Poker maths question

God, for some reason my brain just doesnt work for poker maths. Can someone walk me through not only the answer, but how to calculate this spot:


Early

$0.25/$0.5 No Limit Holdem
4 players
Converted at weaktight.com

Stacks:
BTN JUplay ($45.99)
SB Hero ($49.00)
BB velkaman ($83.05)

Pre-flop: ($1, 4 players) Hero is SB :8s: :qs:
Hero calls $0.25, 1 fold, OMGNevilleAitken raises to $2.50, 1 fold, Hero calls $2

Flop: :kd: :5s: :as: ($5.50, 2 players)
OMGNevilleAitken bets $4, $4 to Hero ($46.5)?



HH is a bit messed up, dunno why. Villain covers, and I think I posted a SB from CO or something, dunno why.

If I raise flop, I pretty much price myself into a pot odds call for my draw if he shoves. So obviously he needs to fold to my raise enough times to make raising the right play here. For instance, if he never folds then raising here is no different to him open shoving flop and me calling (obviously a mistake), likewise if he always folds then raising is always best, so theres a point at which is 0EV and if he folds more than that I should raise, and if he folds less than that I probably should just fold.

Just the maths, I already know the preflop stuff is pretty poor, just using it as an example.

[edit] actually I think I'm wrong in this case that we'd have odds to call (depending of course on how much I raised to), but for the sake of the exercise lets assume that was the case, I always call a push.

Start with a more simple case: suppose you have no pair/no draw. There is $9.50 in the pot, and you'll be betting x to try to win that pot. The 0EV point for betting x to win $9.50 in this spot is x/(9.5+x). For example, if you raised to $12, then you would need villain to fold 12/(12+9.5) = 55.8% of the time to break even. Your particular question isn't much more difficult than that, so I'll leave it to you to solve since what I've given is what seems like the hard part for most players.

Also for what it's worth, I'm not sure if calling a push is +EV with most raise sizes.

Assumptions:
The line is raise / call all-in
If re-raised all-in I am going to assume he has me beat and that I have 9 solid outs.
Opponents stack is not listed. I am going to assume he has you covered
All maths is approximate and I'll ignore rake.
I'm going to assume hero is SB and villain is BB due to the SB being completed followed by a raise pre-flop.
I'm going to assume hero checked on the flop.

The raise/call all-in line is not equivalent to the villain open shoving - it's equavalent to you raise shoving all-in and getting called - his initial bet of $4 is already dead money. If called I have 9 solid outs on the turn and if I miss I have 9 solid outs on the river. If called my chance of winning is:
1 - (38/47 * 37/46) = 35.0%

He has an x% chance of calling, giving him an 1-x% chance of folding. This gives us 3 outcomes:

1-x% chance to win $9.5
0.35 * x% chance to win $52
0.65 * x% chance to lose $46.5

These 3 outcomes/chances add up to 100% of all results. I'm going to define x as a number between 0 and 1 and not use the % notation in the below for clarity. I'll also leave out $ signs. This means that our expectation is:

EV:
(1 - x) * 9.5 + 0.35 * x * 52 - 0.65 * x * 46.5 =>
9.5 - 9.5x + 18.2x - 30.225x =>
9.5 - 21.525x

Our EV is 0 when:
9.5 - 21.525x = 0 =>
9.5 = 21.525x =>
x = 9.5 / 21.525 = 0.4413

We are EV neutral when there is a 44.13% chance that the villain will fold to our raise. This is not so much smaller than Spoon's 55.8% mainly because there's not so much money to be won through folding (the pot is small).

On how much to raise I think there are three questions to ask:
1) Which raise size maximises fold equity against my opponents range (and is that fold equity high enough for it to be profitable to play)?
2) What size raise makes me pot committed if he pushes all in?
3) What size raise makes it possible for me to get away from the hand if I am beat

Before trying to answer these I'd like to talk a bit about the opponents range. SB completes (fie!) which is an invitation for BB to raise him off his hand, knowing he'll have position. Range is basically any two cards, but it can be weighted towards playable and high card hands if the opponent is weak. Similarly on the flop - c-bet into a SB is also an any-two-cards play. In other words, I think the villains range contains a lot of air. There may be some real hands that make under, medium, top pair, two pair, set and flush draw type hands - maybe even a couple of gut shots, but I'd still put the percentage of hands that don't even have that going for them as high as 40-50%. Yeah, I suck at ranges.

Maximise fold equity:
Villain dependant: A raise shove all-in is possibly going to be seen as bluffy and can be called by some bluff catcher type hands. A small/medium sized raise may be called by hands that can improve (gut shots, flushes that are probably behind, sets). Probably the bet size with most fold equity is something near pot sized with the threat of another pot sized bet to all-in backing it. Pot sized is raising to 3 * $4 + $5.5 = $17.5, which would make a $40.5 pot with $29 behind. Maybe tone it down to $15.5 making the pot $36.5 with $31 behind.

Pot committed:
A 35% equity in the hand means that you are neutral EV on a call if you are given 1.86 to 1 odds - which is when the villain makes a 1.16 times PSB. Both the above examples leave the villain with a below PSB. Let me calculate what you would need to raise to, to give yourself a neutral EV call if he shoves (not that you'd want this).

