The passing up small edges debate Doubling your chips=double your EV?

[sirio11]

I know this has been a topic for a long time, and I know some well known players and posters already close the debate with definite statements. I'm for one, have not been convinced of the truth of the statement that says that if you double your chips in the first hand of a tournament, then you double your expected value for the tournament. This looks obvious and as today I think is considered general wisdom in this forum. I was ready to accept it mostly because I respect some of the voices that consider the statement to be true. But keep thinking in the issue, and came with the following scenario.

Lets suppose he have a $1000 tournament with 1000 players

The distribution of expected values for the players is as follows:

200 players ......... 1.9
200 players ......... 1.5
99 players .......... 1.2
1 player (YOU) ... 1.1
1 player .............. 0.9
99 players ........... 0.8
200 players .......... 0.5
200 players .......... 0.1

This is, in the long run you expect to profit 1.1(1000) = $1,100 from this tournament.

A 0.1 player expects 0.1(1000)=$100, that is, he's losing $900 per tournament.

Obviously if you add 1.9(200)+1.5(200)+1.2(99)+1.1+0.9+0.8(99)+0.5(200) +0.1(200) = 1000 the number of players.

You have 100 tables with 10 players. Lets suppose that in the 1st n hands of the tourney (n can be 10), you have 500 all in matchups, and the worse 500 players have been eliminated. So you have now 500 players with T$2000, and their expectations are (or were at the beggining of the tourney)

200 players ......... 1.9
200 players ......... 1.5
99 players .......... 1.2
1 player (YOU) ... 1.1

If we follow that you double your EV then now you must win 1.1(2000)=$2,200, but wait, the 1.9 players now win $3,800 and the 1.5 players $3,000. This is 200($3,800) + 200($3,000) +99($$2,400) +1($2,200) >>> 1000($1000)

Obviouly we have a problem here, with 500 players in the field, this is like a new tournament with T$2000 starting chips and now your EV is not 1.1 anymore, since now you are the WORSE player in the field. Looks like as long as the tournament advances, you need to reevaluate your expected value. For example if your expected value droped from 1.1 to 0.8 after half the field is gone, and you have T$2000, then now you can expect to win 0.2(2000)=$1,600, so your EV increased from $1,100 to $1,600.

This example make me think taht we need to take in account some other variables to know how your EV change when you double your chips.

Can't wait for your comments

David C

[Nottom]

You math looks kind of fuzzy, since you are making some assumptions that shouldn't be made (i.e. everyone doubles up on hand one or goes home and only the good players win). When discussing taking a chance early, it should be with the the thought that you will be one of only a couple people who will now have a double-sized stack and therefore have an edge over all the people with 1000-chip stacks.

Of course the statement you are trying to disprove, " I'm for one, have not been convinced of the truth of the statement that says that if you double your chips in the first hand of a tournament, then you double your expected value for the tournament. " is false. Because additional chips aren't worth as much as your original chips and this is pretty much a mathematical fact.

If you want to look at some numbers you can play with this:
ICM Calculator (http://www.bol.ucla.edu/~sharnett/ICM/ICM.html)

It was made with SNGs in mind but the same ideas will hold true in MTTs. Look at the EVs of a player with 2000 chips vs 8 players with 1000 chips. His EV is only about 18.5% of the prize pool. So his extra chips are only worth about 85% of his starting chips (and as an aside all his opponents chips are now worth 0.19% more just by sitting there.)

[woodguy]

You are not taking into account the increase in your expectation in the tourney due to half the field being eliminated.

Regards,
Woodguy

[valenzuela]

A new point of view...if you are eliminated after taking the small edge...play a profitable ring game.

[sloth469]

This is my line and often weighs into my decision in live tourneys. The live NL ring in AC is just too soft to not concider.

[SossMan]

[ QUOTE ]
A new point of view...if you are eliminated after taking the small edge...play a profitable ring game.

[/ QUOTE ]

let's ignore TVM considerations for now

[SossMan]

[ QUOTE ]
Because additional chips aren't worth as much as your original chips and this is pretty much a mathematical fact.


[/ QUOTE ]

The point that many of us have been trying to prove is that, while addnl chips aren't worth as much as your original chips, it's close enough to not matter.
If your original t1000 chips were worth $1,100, then the next t1000 on the first hand is worth something like $1,080 or something like that.

[valenzuela]

whats TVM?

[sloth469]

Real-time example, in the first 15 mins of the stars 50+5 I've tripled up. But have I really tripled my expectation?

Turning my 1500 into 4600, I would estimate my expectation to be closer to 2.5x greater as opposed to 3x which others here would say. I just can't rationalize it being linear because of the decreasing value of chips and my original expectation already taking into account this possibility.

Any validity to this line or have I just pulled it out of my ass?

-sloth

[SossMan]

[ QUOTE ]
Real-time example, in the first 15 mins of the stars 50+5 I've tripled up. But have I really tripled my expectation?

