Conjecture and Question

[Mason Malmuth]

Hi Everyone:

I have a conjecture that I believe is true and I have reasons for believing that this is true. But I'm hoping that others can explain the reasons a little better than I can, or perhaps explain why my conjecture is wrong. Here it is:

Suppose you are a great tournament player. Perhaps one of the best. You enter a $10,000 buy-in tournament and when the first hand is dealt, since you're such a great player your expectation is $40,000 even though you only have $10,000 in tournament chips. Now a very unusual first hand takes place and you double up. That is you now have $20,000 in tournament chips. My conjecture is that your expectation does not double even though your chip count has. So instead of having an expectation of $80,000 it may only be $78,000, or $75,000, or some other number less than $80,000, but it will definitely be less than $80,000.

So if my conjecture is correct, I would like to hear good reasons why this is the case. If it's not correct, I want to hear those reasons as well.

This should make for a good discussion and thanks in advance for the help.

Best wishes,
Mason

[sirio11]

This has been discussed in the past, and I for one, believe this is true. I came up with one example

in this thread

to try to prove why doubling your chips = doubling your expectation can not be true, but the discussion died in that time with no clear answer.

[ZeeJustin]

This just seems obviously true.

1) Every chip you win in a tournament is less valuable than the chip before it.

If your equity is 40k at the start, that does not even come close to implying that your equity at any point is $4 per chip. I think this is obvious, but if needed I will explain this further.

The 40k equity you start with, that includes play for the whole tournament. For example, maybe you make blind level X 60% of the time. Doubling your chip stack will obviously not double your chances of getting that far.

Also, since you are better than the field, it is safe to assume that you will double up early on more than 50% of the time. If this is something that happens more than 50% of the time, and your resulting equity is 80k, that means your starting equity would have to be more than 40k. This logically doesn't follow since the 40k number is a given, therefore, the 80k number would be an overestimate.

I don't know if I explained myself clearly or not, I'll try again later if needed.

Edit: FWIW, even the 75k figure is an overestimate.

[Exitonly]

This is what Atticus was saying, and also what TPFAP says.. the more chips you have the less each are worth.

I think i agree with this, because i think the closer you get to being knocked out, the more important each chip is.

bleh i'm tired, i'll reply to this again in the morning.

[Pat Southern]

The reason I always assumed this to be true (not sure if it is correct) is because when you assume you have a 40k equity at the start of the tournament because you have a certain probability to finish in every position (ie an A% chance to finish 1st, B% to finish 2nd.... X% chance to finish in Xth place) doubling up does not double the probability of each occurrence. Although your % chance to finish in 1st place may be 2A% (it could be more or less than double also), other money finishes will not neccessarily double, and may decrease to make up for your increased likelihood of finishing higher. Not sure if this is correct but thats why I there is a diminishing return of chips. However, as the tournament field increases, I think the gain in $EV increases as well.

[shaniac]

[ QUOTE ]
Now a very unusual first hand takes place and you double up. That is you now have $20,000 in tournament chips. My conjecture is that your expectation does not double even though your chip count has

[/ QUOTE ]

When I double up early (not sure why you say "very unusual"), my expectation actually MORE than doubles.

Just one of the benefits of not being a math guy.

[NoahSD]

Here's a proof:

Given that you have $40k equity from t10k. Let's say there are 10 people in the tourny for simplicity. Clearly, when you have t100k, you can't have $400k equity because the total equity is only $100k.

If you then apply this argument to all the other stacks at the table, you can see that your equity cannot be linearly related to the equity of your chip stack in a skill-less tournament.

Earlier, I proved that your equity also cannot be simply calculated by your stack, the average stack, and your skill level. In case you care: Go Here.

Edit: I think I proved more than you wanted. ICM calculations already dictate that doubling up shouldn't double your EV. This is easy to see if you look at the extreme case of the bubble. Here, doubling up obviously does not double your odds of simply getting through the bubble (that's almost always more than 50% already), so your equity can't possibly double when you double up. In the early stages of a tournament, you're basically at a much less extreme bubble.

[KneeCo]

I agree that this is fairly irrefutable, I believe it is an implication of Mason's published argument that due to tournament payout structures the more chips one holds, the less each chip is worth, and vis-versa. If we accept that as true (and I think the majority here do), the conjecture necessarily holds by extension.

Given that a player's expectation is contingent on his chip stack and his relative skill to the field the conjecture must hold for the same reason that in mathematics any factor which weighs on an argument of an equation must weigh on the equation's result.

(In this case, the result is of course the expectation, and the arguments of the equation that derives that result are a player's relative skill and his chips, the latter being conditional on their number but not in a straightforward way which by extension means the result is also not correlated in a linear way with the amount of chips).

Or perhaps I'm missing something. I would like to see someone try and tackle the nay side of this argument.

[ZeeJustin]

Let's say instead of poker, you're playing the coinflip game. You have a really good coin that wins 60% of the time. This is where your edge comes from.

Your equity is 40k. 60% of the time, you will win your first flip and double up. 40% of the time you will lose your flip and be out.

Since your initial equity was 40k, your equity after the first flip (with undetermined results) will also be 40k. If you lose the flip, your equity is 0.
.4x+.6y = 40,000
x=0
.6y=40,000
y= 66,667

In this scenario, after your first flip, assuming you win, your equity is $66,667

Although this is not poker, the parallels to the given example should be obvious.

[rbear]

I really like this example. I think that atticus is close to an exponential model that may give correlation to skill/chip amount/ chip average.