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Bluff Maths

A pure bluff of size B into pot P risks B to win P. It breaks even if the opponent folds often enough. Turn it around and you get the defender's rule: fold more than that and any two cards can bluff you profitably.

bluff needs folds ≥ B ÷ (P + B) minimum defence frequency = P ÷ (P + B)
  • Position: acting last is a structural edge, because you see what others do first. Play more hands on the button, fewer under the gun.
  • Ranges, not hands: put the opponent on every hand they would play this way, weighted by likelihood. This is Bayes again: each action is evidence with a likelihood ratio.
  • Balance: bet your strong hands and some bluffs with the same size, in a ratio that makes the opponent indifferent. A pot-size river bet wants about 1 bluff for every 2 value bets.
  • Exploit: once you know a player folds too much, bluff them more. Game theory optimal (GTO) play is the unexploitable baseline; profit comes from deviating against mistakes.
  • Variance and tilt: a winning player can lose for weeks. Track results in big blinds per 100 hands (bb/100) over tens of thousands of hands, and never chase losses.
Shorthand
  • Pot-size bet: bluff needs 50% folds; defend 50%; value:bluff = 2:1.
  • Half-pot bet: bluff needs 33% folds; defend 67%; value:bluff = 3:1.
  • Same maths as quoting: an informed counterparty is a player who only bets the nuts.
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