Capped buy-ins and EV: (an attempt at) some math

[mason55]

Now I'm wondering if there is some sort of relationship between stack size, current ev, and actual long term ev. For example, if both opponents are at or below the max buy-in, then the EV for the play will just be equal to the EV of the actual play as it's currently being made, based on the odds of the cards still to come, folding equity, etc etc.

It seems as though there should be a way to factor in stack sizes above the max buy-in. It wouldn't be possible to factor in anything super long term, but maybe some sort of "earning potential" variable.

Say you're deciding to make a call on the turn. You have $100 and your opponent has $100 in a $50 capped buy in at the start of the hand. Your opponent bets $50 on the turn, putting $60 in the pot total. If you call you'll be left with $45. Assume no more money goes in on the river, you either make your hand and win or miss and lose. If it's 50/50 making your hand that means your EV on the turn call is

-$50 + .5*$110 = $5 in EV.

So calling is obviously a +EV play. Here's the thing though, if you lose the hand you're left with a stack of $45, which you rebuy to $50. So when you make that call, 50% of the time you lose $45 (not $50 since you can rebuy) in "future earning potential." Notice that even if he put you all in, you're still only losing $45 in future earning potential.

My thoughts are that if you figure out the average win percentage that you personally have when you put money in (ie whether you push small edges or wait for big ones), you can incorporate how much that $45 in future earning potential is worth to you personally and include it in your EV calculation.

For example, let's say that you won the last hand, so you now have $155. You sit out a few hands and now you have another opponent who has $100. That previous call you made wasn't worth just the money you made from that hand, but it's also given you ammo to win more money. Your opponent pushes all in and you look down at the "computer hand" and know you're 50/50. You're in a gambling mood, so you call. 50% of the time that money you won in the previous hand is worth twice as much and 50% of the time it's worth 0. So you can subtract .5 * $45 because if you lose that money it would have been worth 50% in the future, but you also have to add .5 * $45 for the same reason. So if you put your money in, on average, as 50/50 then future earnings have no effect.

Say instead that you will only call with the top 75% instead of the computer hand. Now that $45 is worth .75 * $45, so your future potential earnings are more valuable. Meaning that $45 is MORE valuable to you because you're more likely to capitalize on it, so you want to be less likely to lose it.

Hang with me here, I'm kind of thinking out loud. I'm starting to confuse myself.

So now your EV calculation becomes

-(50 + .75*45 - .25*45) + .5 * (110 + .75 *95-.25*95)

Explanation:
It costs $50 to call, plus $45 in 75/25 future earning potential. Half the time you gain $110 and you gain $95 in 75/25 future earning potential. Adjust these percentages for however you get your money in on average. Against a lag you might push smaller edges so that future earning potential could be worth less. Or you might wait until you have the nuts in which case that future earning potential would be worth a lot more because you'd be guaranteed to make more money off the money you don't lose. Also, once you have everyone covered, this no longer becomes a consideration.

In closing
I realize this post is long and confusing. I definitely confused myself while writing it and I didn't try out all the math. I KNOW a lot of things in here are wrong, but it seems like there's got to be a way to factor in the size of your stack ABOVE the fixed buy-in when considering whether to make calls.

I'm hoping that some others can weigh in here and I didn't just waste 20 minutes writing all this. As I said, I'm sure there's errors, but I think there's a lot of room for discussion here.

[Ghazban]

I've thought about this myself (though not to the point of actually doing any math) and I don't think its quantifiable. For one thing, the other stack sizes matter. If ONLY you and the opponent in the given hand are above the cap and you win the hand (thereby knocking him below the cap), all that extra money doesn't really help you as you can still only win the cap from a given player in a given hand. However, if everyone else at the table is awful and way over the cap, getting knocked below the cap by losing the hand is almost a catastrophe (as you can't take full advantage of those other poor players). Do you see what I mean? I don't think its a calculation that simplifies usefully at all; rather, its something that has to be thought about on a more situational basis.

