  Financial markets, with their vast range of different investment
opportunities, can be seen as a system of many different simultaneous games
with diverse and often unknown levels of risk and reward. We introduce
generalizations to the classic Kelly investment game [Kelly (1956)] that
incorporates these features, and use them to investigate the influence of
diversification and limited information on Kelly-optimal portfolios. In
particular we present approximate formulas for optimizing diversified
portfolios and exact results for optimal investment in unknown games where the
only available information is past outcomes.
