  Kelly betting is a prescription for optimal resource allocation among a set
of gambles which are typically repeated in an independent and identically
distributed manner. In this setting, there is a large body of literature which
includes arguments that the theory often leads to bets which are "too
aggressive" with respect to various risk metrics. To remedy this problem, many
papers include prescriptions for scaling down the bet size. Such schemes are
referred to as Fractional Kelly Betting. In this paper, we take the opposite
tack. That is, we show that in many cases, the theoretical Kelly-based results
may lead to bets which are "too conservative" rather than too aggressive. To
make this argument, we consider a random vector X with its assumed probability
distribution and draw m samples to obtain an empirically-derived counterpart
Xhat. Subsequently, we derive and compare the resulting Kelly bets for both X
and Xhat with consideration of sample size m as part of the analysis. This
leads to identification of many cases which have the following salient feature:
The resulting bet size using the true theoretical distribution for X is much
smaller than that for Xhat. If instead the bet is based on empirical data,
"golden" opportunities are identified which are essentially rejected when the
purely theoretical model is used. To formalize these ideas, we provide a result
which we call the Restricted Betting Theorem. An extreme case of the theorem is
obtained when X has unbounded support. In this situation, using X, the Kelly
theory can lead to no betting at all.
