1. A useful thought experiment and the value of a polarised range

Balance is something that gets talked about a lot in poker. I'm not going to go into a lot of detail about balance just now, save to define a balanced play as one where across the range of hands with which you make this play you will make the same amount of net profit irrespective of your opponents actions. ie. on average you don't care what your opponent does. You are therefore unexploitable (but will not fully exploit your opponent's mistakes).

For those of you who are not totally sure what people mean by the idea of balance the following example should help:

Imagine a river situation where your range is split into 50% nut hands which are ahead of your opponents entire range and 50% air which are behind your opponents entire range. Both of you are exactly aware of each other's ranges. You are first to act and have two choices; bet pot or check. When you check your opponent will always check back since you will never call him with worse. If you always checked here then you would each win the pot 50% of the time. I'll use this as an EV base of 0. First thing to do to change this is to value bet our nut range of course. However when we do this our opponent should always fold to our value bets, meaning we don't actually make any more money. So we decide to throw in some bluffs to take advantage of this. If we bluff too much then our opponent should call, if we bluff too little then he should fold. However there is a mid-way point where it doesn't actually matter whether our opponent calls or not. This is the point of balance.

So what is this point? How often should we bluff? Well when our opponent folds we win a pot-sized bet from every bluff, and win nothing from every value bet. When our opponent calls we win a pot-sized bet from every value bet and lose a pot-sized bet from every bluff. Since we're balanced, we don't care whether or not our opponent calls, meaning we must make the same amount whether he calls or folds. Therefore

(profit from folds) number of Bluffs x 1psb = (profit from calls) number of valuebets x 1psb - number of bluffs x 1psb
Therefore number of Bluffs = (1/2) x number of valuebets.

So to be balanced you need to bluff half as often as you value bet.

So how much EV have we gained by doing this? Well, since you're balanced, you don't care whether or not your opponent calls. So for simplicity, we can just assume he folds every time. This means by bluffing half as often as value betting, we have increased our expected value by 50%. This is equivalent to us winning 75% of the pots And this result is independent of what our opponent does. Even if our opponent is the best player in the world, he only has 25% equity here despite having the best hand half the time! Pretty neat eh?

So why do we have so much equity here? It's not really to do with the whole balance idea, that's just a useful way of explaining things. The reason is that our range is polarised, while our opponent's is well-defined. The value difference is real and it's huge. Think a bit about the implications of this when it comes to check-calling multiple streets. That's what my next entry is going to discuss.

For now though there is another interesting application of the idea of a balanced river betting range. The set-up for this experiment is going to be exactly the same river situation as before, except this time you can choose your bet-size. Given that we are going to be consistent about our bet-size (otherwise we are not balanced), what is the best bet-size to use, and how can we work this out? Well earlier on we calculated the balanced number of bluffs for a potsized bet. Given that we are bluffing a balanced amount, we don't care whether or not we get called. So for the purposes of expected value calculations, we can assume that our opponent always folds, and our gain in equity is simply equal to the pot multiplied by the balanced number of bluffs. Therefore the more we can bluff (in a balanced way) the more money we make.

We therefore need to choose the bet-size which allows us to bluff with the greatest frequency (and be balanced). Let's try working out the balanced number of bluffs for a half psb. If we do this then profit from folds is still number of bluffs x 1psb but profit from calls is (1/2)psb x no. value bets - (1/2)psb x no. bluffs.

Equating these as before we find that number of bluffs = (1/3) number of value bets. So betting half-pot, even in a totally balanced way, is simply less profitable than betting full pot. Everybody already knows that it's better to bet something comparable to the pot rather than to make very small bets, but hopefully this will give you a better understanding of why.

So what happens when we bet more than the pot? Well actually, in this particular situation you'll find that the larger your bets the more bluffs you can get away with, and so in this situation you should in fact overbet-shove all your nut range and nearly as many bluffs. (Note that there are diminishing returns with increased bet-size. Betting 1/2 pot allows you to bluff with 1/3 as much as your value range. Betting full pot takes that up to 1/2. You need to bet an infinite amount for that figure to approach as many hands as your value betting range.)

Here we seem to have parted from the realms of realistic poker; how can overbetshoving be the optimal play here? Well the answer is because your range is totally polarised. This won't usually happen in poker. Usually your opponent's range overlaps with your value range, and overbetting will just mean that he'll only call when he's beating you. But occasionally your range will be very polarised and you should start to bet more; overbetting is the optimal play if your range is entirely air or stone cold nuts.

So what can we learn from all this when it comes to actually playing poker?

1. Polarised range good, condensed range bad. Just because you are ahead half the time doesn't mean you have 50% equity in the pot.

2. The amount of bluffs you can get away with making is dependent on your bet-size. If you bet larger you should bluff more.

3. Against an opponent who is well-adjusted to your play you can never bluff more hands than you value bet on the river.

4. You should bet larger with a highly polarised range (relative to your opponent's range) and smaller with a less polarised one.

There may be more things to be drawn from this aswell, I'd be very interested to hear any ideas.