Jane Street·Data Scientist·Technical Phone Screen·Intermediate

Intermediate

Jun 2026

Summary

Jane Street Data Scientist interview with a game theory problem involving a 21-sided die. The core question was deceptively clean but the optimal strategy took some real work to pin down, especially resisting the urge to just accept anything nonnegative.

Questions Asked (1)

Two players take turns rolling a fair 21-sided die (faces labeled -10 through 10). On each turn, the roller can accept the face value as a payment from the opponent, or reject and pass the die. Both play optimally. What is the optimal strategy for each player, and what is the expected payoff to the first player?

Algorithms & Data StructuresTechnical Trade-offs

Author's notes

My first instinct was 'accept anything >= 0' and I was pretty confident about it for an embarrassingly long time.

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AI HintsAI Generated

Suggested Approach

Model the game as a Markov decision process with a value function representing the expected payoff to the current roller. Use dynamic programming to find the optimal stopping threshold: the roller accepts if the roll exceeds the continuation value, otherwise passes. Then solve for the fixed point where the value function equals the expected payoff under this threshold strategy.

Pro tip: Frame the problem as an optimal stopping problem and emphasize that the threshold is determined by the value of passing, which itself depends on the threshold. This shows you understand the recursive nature and can solve it analytically or numerically.

1. Define the value function

Let V be the expected payoff to the current roller when both play optimally. By symmetry, the game is zero-sum and the value is the same for both players when it's their turn.

2. Set up the optimal stopping condition

The roller will accept a roll x if x >= V (since passing gives expected payoff V). Thus the optimal strategy is a threshold strategy: accept if x >= t, where t is the threshold.

3. Express V in terms of the threshold

Given threshold t, the expected payoff is V = (1/21) * sum_{x >= t} x + (1/21) * sum_{x < t} V. Solve for V as a function of t.

4. Find the fixed point

The optimal threshold t* satisfies t* = V(t*). Solve for t* and compute the corresponding V. Since the die is discrete, check integer thresholds around the continuous solution.

5. Verify and interpret

Confirm that the threshold strategy is indeed optimal (e.g., by checking that accepting at t* is better than passing). State the expected payoff to the first player.

Key Points to Mention

Optimal stopping and threshold strategies
Dynamic programming / Bellman equation
Zero-sum game symmetry
Fixed-point equation for the value function
Discrete vs continuous threshold (integer die faces)
Expected payoff calculation and verification

AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.