The traditional definition of PCPs have perfect completeness -- If $x\in L$, then the prover can give a proof on which the verifier (on reading constantly many bits) always accepts. Suppose we modify the definition as follows:A language $L$ is in $iPCP(r,q)$ such  that there exists a verifier who  tosses $r$ coins and queries $q$  locations of a proof $\pi$ such that:If $x\in L$, then there exists a $\pi$ that makes the verifier accept  with probability at least $3/4$. If $x\notin L$, then for every proof $\pi$, the verifier accepts with  probability at most $1/4$.There is an easy reduction from $3SAT$ to $\text{Max}2SAT$ (problem 5 in this) that gives a simple $iPCP$ for $3SAT$. And by definition, the class is closed under complement so it includes $coNP$ as well. Therefore, it is unlikely that there is a way to convert an $iPCP$ to a $PCP$ with the same parameters (well, that would at least mean $coNP = NP$).  (thanks to Peter Shor for pointing this out)Have such incomplete PCPs been studied earlier? 
PCPs with imperfect completeness
cc.complexity theory;reductions;pcp
Yes, PCPs with imperfect completeness have been studied before. The main motivation is that for some natural and interesting problems, finding whether there is a perfect solution is actually easy (polynomial-time), while approximating the best solution, if this best solution is not perfect, is (or believed to be) hard.  Here are some examples:The Unique Games Conjecture. Unique games with perfect completeness are easy, so the Unique Games Conjecture focuses on unique games with imperfect completeness.Feige and Reichman showed NP-hardness for unique games for some constant completeness and soundness values (not c=1-delta s=epsilon as conjectured).Linear equations. Linear equations with perfect completeness are easy (using Gaussian elimination), so hardness results for linear equations (e.g., Hastad's) focus on imperfect completeness.