Advantage of attacker in CPA secure IBE In CPA secure IBE systems, the advantage of attacker (A) is defined as  absolute value of 

$$Pr[b=b^{'}]-\frac{1}{2}$$

Can someone please explain this to me in a few words? Where exactly does that absolute value come from? And how common is this advantage in the realms of cryptography; meaning: how often is this attacker advantage to be considered in general?
Sometimes, to make it easier to understand, the advantage is multiplied by a factor of 2: $Adv = 2|\Pr[b' = b] - 1/2|$.  

Note that when the adversary has no advantage in breaking the protocol, the only thing s/he can do is guessing $b'$ at random.  In this case, $\Pr[b' = b] = 1/2$ and thus $Adv = 0$.  

When the adversary always correctly finds the value of $b'$, we have $\Pr[b'=b] = 1$ and thus $Adv = 1$.  

Similarly, when the adversary always incorrectly guesses the value of $b'$, we have $\Pr[b'=b] = 0$ and thus $Adv = 1$.  Note that such an adversary is as powerful that an adversary who always guesses the correct the value.  It suffices to flip the result to get the same answer.

To sum up, the advantage is a value between 0 and 1 (or 0 and 1/2 with your definition).  The higher the value is, the more powerful the adversary is.

In practice, typically, a cryptographic construction is said IND-CPA secure if $Adv &lt; 2^{-128}$.