Fair Coin Flip First Move Price

This quant interview question is about pricing the advantage of moving first in a symmetric, probabilistic game with cash payoffs. It sits at the intersection of expected value, game valuation, and basic stochastic processes, making it a classic for quant prep and finance interviews.

It trains your ability to formalize a seemingly open-ended game into a clean expectation framework, recognize when the situation effectively "restarts," and capture that using a self-consistent value. It also checks your comfort with random cash flows, risk-neutral valuation, and interpreting probabilistic payoffs as a single number.

This matters for quant interviews because many trading, market making, and derivatives problems have the same structure: repeating opportunities, embedded optionality, and path-independent states. Interviewers want to see you translate an informal description into a precise quantitative object and value it quickly.

What it tests

This problem class is governed by the principle of recursive expectation, where the value of a decision point is expressed in terms of the possible immediate outcomes and the expected value of the game if it resets or continues. The core insight is that when a process can return to its original state with some probability, the expected value at that state can be solved as an unknown in a self-referential equation. This is possible because the Markov property holds: the future evolution of the game depends only on the current state, not on the history. The recursive structure allows us to break down complex, potentially infinite games into a finite, solvable equation. This pattern holds because each round either ends the game, produces a payoff, or resets the situation, so the expected value at each reset must be consistent with itself.