  This paper proposes a uniqueness Shapley measure to compare the extent to
which different variables are able to identify a subject. Revealing the value
of a variable on subject $t$ shrinks the set of possible subjects that $t$
could be. The extent of the shrinkage depends on which other variables have
also been revealed. We use Shapley value to combine all of the reductions in
log cardinality due to revealing a variable after some subset of the other
variables has been revealed. This uniqueness Shapley measure can be aggregated
over subjects where it becomes a weighted sum of conditional entropies.
Aggregation over subsets of subjects can address questions like how identifying
is age for people of a given zip code. Such aggregates have a corresponding
expression in terms of cross entropies. We use uniqueness Shapley to
investigate the differential effects of revealing variables from the North
Carolina voter registration rolls and in identifying anomalous solar flares. An
enormous speedup (approaching 2000 fold in one example) is obtained by using
the all dimension trees of Moore and Lee (1998) to store the cardinalities we
need.
