  An algorithm is developed to gradually relax the Differential Privacy (DP)
guarantee of a randomized response. The output from each relaxation maintains
the same probability distribution as a standard randomized response with the
equivalent DP guarantee, ensuring identical utility as the standard approach.
The entire relaxation process is proven to have the same DP guarantee as the
most recent relaxed guarantee.
  The DP relaxation algorithm is adaptable to any Local Differential Privacy
(LDP) mechanisms relying on randomized response. It has been seamlessly
integrated into RAPPOR, an LDP crowdsourcing string-collecting tool, to
optimize the utility of estimating the frequency of collected data.
Additionally, it facilitates the relaxation of the DP guarantee for mean
estimation based on randomized response. Finally, numerical experiments have
been conducted to validate the utility and DP guarantee of the algorithm.
