Laws of Growth, Utility, and Finite Streams 213
time, thefthat is optimal is that which maximizes the geometric mean HPR.
In a negative expectation game, thefthat is optimal simply stays at zero.
However, thefthat you use to maximize growth is always uniform, and
that uniform amount is a function of where you intend to quit the game. If
you are playing the two-to-one coin-toss game, and you intend to quit after
one play, you have anfvalue that provides for optimal growth of 1.0. If you
intend to quit after two plays, you have anfthat is optimal for maximizing
growth of .5 on the first toss and .5 on the second. Notice that you do not bet
1.0 on the first toss if you are planning on maximizing the EACG by quitting
at the end of the second play. Likewise, if you are planning on playing for an
infinitely long period of time, you would optimally bet .25 on the first toss
and .25 on each subsequent toss.
Note the key word there isinfinitely, notindefinitely.All streams are
finite—we are all going to die eventually. Therefore, when we speak of the
optimalfas thefthat maximizes expected average compound return, we
are speaking of that value which maximizes it if played for an infinitely long
period of time. Actually, it is slightly suboptimal because none of us will be
able to play for an infinitely long time. And, thefthat will maximize EACG
will be slightly above—will have us take slightly heavier positions—than
what we are calling the optimalf.
What if we were to quit after three tosses? Shouldn’t thefwhich then
maximizes expected average compound growth be lower still than the .5 it
is when quitting after two plays, yet still be greater than the .25 optimal for
an infinitely long game?
Let’s examine the tree of combinations here:
Toss#
123Heads
Heads
Tails
Heads
Heads
Tails
Tails
Heads
Heads
Tails
Tails
Heads
Tails
Tails