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

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 $$x \in {0,1}^n$$

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

1. (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?

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 $$\mathbf{BPP}$$.