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  1. 071We play a matching game: each of us shows heads or tails at the same moment. If both show heads I pay you 3; if both show tails I pay you 1; if we mismatch you pay me 2. What mix should I use to make you indifferent, what mix should you use, and what is the game worth to you per round?Games and strategic reasoningCoreQuant tradingQuant research

    Try it first

    What is the game worth to you per round?

    Show the worked solution

    I show heads 3/8 of the time; so should you; and the game is worth -1/8 to you per round. At my mix q, your heads pays 3q - 2(1 - q) and your tails pays -2q + (1 - q). Setting them equal gives q = 3/8, where both pay -1/8. By the same algebra your 3/8 mix makes my choices equal, so neither of us can improve: the table looks fair and is not.

    Why do I randomise to make you indifferent, not to help myself?

    Think of a penalty taker and a goalkeeper. If the taker shoots left more often than he should, the keeper dives left and gains; any pattern is exploited. In a zero-sum game, the mix that protects you is the one that leaves your opponent with nothing to exploit, which means making every one of their choices pay the same. So I do not pick my heads probability by looking at my own payoffs in isolation; I pick it so that your heads and your tails earn you equal amounts. Then it no longer matters what you do.

    My mix of 3/8 heads makes your two choices pay the same: -1/8Payoff to youI showHTH+3-2T-2+1You showWins 3 + 1 = 4, losses 2 + 2 = 4:looks fair, is not-2-1+1+2+30you show Hyou show Tcross at q = 3/8,both pay -1/800.250.50.751q = my probability of showing heads
    Your payoff from heads rises and your payoff from tails falls as my heads probability grows; they cross at 3/8, where each pays -1/8, so that is my equalising mix and the most you can secure per round.

    How do the two equations give 3/8 and -1/8?

    Let q be my chance of heads. Your heads earns 3q - 2(1 - q) = 5q - 2 and your tails earns -2q + 1(1 - q) = 1 - 3q, and they are equal when 8q = 3, at q = 3/8. Plug back in: 5(3/8) - 2 = -1/8. For your side, let p be your chance of heads; my payoffs are the negatives of yours, and because the table is symmetric the same algebra gives p = 3/8. At those mixes, each cell's frequency is HH 9/64, TT 25/64 and the two mismatches 15/64 each: 3(9) + 1(25) - 2(30) = -8, over 64, is -1/8.

    The relationship
    5q−2=1−3q  ⇒  q∗=38,V=5⋅38−2=−185q - 2 = 1 - 3q \;\Rightarrow\; q^* = \tfrac38,\qquad V = 5\cdot\tfrac38 - 2 = -\tfrac18
    qmy probability of showing heads
    5q - 2your expected payoff from showing heads
    1 - 3qyour expected payoff from showing tails
    Vthe value of the game to you per round
    What it says in wordsMy heads probability is set where your two choices earn the same, and that common amount is what the game is worth to you.

    Why does a table with equal wins and losses favour me?

    Because the cells are not played equally often. Your big win needs both of us on heads, and I can make that cell rare; the mismatches, where I win, happen 30 times in 64 at equilibrium. If I played heads half the time you could earn +1/2 by always showing heads, so a naive 50/50 from me would be a gift. In trading the same idea sets a market maker's quotes: they are placed so that the informed side has no choice that beats the others, not so that the market maker profits on any one trade.

    Where candidates lose it

    The trap is adding up the table, 3 + 1 against 2 + 2, and calling the game fair. Equilibrium frequencies are not a quarter each, so a table's totals tell you nothing about its value.

    The second slip is solving for the mix that maximises your own payoff against a fixed opponent. In a mixed equilibrium each player's mix is pinned down by the other player's payoffs; say that sentence before the algebra.

    What the interviewer asks next

    • If I play heads half the time, what should you do and what do you earn?
    • What payoff for both tails would make the game fair?
    • How does the answer change if the mismatch payment is 3 instead of 2?