Advanced poker strategy: balance, blockers and when to leave the book

A game theory optimal strategy is one your opponent cannot exploit, whatever they do. It bets and checks, bluffs and value bets, in proportions that make every option of theirs equally good. It does not mean the most profitable strategy. Against a player who folds too much, the most profitable play is to bluff more than the balanced amount, and a balanced strategy leaves that money on the table.

Use it as the measuring stick. When you know what a balanced range of bets looks like on a river, you can see that an opponent who never bluffs there is a player to fold against, and that one who bluffs every missed draw is a player to call.

On the river, a bet of a given size should contain a certain share of bluffs to make the caller indifferent. The caller is getting pot odds, and the share of bluffs should equal the odds they need.

Earlier streets allow more bluffs, because a bluff on the flop still has cards to come and can improve. The river numbers are the ones to know by heart.

Facing a bet, if you fold too often, any bluff shows a profit. The minimum defence frequency is the share of your range you must continue with, by calling or raising, to stop that: the pot divided by the pot plus the bet. Against a bet of half the pot, continue with about 67% of your hands. Against a pot-size bet, 50%. Against a bet of twice the pot, 33%.

That is the balanced figure against a player who bluffs the right amount. Most players at low and middle stakes bluff the river less than that, so folding more than the minimum defence frequency on the river is usually right against them. Use the number to notice when you are folding far too much, not as a rule to call.

On every flop, ask two questions. Who has more equity with their whole range? And who has more of the very strongest hands? The first is range advantage, the second is nut advantage, and together they set your bet size.

• Range advantage without much nut advantage, such as the raiser on A-7-2 rainbow: bet small and often, with most of the range.
• Nut advantage, such as the big blind caller on 7-6-5 two-tone who has all the straights and the sets of small pairs: bet bigger and less often, with strong hands and strong draws, and check the rest.
• Neither, such as a paired board: small bets or checks, and the pot stays small.

A card in your hand cannot be in your opponent's. When the flush completes on the river and you hold the ace of that suit without a flush, they cannot have the nut flush, which makes your hand a better bluff than a hand with no such card. When you hold a pair that matches the board, there are fewer trips for them to hold. Picking bluffs by what they block, and calls by what they do not, is most of what separates a good river from a guess.

The cheap version at the table: bluff with the hands that make their strong hands less likely, and call with the hands that make their bluffs more likely, such as holding one of the cards that would have completed their missed draw.

Raw equity before the flop is not what a hand earns. Out of position, a hand gets checked and bet off its equity and realises less than its share. In position, with the chance to take free cards and bet when checked to, it realises more. Suited and connected hands realise well in position and badly out of it; weak offsuit hands realise badly everywhere. This is why a hand that is a call on the button is a fold in the small blind at the same raw equity.

With the best hand, pick the size that gets called by the most hands that are worse than yours, which is often not the biggest. With a bluff, pick the hands with no showdown value that block the calls, and make the size the same as your value bets, so the size itself says nothing. When you face a bet, do not ask whether you have a good hand. Ask how many bluffs this player has in this spot, and compare it with the odds the bet gives you. Against most players, the answer is fewer bluffs than balance would require, and the fold is right more often than the math of a balanced opponent suggests.

With three or more players in the pot, someone usually has something. Bluffs work less, thin value bets get raised, and one pair is rarely good by the river. Play tighter before the flop when several players are already in, bet strong hands rather than medium ones, and bluff only when the board and your blockers both line up.

Balance protects you against an opponent who is looking for your leaks. Almost nobody at the tables you play is doing that. So as soon as you have a read, use it. A player who folds too much to the second barrel gets a second barrel every time. A player who never folds a pair never gets bluffed, and gets value bet thinner than feels comfortable. A player who 3-bets too much gets 4-bet lighter and called wider. The balanced strategy is where you return when a read is wrong or an opponent is unknown, not where you live.

Late in a tournament the value of a chip you win is less than the value of a chip you lose, because the prize money is paid for surviving, not for the chip count. The independent chip model, ICM, puts a number on it, and the number changes decisions: the medium stack folds hands near the bubble that would be clear calls in a cash game, and the big stack raises more because the others cannot afford to fight. With ten big blinds or fewer, preflop becomes push or fold. The tournament calculator turns stacks and payouts into money.

Solvers, programs that compute balanced strategies for a given spot, are how these numbers were found, and there are commercial ones for players at every budget. Use them to learn the shape of balanced play: which hands bet, which check, how size changes with the board. Then watch the players you face and write down where they differ from it. The second notebook is worth more than the first.