OTB #049: What is geometric bet sizing and when should you use it?

If you've ever heard the terms "geometric bet sizing", "pot geometry", "3e" or "2e" you might have been left scratching your head wondering what they mean.

Today I want to define geometric bet sizing, show you when you should use it and explain how to work it out.

Let's dive in...

What is geometric bet sizing?

Here's a great definition from The Mathematics of Poker by Bill Chen and Jerrod Ankenman:

“When the game is static and one player is clairvoyant, the optimal bet size is to make the pot grow by the same amount on each street, such that on the last street the entire stack has been bet.”

Geometric bet sizing is using the same percentage of the pot on each such that we have the same percentage left on the river to shove all-in.

It's the optimal bet sizing when your range is perfectly polarised (i.e. value bets and bluffs), as it forces the widest overall minimum defence frequency that gets stacks all-in by the river.

No range is truly polarised until the river, so generally polarised bets are used on turns and rivers. But there are examples where you can use a big bet sizing on the flop that is essentially polarising your range straight away.

On the flop and turn, things can change - draws can become the nuts and the nuts right now can be outdrawn by the river.

When should you use it?

If your goal is to get all in by the river with your best hands, then using geometric bet sizing forces the widest defence from your opponent.

It maximises how much money your opponent puts into the pot.

Geometric bet sizing in action

You raise to 2.3bb from the CO and just the BB calls.

On the As Ts Jc flop you have a huge nut and equity advantage and so you'll want to use a big bet most of the time.

I'm a big fan of using a 2 bet size approach, so if I gave the solver two sizes in this spot it would be 25% and 3e.

3e means 3 equal bets... and this means betting the same size of the pot on all three streets such that I'm all in by the river.

Most of your best hands on the flop will still want to put all the money in by the river.

You bet 3.99bb into 5.9bb (~67% pot) and the BB calls.

Turn strat

Using a big bet or check strategy is common on many turn cards.

You can see the overarching strategies for every turn card in the table below - notice that it's big bet and check for all cards apart from an Ace or any card that completes a draw (spade, Queen and King).

Even though I gave the solver the small bet size (25% pot), it only uses the bigger, geometric sizing. The 8h is a blank and so we want to play a big bet or check strategy - polarising our betting range to value bets and bluffs.

You bet 9.37bb into 13.88bb (~67% pot) and the BB calls.

River strat

On the river, your betting range will be truly polarised: you have hands that want to value bet and hands that want to bluff.

All the hands in the middle should just check back.

On the river you have 22.04bb back and the pot is 32.62bb, so you have... yep, you guessed it... 67% pot to shove all-in.

What happens after a small flop c-bet?

You raise to 2.3bb from MP and just the BB calls, playing about 28bb at the start of the hand.

On the Ac 6h 2d flop, your default strategy should be to c-bet small with almost all of your range this time.

You bet 1.52bb into 6.06bb and the BB calls.

Note: The geometric sizing here is 3.4bb or 56% pot.

When the BB calls the flop c-bet, you are going to want to use a big bet and check strategy on most turn cards if the BB checks again.

Despite giving the solver the option to use a smaller turn c-bet size again, it almost always uses the bigger, geometric size of 77% pot.

On the Qs river, Hero has a decision between jamming for 77% pot or checking back.

How do you calculate the geometric sizing?

This is the tough part.

The formula for working out the geometric bet size is impossible to do in-game.

Here's the formula for the flop:

• 0.5*(((Pot size+2*Effective stack size)/Pot size)^(1/3)-1)

Here's the formula for the turn:

• 0.5*(((Pot size+2*Effective stack size)/Pot size)^(1/2)-1)

Rather than trying to calculate the geometric sizing on the fly, here's a table to help.

The easy ones to remember are:

• If the SPR is 1.5 on the turn, the geometric sizing is 1/2 pot

• If the SPR is 4 on the turn, the geometric sizing is pot
  

Summary

While in theory the geometric bet sizing only works for perfectly polarised ranges vs bluff catch ranges, in practice we can use the geometric size any time we want to use a big bet.

If we go back to my reasoning for using just two bet sizes on each street, having only two bet sizes on each street makes it easier to see what a solver wants to do and also for us mere mortals to actually implement the strategy in game.

That's it for this week.

See you next time.