# Definition of cryptographic advantage vs. probability of success

In game-based security definitions, like for example the one defining IND-CPA security, a given cryptosystem is said to be secure if any probabilistic polynomial time adversary has only a negligible advantage of winning the game, where the advantage is defined to be

$|Pr[b^{\prime}=b] - \frac{1}{2}|$,

being $b$ the attacker guess, and $b^{\prime}$ the challenger flipped coin.

However, sometimes I run into papers, like for example this one, where they write something like this (see page 30):

$Adv^{CPA}(B) \geq \frac{1}{2} Adv^{CCA1}(A)-Adv^P(A).$

Clearly, they seem to be treating the adversary advantage as if it was just a probability, and not as the advantage that follows from the definition I just wrote (and it's also given at the beginning of the paper). Is it correct to mix the terms of advantage and probability in this way? Can you just divide the advantage by $2$, like if it was a probability?

I just mentioned this paper as an example, but I have found many others where the same thing seems to be happening.

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Why do you think you can't divide an advantage by 2? If I can guess $b$ with probability 0.50002, and you can guess it with probability 0.50001, then your advantage is $\frac12$ of mine. – Ilmari Karonen Oct 8 '13 at 20:31