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/

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