Hi,

The following is a toy game theory model taken from Phillip Newall's terrific book Further Limit Hold'em (but it should work for no limit as well). I am only interested from player 1's POV for now.

The rules of the game are simple and as follows:

The pot starts out with 1 betting unit.

2 players are each dealt a number 1 thru 100. Play 1 acts first and can either check or bet. If check, then player 2 can either check behind or bet. If player 2 checks behind, there is a show down and the highest number wins.

If player 2 bets, then player 1 can call or check/raise and now player 2 can only fold or call (there is no 3 betting in this game).

If player 1 bets, then player 2 can either fold, call, or raise. If call, there is a show down and the highest number wins. If raise, then player 1 can only fold or call because once again, there are no 3-bets or re-raises.

The following is a game theory model strategy graph for both players. Again, for now, I'm only interested in player 1.

[IMG][/IMG]

I understand that player 1 needs a 2:1 value bet to bluff ratio. Therefore, if he has 11.8 value bets, he needs 5.9 bluffs, which will be taken from the bottom of his range. I also get that if player 1 pulls 6.1 check/raises for value, then he'll want a 4:1 bluff ratio and needs 1.5 check/raise bluff and that these bluffs are best taken from the top of his folding range.

What I cannot for the life of me figure out is how they derive the 11.8 value bets or the 6.1 value check/raises. I think I'm close. We are a favorite when holding numbers 51 thru 100 for a total of 49 numbers. Since we'd be giving our opponent 2:1 odds if we bet, does that mean 50% of 49 (our highest 24.5 numbers) is our value range? If so, do we construct our value check/raise first? 24.5/4 would give us 6.1 value check/raises with a corresponding bluff ratio of 6.1/4 = 1.5 check/raise bluffs.

But this must be wrong, because if we take 6.1 hands out of our value range to check/raise for value. that leaves 18.4 value bets, which would correspond to 9.2 bluffs, not 5.9. I also can't get any of this to jive with different pot sizes so it must be wrong.

This is all very complicated for me and while I can just accept their distributions, I'd like to understand how this range is being bifurcated across the 1 thru 100 distribution set, so I can do this on my own. I'd appreciate any help!

Thanks,

I don't think you'll be able to look at just Player 1's strategy to derive this. The strategies are interdependent. I don't own the book but assume if it's like most game theory problems both player's are aware of the other's strategy so ranges are based on what the other player will do.

I'm also surprised if you could derive this without some sort of computer simulation.

I think you should get permission from the author/publisher. I know it's seemingly only one chart, but it's one of the more important parts of the book.

Whoops! Super sorry if I did anything wrong! That's why I was sure to credit to the author and I also meant to mention that the author in turn, states "all the numbers in this section have been adapted from Chris and Tom Fergusen, and Cephus Gawargy - Uniform [0.1] Two person poker models" and gives a UCLA link. It's a 2p2 book, but if it's inappropriate any mod can feel free to delete my entire post.

I don't think I'm going to get anywhere with it anyway. Thanks for the warning.

Thanks for responding. It makes sense that game theory is based on the assumption that both players are aware of other's strategy.
I thought it was just some probability based math that I couldn't figure out.

I find it somewhat surprising you'd need a computer sim to find equilibrium in such a simple toy game. But maybe it's not as straight forward as I thought.

Thanks again,

It's not really that it's difficult it's just really time consuming.

Here you are trying to work backwards from the results of the analysis but if you started naively with no idea where to start you'd have to iterate through different potential strategies and eventually arrive at the solution. The math would be tedious and prone to errors. A computer algorithm could do all of that for you.

It's just a guess any way. I am not an exper by any means, but just looking at the final solution I would be surprised if anyone could do it by hand.

I've informed a mod to delete this thread. Thanks again for the heads up.

Right. I'm trying to reverse engineer the frequencies. I just thought if we know some of the common calculations like p/(p+1), etc., we should be able to derive the proper betting and bluffing frequencies from a given set of combos.

