2. A polarised range with multiple streets of action

Today I'm going to extend the model of balanced play with a polarised range over multiple streets. Let's imagine that you again have a range that contains a certain number of nut hands along with a bunch of absolute air (far more air than nut hands), while our opponent's hands are all of medium value. This time however, action starts on the flop. For the purposes of this model we're assuming that hands will not change in value across later streets, the only real purpose of having multiple streets here is to allow multiple betting rounds. For simplicity we will also assume that you always bet the size of the pot.

Since your opponent is never going to bet his bluff-catchers, you obviously should bet all your nut range at every opportunity. But how much should you bluff to be balanced? Remember first that to be balanced you are bluffing with a frequency such that it makes no difference whether your opponent calls (on average). The way to work out how often you should be bluffing on the flop is to work backwards from the river.

River: As discussed last time, you bet pot with 100% of your value range and 1/2 as many bluffs. Against this range it makes no difference whether your opponent calls or folds. They therefore have zero equity against a river range this size. However, for your turn bet to have been balanced, you need to not care whether your opponent calls it. so given that your opponent is calling a pot-sized-bet on the turn, he needs to have exactly 1/3 equity at this point. This means that your value range and half as many bluffs only accounts for 2/3 of your range on the river. The other 1/3 comes from more air which you check and give up with. This air you give up with comes to a range 3/4 the size of your value range.

Therefore on turn you are betting 5/4 times as many bluffs as value bets. Since it doesn't matter whether opponent calls flop bet either, he must have 1/3 equity on calling flop bet. Since it doesn't matter whether he calls the turn bet, we can assume he folds every time, therefore we must give up 1/3 of the time, so that it doesn't matter whether he calls the flop. Since you're betting 9/4 of your value range on the turn (value range + 5/4 bluffs) you must give up turn with a further 9/8. Therefore on the flop we have to bet that 9/8 on top of the 9/4 (18/8) of our value range. Therefore we bet a total of 27/8 of our value range.

Once again, it doesn't matter whether he calls this bet or not, so we can assume that he never does. That means we win the pot with 27/8 times our value range. We've more than tripled our showdown value in equity!

Now let's look at the flipside. We're now the guy calling down. What we've just said is that your opponent can bet 27/8 times his value range on the flop and be perfectly balanced. ie. It makes no difference whether we call. So even though he's bluffing more than twice as often as value betting, a fold is just as good as a call. Now imagine that your opponent is not balanced and he's only bluffing exactly twice as much as value betting. The fewer bluffs in his range mean that even though we're ahead 2/3 of the time it becomes an easy fold. Strange but true...

So here we've shown that a polarised range is far more valuable in relation to the pot size with multiple streets of action remaining. This has a result for bet-sizing which is more intuitive than for the river decision discussed last time. Because overbetshoving a polarised range on one street can not be done with more bluffs than value-bets, doing this can not do any more than double your expected value (see last entry). As we've just shown, with multiple betting rounds we can more than triple our expected value. Therefore we should size our bets such that we can get a decent sized bet in on each street. And it just so happens that the common practice of betting around about the pot on each streets is usually about the right amount to do just that when 100bb deep.

Another application of this discussion is in thinking about the number of hands you can get away with bluffing on the flop. Assuming pot-sized bets, we've shown that you can bet 1/2 as many bluffs as value bets on the river, 5/4 as many on the turn, and almost 2.5 times as many on the flop. But this is assuming that all your value betting hands are strong enough to bet 3 streets. In real poker your range is not this polarised and some hands are only worth one or two streets of value. I'm going to hypothesise that the number of streets of value you can try to get from each hand in your range impacts directly on the number of bluffs you can afford to make (in a balanced way). So I will suggest that for every 1 value street hand you can add 1/2 a bluff. For every 2 street value hand you can add 5/4 bluffs. For every 3 street value hand you can add 19/8 bluffs.

As an example let's say we have a flop where we hit 5% nut hands, 10% 2 value-street hands, 15% 1 value-street hands and 70% air. Assuming for argument's sake that we bet all of our value range on the flop then that allows us to bet with 5x(19/8) + 10x(5/4) + 15x(1/2) = 32% of hands as bluffs. So we bet this flop a total of 62% of the time.

Of course the exact numbers I've got in here are derived using a simplified model that does not factor in draws, opponent's aggression etc, so this isn't going to tell you the exact number of bluffs you should make. But the central point holds and is that:

1. For every value betting hand in your range you should add bluffs.
2. You should add more bluffs for every stronger value betting hand than you should for each weaker value betting hand.

Stack size also effects the equity to be gained from our polarised range in a similar way. If there are only 1 or 2 psbs left in effective stacks then you can't afford to bluff as often, since you won't have as many betting rounds to exploit your advantage.

In summary:
Polarised range even more important with multiple betting rounds remaining
Selective double and triple barrelling is very profitable and allows you to raise bluff frequency several fold
Just because you're usually ahead doesn't mean you should call
More and stronger hands in your value range mean you can bluff more
You should bluff more on early streets with greater stack/pot ratio.