if you aren't nearing the final table or nearing the initial start of payouts in these type of tournaments.

In non bubble situations I believe that the tourney should be played similarly to a cash game, thus it should still be possible for a smaller stack to be able to play well enough to be able to double up over half the time. Now if a player is truly worldclass in any reasonable situation his chances of doubling up before going broke should be above .5 unless he is playing with other worldclass players. While this world class player may only have a chance of doubling up with the smaller stack of .501, while with the larger stack above the average stack it is .6.

Of course it is going to be possible for a player to have a chance of doubling up with the small stack of under .5 and with the large stack of over .5, and in this case obviously what you say will be correct. I'm just saying that this player might think there is nothing he can do about this fact but as long as the blinds are still reasonable in comparison to his stack and we arn't in a bubble situation with work his game should be able to improve to get this probability of doubling up with a smaller stack to above .5.

[pzhon]

[ QUOTE ]

My heuristic proof is that if your chip count is quite low relative to the average stack size, ... 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).

[/ QUOTE ]
I disagree. It's a good thought experiment, but I think your "heuristic proof" is specious.

If your chip count is extremely low, you don't have much bluffing power. However, it is very easy to value bet, and if you expect to be all-in preflop you don't have to worry about getting pushed off a weak draw later. Your chance of winning the next pot may be lower than average (that is not clear), but when you win you may more than double up. It is not uncommon for a small stack (say, with 1-3 BB) to 5-tuple up by pushing after limpers.

If you have a small stack, you can play optimally to take everyone else's chips. The large stacks have to worry about each other, so they can't act as accurately to take your stack. You can play hands that do well with short stacks, e.g., large unsuited cards gain in value and small pocket pairs lose value. The large stacks can't make the same adjustments or else they can lose to the other large stacks.

Imagine that you have a small stack, and observe the other players while ignoring any chips in excess of your own. The other players play terribly! They often fold while all-in, even with clear chances to win. They call based on implied odds that clearly are not present. Sometimes these "mistakes" only help the other big stacks, but often you benefit.

If you don't ignore the extra chips, you get an information advantage: Large stacks have the option to bet or raise more than the size of your stack. That they make large bets or fail to do so gives you extra hints about where you stand.

If you have a small stack, you should expect to gain chips on average. This suggests that the probability of winning is a sublinear function of your stack size. In addition, with a small stack you should get a disproportionate share of second place and lower prizes. This also argues that the chip value is sublinear.

If you have a large stack, you have the ability to bluff other large stacks off small pots. However, that comes with the weakness that you can get bluffed off a small pot. You can't value bet as easily, and you suffer from implied odds. You may win more pots, but lose a lot when you lose.

Though the probability of winning the tournament may be a superlinear function of your stack size when your stack is large, you get a disproportionately small share of second place and lower prizes. Since these effects are in opposite directions, it is not clear whether chip value should be superlinear or sublinear when you have a large stack.

[GimmeDaWatch]

Wow, this is a pretty long thread for a simple logical exercise. David's post was just a reply/contradiction to someone else's. Anyway, his point is about a "good player", i.e. someone with over a 50% chance of doubling through to 40K. How much more likely the player is to win the tourney once he gets to 40K is simply 1/% chance of his making it there. If he were poker god #1 and was 100% to double through, then he would be equally likely to win the tourney at 40K as he was at 20K (and he would be the inevitable winner of course). If he was 51% to double up, then he would be slightly less than twice the favorite once he got to 40K, and so on.

[SossMan]

[ QUOTE ]
If he were poker god #1 and was 100% to double through, then he would be equally likely to win the tourney at 40K as he was at 20K (and he would be the inevitable winner of course). If he was 51% to double up, then he would be slightly less than twice the favorite once he got to 40K, and so on.


[/ QUOTE ]

That's pretty much how I thought about what DS said. Take it to the extremes, and it makes it easier.

[tpir90036]

[ QUOTE ]
Thus, you have not doubled your winning chances by doubling your stack; you have increased them from 0.6Y to Y.


[/ QUOTE ]
fine. the part that i do not understand is why we stop the process here.

so now we are at Y...and we want to get to Z.
and we are going to go from 0.6Z to Z since we are still 60% badass.
but Z = 2Y. so 0.6Z != Y which is where we thought we were after the first double up.

something has to give, right?

or to try to put it in words. why are we restricting a player from increasing their chances of winning the torunament since they will have the same double-up opportunity at the next level? or is this the flaw in my thinking since we are not assuming they will have this same opportunity

-tpir