For instance, if I have 87 combos left in my range on the river with x bets in the pot, what good does it do me if I can't break down my distribution as outlined in the graph? If the pot contains 4 bets, I know I'll need 5 value bets for every bluff. If I check and my opponent bets, I know I'll need to call with the upper 80% of my range. But if I'm using some of my checking range for check/raise bluffs, they need to be subtracted from my check/calling frequency and so on.

That's good to know. I was getting very frustrated trying to work it out and failing. Guess I'll just have to move on and read the rest of the book lol.

Thanks,

I think there is a book/publications sub forum where authors are known to answer questions. I don't frequent it so I don't know if this author is in there but there maybe a thread about the book already.

I guess I don't know all the rules of the toy game but if you have a distribution of hands with varying equity and different betting options even if you could derive the betting frequencies directly (which would involve accounting for the multiple betting lines) you wouldn't know the most +EV for each hand without testing all options.


Well normally that is the point of having the results of the toy game. The results give you insights about how to think about things in game, without having to go through the work of actually solving them.

I can definitely appreciate your willingness to work with the techniques to understand the process, I just don't think in this instance the results were determined without computer assistance. I would be happy to be proven wrong and shown the technique for solving by hand though.

Hi Everyone:

I think it's okay to leave this up.

Best wishes,

Mason Malmuth
Publisher
Two Plus Two Publishing LLC

If there are no 3-bets then the ratios should be solvable by hand. Allowing unlimited bets would need a computer sim. I cant do it, but from reading these types of proofs it should fit into a payoff matrix and the allowance of mixed strategy is where these ratios come from. Just my 1 cent...

I got to thinking about this. I guess I don't know what the legal betting values are. If they are just in 1 betting unit increments then I agree you can solve this by hand and it wouldn't be too crazy but still tedious and error prone.

If they are any value for each betting action then I think you will need a computer.

Should be a fixed limit bet. Arggh! The answer is surely published, but I am just starting out with these proofs such that I can only sorta understand wth is going on in the proof.

Obsession engaged.... this will be fun (no this will hurt my brains).

Thanks for trying! The betting values are fixed at 1 unit (with no 3 bets!). I feel as you do. I am also obsession engaged and think this should be quite solvable.

This thread is a prime example of how helpful it can be to cite sources, so we can all read along and learn together.

This particular one is quite complicated and way more than a simple payoff matrix (it does have payoff graphs).

Here is the original work by the Chris Ferguson, Tom Ferguson, and Cephas Gawargy

https://www.math.ucla.edu/~tom/papers/poker2.pdf

This paper is simply phenomenal, and lays a basis for several key understandings of the complicated math of poker.

Anyone interested in this thread should read this paper over and over until it sinks in, imo!

.....Optimal bet size is the size of the pot, other bet sizes give away exploitable information.

......The first bet made in a heads up fixed limit poker hand is often pot size, and as such should be treated very similarly to a pot limit poker hand (most limit players don't do this, imo). The more raises that are made in fixed limit which bloat the pot dramatically change the optimal continuation strategy of the players. The more bets that happen, the less and less the hand plays like pot limit. Bloating the pot could be a mistake if it makes the rest of the hand a more simple exercise by your opponent.

This thread is about the more complicated version of the toy game that involves player 1 check raising, with a fixed limit size bet which is where all the ranges come from.

Section 4 of the paper by Ferguson/Ferguson/Gawargy deals with the specific toy game in this thread and provides the ranges.

The key concept is the understanding that each player will have the ranges that are described in the photo in Phillip Newall's, Further Limit Hold'em, which he correctly credits to Ferguson/Ferguson/Gawargy. Once you understand this, the ranges are bifurcated by a series of twelve points. The twelve points go into a set of twelve equations with 12 unknowns, and are solved.

***THIS IS COPIED FROM THE PAPER, I DID NOT FIGURE THIS OUT ON MY OWN****

Here are the equations for pot limit where the first bet is 1 bet and the raise is 3 bets. (actually it has an initial pot of two bets and the first bet is two and the raise is 6)

The ranges are based on indifference and logically explained and defined by:

Player II makes Player I indifferent to:
1. bet-fold and check-fold at a: 2k = 1.
2. check-fold and check-raise at b: g − h + 3j = 2.
3. check-raise and check-call at c: −h + 2j = 1.
4. check-call and bet-fold at d: g − h − 2k + 3m = 0.
5. bet-fold and bet-call at e: −2k + 2m + n = 1.
6. bet-call and check-raise at f: g − h − 3j + 4k − 3m + 3n = 0.


