Part 3 · Lesson 10 of 25 · Free

Poker game theory: optimal and exploitative strategy

Course by Hernan Alejandro Gajardo Morales

Most players you meet can be read and beaten. This lesson is about the one who reads you. He has watched 50,000 of your hands; he cannot see your cards, but he knows exactly how often you bluff. Call him the nemesis. What do you play against an opponent who knows your strategy?

The hand

The river completes the flush. The pot is 10 bb and you act first. Half the time you hold a flush, which wins; half the time a missed straight draw, which loses to anything. The nemesis holds top pair, a bluff catcher: it beats your misses and loses to your flushes.

Predict first

You bet the pot, 10 bb. Against the nemesis, which earns more?

1. Never bluff
2. Always bluff
3. They earn the same

Working it out

Bluff with only some of your misses, and he has two answers. Bluff 2 of your 10 misses and he folds everything: (10 + 2) × 10 ÷ 20 = 6 bb a hand. Bluff 5 and something changes: calling and folding earn him exactly the same, so he is indifferent, and you earn 7.5 bb a hand whatever he does. Your bets are then one third bluffs, exactly what his price needs: he calls 10 bb to win 20 bb, so he must be right 10 ÷ 30 = 33.3% of the time.

When neither player can gain by changing his own strategy, the two strategies form an equilibrium: the strategy whose worst case is best. The tool that finds it is indifference, making each of the opponent's choices pay the same. Toy games, small versions of poker, make this exact; the next lessons solve them.

Formula

Equilibrium: each strategy is a best reply to the other. Pot-sized bet: one bluff for every two value bets, so bluffs are bet ÷ (pot before the bet + 2 × bet) = 1 ÷ 3 of the bets

Check your understanding

1. At equilibrium, what share of your half-pot bets should be bluffs?

   1. 25%
   2. 33%
   3. 50%
   Show the answer

   Answer: 25%

   His call costs 0.5 to win 1.5, so it needs 0.5 ÷ (1.5 + 0.5) = 25%: make a quarter of your bets bluffs, one bluff for every three value bets.

2. Against a calling station who calls every bet, how often should you bluff the river?

   1. Never
   2. One third of your bets, as at equilibrium
   3. With every missed hand
   Show the answer

   Answer: Never

   Every bluff gets called and loses. Against a known leak, leave balance behind: value bet only.