  In a pathbreaking paper, Cover and Ordentlich (1998) solved a max-min
portfolio game between a trader (who picks an entire trading algorithm,
$\theta(\cdot)$) and "nature," who picks the matrix $X$ of gross-returns of all
stocks in all periods. Their (zero-sum) game has the payoff kernel
$W_\theta(X)/D(X)$, where $W_\theta(X)$ is the trader's final wealth and $D(X)$
is the final wealth that would have accrued to a $\$1$ deposit into the best
constant-rebalanced portfolio (or fixed-fraction betting scheme) determined in
hindsight. The resulting "universal portfolio" compounds its money at the same
asymptotic rate as the best rebalancing rule in hindsight, thereby beating the
market asymptotically under extremely general conditions. Smitten with this
(1998) result, the present paper solves the most general tractable version of
Cover and Ordentlich's (1998) max-min game. This obtains for performance
benchmarks (read: derivatives) that are separately convex and homogeneous in
each period's gross-return vector. For completely arbitrary (even
non-measurable) performance benchmarks, we show how the axiom of choice can be
used to "find" an exact maximin strategy for the trader.
