  Following the paper by Algoet--Cover (1988), we analyse log-optimal
portfolios where return evaluation includes `weights' of different outcomes.
The results are twofold: (A) under certain conditions, logarithmic growth rate
is a supermartingale, and (B) the optimal (martingale) investment strategy is a
proportional betting; it does not depend on the form of the weight function,
although the optimal rate does. The existence of an optimal investment strategy
has been established earlier in a great generality by Kramkov--Schachermayer
(2003) although our underlying assumptions are different.
