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  1. 045Simplified poker with three cards, A, K and Q: each player antes 1, you are dealt one card and I am dealt another. You may bet 1 or check; if you bet, I call or fold, and if you check the higher card wins the antes. How often should you bluff with the Q, and how often should I call with the K, in equilibrium?Games and strategic reasoningHardOld Mission CapitalNew York · 2022

    Try it first

    How often should I call a bet when I hold the K?

    Show the worked solution

    Bluff with the Q one time in three, and call with the K one time in three. You always bet the A and always check the K; I always call with the A and fold the Q. A Q bluff risks 1 more to win the 2 antes, so I must defend two thirds of hands facing it: the A gives half, the K calling one time in three gives the rest. The game is worth 1/18 per hand to you.

    Which hands are easy, and where is the real decision?

    Clear away the obvious hands first. With the A you always bet, because you win whether I call or fold. With the K, betting only gets called by the A and folds out the Q, which you beat anyway, so you check. As the caller, I always call with the A and always fold the Q. The whole game comes down to two mixed choices: how often you bluff with the Q, and how often I call with the K.

    Equilibrium: bluff the Q one time in three, call with the K one time in threeYour cardeach 1/3AKQBet, alwaysvalue betCheck, alwaysshowdown for 1Bluff 1/3worth -1Check 2/3worth -1Caller facing a betA: call alwaysK: call 1/3, fold 2/3Q: fold alwaysDefends 1/2 + 1/2 x 1/3 = 2/3 vs a bluffWhy 1/3 eachQ bluff: 1/2(-2) + 1/2(c(-2) + (1-c)(+1))equals checking, -1, when c = 1/3K call: (-2 + 2b) / (1 + b)equals folding, -1, when b = 1/3Game value to the bettor: +1/18 per hand
    In equilibrium the bettor always bets the A, always checks the K and bluffs the Q one time in three, while the caller always calls the A, folds the Q and calls with the K one time in three; together the A and the K calls defend two thirds of hands facing a bluff.

    How do the indifference conditions fix both frequencies?

    A teacher who spot-checks homework faces the same logic: check every paper and time is wasted, never check and everyone copies, check at the right rate and copying stops paying. In equilibrium each player mixes at the rate that makes the other indifferent between their two options. Your Q loses 1 by checking. Bluffing loses 2 against my A, and against my K loses 2 if I call and wins 1 if I fold. The two are equal only when I call with the K one time in three. My K loses 1 by folding; calling loses 2 against your A and wins 2 against a bluff, which is worth -1 only when you bluff one time in three.

    The relationship
    12(−2)+12(c(−2)+(1−c)(1))⏟Q bluffs=−1⇒c=13−2+2b1+b⏟K calls=−1⇒b=13\underbrace{\tfrac12(-2) + \tfrac12\big(c(-2) + (1-c)(1)\big)}_{\text{Q bluffs}} = -1 \Rightarrow c = \tfrac13 \qquad \underbrace{\frac{-2 + 2b}{1 + b}}_{\text{K calls}} = -1 \Rightarrow b = \tfrac13
    chow often the caller calls with the K
    bhow often the bettor bluffs with the Q
    -1the payoff of the alternative: checking the Q or folding the K, losing the ante
    What it says in wordsEach frequency is set so the opponent's two choices are worth the same.

    Check against the pot-odds rule. A bluff risks 1 extra to win the 2 antes, so the caller must defend 2/(2 + 1) = 2/3 of the hands facing it; the A already covers half, and the K calling one time in three covers the other sixth. That two thirds is the number people misremember as the K's calling rate. Put the strategies together and the bettor, who acts first with more information about their own hand, earns 1/18 of a chip per hand. The limitation: with a bigger bet or more cards the ratios change, but the method, indifference on both sides, carries over.

    Where candidates lose it

    The common loss is setting the K's calling rate to two thirds. Two thirds is the total defence the caller needs against a bluff, and the A already provides half of it, so the K calls only one time in three.

    The second is never bluffing because the Q cannot win a showdown. A player who never bluffs lets the caller fold every K to a bet, and the A's bets stop earning. The bluff is what gets the A paid.

    What the interviewer asks next

    • What is the value of the game to each player?
    • The bet size doubles to 2. How do the bluffing and calling frequencies change?
    • Now the caller may also bet after a check. What changes?

    Asked at Old Mission Capital, Quantitative Research, New York, 2022 (Wall Street Oasis): Asking to find the game theory optimal strategy in a simplified poker game