is symmetric. Through some heavy lifting in calculus, we find that the optimal density is proportional to:
When translating this to code, we need to handle the accuracy function dynamically. Most models use a linear accuracy
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
is the accuracy function, the "value" of the game is determined by finding a threshold (the earliest possible shot) and a density function for all times
The goal is to make the opponent's payoff constant regardless of when they shoot. This leads to an integral equation where the payoff
, but real-world simulations might use a sigmoid or exponential curve.