Turning my 1500 into 4600, I would estimate my expectation to be closer to 2.5x greater as opposed to 3x which others here would say. I just can't rationalize it being linear because of the decreasing value of chips and my original expectation already taking into account this possibility.

Any validity to this line or have I just pulled it out of my ass?

-sloth

[/ QUOTE ]

it's close enough to 3x that it shouldn't effect the decision in the hand that tripled you up (assuming it was close...i.e. you shouldn't be giving up a 'known' 40% equity for a chance to triple up)

[MLG]

You are right that your EV as compared to the field should constantly be revalued, as it is constantly changing. I think the situation where the entire bottom half of the field is eliminated without a top player going out is somewhat unrealistic, however. I'm not sure you can ever know your precise advantage over every player to the specific level of evaluating it the way you do.

The other thing that I would say is that if you do not double early on, and you find the field shrinking from 1000 to 500 with the same players and you haven't doubled, then hasn't your EV decreased to 800? So, I guess my reaction to your point would be to add the phrase, given that your edge remains constant to all these EV calculations.

[fnord_too]

I have not read the replies yet, but the cashing equity you receive from doubling your chips is far from a straight forward relationship. If all players are equal, doubling your chips will not double your cashing equity in normal tournament payout structures if you ignore the benefits of having a big stack. (The simple case of everyone being equal is hard enough for me to get my mind around!) I have some thought on how stack size (in relation to average stack size) increases your cashing expectation, but I am nowhere near done fleshing them out. Suffice it to say, if you are expert at weilding a big stack, doubling your chips will be worth more to you than someone who does not know how to play a big stack, and even if you are only ok at playing a big stack, it will probably be worth more in terms of cashing expectation than just the extra chips would suggest.

[Nottom]

[ QUOTE ]
The other thing that I would say is that if you do not double early on, and you find the field shrinking from 1000 to 500 with the same players and you haven't doubled, then hasn't your EV decreased to 800?

[/ QUOTE ]

If anything your EV has incresed slightly just by watching half the field get knocked out. It might not be much, but its definately gone up.

** This is assuming pretty even players, not that only players worse than you are gone as was the case in the OP. If that was the case, it could be posible that being the shortstack as the worst player in the field might take away some of you EV.

[slickpoppa]

Although this discussion is somewhat interesting, it is not really relevant to the concept of folding small edges. Even if it were true that doubling your stack only multiplied your EV by .8 that does not mean that you should pass up a 55/45 edge. If a 55/45 edge was the best tht you could possibly do, then you would have to call regardless of the fact that you do not double your EV.

The relevant question when deciding whether or not to pass up a small edge is whether you can exploit a greater edge later. Of course it is not that simple because you have to somehow figure how long it will take for you to get that edge, how many chips you can get into the pot with the greater edge, etc. Nevertheless, the essential question is whether or not you can find a greater edge. How much the value of your stack increases when you double up is a side issue, so I don't understand why we are getting bogged down by this.

[MLG]

I was assuming everything Sirio did about who got knocked out. So now you are the worsed player in the field, with a stack .5 the size of everybody elses. I think your EV has gone down.

[sirio11]

[ QUOTE ]
You are not taking into account the increase in your expectation in the tourney due to half the field being eliminated.

Regards,
Woodguy

[/ QUOTE ]

With 500 players with T$2000, its exactly like a new tourney, so you have a new expectation relative to this field, and is going to increase somehow, because the "new" buy in would be $2000. Problem is when you multiply your new expectacion by $2000, is not going to be twice (original EV) *( the original $1000).

[sirio11]

[ QUOTE ]
Of course the statement you are trying to disprove, " I'm for one, have not been convinced of the truth of the statement that says that if you double your chips in the first hand of a tournament, then you double your expected value for the tournament. " is false. Because additional chips aren't worth as much as your original chips and this is pretty much a mathematical fact.

If you want to look at some numbers you can play with this:
ICM Calculator


[/ QUOTE ]

But thats because of the 1-2-3 prize structure, if you change to wiinner take all and make the calculations, you'll see that if you double your chips, your expectation is twice the original. This is the point of most of the posters, since the payments are really far at the beggining you can consider it like a winner take all tournament. And your expectation depends on your chips. But this is only if we consider everybody to have the same expectation at the beginning. My point is that since everybody has different expectation, and when half the field is eleminated usually your expectation (relatively to the remaining field) goes down, then doubling your chips, do not double your expectation.
Of course I'm using an extreme example, but just because I want the example to be as easy to apply as possible (for the calculations involved). But the idea I think can be equally appied to an example closer to reality.

David

[MLG]

Its an intersting line of thought to take. I think that your considerations are right, however, I think they are outweighed by another factor. As the field gets better, and thus your edge over the field shrinks, it will be harder and harder to find edges to take. This problem is exascerbated by the rising blinds making post flop play rarer. Thus shouldn't you push any edge you know you can get now because you will be less likely to find edges later, not more likey?

