Dynamic Programming / Continuation Value
Solve decision problems backwards: act when the current value beats the continuation value.
What is it?
Starting from the last decision, compute the value of playing optimally, and use it as the threshold at the previous step.
Why is it useful?
Optimal stopping, reroll games and option-like choices appear frequently in trading interviews.
When should you think of it?
Use it when you can stop or continue and future choices are also yours.
Think of this method when…
- "You may reroll", "you may stop at any time"
- A finite number of rounds or a clear last step
- Choosing the optimal strategy, not just computing a mean
- Overlapping subproblems in an algorithm
Core idea
Value now , computed from the end.
Step-by-step playbook
- 1Find the value of the final round.
- 2At the previous round, stop when the draw beats that value.
- 3Compute the expected value under that rule.
- 4Repeat backwards to the start.
Worked example
Worked example
Roll a die; you may reroll once and must keep the second roll. What is the expected payoff under optimal play?
- 1
The reroll is worth , so keep 4, 5 or 6 and reroll 1, 2 or 3.
- 2
Average.
- Answer:
Common mistakes
- Solving forward instead of backward.
- Using the unconditional mean where the continuation value is needed.
- Keeping a draw equal to the continuation value — ties are indifferent.
Would you use this method?
You draw a card and get paid its rank. Nothing else can happen. Do you need DP?
Practice this method
17 QuantSolve questions are tagged with this method.
Practice this method