The fixed-jackpot slot, press your luck

Question

A game pays a fixed \3 each round with probability 2/3; with probability 1/3 it busts, wiping your accumulated pot and ending the game. After each win you may stop and bank the pot instead of pressing on. What is your stopping rule, and what is the game worth?

Accepted Answer

One-step reasoning. With pot s, playing one more round: - With probability 1/3 you bust and lose s (change = -s). - With probability 2/3 you win the fixed \3 (change = +3). Expected change: E[\Delta] = \frac{2}{3}(3) + \frac{1}{3}(-s) = \frac{6 - s}{3}. Positive when s < 6, so keep playing while your pot is below \6; stop at \6 or more. Since the prize is a constant \3, the pot climbs 0 	o 3 	o 6, and you stop right when you first reach \6 (two wins). The general rule. For any fixed prize g and bust probability p, one more round has E[\Delta] = (1-p)\,g - p\,s, which flips sign at s^{} = \f…

The fixed-jackpot slot, press your luck

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