Alice's Coin Flip Edge

This quant interview question is built around asymmetric information and conditional probability in a simple-looking betting game. One participant must pre-commit to a probabilistic choice and reveal it, while the other reacts optimally after observing it. The setup forces candidates to think in terms of expected value under strategic interaction, not just standalone probability calculations, which is typical of modern trading games and market microstructure situations in quant prep.

It trains facility with conditional expectation, linearity of expectation, and reasoning about worst-case responses in a zero-sum setting. You must translate a verbal game into payoff expressions, identify how one player's parameter affects the other's expectation, and reason about ranges of parameters that neutralize an opponent's edge. This blend of probability, algebra, and game-theoretic thinking is core to strong quant prep.

This matters for quant interviews because many trading, market making, and structuring roles expect you to defend yourself against an adversarial counterparty. Interviewers want to see whether you can design robust strategies that are profitable or at least break-even against any rational response. Being able to quickly evaluate such conditional probability games shows you can handle real-time decision making under uncertainty, a central requirement in high-stakes quant interviews.

What it tests

This problem class is governed by the principle of adversarial optimization in two-player games with sequential moves and adjustable probabilities. When one player chooses a probability first and the other responds, the second player can always pick an extremal value (here, 0 or 1) to maximize their own expected value, unless the first player chooses their parameter so that the expected value is non-positive for all possible responses. The core structure is to express the expected value as a linear (or affine) function in the second player's parameter, then ensure that its maximum over the allowed range is non-positive. This is achieved by making both the value at one endpoint and the value at the other endpoint (or the maximum, if the slope allows) less than or equal to zero. The principle holds because, in zero-sum or adversarial expectation settings, the optimal defense is to minimize the opponent's best possible outcome, which always occurs at a boundary when the function is linear in their choice.