Before revision:  Motivated by the complexity of real investments, we propose several modifications of this game  to investigate the influence of diversification and limited information on investment performance. Analytical and numerical results obtained from these toy games are well related to their real-life counterparts .
After revision:  Motivated by the complexity of real investments, we propose several modifications of this game 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 .