There is a total of $98.5 to be wagered. The pot is already at $5.5 and the villain has already put $4 into the pot. You will now raise to an amount, which he will call (either interim call for calculation purposes or call and shove on next street), leaving an amount behind that is larger than the pot by 1.16 times.

The easiest way to figure this I think is by means of shortcut. You are interested in the ratio between the stacks behind and the pot (SPR! Nah). I am going to pretend that you both started with $49.25 (this is the shortcut that works for this calculation), call the size of your raise y and throw it all together like this:

49.25 - y = 1.16 * 2 * y =>
49.25 = 2.32y + y =>
3.32y = 49.25 =>
y = 49.25 / 3.32 = 14.83

Thus, if you raise to $14.83 you create a situation where it is neutral EV for you to call an all-in (based on your 9 solid outs etc) if he shoves on you. If you raise higher it's +EV to call if he shoves, if you raise less it is -EV to call if he shoves.

Note that by betting a larger amount we are not making our overall play on this street +EV - we are just pot committing ourselves. If the villain range is not full of air and we have low or no fold equity making the raise in the first place is a mistake, even if we can size the mistake in such a way that the call when he (predictably) shoves on us is not a mistake in isolation.

Raise/fold:
Is it possible to make a raise that is small enough to fold out all (or almost all) the hands the villain has that are worse than ours? This way when he re-raises (shove or not) we know he is 1) not full of air and we have exactly 35% pot equity and 2) we've created a situation where we are not pot committed and can fold instead of going deeper into a -EV situation. This raise would need to be lower than $14.83, and to allow a modicum of air in his range (still) if he raises it should probably be no bigger than $10-$11. The question now becomes, if we raise to $10 will he fold all worse hands and only continue with better? To such a small raise I guess the answer is no.

My conclusion is that if we raise it HAS to be for fold equity. Any raise that would allow us to get away doesn't have enough fold equity to define the villains hand when he shoves. We need to raise a size that has 44.13% fold equity and the best bet for that is probably around a PSB raise (like raise to $17.5 - or my slightly preferred $15.5).

Is this the best play? I don't personally think so.

Saturday I read Professional No Limit Holdem in bed from cover to cover, and I remember something about playing drawing hands, facing a bet. I don't remember if it was in or out of position, and I'm not sure it's too important here.

The basic decision tree suggested in that book was this:
If you have implied odds - call
If you do not have implied odds and you have solid pot equity (many outs) - raise (fold equity and pot equity is a profitable semi-bluff)
If you to not have implied odds and you have poor pot equity (few outs) - fold.

In the present situation I do the following approximate maths. The bet is $4 and I have 9 outs to improve on the turn. 9 outs mean I improve somewhere between one in 5 and one in 6. For calling to be profitable I should expect to win $22 ($4 * 5.5). At this time I am guaranteed to win $9.5, so I will need to win an additional $12.5 out of my opponents stack on average (he'll fold some) for calling to be profitable from an implied odds perspective. Keeping in mind that the pot on the turn will be $13.5 that's an additional full PSB I will need to win on a later street if I improve.

If a spade falls he might be inclined to give me credit for the flush and fold if I bet. That reduces my direct implied odds, but it suggests the presence of steal equity. What other cards can fall that will scare the opponent into folding if I bet them? While perhaps unrealistic in this example I'll just pretend that if an ace comes and I bet it he'll give me credit for trip aces and fold - this would give me 3 'steal' outs - effectively 12 outs, meaning I need to win $16 - an additional $7.5 - improving my implied odds. I can't exactly say that an ace is a good card however, since he may hold it, but if the villain will believe I've floated a gut shot I could fire any of the 11 queens, jacks and tens representing a straight etc. Between my actual outs and giving a non-zero value to steal outs (and opponent tendencies are very important to consider) - as well as discounting the flush outs in case he's holding Ks (with or without another s) I can estimate an amount of effective outs - like 10 - and base my implied odds calculation on that.

To get slightly back on track - do we have implied odds? That depends on the opponents range. If he has air, he's not going to be calling any more bets whether your hand completes or not. It'll reduce your straight implied odds, but improve your steal equity. If he does not have air but has a semi-solid hand he'll possibly call a bet even if the flush completes giving you the implied odds you need, but may not give you much in the way of steal equity if you try to bluff on a J or T. I could say I bet any A, K, 5, Q, J, T, 4, 3, 2, spade - and sometimes I get in trouble. Because of the times I get in trouble (and the times I only get a fold when I successfully bluff on a later street) I'll call it 10 effective outs on average. If I do bluff with a 3 and get raised I fold - if I get called - I'd possibly only bet spades on the river.


This means in this hand you have three types of equity:
Fold equity on the flop if you raise
Implied odds equity if you call and hit your hand
Steal equity if you call, scare cards come, you bet them and he folds

Given my belief in the opponents range (lots of air) I think the most profitable play in this situation is to call for steal equity. That said, I think it's a very marginal spot and I wouldn't complain about folding either. Raising on the flop is only +EV if I know the villain and knows he is very likely to make this play with air.

I know my steal equity calculations are incomplete.

If you want to understand what I did here - go through all the calculations and adjust them to take rake into account.