From: John Conover <john@email.johncon.com>
Subject: Tutorial on fractal speculative games
Date: Tue, 23 Apr 1996 23:11:02 -0700
Tutorial on Fractal Time Series
This chapter presents a remedial tutorial on the optimization of
betting strategies in speculative markets. It is offered in academic
perspective, and under no circumstances would it be appropriate to
consider it financial advice. It can serve, however, as an
introduction to the contemporary economic theory of speculative
markets. Rigorous and sophisticated approaches that address the issues
of investing in speculative markets are contained in the
bibliography. This section begins with the analysis of a very simple
speculative game, that of tossing coins. The analysis will then be
expanded by permitting the use of unfair coins in the game, roughly
following [Sch91, pp. 128]. An optimal betting strategy will be
developed for the game, and this strategy will then be generalized and
extended to include remedial betting strategies in certain speculative
markets.
A.1 The Coin Tossing Game
Consider a coin tossing game, where a player makes a wager, and then,
flips a coin. If the coin comes up heads, then the player wins twice
the original wager, (ie., makes back the original wager plus an amount
equal to the original wager from the "bank.") But if the coin comes up
tails, then the player looses the wager to the bank. The game is
iterated, many times, until the player decides not to play any more,
or goes "bust."
A.1.1 Strategic Considerations in the Iterated Coin Tossing Game
The player has an initial cash reserve, to which are added the
cumulative returns, and from which each wager is made, and the
cumulative returns will increase by the amount of the wager each time
the player wins, and, likewise the cumulative returns will decrease by
the amount of the wager each time the player looses. The objective of
the player is, obviously, to maximize the magnitude of the cumulative
returns, over time. Note that this is a speculative game, in that the
player speculates on the likelihood that the coin will come up heads
on next iteration of the game, and adjusts the wager accordingly,
betting zero if the outcome of the next coin toss is anticipated to be
tails.
Description of "Time Series" and "Fractal"
Time Series: If the player makes a list, recording the time, and
magnitude of cumulative returns, for each iteration of the game,
such a table is called the time series of the cumulative
returns[Sch91, pp. 223], [C, 93, pp. 199].
Fractal: If the player plots a graph of the time series, time on
the X--axis and magnitude of cumulative returns on the Y--axis,
this graph will exhibit fractal characteristics, which means that
it represents a system that has the characteristics of a
cumulative sum, (ie., integrative process,) of random events, coin
tosses in this case[Cro95, pp. 229], [Fed88, pp. 163].
A.2. OPTIMAL BETTING STRATEGY IN THE ITERATED COIN TOSSING GAME
It is an important concept that the magnitude of the cumulative
returns, at any time during the iterated game, is the cumulative sum
of all of the wins and losses of wagers made in the previous
iterations of the game, starting with the initial cash reserves, ie.,
it is an integrative process. If the player does not have "a priory"
knowledge of the outcome of the future coin tosses, then optimizing
this integrative process is the strategic objective of playing a
rational game. For example, it would be foolish for the player to
always wager zero, since, although there would never be a loss, there
would never be a win, either. Likewise, it would be foolish for the
player to wager a large percentage of the cumulative returns, since a
few losses in succession would deplete the cash reserves and
cumulative returns to zero, and the player would "go bust," thus
ending the game. Obviously, the player could wager too much, or too
little of the cumulative returns on a single game iteration. The
series optimum wagers, is termed the optimal betting strategy, [Sch91,
pp. 128], [Rez94, pp. 450].
