Silent Duelsвђ”constructing The Solution Part 2 Вђ“ Math В€© Programming May 2026

For a symmetric duel (equal accuracy and one bullet each), the boundary condition is: ∫a1f(x)dx=1integral from a to 1 of f of x d x equals 1 2. Solving the Integral Equation

import numpy as np from scipy.integrate import quad def construct_strategy(accuracy_func, derivative_func): # 1. Find the starting threshold 'a' # For a symmetric 1-bullet duel, a is found where # the integral of f(x) from a to 1 equals 1. def integrand(x): return derivative_func(x) / (accuracy_func(x)**3) # We solve for 'a' such that integral equals 1/h # (Simplified for demonstration) a = 0.33 # Derived from solving the integral for A(x)=x return lambda x: integrand(x) if x >= a else 0 # Example: Linear Accuracy A(x) = x f_optimal = construct_strategy(lambda x: x, lambda x: 1) Use code with caution. Copied to clipboard 4. Programming Challenges: Precision and Normalization For a symmetric duel (equal accuracy and one

is symmetric. Through some heavy lifting in calculus, we find that the optimal density is proportional to: Through some heavy lifting in calculus, we find

In a silent duel, the core challenge is that neither player knows when the other has fired. This lack of information forces us to rely on a rather than a single "best" time to shoot. 1. The Strategy Profile To construct the solution, we define a strategy as a distribution of firing times. If is the probability of hitting the target at time we will look at

The goal is to make the opponent's payoff constant regardless of when they shoot. This leads to an integral equation where the payoff

In Part 3, we will look at , where one player is more accurate or has more bullets than the other.

, but real-world simulations might use a sigmoid or exponential curve.