Pre shove range

Question

Say you want to play mathematical poker and you want to balance your pre shove range. A range where your opponent has shown strength. Either they came in for a raise and you are in position or they raised you when you opened out of position. Assume you always open for 2.5 BB. Assume you are medium stack 30-40 BB (inside the Harrington green zone).

The object here is how wide to make your range so you can get optimal action when you do have AA KK. The bottom of your range will likely be behind but it is sacrificed to get more action when you are ahead.

If you only shove with QQ+ there are not very many hands (18). If your opponent knows you only shove with QQ+ then they should only call with KK+. KK is 50/50 but there will be some overlay money on the table. Not enough balance to get action when you do have AA.

If you shove with TT+ now you should mathematically get action from QQ+. Is that enough balance. So now AA and KK can get action from KK and QQ.

TT+ feels kind of tight to me and I would add in AKs. Still only 34 hands.

If I add 9Ts, 89s, 78s, 56s like 1/4 of the time to the raise range so if I call and the board comes up to hit those hands they have a harder time putting me on the hand. But then only 40% equity against JJ+. I think a better question would have been to put the opponent on a range but I already asked it.

What I kind of like about a tight raise range is now your call hands can include some strong hands.

I know should mix up play. And should mix in bluffs. This a somewhat hypothetical question. In real life I would add in some

What would be your shove range in that situation?

I would like to get the math but just your opinion is good.

I don't think position matters here as this is limited to a re-raise and there is no position after an all in.

For example 2/1 blind so you have 30 BB. You come in for 5 and get raised to 10. There would be like 16 in the pot. You have 25 behind so it is not that much of an over-bet. If you have 60 you would have 55 behind so it is a big over-bet. But if you raise to 30 you are pot committed and they know it.

Answer 1

Consider this just a draft we could improve iteratively

Let's see. This would require you somewhat analyze the opponent's range. You will categorize the ranges between Loose or Tight.

This kind of strategy is somewhat hard and better suited if you have the guarantee the match is a heads up, or both you are in the final part of the tournament and so it turned into a heads up.

For this, I will assume this is an online tournament and you cannot predict or infer other type of behaviours but just the range.

Your final goal here is to get action from the other player, playing you tight, and playing him loose. Still you have absolutely no way to predict whether you will lose or win in an deterministic way, and having to shovel is quite decisive to be analyzed in only one tournament since the final matter is random and statistical. But let's assume the EV becomes somehow the deterministic answer (this means: the better EV, you instantly win, just for this theoretical model).

Since this is heads up, you're always in the best positions (dealer or cut-off) and so by position (or perhaps the closeness to a short-stack amount and so the strategy) you will also play, under certain circumstance, down to 5-4 connectors.

So let's define two sets FOR SHOVELING:

• Upper ranges: AA, KK, QQ, JJ, AK.
• Lower ranges: TT .. 22, A[K,Q,K,T], [K,Q,J,T]C2, A9..A2, KQ, QJ, JT, ... 54.

This is just an opinion and perhaps you'll put the line between the "upper" and "lower" by estimating the equity from the enemy's side (since you will definitely go all-in), sorting the hands by % of winning and capping (drawing the line) when you add up the % to that metric you calculated beforehand.

So these are the premises:

• You want to be all-in. There's no change here.
• You expect Villain goes all-in. He may or may not go. Our target is he goes, and we find he is playing a lower range. In Prisoner's Dilemma language, you want to betray him while he cooperates.

Why would he want to play your all-in? Let's assume:

• He raised up to a total of $X.
• You went all-in with $Y, where $Y > $X (other cases become trivial because the question is whether he'll call your all-in).

So he says: Should I go or fold?

• Folding makes him lose $X already in table.
• He will have a certain hand.

