Derivatives Foundation puzzles, solved step by step

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All topicsMental maths and estimation9Random walks and Markov chains7Conditional probability and Bayes7Volatility and correlation7Option pricing intuition7Expected value and optimal stopping10Market making11Option payoffs and no-arbitrage10Probability and counting11Distributions and statistics8Games and logic8Betting and sizing5
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  1. 061You and I each show heads or tails at the same time. You win Rs 3 if we both show heads, Rs 1 if we both show tails, and you lose Rs 2 if we show different faces. What mix should you play, and is the game worth playing?Games and logicCoreQuant tradingProp trading firms

    Try it first

    Two wins and two losses in the table. Is this game good for you?

    Show the worked solution

    Show heads 3/8 of the time, and do not play unless you are paid at least Rs 0.125 a round. If you show heads with probability p, your expected payoff is 5p - 2 when I show heads and 1 - 3p when I show tails. I will pick whichever is lower, so you choose p to make the lower line as high as possible, which is where they cross: p = 3/8, value - 1/8. Any other p lets me push you below that.

    Why is the answer a mix rather than a single face?

    Two children playing odds and evens learn fast that any pattern is punished: show heads every time and the other child shows tails every time. In a game where my best reply depends on what you do, any fixed choice is exploited, so you protect yourself by randomising in a ratio that leaves me with nothing to exploit. That ratio is found by making me indifferent between my two replies. If you show heads a fraction p of the time, my heads earns you 3p - 2(1 - p) = 5p - 2 and my tails earns you - 2p + (1 - p) = 1 - 3p. They are equal at p = 3/8.

    Whatever you do, I pick the lower line; you pick the point where the lower line peaks0-2-1+1+2+300.250.50.751p, how often you show headsyour expected payoff per round, RsI show heads: 5p - 2I show tails: 1 - 3pp = 3/8crossing: value = - 1/8 = - Rs 0.125the lower line: what I can hold you toYour payoff, RsI: HI: TYou: HYou: T+3-2-2+1Both of us mix 3/8 heads.You cannot do better thanminus 1/8 a round if I playwell, and any other mix letsme push you lower.Do not play without a fee of Rs 0.125.
    Your expected payoff is 5p - 2 if I show heads and 1 - 3p if I show tails, and since I will always pick the lower line, the best you can do is the crossing at p = 3/8, where both lines give minus 1/8, so the game is worth minus Rs 0.125 to you per round.

    How do you know minus 1/8 is the most you can guarantee?

    Look at the lower of the two lines across all p. To the left of 3/8 the heads line is lower and rising; to the right the tails line is lower and falling, so the lower envelope peaks exactly at the crossing, and that peak is your guaranteed value. I have the same calculation from my side: if I show heads a fraction q of the time, you are indifferent when 3q - 2(1 - q) = - 2q + (1 - q), which again gives q = 3/8, and at that mix I hold you to - 1/8 whatever you do. Both sides landing on the same number is the minimax theorem, attributed to von Neumann, at work in a two by two table.

    The relationship
    5p−2=1−3p  ⇒  p=38,V=5⋅38−2=−185p - 2 = 1 - 3p \;\Rightarrow\; p = \tfrac{3}{8}, \qquad V = 5\cdot\tfrac{3}{8} - 2 = -\tfrac{1}{8}
    pyour probability of showing heads
    5p - 2your expected payoff when I show heads
    1 - 3pyour expected payoff when I show tails
    Vthe value of the game to you per round
    What it says in wordsEqualising your payoff across my two replies gives a three-eighths mix and a value of minus one eighth of a rupee per round.

    What is the desk version of this question?

    Quoting against a counterparty who sees your pattern. A market maker who always leans the same way after a fill is the child who always shows heads, and the counterparty who notices earns the difference, so randomised sizing and skew are the trading-floor form of the 3/8 mix. The limitation of the puzzle answer is that it assumes I play optimally; against an opponent who shows heads half the time out of habit, your best reply is pure heads, with an expected 0.5 x 3 - 0.5 x 2 = + Rs 0.50 a round, and the game becomes worth playing. Ask who you are playing before you quote the value.

    Where candidates lose it

    The common answer is that the game is fair or favourable, from summing the four cells. The sum of a payoff table says nothing when the opponent chooses the column. Set up the two lines and find where they cross.

    The second loss is solving for the right p and then saying the game is fine because 3 and 1 are bigger than 2. State the value, minus 1/8, and say you need a fee of at least that to play.

    What the interviewer asks next

    • What is my optimal mix, and what does it earn me?
    • Change the heads-heads payoff to Rs 4. Does the game become worth playing?
    • I am known to show heads 60% of the time regardless. What should you do now?
    • Why do both players end up with the same 3/8 here, and is that a coincidence?