Week 7: Mixed strategies and the algebra of bluffing

Deriving optimal bluffing frequency and minimum defense frequency from indifference conditions, in the simplest toy game that has both

5 April 2027

Why a strategy needs randomness at all

Every decision so far this course has had a single correct answer: call, fold, whatever the numbers said. Mixed strategies show up when always doing the numerically-best thing becomes exploitable — if villain could predict “hero only ever bluffs with nothing,” villain just calls everything. Game theory’s answer is to randomize, and the question this week formalizes is: randomize how much?

The smallest toy game with a real bluff

Simplify a single betting round to its bones: villain either has a value hand (always wins) or is bluffing (always loses) — no in-between. Hero holds a bluffcatcher: loses to the value hand, beats the bluff. Villain bets; hero calls or folds. There is no more betting after this.

Let pot be the amount already in the middle before villain’s bet (unlike week 3, where “pot” included the bet already — here the bet needs to be separated out to track each player’s own contribution). Let bet be villain’s bet size, which hero must match to call.

Hero’s side: when is calling worth it?

If x is the fraction of villain’s betting range that’s a bluff, hero’s equity when calling is exactly x (call only beats a bluff). Using the same two-outcome EV formula from weeks 3–4:

EV(call) = x · (pot + bet) − (1 − x) · bet

Setting this to zero — the fraction of bluffs that makes hero exactly indifferent between calling and folding:

x* = bet / (pot + 2 · bet)

This is week 3’s required-equity formula, just read from the other player’s seat: hero risks bet to win a pot that villain’s own bet has already grown to pot + bet. There, it told hero the equity needed to call. Here it tells villain the bluff frequency that keeps hero from being able to profitably exploit either “always call” or “always fold.”

Villain’s side: how often must hero defend?

Now flip it. Suppose hero calls with probability c. Is villain’s bluff profitable? Relative to checking (worth $0, since a bluff loses at showdown either way):

EV(bet as bluff) = (1 − c) · pot − c · bet

Setting this to zero — the calling frequency that makes villain exactly indifferent between bluffing and giving up:

c* = pot / (pot + bet)

This is the minimum defense frequency (MDF): the least often hero must call to stop villain’s bluff from being automatically profitable.

Plugging in week 3’s own numbers

Pot = $100, bet = $50 — a half-pot bet, in the same dollar amounts week 3 used, so the arithmetic stays familiar:

These are not two views of one number, and it is worth being careful about which pairs with which. The quantity that completes MDF is hero’s own folding frequency, 1 − c* ≈ 33.3%. Villain’s bluffing frequency is the separate 25% — one bluff for every three value hands, the standard ratio for a bet of half the pot.

Kuhn poker, named but not fully derived

This two-hand-type toy game is a simplification of Kuhn poker — a three-card game (Kuhn, 1950) usually credited as the smallest poker variant ever completely solved. Kuhn poker’s real equilibrium lets a player hold a middle hand too, which must sometimes call, sometimes fold, sometimes even bet — a richer system of indifference equations than the one just solved here. Deriving it fully is beyond this week’s scope; naming it correctly, and knowing it’s the next rung up from what was just built by hand, is the goal for now. Week 1 catalogued folk claims to be cashed in later; this is the same move for a method, not a claim — the capstone in week 12 is where Kuhn poker’s real equilibrium finally gets solved, checked against the published result, and reused on a game bigger than anything solved by hand in this course.

Before next week

Both numbers above came from a single simplifying assumption: exactly two mutually exclusive hand strengths, and one opponent. Week 8 asks what “optimal” even means once a strategy has to hold up against any opponent strategy, not just the polarized one assumed here — the actual definition of a GTO strategy.

Teaching team