Conjecture and Question

[gergery]

[ QUOTE ]
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.

[/ QUOTE ]

I’m not sure I agree with this. If it’s early in a poker tournament, wouldn’t the correct analogy be “I’m going to flip my 60% favorite coin for 5% of my stack repeatedly with you, and after 100 flips, lets see what the average chip count is for both of us”?

I don't think you can just freeze your equity at one point in time because you are not accounting for your future chip stream in that model.

For example, let's say you and I invest in 2 companies. They are worth $100 now. Your company earn $1 a year, mine earns $2 a year. After one year (ie. flip), you've made $1 and I've made $2. But you can't then say that your That doesn't mean our equity in the companies is now $101 and $102.

-g

[ZeeJustin]

[ QUOTE ]
[ QUOTE ]
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.

[/ QUOTE ]

I’m not sure I agree with this. If it’s early in a poker tournament, wouldn’t the correct analogy be “I’m going to flip my 60% favorite coin for 5% of my stack repeatedly with you, and after 100 flips, lets see what the average chip count is for both of us”?

I don't think you can just freeze your equity at one point in time because you are not accounting for your future chip stream in that model.

For example, let's say you and I invest in 2 companies. They are worth $100 now. Your company earn $1 a year, mine earns $2 a year. After one year (ie. flip), you've made $1 and I've made $2. But you can't then say that your That doesn't mean our equity in the companies is now $101 and $102.

-g

[/ QUOTE ]

Your analogy does not apply. I am factoring in future scenarios, while you are not. I am including the original 40k estimate in my formula, while you are not using any such original estimate in your formula.

[A_PLUS]

The really simple answer is that our skill edge doesnt multiply along with our chips.

Basically, we calculate EV by (for an average player)

The sum of:
% of total chips * payout for 1st place
% of total chips * payout for 2nd place
.....

= Our Expected Value of playing the tournament


So, something else has to be at play here for a better player.

They calculate their EV by:

% of total chips + SKILL ADJUSTMENT * 1st place payout
% of total chips + SKILL ADJUSTMENT * 2st place payout
..........

So for Mason's example, the player has a skill advantage of
X. Which is just how much more on average they expect to finish in certain spots.

So, when we double our chips, we double the portion of the above calculation that is from % of total chips. If we do not also double the "SKILL ADVANTAGE" Factor, the EV does not double along with it.

Intuitively, this makes sense. The further we progress through a tournament, the more our % of total chips effects the outcome. If we did in fact try to double or "skill adjustment" along with total chips, we would quickly reach an unreasonable EV.

So, it comes down to the decay of the skill adjustment factor. It is mathematically impossible for it to double, but if it were to remain constant, It quicky becomes unimportant in decesion making.

Actually, that doesnt feel very different from how many pros play. The edge they require before risking all of their chips getting smaller and smaller as the field condenses, converging to anything greater than 0, at a point.


That being said, I find it hard to believe that many people have a 4x edge in a large MTT, and 99% of the players on this board are going to use this as an excuse to play weak tight

[DonT77]

I agree with Durron, in that having a large stack has its advantages (blind stealing, blind protecting, being a bubble bully, continuation betting, 'putting a man to a decision for all his chips', etc.) - so while your % of total chips increases, the "effectiveness of your stack" (rather than your skill level) also increases. This goes back to the ("Gigabet") discussions that sometimes it maybe be +$EV to take a slightly -cEV play because of the relative worth of various stack sizes.

I do think that someday some 2+2 calculus wiz will figure out the cEV/$EV relationship across the multi-dimensional curve that includes (as a minimum) the variables: #players, stack sizes, and payout structure.

Probably the biggest difficulty in solving this problem (aside from different players having different skill levels) is trying to quantify the advantages of a big stack (which will vary from player to player).


To Mason's conjecture-

In the MTTs where I never double-up my $EV is extremely low - quite possibly 0.

It seems that the earlier a player doubles-up the better his chances are of making the final table, and the longer it takes to double-up the more his chances of making the final table diminish as he is fighting the battle of having a less than average stack (and not having the afore-mentioned big stack benefits) and the vulnerability of being taken out by a larger stack for most of the tournament.

So empirically, I think ZJ's coin-flip example has some merit and that a person's $EV after doubling up early may actually be greater than 2x his starting $EV - although I don't have the mathematical wherewithal to prove it.

[Jman28]

Okay, I'll try the other side since nobody else has, and because it's what I believe. First let me state my conjecture: In the situation Mason described, your expectation probably does not double if your chip count does. However there are times when it can double and more. Therefore, Mason's conjecture that it definitely does not double is incorrect.

