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)
such that the total probability of action equals 1. In a simple linear case where , the optimal strategy is to fire at exactly . 2. The Programming Challenge: Discretizing the Continuous
, which represents the probability of hitting a target at time goes from 0 to 1). To find the optimal time to fire ( t*t raised to the * power We iterate through the time steps until we
def solve_silent_duel(accuracy_func, steps=1000): # Backward induction to find the 'tipping point' for t in reversed(range(steps)): prob_hit = accuracy_func(t / steps) # If the risk of the opponent hitting us next # is higher than our current hit chance, we wait. if prob_hit >= calculated_threshold(t): return t / steps Use code with caution. Copied to clipboard 4. Why This Matters
In a symmetric duel, both players share the same accuracy function, In a simple linear case where , the
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
A(t)=∫at1P(x)dxcap A open paren t close paren equals integral from a to t of the fraction with numerator 1 and denominator cap P open paren x close paren end-fraction d x The goal is to find the lower bound To find the optimal time to fire (
Deciding when to "patch" a system versus waiting to gather more data on an exploit.