Spin for a uniform payout, pay to re-spin

Question

A wheel pays a payout drawn uniformly from $[0, 100]$ dollars. After any spin you may keep the current payout and stop, or pay $\$ 10 $to spin again, replacing the current value. Re-spins are unlimited and the$ $10$ costs come out of your winnings.

What is your optimal cutoff, and what is the game worth?

Accepted Answer

Let G be the value of holding a spin and playing optimally. Re-spinning is worth G - 10, so holding a payout x is worth \max(x,\ G - 10), and G = E\big[\max(X,\ G - 10)\big], \qquad X \sim 	ext{Uniform}[0, 100]. Let R = G - 10 be the cutoff: keep any x \ge R, re-spin below. For X uniform on [0, M], a short integral gives E[\max(X, R)] = \frac{R^2 + M^2}{2M}, since draws below R (a fraction R/M) each contribute R, and draws above integrate to (M^2 - R^2)/(2M). With M = 100 and R = G - 10: G = \frac{(G-10)^2 + 10000}{200} \implies 200G = (G-10)^2 + 10000 \implies G^2 - 220G + 10100 = 0. The in…

Spin for a uniform payout, pay to re-spin

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