Conjecture and Question

[AtticusFinch]

By the way, the easiest way to see that cEV does not equal $EV, even in a winner-take-all tourney, is to look at an extreme example. Let's say you're in a 100-person tourney, winner-take-all. Your skill is such that you have a 2% chance of winning (Twice the odds of an average player.)

On the first hand you're offered a 100-way allin, where you have precisely a 1.5% chance of winning. Furthermore, the other players won't take the bet unless you come along. Clearly this is a highly +cEV opportunity. It's even a +$EV play (meaning you'll profit over time from your buyin). However, based on your skill expectation, you should decline, as its $EV is 25% less than the $EV you'd get from playing. (This ignores hourly rate, of course, but you get the idea.)

[Jason Strasser (strassa2)]

Gregery,

I want to say that I think that according to this formula there are only two relevant values. The value of your chips, and the value of the chips you don't have. I guess you'd take into account someone like Negreanu in the field (IE, your chips worth less, chips you don't have are worth more), and that this value can obviously change throughout the competition. However, any chip you make is a chip you make, regardless of where you got it from. If you are in a field of 1000 players and you stack Negreanu, you probably will not be able to change the value of the chips you now hold because the difference in the field is basically unchanged.

However, if you are three handed with Negreanu and he goes broke to a complete fish, the value of your stack goes up significantly.

That's how I think you should approach it.

This was a really really really really good post.

-Jason

[gergery]

[ QUOTE ]
Tiger Woods analogy....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.


[/ QUOTE ]

The problem with this analyogy is that in golf, once you finish one hole, the next one you start from scratch and you are both just as likely to get a good score as on the last hole.

But in poker, if you just doubled your chips, then you have more weapons in your arsenal. It would be more like if Tiger got an eagle on the first hole, then he gets to use 1 additional club in his bag until someone else catches up to him on the leaderboard.

Basically, having a big stack would not be any better if you just redeemed your chips at that point in time. But you don't -- you play on, and having a big stack ends up in reality meaning your ability to acquire more chips goes up faster than linearly.

-g

[Slow Play Ray]

[ QUOTE ]
It would be more like if Tiger got an eagle on the first hole, then he gets to use 1 additional club in his bag until someone else catches up to him on the leaderboard.

[/ QUOTE ]

At the risk of sounding like a complete nit, I would like to offer that a better analogy would be he gets to play from the ladies' tees until someone catches up to him.

[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.

[locutus2002]

Hi,

Your conjecture that expectation does not double even though your chip count has is true and is probably understated. It’s more likely that twice the chips has the same EV for a player early on, than twice the EV.

Every players plot of EV to chips is going to have several characteristics that are identical.
1) EV(zero chips) = 0
2) EV (55M chips or 100%) = $7.5M or 100% of the chips
3) EV(chips) is a convex function: this is intuitively obvious since the payout structure is goes to the top 10% in increasing amounts, chips won at the end are much more valuable in $ than chips won in the beginning (percentage wise). The proof for this is like the proof for iso-utility lines in economics.

The implications are far reaching:

1) Players EV functions are ALWAYS convex, if they weren’t they would cross other players EV functions at some point and the better player would have lower EV for the same number of chips.

2) All players are below the WOTMOG line ~ 1 (for large field size). Definition of convex. This implies that even the greatest player in the world; lets say Greg Raymer, has an EV less than $1363 with a stack of T10,000 at the beginning of the tournament. (based on 55M chips and $7.5M 1st place.)

3) Somewhere around the bubble is where the slope of the convex function > 1 for most players when chips won are more valuable than chips lost EV(2XT) > 2 X EV(T)

Maybe someone talented with math can draw some graphs.

[ilya]

Quick aside....

It seems pretty obvious that in a winner-take-all tournament with Mason's parameters, as

total_prize_pool --> $40,000 ==> value_of_doubling_up --> $0,

but isn't it also true that as

total_prize_pool --> $INF
==> value_of_doubling_up --> $40,000 ?

My point is just that Mason's conjecture may seem intuitively correct because $40,000 equity in a $10,000 buy-in seems like a large edge.

What seems intuitively correct to me is that your expectation comes closer and closer to doubling as your edge represents a smaller and smaller % of the winner-take-all prize, no matter how huge that edge seems relative to the buy-in.

[EverettKings]

If your expectation did not go down, then you could theoretically keep doubling up and end up cashing 4x the tournament prize pool. The smaller field and the closer to the money, the less your expectation increases with each chip.

Everett

[DonT77]

Quick (& extreme) counter example-

Let's say you're playing heads-up and you both put in $50 and play winner takes all. Your opponent is stronger than you, so his expectation is $70 and yours is $30. You double-up the first hand. Now your $EV went from $30 to $100 in one hand (over 3x$EV, by doubling).

Okay you say, but the guy who started with an $EV of $70 can never get an $EV over $100 because $100 is the most he can win - so are we saying then that because a player is better his $EV can never more than double when his chip EV doubles - even in a 1000 person tournament with escalating payouts and given the advantages that a big stack has?