Prophet vs Gambler

You have one asset to sell over a few days. Each day an offer arrives, and looking at it costs you. Sell now, or hope for better tomorrow?

The prophet sees all the offers in advance and takes the best one. The gambler sees them one at a time and must decide on the spot, with no going back. Selling on day k pays the offer minus k·c, since each day of waiting costs c. Prophet inequalities bound the gap between them — how much foresight is worth. Everything below is computed exactly, not estimated, except where it says otherwise.

The optimal rule

Working backwards from the last day gives a threshold for each day: sell if today's offer clears the bar, otherwise wait. The bar drops as days run out — and every day of waiting has already cost c.

sell region possible offers, sized by probability

Sampled runs

Playing that rule against sampled offers. The average gap over many runs converges to the exact number above — this is the one place on the page showing estimates rather than exact values.

prophet payoff gambler payoff

Gold marks the day the prophet took; blue marks the day the gambler sold.

How the gap moves with cost

The same offers, swept across every cost from free to prohibitive. Costly observation cuts both players down, but not equally.