A.2 Optimal Betting Strategy in the Iterated Coin Tossing Game
If the coin is a fair coin, ie., it has a 50% chance of coming up
heads and a 50% chance of coming up tails, then the player should
elect not to play. The rationale for this statement is that, in the
long run, some iterated games will be won, and some lost, with the
amount of money won equal to the amount of money lost--so there is no
financial incentive to play the game. However, suppose that there is a
60% chance for the coin to come up heads, on any single iteration of
the game, and a 40% chance of coming up tails. It turns out that the
fractal characteristics of the game can be exploited to determine the
optimal betting strategy. The optimal betting strategy, in this case,
is for the player to wager 20% of the cumulative returns, every
iteration of the game. As it turns out, this will maximize the growth
of the player's cumulative returns[Sch91, pp. 128], [Rez94,
pp. 450]. The way that this was computed was from the formula:
F = 2P - 1 (A.1)
where F is the fraction of the player's cumulative returns that should
be wagered on an iteration of a game with a P chance of winning the
game[Sch91, pp. 151], [Rez94, pp. 450]. In the above case, with a 60%,
(ie., p = 0.6,) chance of winning:
F = (2 * 0.6) - 1 (A.2)
F = 1.2 - 1 (A.3)
F = 0.2 (A.4)
or F is 20% of the player's cumulative returns. Playing this betting
strategy, the player can expect an average of 2% increase in the
magnitude of cumulative returns on each toss of the coin. For those
wishing to experiment with optimal betting strategies, the unfair coin
can be simulated with a six sided die. After the wager, the die is
rolled, and if the die comes up 1, 2, 3, or 4 the player wins. But if
it comes up 5 or 6, the player looses.
A.3. IMPORTANT INTUITIVE CONCEPTS OF SPECULATIVE GAMES
The probability of winning, P, in this case is 0.66 since the player
will win 4 times out of 6, on average. The optimal wager will be F = 2
* 0.66 - 1 = 0.33, or 33% of the cumulative returns should be wagered
on each iteration of the game. It is interesting to play many
iterations of the game, particularly using different betting
strategies--for example change the wager fraction to 20%, or 40% of
the cumulative returns--and see how the long term cumulative returns
change in response to the different betting strategies.
A.3 Important Intuitive Concepts of Speculative Games
The unfair coin tossing game is probably one of the simplest
speculative games. It is important to develop an intuitive concept
based on the fundamentals of this simple game. Speculative games have
the following characteristics:
*Speculative games are iterated. A wager is made from the player's
cumulative returns for the game, and depending on the outcome of
the iteration of the game, the player either wins or looses the
wager for that iteration. The winnings or losses, for each
iteration, are summed to the player's cumulative returns.
*The outcome of a particular iteration has random characteristics,
ie., the outcome of a particular iteration is not "predictable."
*The objective of the game is to maximize the value of the
player's cumulative returns.
As it turns out, these simple concepts have many applications, for
example, they can be used to model and analyze the capital markets
[Pet91, pp. 81].
A.4 An Analytical Approach to the Iterated Unfair Coin Tossing Game
In section A.2 it was assumed that the player had knowledge about the
probability of a tossed coin coming up heads. In most speculative
games, knowledge of the random mechanism is not available. For a
simple game, like tossing an unfair coin, the coin could be tossed
many times, and the probability of it coming up heads measured. The
methodology would be to toss the coin, say, 100 times, and count how
many times it came up heads. Say it comes up heads 60 times out of the
100 tosses. Then the probability that the coin will come up heads on
any particular iteration of the game would be 60%, and the player
could arrange a betting strategy, accordingly. It turns out that this
concept is very extensible. In many speculative games, there is no
knowledge available about the characteristics of the random process of
the game. As a simple example, assume that no knowledge is available
about the underlying random process of the unfair coin tossing
game. Like the capital markets, we have only historical data about the
wagering process, ie., what was won, and what was lost during each
iteration of the game. If we look at the historical time series of the
game, we would observe that since the cumulative returns are
increasing, that the game is unfair. It would be desirable gain some
insight into the random process that controls the outcome of an
iteration of the game, so a betting strategy can be
formulated. Referring to the preceeding paragraph, when the coin was
tossed a hundred times to count how many times it came up heads, it
should be realized that this was a cumulative sum of number of times
the coin came up heads over a hundred iterations. Being formal, in n
many tosses of the coin, it would be expected that the coin came up
heads, P * n many times, and come up tails, (1 - P) * n many times,
where P is the probability of the coin coming up heads.