Basically, I think that the SKILL ADJUSTMENT factor in A_PLUS's equation can change value when your chips stack does. I don't think anyone will argue that having a bigger stack allows you to make more +EV plays, especially on the bubble.

If the game is such that the gain you get from having a big stack at bubble time (or at other times) is sufficiently large, your expectation can double (or even triple, theoretically) when you double up. This gain may be different depending on tourney structure, or the tendencies of your opponents.


As an extreme example, let's say we're 25 handed, with 20 spots paying. Blinds 100/200.

You have 6000 chips and everyone else's stack is between 3500 and 4500.

Everyone is trying to make the money desperately. So desperately, in fact, that if the pot is raised in front of them by someone who covers them, they will fold any hand but AA. If they cover the raiser, they will play a much more 'standard' game.

Of course this is a huge exaggeration of most real players' tendancies, but think of the massive advantage you gain in having 6000 chips compared to having 3000. This hopefully shows that a much smaller but very real advantage exists when you have a big stack in more realistic scenarios. This advantage may sometimes be enough to counter the effects of the standard equity assumptions most of us have.

Edit: 2 more points.

Your edge doesn't stop when the bubble bursts either, as people are attempting to climb the prize ladder and will still fear you.

If what I'm saying is true, the implications would be strange and counter intuative. In a situation where doubling up would more than double your equity, it would be +EV for two players next to each other to decide to go all in blind the first time it was folded around to them in the blinds. This makes me doubt my conclusion a little bit, but I'd like to hear what others have to say.

[betgo]

I agree with this. Your expectation less than doubles.

However, in most situations where you have an average to twice average stack from the point where half of the field has been eliminated onward, doubling up results in much more than a doubling of your expected payout, regardless of your skill level.

[Spee]

[ QUOTE ]
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.

[/ QUOTE ]

Hello Mason,

With respect to the start of the tournament, IMHO, it is intuitively obvious that your conjecture is correct. Typical tournament payouts are more or less linear whereas doubling up is exponential. At some point, the maximum expectation will be reached regardless of the amount of doubling. Stack size will increase at a faster rate than expectation.

With respect to latter stages of the tournament, I don't think this is so much the case. For example, does doubling up further increase expectation once into the money? I would think that it most definitely would (or is that stating the obvious??).

When I was thinking about your conjecture, the immediate analogy that popped into mind was Tiger Woods in golf. After all, he is the greatest tournament player of the current generation, if not all time.

For 2005, he played in 21 PGA events and made 13 top 10s and 6 wins. As an aside note, he also dogged it and missed the cut in 2 events.

So let's say Tiger is roughly 2:1 or 2.5:1 to win the event (as he usually is rated by the bookies here in the UK). He then promptly goes out and eagles the first hole to take a two shot lead on the field. Does that increase his expectation? Maybe a little but not too much. He is after all quite a bit better than even money to finish in the top 10. But it is also very early in the tournament.

Now lets say it is Saturday afternoon, and he has already made the cut (i.e., in the money) and is now tied with 9 other players for the lead. These 10 players together are 4 shots clear of the rest of the field. Now Tiger goes out and eagles the first hole to take a two shot lead over his 9 closest competitors and 6 on the rest of the field. Does that increase his expectation? I would say yes, proably by a lot.

So now back to poker, the greatest tournament player is now already somewhere in the money. Does a double increase his expectation more so than at the beginning of the tournament? Yes, I think intuitively that it would have to. Or maybe this is restating the obvious as well??

Just my own wrong opinion FWIW ...

Cheers,
Spee

[Ansky]

lots of golf analogies on this forum lately.

[Neuge]

Consider this extreme situation. Let's say that you miraculously acquire someone's entire starting stack on every hand you play and no one else gains or loses chips (ignore blind and ante money for simplicity). You must knock out X% of people, minus 1 for yourself, to have equal equity as the X% payout for first place if you assume that your equity doubles with your chip stack. This essentially equates to 1st place payout equity if you start with X% of chips in play. Even in such an advantageous position you surely cannot be guaranteed to win the tournament 100% of the time.

This only assumes that you have $10,000 equity in a $10,000 tourney. Suppose you have Y x buyin equity (in your example Y=4), then if the conjecture is true you only have to knock out X/Y - 1 people before you are "guaranteed" to win.

[Spee]

[ QUOTE ]
lots of golf analogies on this forum lately.

[/ QUOTE ]

Yeah I saw Jason's earlier post, too (liked it). Couldn't help it here. It just seemed like a really strong analogy with respect to the OP.