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

When constructing the solution programmatically, two hurdles often arise: If your accuracy function starts at zero, the term explodes. We must enforce a lower bound to ensure the strategy is valid.

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 f(x)=A′(x)A(x)3f of x equals the fraction with numerator

: In the actual game loop, sample from this distribution to decide the exact frame of the "Silent" shot. When constructing the solution programmatically

f(x)=A′(x)A(x)3f of x equals the fraction with numerator cap A prime open paren x close paren and denominator cap A open paren x close paren cubed end-fraction f(x)=A′(x)A(x)3f of x equals the fraction with numerator

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