Poker Theory

Poker Player
All In
Poker Player
All In

My name is Peter Allen, I’m a part time poker player and a full time software engineer. I am not a writer. What I am is someone who has played poker for most of his life and has obsessed over the game for at least half of it. When a good friend offered me an opportunity to write a column on poker theory, I flipped at the chance to subject an audience to my rantings.

Everything was good until I submitted the first draft.

The notation is scary

Um, do you think you could make this more consumable by the general public?

Forgive me, dear reader, if I sometimes forget there are others out there who do not yet supplicant themselves before the altars of logic & mathematics. But I’m not interested in writing a beginner’s guide to poker or game theory. That’s what the internet is for. My only job is to blow your f***ing mind with science. I’ll try to keep it simple while we explore modern poker theory, but I can’t promise it won’t get weird. Like multiverse weird.

How to Bluff All-In on the River

One of the most exciting plays in no-limit hold’em is shoving all of your chips in the middle as a bluff. Poker is a game of hidden information and no-limit means unlimited bet sizes. What could be more fun than risking every chip you have to scare your opponent away from a huge pot that you don’t deserve? If you apply a little method to your madness, you can turn an apparently suicidal move into a long term money maker. Besides, if you never shoved all-in as a bluff why would anyone ever call your big bets when you have a monster hand?

When you pull that move off successfully it can seem like your two cards don’t matter. They do, sort of, but what’s more important is your range of hands –  all the hands you could possibly have in that spot, as if every Peter Parker across the entire Spiderverse was dealt one of every possible poker hand. Then you remove the hands you would have folded and what you’re left with is your range, like some sort of quantum superposition of every possible hand. If you know your range, you can calculate which exact hands can profitably make a huge all-in bluff and pull the trigger whenever your two cards are on that list. The fate of any single Peter Parker isn’t that important. To save the Spiderverse, you must try to get a positive number when adding up the wins & losses of every Peter Parker across existence.

Coming up with a perfect list of hands to bluff with is practically impossible to do while playing at the table. What many pros do is study different situations at home using a poker calculator, where they can come up with a perfect answer. The more time you spend doing that, the easier it is to make an educated guess at the table. And whenever you run into a situation you’ve studied before, you’ll absolutely crush it.

I’m going to show you how to examine one very specific situation where we’ll construct an all-in bluffing range. We’ll have an all-in value range too, but what I really want to know is the maximum number of hands we can bluff all-in with.

We’re playing no-limit hold’em with six players and each player has $100 in chips. Every player is dealt two cards face down that only they can look at. By the end of the hand there will be five cards face up on the board which all players share. You can win by having the best five card poker hand using any combination of your two cards and the five cards on the board, or you can win whenever every other player folds.

In this situation, almost all of the action has already taken place and only one decision remains: will we check and let the best hand win at showdown? Or will we shove all-in with the last of our chips and let our opponent decide whether to call or fold? The actual two cards we’ve been dealt don’t matter yet, because we’re going to figure out what every Peter Parker across the entire Spiderverse is going to do. We are Player 5, the Hero in this hand. The Villain is Player 2.

Toby McGuire

PREFLOP

  1. Player 1 is in the small blind and posts $0.50.
  2. Player 2 (Villain) is in the big blind and posts $1.
  3. Each player is dealt two cards.
  4. Player 3 folds.
  5. Player 4 folds.
  6. Player 5 (Hero) raises to $3.
  7. Player 6 folds.
  8. Player 1 folds.
  9. Player 2 (Villain) re-raises to $9.50.
  10. Player 5 (Hero) calls.

Three shared cards, the flop, are dealt to the board: Jh Th 3s

FLOP

Pot: $19.50

  1. Player 2 (Villain – stack size: $90.50) bets $9.
  2. Player 5 (Hero – stack size: $90.50) calls.

One more shared card, the turn, is dealt to the board: 5d

TURN

Pot: $37.50

  1. Player 2 (Villain – stack size: $81.50) bets $15.
  2. Player 5 (Hero – stack size: $81.50) calls.

The final shared card, the river, is dealt to the board: 9d

RIVER

Pot: $67.50

  1. Player 2 (Villain – stack size: $66.50) checks.
  2. Player 5 (Hero – stack size: $66.50) bets all-in for $66.50.

It’s a pretty exciting hand! We were the first one to raise preflop, but got re-raised and decided to call. On the flop the Villain fired a bet a little less than half the size of the pot, then did it again on the turn. On the river they checked and we shoved all-in with our entire stack. We’d lose money fast if we did that move with every hand, but we can use a poker calculator to choose the exact hands that could make a profit with this play.

