The toll booth

When taking an additional roll incurs a search fee or continuation cost c, the effective continuation value is reduced by c. The stopping hurdle for stage k becomes the net continuation value V_{k+1} - c. Paying a search cost lowers the reservation threshold, making it rational to accept lower current outcomes to avoid paying the continuation fee. On a fair six-sided die with a 0.50 reroll cost, net continuation is 3.50 - 0.50 = 3.00, raising the stopping set to include 3 and giving a net expected payout of 4.00.

Words you need

Threshold: The cut-off where you switch from continuing to stopping.e.g. Keep a die roll of 4 or more, reroll 1 to 3.
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

Rerolling requires paying cost c, so the hurdle threshold equals future expectation minus the cost.

Threshold = V_{next} - c, Net EV = P(stop) * E[stop] + P(reroll) * (V_{next} - c)

Spot it: Each optional reroll or search step incurs an explicit transaction fee.

Technique reference and sources

Every time you choose to reroll, you must drive through a toll booth costing c. If the destination past the toll is worth $3.50, driving through is only worth $3.50 - $0.50 = $3.00 net. You will accept any first roll of $3.00 or better to avoid paying the toll.

When this method fits

Each optional reroll or search step incurs an explicit transaction fee.
You need to price options with exercise or carry costs.
Evaluating search models where acquiring new offers costs time or money.

Terminal gross value. Find the unpenalized expectation of the final roll: V_2 = (S + 1) / 2.
Net continuation hurdle. Subtract cost c from terminal expectation: Hurdle = V_2 - c.
Determine stopping set. Stop if X_1 > Hurdle. If X_1 = Hurdle, stopping and continuing have identical net EV.
Compute net expected game value. Weight stopping outcomes and net continuation value (V_2 - c) by their branch probabilities.

Six-sided die with 0.50 reroll cost: Final roll averages 3.50. Net continuation is 3.50 - 0.50 = 3.00. Hurdle is 3.00. Net game EV is exactly 4.00.

Hurdle: 3.5 - 0.5 = 3.0
Net EV: (2/3) * 4.5 + (1/3) * 3.0 = 4.0

Cost indifference at boundary: When a roll exactly equals the net continuation value, stopping for X or paying c to roll again yields the exact same expected net payout.

On d6 with c = 0.5, roll 3 equals net continuation 3.0
Stopping gives 3; rerolling gives 3.5 - 0.5 = 3

Common traps

Ignoring the reroll cost when deciding whether to keep a first roll. Subtract the reroll cost from future expectation before deciding whether to stop. Failing to subtract cost leads to paying fees for negative-EV rerolls.

Confusing gross expected payout with net expected profit. Gross payout is the face value received; net profit subtracts all fees paid. When roll 1 is free, net EV is gross payout minus P(reroll) * c.

Deducting cost from the first roll when only the reroll incurs a fee. Only deduct costs for the rolls where fees are incurred. If the initial roll is free, no fee is subtracted from stage 1.

Another example: Fair eight-sided die with free first roll and 0.50 reroll cost

Roll 2 has gross expected value V_2 = 4.50
Net value of rerolling is 4.50 - 0.50 = 4.00
Stop on {5, 6, 7, 8} (probability 4 / 8 = 0.5, average 6.50)
Reroll on {1, 2, 3, 4} (probability 0.5, net continuation 4.00)
Net expected value: 0.5 * 6.50 + 0.5 * 4.00 = 3.25 + 2.00 = 5.25

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.