Player I makes Player II indifferent to:
7. bet-fold and check at g after a check: 2b − d + f − g = 1.
8. check and bet-fold at h after a check: 2b − 3c − d + f + 2h = 1.
9. bet-fold and bet-call at j after a check: −2b + 2c + f = 1.
10. fold and raise at k after a bet: a − d + 3e − 2f = 0.
11. raise and call at m after a bet: d − 2e + f = 0.
12. call and raise at n after a bet: e + f − 2n = 0.

The solution of these 12 linear equations in 12 unknowns is as follows. All numbers
are to be divided by 150:

a = 8, g = 20, b = 77, c = 80, h = 110, d = 128, j = 130, f = 144
k = 75, m = 80, e = 136, n = 140.

So, we see that the initial question asked by the original poster in this thread is:

Where do a and f come from?

a is calculated logically as making player 2 indifferent to:

fold and raise at k after a bet

where a is described as: a − d + 3e − 2f = 0

so Player 1 has a bluff-fold bottom range of 2f-3e+d which is

twice the (value check raise range) minus three times the (bet call value range) plus one times the (bet fold value range)

So a and f are related by the description :

Player 1 Bluff-Fold range = 2(value check raise range) - 3(Bet-call value range) + 1(bet fold value range)

The actual numbers for fixed limit bet where the bet is the size of the initial pot are slightly different and given in the paper as:


a .059 (the bluff check raise range)
b .511
c .526
d .821
e .850
f .939 (the value check raise range)
g .140
h .719
j .790
k .500
m .526
n .895

The value of f is present in every equation of player 1, so it is used to derive every other equation, so putting that into words is kinda pointless.

When the bet is fixed at the size of the initial pot, the ranges are computed by the twelve equations and the ranges in the photo are computed (this is near the end of the paper).

FOLLOW UP QUESTION:

This game is rigged! As in many toy poker games, optimal strategy will favor one of the two players. Which one do you think this one is? Make your guess before reading the paper and remember to try to figure out why...


It is not long, and this paper cites several previous works by pillars of game theory such as:

Emile Borel (1938) ´ Trait´e du Calcul des Probabilit´es et ses Applications, Volume IV,
Fascicule 2, Applications aux jeux des hazard, Gautier-Villars, Paris.

William H. Cutler (1975) “An optimal strategy for pot-limit poker”, Amer. Math. Mo.,
82, 368-376.

William H. Cutler (1976) “Poker Bets on a continuum”, Preprint.

Chris Ferguson and Thomas S. Ferguson (2003), “On the Borel and von Neumann Poker
Models”, Game Theory and Applications, 9, 17-32, NovaSci. Publ., N.Y.

Donald J. Newman (1959) “A model for ‘real’ poker”, Oper. Res., 7, 557-560.

John von Neumann and Oskar Morgenstern (1944) Theory of Games and Economic Behavior, John Wiley 1964, 186-219.

CORRECTION: The first bet in heads up limit poker is often 1/2 or 1/4 pot and should be treated as such.

Thanks a ton! It didn't even occur to me to Google the cited source. Seriously, thanks!

The paper while short, is going to be a very difficult read for my math level, but I will take your advice and try to make sense of it.

My very sophomoric answer to the follow up question without having read anything or fully thinking it through, is that the game is rigged in Player II's favor, since he is assured of deriving value from the top of this range. Player I has no defense for this, since the rules state he can neither re-raise as a bluff or for value. In addition, Player I cannot get value from the lower end of the range in which he's a favorite, while Player II makes use of his whole distribution as a favorite.

I'm sure it isn't that simple and is probably wrong. Hopefully, my understanding of game theory will improve after reading this paper and finishing the book!

Really appreciate your post.

Thanks,