The shuffle model of differential privacy sits between local and central privacy: users apply local randomization, then a shuffler permutes the messages before the analyst sees them. The shuffling provides amplification — privacy improves with the number of participants. A natural expectation is that growing the alphabet size automatically amplifies shuffle privacy, since more possible messages means more ambiguity.
This paper proves the expectation is wrong. Growing alphabets do not automatically amplify shuffle privacy. Explicit obstruction families exist for which the shuffled privacy curve equals binary randomized response for all alphabet sizes — no improvement regardless of how many symbols are available. The shuffled histogram experiment depends only on the pushforward law of pairwise likelihood ratios, not the alphabet size.
The optimal mechanism turns out to be augmented GRR: a fraction of users applies aggressive generalized randomized response with a specific parameter, and the rest sends a null symbol. This thinning principle — some users actively randomize while others stay silent — is specific to the shuffle model and has no local-DP counterpart. In local DP, everyone participates identically. In shuffle, strategic silence by some users improves everyone's privacy. Doing less is provably optimal.