If a counting process is started, tallied in C, by which, if the coin
is tossed, and it comes up heads, we increase the count by one, and if
it comes up tails, we decrease the count by one, then it would be
expected, after n many tosses:
C = p * n - (1 * P) * n (A.5)
Notice that C was derived empirically, and from C, we can compute the
probability, P , of the coin coming up heads in any iteration of the
game. Rearranging:
C = P * n - n + P * n (A.6)
and dividing both sides of the equation by n:
C/n = P - 1 + P = 2 * P - 1 (A.7)
and solving for P :
P * (C/n + 1) / 2 (A.8)
noting that Cn is the "average" C. The same methodology can be used in
general. Access to the unfair coin to measure the probability of it
coming up heads on any iteration is not necessary--this information
can be deduced from the historical files of a game where the coin was
used. For example, we can take the historical time series of a unfair
coin tossing game, and for each iteration, subtract the value of the
cumulative returns of the previous iteration from the value of the
cumulative returns of the next iteration, dividing the result of the
subtraction by the value of the cumulative returns in the previous
iteration, making a new time series. This is a very powerful concept
in the strategy of speculative games. The new time series contains the
fraction of the cumulative returns that was won or lost on each
iteration of the game. Using our example of the unfair coin tossing
game, we would observe that the new time series would be a list of
numbers, containing either +F , if the wager was won in an iteration,
or -F , if the wager was lost, (assuming that F was constant
throughout the game.) The important concept here is that, given a
specific iteration, the fraction of the cumulative returns wagered can
be deduced, and whether the wager was won or lost. It is an important
concept that we can reconstruct the characteristics of the random
mechanism, and the fraction of the cumulative returns wagered from the
historical data of a speculative game, without having knowledge of the
random mechanism 6 . As before, we do a cumulative sum on the random
game's process, only instead of it being a tossed coin, it is the new
time series that contains the fraction of the cumulative returns that
was won or lost in each iteration of the game. Formalizing, using
equation A.8, and replacing C/n , the average value of C, with the
"average" value of F , found by summing all of the values in the new
time series, and dividing by the number of iterations:
n
P = 1 / 2 + 1 / (2 * n) * sum F(i) (A.9)
i=0
For computational reasons, it is advantageous to implement counting of
the number of heads in a series of coin tosses in this manner, which
finds the "average" C by summing both heads and tails, with differing
signs. In the unfair coin tossing game, the random mechanism can be
analyzed by simply counting the number of times heads comes up in
series of tosses. However, in speculative games, in general, the
random mechanisms are much more sophisticated, requiring an "average"
to be taken. This methodology provides a means of extensibility to
these types of systems. It is not a complicated concept, actually, if
you look at the process by which the historical time series was
made. A wager is made, that is a fraction of the cumulative returns,
and the wager was either added or subtracted from the cumulative
returns for the game, depending on the results of a random
process. When we subtract the value of the cumulative returns of a
previous iteration from the value of the cumulative returns of the
next iteration, and dividing by the value of the cumulative returns in
the previous iteration, we are actually "undoing" the cumulative
returns process of the game--kind of working backward to create the
underlying random process and betting strategy.
Interestingly, if we want to find out the fraction of the cumulative
returns that was wagered each iteration of the game, the absolute
value of F *i*can be taken in equation A.9. In the simple case of the
unfair tossed coin, it is simply F , since F *i*is either *F or *F ,
ie., we simply remove the signs, and the equation reduces to:
F * 2P - 1 (A.11)
which is the same as equation A.1. Although this is a generalization,
this derivation has not shown that this is indeed an optimal
solution. See section 2.3.3 in chapter 2 for a presentation on the
optimal solution--it turns out that F = 2P - 1 is, indeed, the optimal
solution.
--
John Conover, john@email.johncon.com, http://www.johncon.com/