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Optimal Stopping: When to Stop

"Roll a die. You can take the number in dollars, or roll again. What is the game worth, and when do you stop?" is one of the most common questions in trading interviews, and it has many relatives: draw cards until you choose to stop, accept or reject offers as they arrive, quit a game while you are ahead. They all share one structure, and once you see it the answers come quickly.

The one idea

At every point where you may stop, compare two numbers: what you get by stopping now, and the expected value of carrying on if you play the rest of the game well. Take the larger. The value of a state is

V(state)=max⁡(payoff from stopping,  E[V(next state)])V(\text{state}) = \max\big(\text{payoff from stopping},\; E[V(\text{next state})]\big)

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