Interactive Proofs: Why $\delta \lt \frac 13$ for Soundness & Completeness?

Question

From a text on Interactive Proofs

$x \in {0,1}^n$ is input
$V$ is verifier
$P$ is prover
$r$ is $V$'s internal randomness

$P$ provides a value $y$ which is claimed to be equal to $f(x)$

1. (Completeness) For every $x \in {0,1}^n$

$Pr[out(V, x, r, P) = 1] \ge 1 - \delta_c$

2. (Soundness) For every $x \in {0,1}^n$ and every deterministic prover strategy $P'$, if $P'$ sends a value $y \ne f(x)$ at the start of the protocol, then

$Pr[out(V,x,r;P') = 1] \le \delta_s$

An interactive proof system is valid if $\delta_c, \delta_s \le \frac 13$

What is the significance of $\frac 13$ here? Why has $\frac 13$ been chosen as the what the 2 $\delta$s have to be lesser than? I mean why not $\frac 14$ or $\frac 12$ or something else?

Answer 1

This is arbitrary. The definition of the class does not change as long as the $\delta$s are sufficiently away from $1/2$, where 'sufficiently' means by a non-negligible value in $n$, as they can be amplified (by, e.g., parallel repetition) to be overwhelmingly close to $1$. $1/3$ satisfies this criterion. See the discussion after the definition in the Wikipedia article about the class $\mathbf{BPP}$.