STOCHASTIC

ME-24 the reader is invited to consider the relationship between properties 1
and 2, to attempt to verify the validity of property 2 for Breiman's model
reprinted here, and to consider the discrete asset allocation case.
Thorp's paper provides a lucid expository treatment of the Kelly criterion
and Breiman's results. He also discusses some relationships between the max
expected log approach and Markowitz's mean-variance approach. In addition
he points out some of the misconceptions concerning the Kelly criterion, the
most notable being the fact that decisions that maximize the expected log of
wealth do not necessarily maximize expected utility of terminal wealth for
arbitrarily large time horizons. The basic fallacy is that points that maximize
expected log of wealth do not generally maximize the expected utility of wealth
if an investor has nonlogarithmic utility function; see Exercise ME-25 for one
such example and Thorp and Whitley (1973) for a general analysis. See
Markowitz (1972) for a refutation, in a limited sense, of the fallacy if the
investor's utility function is bounded. For some enlightening discussion of this
and other fallacies in dynamic stochastic investment analysis see Merton and
Samuelson (1974) and Exercise ME-26. Miller (1974b) shows how one can
avoid the fallacy altogether by utilizing what is called the utility of an infinite
capital sequence criterion. Under this criterion, the investor's utility is assumed
to depend only on the wealth levels in one or more periods infinitely distant
from the present; that is, capital is accumulated for its own sake, namely its
prestige. In this formulation the time and expectation limit operators are
reversed (from the conventional formulation) and hence the improper limit
exchange that yields the fallacy does not need to be made. One disadvantage
of this formulation is that the admissible utility functions generally are variants
of the unconventional form: limit infimum of the utility of period ?'s wealth.
The reader is invited in Exercise CR-21 to consider some questions concerning
Thorp's paper. In Exercise ME-27 the reader is asked to determine whether
or not good decisions obtained from other utility functions have a property
"similar" to the expected log strategy when they produce infinitely more
expected utility.
For additional discussion and results concerning the Kelly criterion the
reader may consult Aucamp (1971), Breiman (1961), Dubins and Savage
(1965), Goldman (1974), Hakansson (1971a,b,d), Hakansson and Miller
(1972), Jen (1971, 1972), Latane (1959, 1972), Markowitz (1972), Roll (1972),
Samuelson (1971), Samuelson and Merton (1974), Thorp (1969), Young and
Trent (1969), and Ziemba (1972b).
For an elementary presentation of the theory of martingales the reader may
consult Doob (1971). More advanced material may be found in the work of
Breiman (1968), Burrill (1972), and Chow et al. (1971).
The highly technical Merton paper discusses the optimal consumptioninvestment
problem in continuous time. Because of its heavy reliance on
INTRODUCTION 449