  For a gambler with side information, Kelly betting gives the optimal log
growth rate of the gambler's fortune, which is closely related to the mutual
information between the correct winner and the noisy side information. We show
conditions under which optimal Kelly betting can be implemented using
single-letter codes. We show that single-letter coding is optimal for a wide
variety of systems; for example, all systems with diagonal reward matrices
admit optimal single-letter codes. We also show that important classes of
systems do not admit optimal single-letter codes for Kelly betting, such as
when the side information is passed through a Z channel. Our results are
important to situations where the computational complexity of the gambler is
constrained, and may lead to new insights into the fitness value of information
for biological systems.
