Jane Street Puzzle: Robot Javelin (December 2024)

Problem Statement

It's the Robot Javelin finals! Two robots compete:

• Each throws a javelin, with distance drawn uniformly from [0, 1]
• After the first throw, each robot decides to keep it or rethrow (must keep second throw)
• The robot with larger final distance wins

The Twist: Spears Robot cheats by receiving a bit of information: whether Java-lin's first throw was above or below some threshold d of their choosing. Spears assumes Java-lin plays the Nash equilibrium strategy.

The Question: If Java-lin knows about the cheating and adjusts its strategy optimally, what is Java-lin's winning probability?

Solution

Key Insights

1. Nash Equilibrium: In the fair game, both players use threshold t* = (√5 - 1)/2 ≈ 0.618 (the inverse golden ratio). Keep if first throw ≥ t*, rethrow otherwise.

2. Spears' Optimal Cheating: Spears sets d = t* to perfectly distinguish Java-lin's keep/rethrow decision:

   • b=0 (X < t*): Java-lin rethrows → Spears uses threshold 1/2
   • b=1 (X ≥ t*): Java-lin keeps → Spears uses threshold (5-√5)/4 ≈ 0.691

3. Java-lin's Counter-Strategy: Java-lin uses a lower threshold t_j = 7/12 ≈ 0.583:

   • When t_j ≤ X < t*: Java-lin keeps the throw, but Spears sees b=0 and incorrectly expects a rethrow
   • This mismatch allows Java-lin to exploit Spears' incorrect response

The Answer

Java-lin's optimal winning probability:

$$P(\text{Java-lin wins}) = \frac{229}{192} - \frac{5\sqrt{5}}{16} \approx 0.4939370904$$

Mathematical Derivation

For Java-lin using threshold t_j = 7/12:

• Region 1 (X < 7/12): Java-lin rethrows, Spears sees b=0

  • Win rate: 3/8
  • Contribution: (7/12)(3/8) = 7/32

• Region 2 (7/12 ≤ X < t*): Java-lin keeps, but Spears expects rethrow

  • Spears uses wrong threshold (1/2 instead of 0.691)
  • Win rate: 3x/2 - 1/2 for x ≥ 1/2
  • Contribution: (11 - 5√5)/8 + 7/192 = (271 - 120√5)/192

• Region 3 (X ≥ t*): Java-lin keeps, Spears responds correctly

  • Split at Spears' threshold (5-√5)/4
  • Contribution: (-7 + 5√5)/16

Total: 7/32 + (271 - 120√5)/192 + (-7 + 5√5)/16 = 229/192 - 5√5/16

Files

• final_solution.py - Complete symbolic solution with derivation
• corrected_analysis.py - Numerical optimization and verification
• solution.py - Initial exploration (superseded by final_solution.py)
• monte_carlo.py - Monte Carlo simulation to verify the answer

Verification

Run the Monte Carlo simulation to verify:

python -m venv venv
source venv/bin/activate
pip install numpy scipy sympy
python monte_carlo.py

Answer

0.4939370904 (rounded to 10 decimal places)

Or in exact form: 229/192 - 5√5/16