In Part 1, we defined the "Silent Duel" as a game of timing and nerves. Two players, each with one shot, approach each other. A miss gives the opponent a guaranteed hit at point-blank range. In Part 2, we move from the abstract game theory to the actual construction of the solution —where the math meets the code. 1. The Mathematical Foundation: The Symmetric Case
, the probability of hitting is 100%. We use this boundary condition to calculate the "Expected Value" (EV) of firing at tn−1t sub n minus 1 end-sub In Part 1, we defined the "Silent Duel"
In the final part of this series, we will look at , where one player is faster, but the other is more accurate. In Part 2, we move from the abstract
), we look for the . If I fire too early, my accuracy is low; if I fire too late, you might preempt me. The solution is derived from the differential equation: We use this boundary condition to calculate the
We iterate through the time steps until we find the point where the EV of firing equals the EV of waiting. 3. Implementation Logic (Pseudocode)