Game theory for poker players, without the equations

Game theory as a field began with a 1944 book, Theory of Games and Economic Behavior, by the mathematician John von Neumann and the economist Oskar Morgenstern. Six years later John Nash proved the result that gave the subject its central idea. In any game with a finite number of players and choices, there is at least one set of strategies where no player can do better by changing their own strategy alone.

That set is a Nash equilibrium. The equilibrium Nash proved always exists may need players to mix: to choose between actions at random, with set frequencies. In poker the equilibrium strategy is what players call GTO, and it is what a solver computes.

Rock, paper, scissors has one equilibrium: play each throw a third of the time, at random. Any pattern can be spotted and beaten, and a perfect random third of each cannot. Against it, every strategy wins and loses equally often on average.

Against a predictable opponent, you can do better than breaking even. Suppose your opponent throws rock 50% of the time and splits the rest evenly. The equilibrium still only breaks even against them. Always throwing paper wins:

Always throwing paper is the best response to this opponent. It is also easy to exploit in turn, as soon as they notice. Poker works the same way, with more choices and hidden cards.

Consider a simplified river game with enough weak hands available to supply the required bluffs. On the river, the pot is 100 chips. The bettor holds either the best hand or a hand that cannot win, a polarised range. The caller holds a bluff-catcher: it beats every bluff and loses to every value hand. The bettor bets 100.

The caller should be indifferent between calling and folding, or the bettor could do better. Calling risks 100 to win the pot plus the bet, 200. The bettor's range has to hold just enough bluffs that calling breaks even:

So one hand in three is a bluff: one bluff for every two value hands. The bettor's bluffs must also break even, which sets how often the caller has to call. That is the minimum defense frequency: pot ÷ (pot + bet) = 100 ÷ 200 = 50%.

The lesson a ratio and a share describe the same bets works through the same math with the cards on the table.

Real poker is far bigger than the toy game, but the same rule holds: when a strategy mixes, the actions it mixes between are worth the same. Here is a hand from the trainer's solved packs: the button holds the queen and ten of hearts on a flop of K♠ 8♦ 3♥, and the big blind has checked.

The three actions are within 0.06 big blinds of each other. In an exact equilibrium they would tie; the small differences here reflect a numerical solve stopped at a tolerance. An action that remained strictly worse against the equilibrium opponent would not belong in the mix. The mix is what keeps the button's checks from being all weak hands and its bets from being all strong ones, so the big blind cannot read the hand from the action. The lesson a mix means the actions tie covers more examples.

In heads-up zero-sum play before rake, an equilibrium strategy protects the game's equilibrium value against any opponent. It does not necessarily earn the most against an opponent who makes mistakes. Go back to the river game and suppose the caller only calls 30% of the time against a pot-sized bet, well short of the 50% the equilibrium needs.

Against this player the best response is to bluff with every hand that cannot win. Against a player who calls too much, it is to stop bluffing and bet only for value. Both moves give up balance for profit, and both can be punished if the opponent adjusts. Equilibrium play tells you where the balanced line is, so you know what you are giving up when you leave it.