Diversification and limited information in the Kelly game

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 [J.L. Kelly, IEEE Transactions on Information Theory 2 (1956) 185-189] 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.