The reservation hurdle

In sequential decision problems, earlier choices depend directly on the value of future choices. Backward induction determines optimal play by evaluating the final unconstrained stage first. The expected value of that terminal stage serves as a strict reservation hurdle: stop if the current offer exceeds future expectation, and reroll otherwise. For a fair six-sided die with one optional reroll, the second roll averages 3.5, making it optimal to keep only 4, 5, or 6, yielding a fair game value of 4.25.

Words you need

Backward induction: Solve the last decision first, then use its value to solve the decision before it.
Continuation value: What carrying on is worth on average. Stop when what you have beats it.
Expected value (EV): The average result if you could repeat the bet many times: each outcome times its probability, added up.e.g. A die pays its face in dollars: EV = 3.5.

Why it works

The terminal stage has an unconstrained expected value that serves as the reservation price for the prior decision.

EV = P(X > V_terminal) * E[X | X > V_terminal] + P(X <= V_terminal) * V_terminal

Spot it: You have a single optional reroll on a die or random payout.

Technique reference and sources

Think of the final stage expected value as a cash offer already sitting in your back pocket. On the first roll, any outcome above that pocket cash is worth keeping and any outcome below it should be rejected. At exact equality both choices pay the same on average, so keep above, continue below, and break ties by convention (for even dice the hurdle is a half-integer, so ties never occur).

When this method fits

You have a single optional reroll on a die or random payout.
You need to price an option to reject a first offer in favor of a redraw.
An interviewer asks for the fair market price of a die roll with one reroll.
You want to quantify the value of optionality in sequential decisions.

Find terminal expected value. Calculate the average of all possible outcomes on the final roll: V_2 = (1 + S) / 2.
Establish stopping hurdle. Stop on roll 1 if X_1 > V_2; reroll if X_1 <= V_2.
Calculate stopping probability and conditional mean. Count faces above the hurdle and find their arithmetic mean.
Weight by probabilities. Compute V_1 = P(stop) * E[stop] + P(reroll) * V_2.

Even die sides with two rolls: For any fair die with an even number of sides S, the terminal expectation is (S + 1) / 2. You keep the upper S / 2 faces with average (3S + 2) / 4, yielding game value V_1 = (5S + 4) / 8.

d6: (5 * 6 + 4) / 8 = 34 / 8 = 4.25
d4: (5 * 4 + 4) / 8 = 24 / 8 = 3
d8: (5 * 8 + 4) / 8 = 44 / 8 = 5.5

Optionality value: The option to reroll an even die once adds exactly S / 8 to the baseline single-roll expected value (S + 1) / 2.

d6 adds 6 / 8 = 0.75: 3.5 + 0.75 = 4.25
d8 adds 8 / 8 = 1.0: 4.5 + 1.0 = 5.5
d20 adds 20 / 8 = 2.5: 10.5 + 2.5 = 13

Common traps

Keeping a 3 on the first roll of a d6 because 3 is close to average. Roll 2 averages 3.5, which is strictly higher than 3. Keeping a 3 surrenders 0.5 in expected value compared to rerolling.

Averaging the stopping values with all first-roll values rather than weighting by their respective probabilities. Always weight conditional payoffs by branch probabilities: EV = P(stop) * E[stop] + P(reroll) * V_2.

Assuming you can return to the first roll if the second roll is worse. The problem specifies that the second roll is final and supersedes the first roll. Once you reroll, the first roll is gone.

Another example: Fair eight-sided die with one optional reroll

Roll 2 has expected value (1 + 8) / 2 = 4.5
On roll 1, stop if face > 4.5: keep {5, 6, 7, 8} with average (5 + 8) / 2 = 6.5
Reroll {1, 2, 3, 4} with probability 4 / 8 = 0.5 and continuation value 4.5
Expected game value: 0.5 * 6.5 + 0.5 * 4.5 = 3.25 + 2.25 = 5.5

Source notes

These explanations and questions are written for QuantArc. Links show the underlying mathematics and learning design.

Jane Street: Probability & Markets Trading Interview Guide
Interview preparation guide (page 9) testing optimal stopping and market making on a d6 with one optional reroll. Lesson examples and questions are original.

Brainstellar: Probability interview puzzles
Free step-by-step walkthroughs of classic probability interview puzzles by difficulty; companion practice. Lesson examples and questions are original.

IES: Organizing Instruction and Study to Improve Student Learning
Supports worked examples, retrieval, explanations and spaced study. Does not establish QuantArc's exact lesson length, mastery threshold or review intervals.