St Petersburg Coin Toss Payoff

This question is built around the classic St Petersburg-style coin-flipping game, where the payoff grows explosively with the waiting time until a particular outcome occurs. The candidate is asked to reason about the "fair value" of a single play and then to translate that theoretical value into a realistic quote under extreme time pressure. It connects an idealized infinite-wealth casino game to the practical constraints of quoting a one-off trade to a demanding client, highlighting the mismatch between mathematical expectation and economic intuition.

On the technical side, the problem leans on geometric distributions, infinite series for expected value, and the behaviour of payoff tails that grow too fast relative to probability decay. Strong answers invoke ideas from risk management and utility, such as risk aversion, capital constraints, and position limits, to justify a finite quote despite an unbounded theoretical expectation. Interviewers watch for quick, structured thinking under time pressure, an ability to separate mathematical value from tradable price, and clear communication about assumptions in an unrealistic but revealing toy model.

What it tests

This problem class is governed by the interplay between geometric probability distributions and exponentially growing payouts. When the probability of an event decreases exponentially with $k$ (like $2^{-k}$ for the first head on the $k$-th toss), but the payout increases at the same exponential rate ($2^k$), their product becomes constant for each $k$. Summing a constant over an infinite sequence leads to divergence, so the expected value is infinite. The underlying structure is that, if the tail of the payout distribution grows as fast or faster than the decay of the probability, the expected value can diverge, regardless of how unlikely large payouts are. This counterintuitive result shows that expectation alone does not always capture practical value, especially when rare, extreme outcomes dominate the sum.