[woodguy]

[ QUOTE ]
With 500 players with T$2000, its exactly like a new tourney


[/ QUOTE ]

Not quite.

The "new" tourney pays 20% of the players, whereas the "old" tourney pays 10%.

This has an effect on your expectation.

Regards,
Woodguy

[sirio11]

[ QUOTE ]
Although this discussion is somewhat interesting, it is not really relevant to the concept of folding small edges. Even if it were true that doubling your stack only multiplied your EV by .8 that does not mean that you should pass up a 55/45 edge. If a 55/45 edge was the best tht you could possibly do, then you would have to call regardless of the fact that you do not double your EV.

The relevant question when deciding whether or not to pass up a small edge is whether you can exploit a greater edge later. Of course it is not that simple because you have to somehow figure how long it will take for you to get that edge, how many chips you can get into the pot with the greater edge, etc. Nevertheless, the essential question is whether or not you can find a greater edge. How much the value of your stack increases when you double up is a side issue, so I don't understand why we are getting bogged down by this.

[/ QUOTE ]

I think is relevant, because it determines (or helps to) the level of risk you should be willing to take.

Lets analyze the 55/45 edge. If you play 100 tournaments with T$1000 initial chips. Then 45 of them you're out and your EV is 0, and 55 you have T$2000. Now lets suppose your expectation factor is 2 at the beginning. This is 2($1000)=$2000, you're suppose to win 1 buy in.
So, you play 100 tournaments with expectation 2, then you win $200,000 for a $100,000 profit. If your expectation when you double your chips changes from 2 to 1.8, then you win 55*$2000*1.8 = $198,000 for a $98,000 profit, but if it changes from 2 to 1.9 then 55*$2000*1.9 = $209,000 and clearly you should take the 55/45 edge.

[sirio11]

You're right, I have been considering expectation in terms of buy ins, regardless of the % of players paid, but obviously this should affect the analysys. Any thoughts how to add this variable?

[slickpoppa]

[ QUOTE ]
[ QUOTE ]
Although this discussion is somewhat interesting, it is not really relevant to the concept of folding small edges. Even if it were true that doubling your stack only multiplied your EV by .8 that does not mean that you should pass up a 55/45 edge. If a 55/45 edge was the best tht you could possibly do, then you would have to call regardless of the fact that you do not double your EV.

The relevant question when deciding whether or not to pass up a small edge is whether you can exploit a greater edge later. Of course it is not that simple because you have to somehow figure how long it will take for you to get that edge, how many chips you can get into the pot with the greater edge, etc. Nevertheless, the essential question is whether or not you can find a greater edge. How much the value of your stack increases when you double up is a side issue, so I don't understand why we are getting bogged down by this.

[/ QUOTE ]

I think is relevant, because it determines (or helps to) the level of risk you should be willing to take.

Lets analyze the 55/45 edge. If you play 100 tournaments with T$1000 initial chips. Then 45 of them you're out and your EV is 0, and 55 you have T$2000. Now lets suppose your expectation factor is 2 at the beginning. This is 2($1000)=$2000, you're suppose to win 1 buy in.
So, you play 100 tournaments with expectation 2, then you win $200,000 for a $100,000 profit. If your expectation when you double your chips changes from 2 to 1.8, then you win 55*$2000*1.8 = $198,000 for a $98,000 profit, but if it changes from 2 to 1.9 then 55*$2000*1.9 = $209,000 and clearly you should take the 55/45 edge.

[/ QUOTE ]

But my point is that if a 55/45 is the best edge that you can get, then you would still call even if your expectation is only 1.8 because that is the best you can do.

[woodguy]

[ QUOTE ]
Any thoughts how to add this variable?

[/ QUOTE ]

Not really, it was just something that jumped out at me.

Regards,
Woodguy

[JaBlue]

200/1000 are not .1 players. You need to base your expectations on a standard bell curve.

[gergery]

[ QUOTE ]
You're right, I have been considering expectation in terms of buy ins, regardless of the % of players paid, but obviously this should affect the analysys. Any thoughts how to add this variable?

[/ QUOTE ]

You would want to adjust your equity in proportion to the number of players left.

In other words, if the best player had equity of 1.9 and the worst had equity of 0.1, then you would expect 19 crappy players to get knocked out every time 1 great one did. So you’d want to add up the expected EV losses, which would be 1 x 1.9 + 19 * 0.1, and then divide by the number of players. As a result, over time you would expect an average amount of EV to be eliminated (in your example, you had only the lowest EV going out).

This makes sense because their EV is really nothing more than the chance they get 1st * the value of first, + the chance they get 2nd * value of second, etc. well, at the end of that, you get the chance they don’t get a place at all and obviously for the good players that will be less, but when they are eliminated, it frees up relatively more % chance of high placings for the rest of you to capture. By definition.

--Greg