I'm not so fresh with the hold'em EV for heads-up only. But let's state this question from his view point:

How strong my hand is, absolutely? In particular you published an article with the strength of 5-card draw hands. The idea would be analogous here but considering an EV of power of possible 5-card hands based on the all possible outcomes in the commie cards. So assume you can compute such power and say In average, the % of winning of my hand is %W. Actually, he will say that. You're definitely on shoveling right now. So let's go:

• He already added up to $X.
• You went all-in with $Y which is > $X.
• Both you and him infer -reciprocally- the usual range of playing is AA .. 54o.
• I will assume he can calculate the %EB of beating you with his hand (see below), and %ET of tying you, and 100 - %EB - %ET = %EL losing against you. At this point, we did not consider your cards, because I'm just focusing on him, initially, with no additional information about your rank but just the good practice of playing up to 54o (again: this is heads up holdem: in regularly cold hands, we are in position to play up to that range).
• He ends -$X when folding. He ends %EB * $Y when winning, while -%EL * $Y when losing. He will pick the best action between -$X (fold) and (%EB - %EL) * $Y (call your all-in).

Ideally, we'd want him to be used to the fact we play a loose range with somewhat big amounts of chips (is it still ethical to show your hand when you previously shoveled and he folded?). To an extent that he says okay, now I estimate his new range, I recalculate the %EB and %EL for my current hand.

Calculation: Usually you estimate a range your opponent plays. Initially, we say it will be 54o under this circunstances. Then, given your hand, you iterate over all the possible 2-cards combinations of the expected opponent's range (say, N), add up the %EB, %ET and %EL or each comparisson (e.g. go here and simulate just 2 hole cards, against other 2 hole cards being iterated, and get the percentages) and finally divide those three accumulated %AEB, %AET, %AEL by N each of them.

Perhaps this calculation is quite complex and long, and you already performed them and have your own shortcut, but you get the idea.

Now he will decide whether fold or call, according to the previous formula when stating his own (%EB - %EL) * $Y vs -$X.

Now go back to the start and let's simplify a while. Both you and Villain will have only two types of ranges: the tight one (reaching AK) and the loose one (reaching 54o).

The villain, based on your previous behaviour, will establish a Hurwicz coefficient H (0..1 - think of a weight in neural networks so there is a chance that Villain will not think like this... conciously) to determine whether you're in a tight (H) or loose strategy (1 - H). So he will:

• Know own hand.
• Know $X of own total bet.
• Know $Y of your shovel.
• Compute own %EB and %EL for tight range. Let me call them %EB_T and %EL_T.
• Compute own %EB and %EL for low range. Let me call them %EB_L and %EL_L.

Now their actions will be valued differently:

folding = -$X * H + -$X * (1 - H)

This means, regardless the faith (which may be an adjustable value he mantains based on your recent plays) of his H coefficient, folding is always -$X.

calling = $Y * ((%EB_T - %EL_T) * H + (%EB_L - %EL_L) * (1 - H))

Then get the action with better value: call or fold. You want him to call, so you want him to perceive the H close to 0. We don't need to know that most of the times %EB_L - %EL_L > %EB_T - %EL_T, since the lower your ranges, the better chances of winning Villain has.

Ideally, this is the idea behind cooperating or betraying: cooperating with him may push H closer to 0, while betraying him will push H closer to 1, but you'll also consider the involved $X (the last status for his live bet) and $Y (your all-in).

This is the naive approach, which involves an appreciation Villain has over your range moods which, instead of being pure random moods (where H = 0.5) you have a bias.

After that, predicting the behaviours of betraying vs. cooperating is hard as f*** but well documented and, assuming you got the idea behind betraying and cooperating, it only involves just a coding or simulation effort. As you see there are a lot of strategies. I have no up-to-date documentation of how do they beat each other. For very long games, I think you should be prepared for complex game theory interactions. However, for short games (perhaps very quick blinds?) the naive approach should be enough.

Answer 2

Was not my intent to answer this question when I posted and will not accept my own answer.

Let say your open to 2.5 bb and call up to 4 bb is top 70 hands. Say your re-raise jam is top 34 hands.

Assume your opponent plays the same.

No matter what your shove range is if you are shoving with the same range as your opponent's raise range then you are slightly ahead as you pick up the overlay.

Opponent should call every time. Even if you raise 2.x the pot they are getting 1:5 to 1 and they should call getting 1:1. With the pot odds they should even be calling with top 70 hands even if they put you on a top 34 hands.

So I proved not much mathematically. You can jam with the range you think your opponent is raising with and you are positive EV.

Big stacks behind is a different story. Re-raise 2x the pot with top 34 hand and get raised back is harder as then you might need to lay down TT and AKs (and maybe more).