April 3, 2010 – 10:41 am | 7 Comments

I’ve compiled a short (just 7-pages) e-book, an introduction to the mathematics of poker. It’s basically covers how to calculate your expected value in a certain spot – starting with explaining what EV is, all …

Theoretical

Discussions about the theory behind the game.

Opponent Modelling

Ideas about modelling opponent’s play, and how to best do that with mathematical tools.

Hand Analysis

Analyses of past hands (my own or otherwise), using math as much as possible to determine how well the hand was played.

Mental Processes

Thoughts about the mental/psychological side of poker – dealing with tilt, etc.

Miscellaneous

Other information – book reviews, contests, announcements, or random spew.

## Small Ball Strategy – Flop Bet Sizing

Submitted by on January 17, 2010 – 10:58 pm5 Comments

In a previous post, I looked at this example from TrueGamble, and studied the preflop action; concluding that the ideal raise would have been between 2.5x to 3x the big blind, and that the 2x the big blind raise was probably too little. Here, I’d like to continue on using that example, taking a look at the action on the flop. Here’s the action until the flop, from the example.

Blinds are 100/200, John has A J and is using the ‘Small Ball‘. He minimum raises from a latter position and Dave calls it. Flop is 7 6 J, good flop for John, Dave checks and John makes just a weak bet of 250 in a 900 pot. Seeing this Dave calls.

So, John hits top pair, top kicker on the flop. There are two connected cards, and the pot is 900. He bets 250 into the 900, giving Dave odds of about 27.8%. For the sake of argument, let’s assume it’s a rainbow flop.

Is this the right bet size? Remember, the main question here is whether the small ball strategy works, or whether it is (as TrueGamble says), “A New Way of Trapping Yourself”.

What could Dave have in this situation? Given the preflop bet of 2 big blinds, Dave could probably have any two cards, any of the 1081 total possible hands. Assuming that, what hands does Dave have that could beat John? At this point in time, John is only losing to a 6-6, 7-7, J-J, Q-Q, K-K, A-A, J-7, J-6, 7-6, a total of 30 hands. There are 6 hands (other A-J combinations) that would tie with John. Assuming all the hands for Dave are equally likely (we assumed that Dave could have any two cards), that gives John’s hand a current strength of 96.85% (ie. if play were stopped at this point in time, John has a 96.85% equity). Even when he’s behind, though, he has about a 11.5% chance of improving to win (if Dave has 6-7, for example, any A or J would give John the winning hand again).

Given that, at this point in time, John has a pretty strong hand, and should definitely bet. The question is then, how much. When Dave is behind, at this point in time, he has at most 8 outs. Let’s assume he has 9-T (the best hand he could have that’s behind). He then has 8 outs, and a 17.8% chance of hitting one of those 8 outs on the turn. If John bets 250 into the 900 pot (as in the example), Dave’s call would yield an EV of 0.178 * (250+900+250) – 250 = -1.11, which are still the wrong odds to call.

But Dave isn’t necessarily sure that John has the J. If John has the 7 or 6, or no pair, Dave has 6 additional outs. If we use 14 outs, then (the straight draw + the draw to any pair), then Dave has about 31.1% of hitting an out, and an EV of 0.311 * 1400 – 250 = 185.56, making it a profitable call. So if Dave believes John is essentially bluffing without TP, he has an EV of 185.56, and if he believes John has TP, he has an EV of -0.8.  That means his overall EV is -0.8x + 185.56 (1-x) = -1.11x + 185.56 – 185.56x = -186.36x + 185.56, with x = probability that John has TP.

To make this profitable, then,

EV > 0
185.56 > 186.36x
x < 185.56 / 186.3 = 0.994

Essentially, this means that unless Dave is very sure that John has the TP, he should be making the call. I don’t think that in this situation, it is ever possible to say about 99.4% certainty that John has TP, so with a hand like 9-T, Dave definitely has to call.

Now, let’s say John bets 450 instead. Using the same calculations, John needs to have TP at least 45.8% of the time for Dave to make the call. This is a much tougher spot, putting Dave in more or less a 50-50 position. If John’s bluffing about half the time, then Dave should not make this call. If the bet is 600, that bluffing percentage then has to be below 19%.

All this is assuming Dave has a hand like 9-T, which is the best case scenario, so we might want to lower the percentages slightly to account for the other hands.

What does this all lead to? This means that Dave is almost definitely going to be right in calling a bet of 250, no matter what hand he has. So it would seem that 250 is definitely too small a bet. If John bets between 450 to 600, however, Dave would need to believe that John is bluffing at least 55%-80% of the time, based on the bet. That makes it a much more marginal call. I think that it’s going to be hard for Dave to say with anything more than 60-70% certainty that John doesn’t have the J, especially with the higher bet sizes.

So, my conclusion from this analysis? I would believe that a bet in the range of 450 to 600 is a better bet, somewhere between half to 2/3 of the pot.

What do you think? How much do you generally continuation bet on the flop, and why?

• Mac says:

Well Small ball is good in tourney where you see more flop cheaper and will get more action because of the blinds going up.

In a cash games are really more about odds and people will bust you more often then not.

In a tourney you bet less and take small pots with good hands. As opposed to making a 3x-4x raise and forced to move in on a middle pair or draw or a flip situation that you would rather avoid.

It also give you more room to make folds, you have more option on flops you feel your opponent hit. So you can check and only lose a small amount..

It's very exploitable strategy and it's better to reduce the variance if you a good player who can read opponents well.

• Derrick Kwa says:

Hm. Yeah, as I gain experience, I'm starting to realize that the small ball approach doesn't quite work for cash games. The question then for me is…what *is* the right strategy for cash games? Haven't quite figured that out yet.

• Mac says:

Mirco stakes if you want to keep it tight the best thing to do is limp a lot people will stack of regardless of the fact you limp and move in.

If you want to be aggressive you could always just raise it up 3xbb+1 for donk lmipers.

I'm playing sngs when there is no juicy tables.

aggression =power poker.

• jason says:

negreneau in his videos about small ball poker recommends what you said, about 1/2 to 2/3rds of the pot.
personally, if i had made those bets and was facing his raise on the flop, i'd probably reraise.

• Stefanie says:

This is excellent. I’ve researched this opinion before but this is a lot more accurate. Kudos