Katkat
Back

Alternating Die-Roll Game: Optimal Accept/Reject Strategy

Jane StreetQuantitative Trader · Technical Screen
HardQuantitative TraderTechnical ScreenSoftware Engineering Fundamentals

Last updated

This question evaluates probabilistic reasoning, dynamic decision-making and zero-sum game analysis, testing competency in expected-value calculations, optimal stopping and symmetric strategic reasoning.

Two players, A and B, play the following game with a fair 21-sided die whose faces are labeled with the integers $-10, -9, \ldots, 9, 10$ (one integer per face, each face equally likely).

Player A moves first. On a player's turn, that player rolls the die, observes the face value $v$, and then chooses one of two actions:

  • Accept. The game ends immediately, and the opposing player pays the roller $v$. (If $v$ is negative, this means the roller pays the opponent $|v|$.)
  • Reject. Nothing is paid, and the turn passes to the other player, who now rolls and faces exactly the same accept/reject choice.

Turns alternate this way indefinitely until some player accepts.

What is the optimal strategy for each player, and what is the expected payoff to A when both players play optimally?

Constraints & Assumptions

  • The die is fair: each of the 21 integer values $-10$ through $10$ has probability $1/21$, and successive rolls are independent.
  • The game is zero-sum: whatever the accepting player receives, the other player pays.
  • There is no bound on the number of turns, no discounting, and no cost to reject.
  • Both players are risk-neutral expected-value maximizers, and both play optimally (each knows the other is rational).

Clarifying Questions to Ask

  • Is every face equally likely, and are successive rolls independent of earlier ones?
  • When B accepts, does A pay B the face value — i.e., is the game fully symmetric under a role swap?
  • Is there any limit on the number of rounds, or any per-round cost or time discounting?
  • Can a player accept a negative value, and does that mean the roller pays out of pocket?
  • Are we maximizing expected value (risk-neutral), or should risk aversion factor into the strategy?

Hints

Hint 1

Where to start: After any rejection, the roles simply swap and the remaining game looks **identical** to the original one. Look for a *stationary* strategy — and ask whether A's and B's decision rules should really differ at all.

Hint 2

Value recursion: Let $V$ be the expected payoff, under optimal play, to whichever player is *about to roll*. Rejecting hands your opponent a position worth $V$ to them (so worth $-V$ to you). Write $V$ as an expectation over the roll of the better of accepting and rejecting, and solve the resulting fixed-point equation.

Hint 3

Common pitfall: "Accept anything $\ge 0$" feels natural but is **not** optimal. Compare taking a small guaranteed loss now against the expected cost of handing your opponent the move.

Follow-up questions

  • How do the strategy and the game value change if the roll is continuous, uniform on $[-10, 10]$?
  • Suppose each rejection costs the rolling player a fee $c > 0$. How does that shift the thresholds and the value?
  • What happens if every face is positive — say the faces are $1$ through $20$? What does that reveal about where the accept/reject tension actually comes from?
  • How would you verify your closed-form answer numerically?