  We consider games of chance played by someone with external capital that
cannot be applied to the game and determine how this affects risk-adjusted
optimal betting. Specifically, we focus on Kelly optimization as a metric,
optimizing the expected logarithm of total capital including both capital in
play and the external capital. For games with multiple rounds, we determine the
optimal strategy through dynamic programming and construct a close
approximation through the WKB method. The strategy can be described in terms of
short-term utility functions, with risk aversion depending on the ratio of the
amount in the game to the external money. Thus, a rational player's behavior
varies between conservative play that approaches Kelly strategy as they are
able to invest a larger fraction of total wealth and extremely aggressive play
that maximizes linear expectation when a larger portion of their capital is
locked away. Because you always have expected future productivity to account
for as external resources, this goes counter to the conventional wisdom that
super-Kelly betting is a ruinous proposition.