First, we need to figure out exactly which hands we could have in this spot. We’ll start with an initial range and remove hands along the way. For this example, I used software called Flopzilla, which makes examining poker ranges much easier. In the notation below, “s” means the two cards are of the same suit (clubs, diamonds, hearts, or spades). “o” means offsuit, two different suits.

We’ll start from the beginning and choose the specific hands we would have raised with when we entered the pot. Each players’ range is different and changes depending on their position, how the other players behave, whether Peter Parker has a symbiote attached to him, etc. Raising 24.8% of all possible hands is a reasonable starting point for our position. That range includes:

  • All pairs
  • All suited hands with an ace
  • Some offsuit hands with an ace
  • Some suited hands with a face card
  • Some offsuit hands with a face card
  • An offsuit connector
  • A handful of suited connectors

There are 330 different hand combinations in that range and we could have had any one of them when we made our raise. Each combination is equally likely in the Spiderverse, but only 6 Peter Parkers have AA while 16 of them have AK. That’s just how the cards in the deck combine together.

https://www.gettyimages.co.uk/detail/news-photo/andrew-garfield-attends-the-world-poker-tour-presents-four-news-photo/616268660

Then the Villain re-raised us and we called their raise, so now we can remove a few hands from our range. If we had a monster hand we probably would re-raise instead of call, so we can remove AA, KK, QQ, and AK from our calling range. We’ll also fold out our weakest hands that don’t play very well against whatever the Villain likely re-raised us with. We end up removing about 50% of the hands from our range and are left with 153 hands that would have called preflop.

FLOP

Pot: $19.50

The Villain bet $9, making the new pot $28.50. When the Villain risked $9 to win $28.50, they would show a profit if we folded more than 31.5% of the time. No matter what their cards were! Without knowing the Villain’s range, our Peter Parkers have a great responsibility to not fold more often than that. We’ll call with any pair or better. We’re also going to call with our flush draws and open ended straight draws. That narrows us down to 105 hands; only folding 31.4% of our range.

To keep things simple, let’s assume we won’t raise the Villain now or on the turn with any of our hands. If we had a raising range we’d have to remove those hands from our calling range, but I want to maximize the number of hands we can shove all-in with on the river. You’ll see at the end that the more Peter Parkers there are shoving all-in with monster hands, the more Peter Parkers there can be shoving all-in with bluffs.

TURN

Pot: $37.50

The Villain bet $15, making the new pot $52.50. If we folded more than 28.5% of the time here the Villain would show a profit with any two cards. We’ll only fold our weakest pairs, 25% of our range, and are left with 78 hands.

Raising at least some of our hands would be better, but our spidey sense is tingling. Maybe the Villain will bet again on the river and we’ll be able to win more chips? Let’s keep all of the good hands in our calling range by never raising.

RIVER

Pot: $67.50

The Villain checked. My spidey sense misfired, so we’ll have to rely on my other powers. In this situation we heroically shoved all-in with our last $66.50 after the Villain showed weakness. Not every Peter Parker did that, of course. Many just checked and let the best hand win. But which ones made that heroic shove with every last chip? And how many of those Peter Parkers were lying their ass off with a stone cold bluff? 

If the Villain called our bet, the final pot would be $200.50. To pay $66.50 for a chance to win $200.50, the Villain has to win the hand at least 33% of the time to break even. To make the absolute most money over time, and not even care what the Villain decides to do, we should ensure our all-in range contains exactly 66% value hands and 33% bluffs.

To accurately construct our value range, we would need to start from the beginning to work out the Villain’s range so we can compare it to ours. In the interest of keeping things simple, let’s cheat and assume we know the Villain so well that we’re sure they’re going to almost always show up with a pair in this spot. If that’s true, we can shove for value with a straight, three of a kind, two pair, and top pair with top kicker. That comes to 30 hands for value.

To get the perfect ratio we need to now include 15 hands from our range as bluffs. If we have a decent pair we‘ll just check instead of bluff because there’s a small chance we can win at showdown. For bluffing, we want to choose our weakest hands that are unlikely to win if we check. Our fifteen Peter Parkers that made it this far with those terrible hands will now try to make the most of it by bravely shoving every chip into the middle and hope it’s enough to scare the Villain away.

Screenshot from this video: https://www.youtube.com/watch?v=RxuAMkRPPYU ]

Because our bluffing frequency perfectly matches the odds we’re offering the Villain, nothing they do from that point on can mathematically improve their results against us. When we shove a perfect ratio of value to bluffs, we are indifferent to whether the Villain always folds or always calls. The entire Spiderverse wins no matter what happens next.

Of course if the Villain does fold, and you happen to be holding one of those fifteen bluffing hands, be sure to show it to the table so everyone can see you’re just a wild maniac willing to bluff off everything you’ve got with any two cards!

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