Tuesday, September 04, 2007

The Indifference Point

If you are trying to do an expectation value calculation that incorporates game theory, you end up having to look at the point of indifference for each player in the game. The point of indifference is the point where a players hand range or calling/folding/raising frequency becomes ideal and variations from this will lead to lower expectation value. I probably did not explain that very well, and I may not even be using the right term. It is where for example in a range of hands that it makes little difference if you fold or push the same range game theory wise. When I did my blind steal and resteal EV calculations a while back, I did not continue on to this important level to further refine the results. I came up with ranges for people, but did not look at what those exact ranges did against each other to further refine the ranges and find the point where the pushing/folding the range made no difference for all players involved.

So I thought it would be great fun to do a bit simpler of a hand, but take it all the way to see what ranges fall out. It would be especially fun to use the hand with Hoy that I recently posted about, since there seems to be some hand range controversy in this one. To make this even more fun, I will show the hand, and list my assumptions and do the actual math later as an edit to the end of this post. I may be way wrong when the results come out on Hoy's range for this hand but we shall see. The hand in question if from a Riverchasers Tourney a few weeks ago. The theme is a big stack pressuring the table with a bunch of preflop raising, and what kind of range do you want to raise/call a shorties push with. From the shorties perspective, what is your range for pushing in over a big stacks late steal attempt, knowing it is highly likely they will call you because of them being "priced in".

FullTiltPoker Game #3206450743: Riverchasers Online Poker Tour (24300118)
Table 3 - 500/1000 Ante 125 - No Limit Hold'em - 23:49:17 ET - 2007/08/09
Seat 1: hoyazo (9,778)
Seat 3: stl_phily (29,045)
Seat 4: jeciimd (7,795)
Seat 6: AlCantHang (16,379)
Seat 8: crazdgamer (12,875)
Seat 9: Blinders (38,128)
hoyazo posts the small blind of 500
stl_phily posts the big blind of 1,000
The button is in seat #9
*** HOLE CARDS ***Dealt to Blinders [Some Range of Hands]
*** HOLE CARDS ***Dealt to Hoyazo [Some Range of Hands]
jeciimd folds
AlCantHang folds
sircrazdgamer folds
Blinders raises to 3,000
hoyazo raises to 9,653, and is all in
stl_phily folds
Blinders calls 6,653

A left out the results of the hand because that is not important. The assumptions here are as follows.

1) Blinders must call any shorties push if he raises preflop, and always must raise 3x initially.
2) Hoyazo may push-in or fold, and knows that Blinders will call his push.
3) The BB must fold preflop.

Question #1: What is Blinders optimal raising range preflop
Question #2: What is Hoy's optimal pushing range preflop

Antes are 125, and Hoy posted the SB of 500
A Blinders fold preflop EV = -125
A Hoy fold preflop EV = -625
There is $2,250 in dead money in the pot.
Hoy has $9,153 left.
Blinders must call $6,653 if Hoy pushes
Total pot will be $21,056 if Hoy goes all-in

Edit to follow with range results if I can pull the calculations off. feel free to throw out guesses for the ranges now if you would like.


OK, I could not wait to do the calcs so here are the results. The first thing is that I am leaving out the case where I fold from Hoy's perspective. In this case it is a "battle of the blinds". For this case which is pretty likely, I just give Hoy 45% equity in the dead money. This seems fair, as he is out of position, and the positional advantage is worth some equity for the BB. So when you start doing trial and error to find ranges, because the BB has some equity, but can be forced out per the assumptions by a Blinders raise, the correct answer is Blinders pushes all hands, and Hoy Pushes all hands as well. This maximizes EV for both Hoy and Blinders. This is not very realistic, so if you look at some of the middle cases, it appears that without taking every last drop of equity from the BB things work best somewhere near the Blinders Raise/Calls top 40% range and Hoy pushes in over the raise with about the top 35% of hands. That is a pretty wide range for both of us, but this is late stage MTT stuff here so the math calls for these types of plays. Optimal ranges for both are shown below.

Hoy's Ideal Range Based on the Assumptions

Blinders Ideal Range

Case #1 were the results for the A8+, 88+ range.



At 11:19 AM, Blogger Hammer Player a.k.a Hoyazo said...

Blinders is on a huge stack and likes his position. He probably raises here with any pair, any Ace, any suited King or Queen, any two cards Ten or higher, and any suited connectors. Even a tighty like you tends to get giddy when you amass a huge stack like that.

For me to reraise allin there, I would not do so with any card below an 8 in my hand under any circumstances since having one card below half of the average cards out there would prevent me from having a great chance to win even if you have exactly the range I think you have. Any pair 88-AA or any A8-AK is probably the extent of my range to push here.

At 10:35 AM, Blogger Hammer Player a.k.a Hoyazo said...

Help me to interpret this please. I can't follow all the charts. I am just a simple unfrozen caveman lawyer.

At 11:52 AM, Blogger Blinders said...

I made a spreadsheet that calculates the EV for each of us (whats left is EV for rest of the table). I pick a hand range for each of us, and run the range through poker stove, to see what the equity is for each of us. The spreadsheet uses the range% and equity% to calculate EV. It is a trial and error process, but it seems to settle at me with a 40% range, and you with a 35% range for us both to maximize our equity from the hand. The actual answer is I raise/call all hands, and you push all hands, but that is not realistic with the BB in the hand.

The hand charts show in yellow the hands in the range (40% or 35% range). The spreadsheet shows specific range match-ups. Your overly tight "nothin lower than an 8" (case 1 range) is one of the poorest strategy choices based on EV.


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