# Why I always obtain this soundness bound in parallel repetition of interactive proof systems

Fix an interactive proof system $$(P,V)$$ and denote by $$(P_k,V_k)$$ an interactive proof system in which the parties play in parallel $$k$$ copies of $$(P,V)$$ and for which $$V_k$$ accepts if and only if $$V$$ would have accepted in all $$k$$ copies. The Parallel Repetition Theorem says that given a prover $$P$$ and input $$x$$ to the proof system: $$\text{If } \Pr[(P^*,V)(x)=1] \leq \epsilon, \text{ then } \Pr[(P^*,V_k)(x)=1] \leq \epsilon^k.$$ However, I do not understand why this should hold true in the (worst) case that $$P^*$$ plays with $$V_k$$ $$k-1$$ "good" copies of $$(P,V)$$ and $$1$$ "bad" copy. Shouldn't the soundness bound in that case be $$\epsilon^2 < \Pr[(P^*,V_k)(x)=1] \leq \epsilon$$?

• What "flavor" of parallel repetition is this? Is the prover required to succeed in all instances? (Which seems to be the case). Dec 22, 2022 at 0:18
• @MarcIlunga Yeah, I forgot to mention that. I edited the text, so you are right. Dec 22, 2022 at 6:54

Assuming we are not too concerned by the constraints of computational proof systems, parallel repetition emulates $$k$$ independent interactions, and the parallel verifier accepts only if all interactions are accepting.
For this validation strategy (contrasting with the worst-case scenario of the question) and assuming the independence condition, the soundness error is similar to the probability of $$k$$ successes in $$k$$ repeated Bernoulli trials, which gives the desired soundness error of $$\epsilon^k$$.