It is routinely assumed that there is a trade-off between police efficiency and the warrant requirement. But existing analysis ignores the interaction between law-enforcement investigative practices and criminal innovation. Narrowing the definition of a search or otherwise limiting the requirement for a warrant gives criminals greater incentive to innovate to avoid detection. With limited resources to develop countermeasures, law enforcement officers will often be just as effective at capturing criminals when facing higher Fourth Amendment hurdles. We provide a game-theoretic model that shows that when law-enforcement investigation and criminal innovation are considered in a dynamic context, the police efficiency rationale for lowering Fourth Amendment rights is often inapt. We analyze how this impacts both criminal activity and innocent communications that individuals seek to keep private in the digital age. We show that both law-enforcement and noncriminal privacy concerns may be better promoted by maintaining the warrant requirement.