What is a second try worth? Price it backwards
A die pays its face; you may reroll. The option to try again is worth money, and the way to price it is from the last decision backwards: keep what beats the continuation, throw back what does not.
- Value a die with no reroll, then with one, by the keep-or-reroll rule
- Chain the rule for two and three rerolls
- Explain why each extra reroll is worth less than the last
- Recover the size of the die from the value of the one-reroll game
The intuition
A fair six-sided die pays its face in dollars: worth $3.50 with no option. Now you may reroll once. See a 4, 5 or 6 and you keep it, because it beats the $3.50 a fresh roll is worth; see a 1, 2 or 3 and you throw it back. Half the time you keep an average of $5, half the time you take $3.50: the game is worth $4.25, and the reroll option itself is worth $0.75.
A second reroll is priced the same way against the one-reroll value: now you should only keep a 5 or a 6, because throwing the die back is worth $4.25. That gives $4.67, and the second reroll added only $0.42. The bar you reroll against keeps rising, so each option is worth less than the last. This is exactly how an American option is priced: at every point, compare what you hold with what waiting is worth.
V₀ = (s + 1) ÷ 2. Vᵢ = (1 ÷ s) × Σ max(k, Vᵢ₋₁) over the faces k: keep a face that beats the continuation value, otherwise take the continuation. With one reroll left, keep any face above V₀. The i-th reroll adds Vᵢ − Vᵢ₋₁, and that falls with i.
Why it works
- The conventions here: a fair die with s faces pays its face in dollars. Each reroll replaces the previous roll, the final roll must be accepted, no fee, no discounting. Dice have an even number of faces, so V₀ is never a whole number and the keep rule never ties.
- Backward induction. The last decision is the easiest: with no rerolls left the die is worth its average face. Each earlier decision compares the face in hand with the value of what comes next, and the value of the stage is the average of those decisions.
- The keep rule is a threshold. With i rerolls left, keep the smallest face above Vᵢ₋₁ and everything higher. Because Vᵢ rises with i, the threshold rises too: more options in hand means demanding more before you stop.
- Diminishing gains. The first reroll rescues the whole bottom half of the faces; the second only improves on cases where the first already did well. With unlimited rerolls the value approaches the top face, but a handful of rerolls captures most of it.
- A fee does not change the rule. Once paid it is sunk. It changes whether the game was worth entering, not which faces to keep.
- Run it backwards. For an even s the one-reroll value has a closed form in s: the top half of the faces averaged, plus half of V₀. Try even candidates until it matches.
| No reroll: (1 + 6) ÷ 2 | $3.50 |
| One reroll: keep 4–6; (4 + 5 + 6) ÷ 6 + 3 ÷ 6 × $3.50 | $4.25 |
| Two rerolls: keep 5–6; (5 + 6) ÷ 6 + 4 ÷ 6 × $4.25 | $4.6667 |
| Three rerolls: keep 5–6; 11 ÷ 6 + 4 ÷ 6 × $4.6667 | $4.9444 |
| Value added by each reroll | $0.75, then $0.4167, then $0.2778 |
| Lowest face kept with one reroll left; with two | 4; 5 |
A twenty-sided die is worth $10.50 with no reroll and $13.00 with one: keep 11–20, (11 + … + 20) ÷ 20 = $7.75, plus 10 ÷ 20 × $10.50 = $5.25.
The formulas
No rerolls: the average face.
Keep k if it beats the continuation; otherwise take the continuation.
The threshold is the smallest face above the continuation value.
Each option is priced against a higher bar.
Worked example
Value the die with no option, split the faces into kept and rerolled, and average. The follow-up prices the option itself.
See it move
Same game. Change the number of faces and the number of rerolls allowed.
- Add faces. Every value rises, both keep thresholds rise, and each reroll's added value rises with them.
- Allow more rerolls. The game's value rises, and each extra reroll adds less than the one before.
- Compare the lowest faces kept. With two rerolls in hand you demand a higher face than with one, because throwing the die back is worth more.
Run it backwards
The one-reroll game is worth a stated amount. How many faces does the die have?
For an even s the one-reroll value is the average of the top half of the faces plus half of V₀, a formula in s alone. Try even candidates: one size smaller is too low, one size larger too high.
It is the same move as every inverse in this course: write the forward formula, then search or solve for the missing input.
Traps
Say it in the interview
“A die pays its face and you can reroll once. What is it worth?”
Check yourself
4 fresh questions, with new numbers. Answer each one correctly to finish the chapter. Get one wrong and you will see the full working, then you can try it again with new numbers.
Answers within 1% are marked right. Type the number; $, %, x and M are fine. First tries count toward Learned: the topic is Learned once every chapter is done and 75% of first tries were right.
- No option: the average face. One reroll: keep what beats it, average the outcomes.
- Each further reroll is priced against the value of the game with one fewer.
- The keep threshold rises with the rerolls in hand; each reroll adds less than the last.
- A fee is sunk. It never changes the keep rule.