From The Mathematics of Poker, page 186:

Example 16.4 - [0,1] Game #7
One full street.
Two bets left.
Check-raising is allowed.
Folding is not allowed.

x2 = 1/9 (size of check-raise region)

How does one solve for an arbitrary bet size where folding is allowed? I'm having difficulty with the higher-level math necessary to produce an equation.

Any help is greatly appreciated. Thanks

Don't they solve it in the book? Iirc they set up the indifference equations and just solve using algebra...

Can you point me to where? Maybe I'm missing something

bump

The solution to the full street of betting in the (0,1) game with two bets is shown in Further Limit Holdem for multiple pot sizes. The powers that be would probably get mad at me if I showed the solution here. However, I think it's ok for me to say this much:

This is how the action regions break down:

[check raise / bet call / bet fold / check call / check raise / check fold / bet fold]

My loose theory is that in games with 3+ bets to play, the action regions will break down differently like this:

[bet 3 bet / check raise / bet call / bet fold / check call / check raise / check fold / bet 3 bet / bet fold]

I think.

The Books and Publications forum is right next to the theory forum. The actual author may possibly answer your question.

It may not be on that particular page, but I can not imagine the method is not in the book somewhere since it is crucial to getting the exact answer.

Here is a start.

The order in which the ranges are divided was proven by John von Neumann such that:

Your most aggressive bluff comes from the bottom of your bluff range.

Therefore, that region is bounded by zero on bottom and some point called an indifference point where we stop taking that action because our opponent becomes indifferent.

Now we look at the entire range of our opponent.

At some arbitrary bet of X$, where will our opponent be indifferent between calling and folding?

That will give a precise numerical value to that portion of the caller range.

So on and so forth....

You wind up with simple linear equations were:

Number of unknowns = Number of equations

CORRECTION

As usual, I was incorrect.

Above should read:

Your most passive bluff comes from the bottom of your bluff range.

This is the Bet-Fold bluff range.

I checked the book mentioned and the method is displayed, but not taught per se, the book is written for an audience familiar with deriving linear equations. Chapter 11 introduces the concept of indifference and the derivation of the equations. Stay in chapter 11 until you are rock solid with it. I did not read the op correctly, per usual, so the amended and further corrected (liable to further amending and correcting) is that our most aggressive check raises (no folding allowed) are from the top of our range (which this author places at the bottom... 0 is the nuts).

-Rob

So I read the book and the author references this paper:
UNIFORM(0,1) TWO-PERSON POKER MODELS

In section 4.2, they solve for the rigid pot limit game. How do you generate the linear equations for the no limit case? Assume same bet size for both players.

Its not in that paper. The no limit solution is called the Newman model from 1959, it is referenced at the end of the paper. Donald J Newman (1959) “A model for real poker” iirc.

A model for ‘real’ poker

Thanks for that. But they only solve for a 1 bet game, so no check-raising region.

Was just about to look at Newman for another reason. But my hunch is that bet size will converge to pot size with n bets as n—>infinity

But let's say I wanna solve for a 1/2 pot bet size with a 2 bet cap. How would I do that?

each player gets 1 chance to bet?

or

each player gets 2 chances to bet?

One full street.
Two 1/2 pot bets left.
Check-raising is allowed.
Folding is allowed.

Same toy game as the OP except no limit and folding allowed.

bump

I am still confused. You say there are two 1/2 pot bets left for each player, but you want to solve for the optimal bet size?

If you want to use the toy game for one full street no limit then you really need deeper stacks.

If you just want to solve for 1/2 pot bets, then just put those numbers in for B in the equations.

Optimal bet size is always at least pot size in these extensions of von Neumann poker.

If you want to know the *theoretical* optimal numbers for no limit deep stacks, it is more complicated since you have to start with the last bet placed and work backwards by setting the derivative = 0 and solving for Betsize. Ferguson only does this for 1-1/2 streets to show that a first bet is optimally slightly more than pot size, and any raises are full pot thereafter.

Thus, an approximate optimal strategy is pot limit anyway, so just using pot limit betting rules makes all of these extra extensions of toy poker so much easier to solve.

The one paper I read with an attempt at an actual no limit first bet size for one full street was in fact slightly more than pot, but was probably slightly incorrect anyway, and the author omitted the proof.

If your model has a pot size and each player has about pot left behind, i would just use the bet size that you would normally make as a poker player, then solve the equations for that size and get an estimation of your bluff and value hands.

I'm trying to solve for optimal check-raise frequency given any fixed bet size. The MOP model only solves for the limit case and doesn't account for folding. The Ferguson model only solves for the rigid pot-limit case. I want to solve for the check-raising region in the full street game where there can be a 1/2 pot bet and raise and players can fold.

The MOP solution is 1/9 ~ 11%. The Ferguson solution is 1 - 144/150 = 4%. How do I adapt these models to solve for a 1/2 pot bet size?

You put bet size into the equations to get the frequency. Take the Ferguson equations and put 1/2 pot bet in for betsize in the equations. That is what was at the end of that paper, a chart of the different frequencies with bet sizes below full pot.

I'm not seeing anywhere where I can just plug in bet size. Can you perhaps show me how you're doing it?

Ahhh, Ferguson simplified his equations for pot limit, and then only summarized the results for smaller bet sizes. Oops! The method is in section 3.2 and 3.3 for the previous section on von Neuman poker extension where the betsize is variable.

I will take a look and maybe post what the equations should be, then you can double check.

Hey just stumbles upon this thread . I have no idea what you guys about lol ( optimal bet sizing equations/ toy game) but very interested . What’s the name of the book? Also if this isn’t for nL Hold’em then just ignore my questions .

Thanks

It's listed in the OP - "The Mathematics of Poker" by Bill Chen and Jerrod Ankenman
https://www.amazon.com/Mathematics-P.../dp/1886070253

The information in the book could be used for any kind of poker