The following paradox was raised by Rann Smorodinsky:
Rann Smorodinsky’s Privacy Paradox
Suppose that you have the following one-time scenario. You want to buy a sandwich where the options are a roast beef sandwich or an avocado sandwich. Choosing the sandwich of your preference (say, the avocado sandwich) adds one to your utility, but having your private preference known to the seller, reduces by one your utility. The prior people have on your preferences is fifty-fifty.
If you choose the avocado sandwich your utility is zero, hence you can improve on this by picking each type of sandwich at random with probability 1/2. In this case your private preference remains unknown and you gain in expectation 1/2 for having the sandwich you prefer with probability 1/2.
But given this strategy, you can still improve on it by choosing the avocado sandwich.