authored by Premmi and Beguène
By definition,
\begin{equation*}
\begin{split}
\textit{Distinguishing Advantage} & = \big|\text{Advantage of Adversary with respect to Challenger}\big| \\
& = \big|\text{Probability that the Adversary wins - Probability that the Challenger wins} \big| \\
& = \big|\text{Probability that the Adversary wins - Probability that the Adversary loses} \big| \\
\end{split}
\end{equation*} For a deep intuitive understanding of advantage refer to this and this.
Why is the advantage an absolute value? It suggests that even when the adversary loses more than it wins, it has an advantage. Why is this so?
Let us consider the following example to answer these questions.
Suppose we have a loaded coin with a bias of \frac{1}{4} towards head. For this coin, the probability of heads, P(H) = \frac{1}{2} + bias = \frac{1}{2} + \frac{1}{4} = \frac{3}{4} and the probability of tails, P(T) = \frac{1}{2} - bias = \frac{1}{2} - \frac{1}{4} = \frac{1}{4}.
Now you play a game in the casino. The casino asks you to choose either Heads or Tails, and you win whenever what you choose turns up. No casino would ever play this game with the loaded coin we just described (as if life is that kind! 😂), but for the purpose of this example let us assume that’s the case.
Let us assume that you play this game several times.
Suppose you pick Heads. Now,
Your advantage with respect to the casino = Probability you win – Probability you lose (or Probability the casino wins) = \frac{3}{4} - \frac{1}{4} = \frac{1}{2}.
Now, suppose you had picked Tails. When you play this game several times, you would notice that you are winning only \frac{1}{4}th of the time. So you would eventually know that the coin is loaded in favor of Heads and you would change your strategy and pick Heads and win \frac{3}{4}th of the time and have an advantage of \frac{1}{2}. This is why the definition of advantage has an absolute sign since irrespective of whether you picked Heads or Tails, you would eventually know that the coin is loaded and hence have the same advantage in both the cases. This is the reason we define your advantage with respect to the casino as \big|Probability you win – Probability you lose (or Probability the casino wins)\big| or more generally, ‘Advantage of Adversary with respect to Challenger’ as an absolute value.