[mason55]

Yeah, I did think about that a little. That's why I said that it doesn't matter once you have the table covered. It's quite possible that you're correct that there's no way to quantify it, which wouldn't surprise me at all. We were just in a discussion about something like this in the SSNL forum and it made me think.

[Stork]

I don't think it makes a huge difference in most circumstances. I think it only really matters when there is a huge fish with a stack well over the buy-in, and the decision is pretty close EV-wise, within like a 60/40 favorite. And as someone else said, it is very hard to quantify exactly how much the extra amount in your stack is worth. Still though, good post.

[Ghazban]

Its definitely important and I know for a fact that I've avoided some marginally +EV situations in capped games because there was a lot of money on the table in the hands of poor players. For example, I folded the nut flush draw in a 3-way pot despite having correct odds because I felt fairly sure I could find better spots against this particular group of players (not to mention the uberfish who was sitting on over $1K in a $100 capped buyin game).

In a sense, the cap issue is akin to tournament strategy as the chips over the cap you lose are worth more than chips under the cap (that can be easily replenished by rebuying). If the players are deep and bad (and you know they aren't about to leave-- don't ask how you can determine that with any certainty), passing on marginally +EV situations may be correct if you expect situations with higher EV down the road.

[mason55]

[ QUOTE ]
And as someone else said, it is very hard to quantify exactly how much the extra amount in your stack is worth. Still though, good post.

[/ QUOTE ]

Thanks. I like thinking about this in a tournament sense, in that each chip up to the cap is worth the same (in this scenario I'm thinking you might say it's worth nothing). So you have $50 of chips worth "nothing" because you can rebuy them any time you want. Then you have your first chip over $50. Those first few chips are worth a lot because they can make a lot more (any player who is over the cap can double you up on those chips). As you get higher and higher the chips become worth less for two reasons. Number one it's harder to get more and more chips into a pot (how often do you find yourself putting, say, 300BB into a pot) and number two it's harder to play against the players who can double you up. Any chips that cover everyone else are once again worthless in this discussion because they can't make you any MORE money, which is what we're talking about.

So it seems as though there's range between the cap and where you have everyone covered that is extremely important. As you get closer to the point where you have everyone covered each chip you win is worth less in future earning potential. Once you have everyone covered, all the chips you win become like putting money under the mattress instead of into the stock market: they're yours but they can't make you any more money unless someone else doubles up.

I'm just trying to figure out if there's a scale where you can decide the importance of chips. For example if you have everyone covered four times over, you can make a 50/50 call. If you lose, no one is even close to your chip count and it will be very easy to get those chips back. However, say you have 4 buyins but you're in a 50/50 situation and have to call off your stack to someone who has 8 buyins. You have the same amount of money but you're in a different position relative to the table so making that 50/50 call is a much worse proposition. This is all obvious and well known deep stack theory, I'm just trying to think of a way to quantify it possibly include it in some sort of EV calculation and I still maintain that it's possible, I just don't think it's easy. It might be one of those things where you have to do the math beforehand for common situations but I still believe it can be quantified.

[TStoneMBD]

im confused by the topic at hand, but i believe i know what you are talking about and feel that i have a pretty good method for coming to the quantifiable answer that youre looking for.

if you were to record an abudance of poker tracker hand histories, you could determine what your long term BB/100 is in regards to the amount of chips you have at a table. at a $25max table, your BB/100 may be 5, when you possess a $25 stack, but when you possess a $50 stack it may be 7.

if that were to be the case, it would become logical to gamble more frequently with a $25 stack size, with a -ev hand, so that you could double up to $50 and expect a greater amount of long term profits. i dont feel like doing the math, but to elaborate, you could theoretically call an allin with a 1:2 underdog, getting 50/50 on your money, and expect a long term profit depending upon your BB/100 when considering stack sizes.

[mason55]

Is there any way to get poker tracker to do this natively or would I have to write my own database calls? I don't have a ton of experience with PT, I use it, but just as a tracking tool and not as a stats analysis tool.