  Grammar-based compression is a loss-less data compression scheme that
represents a given string $w$ by a context-free grammar that generates only
$w$. While computing the smallest grammar which generates a given string $w$ is
NP-hard in general, a number of polynomial-time grammar-based compressors which
work well in practice have been proposed. RePair, proposed by Larsson and
Moffat in 1999, is a grammar-based compressor which recursively replaces all
possible occurrences of a most frequently occurring bigrams in the string.
Since there can be multiple choices of the most frequent bigrams to replace,
different implementations of RePair can result in different grammars. In this
paper, we show that the smallest grammars generating the Fibonacci words $F_k$
can be completely characterized by RePair, where $F_k$ denotes the $k$-th
Fibonacci word. Namely, all grammars for $F_k$ generated by any implementation
of RePair are the smallest grammars for $F_k$, and no other grammars can be the
smallest for $F_k$. To the best of our knowledge, Fibonacci words are the first
non-trivial infinite family of strings for which RePair is optimal.
