Finding Favorites

We investigate a new type of information-theoretic identification problem, suggested to us by Alan Taylor. In this problem we are given a set of items, more than half of which share a common “good ” value. The other items have various other values which are called “bad”. The only method we have for gaining information about the items’ values is to ask whether two items share the same value. We can assume there is an oracle which always answers each such query truthfully. Our goal is to identify at least one good item, i.e., an item which is guaranteed to have a good value, using a minimum possible number of queries. We will establish upper and lower bounds for the number of queries needed for both adaptive as well as oblivious strategies. The practical context in which this problem arose was in connection with trying to identify a good sensor from a set of sensors in which some are non-operational or corrupted, for example, where it was desired to minimize the amount of intercommunication used in doing so.