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This is a simpler version of the trading model, with individual investors replacing pairwise traders. Initially, the wealth of each investor may be equal, or perhaps follows a selected distribution, e.g., exponential, uniform, or triangle.
During each trial, a randomly selected investor either gains or loses an amount invested. We can also view this as flipping a coin to determine whether the investor gains or loses. Over time, what happens to the relative wealth of the investors?
If the bets are small relative to the initial wealth of each investor, and the number of investment periods (trials) is relatively low, and all investors start with equal wealth, there will be a normal distribution of wealth.
This normal distribution occurs because it is relatively unlikely (although not impossible) that a given investor will "achieve a positive return" on all investments or "lose" on every investment, so either extreme of the bell-shaped Gaussian distribution is unlikely: the wins and losses tend to cancel out in accordance with the central limit theorem.
Importantly, however, if the investments (bets) are not "trivially" small, and/or a large number of investment periods pass, some investors will go broke. In this simulation, this is an irreversible state (investors are not allowed to take on debt in the hope that their fortunes change). Consequently, wealth tends to concentrate in fewer and fewer investors, as more and more "go broke." If the investment rule is "double or nothing," it is only a matter of time before all investors are broke, and the system enters a trivial stable state.
This model is based on the so called "yard-sale simulation" model and related variations as are described in the chapter "Follow the Money," in "Group Theory in the Bedroom" by Brian Hayes. The original article appeared in American Scientist. The basis for the model is described in "Wealth Distributions in Asset Exchange Models," by Slava Ispolatov, Paul Krapivsky, and Sidney Redner references. However, it is a simplified version, based on the realization that the "yard sale" model has a winner and loser for each trial. This is roughly equivalent to having a 50% chance of winning, regardless of the counterparty. There may be subtle differences based on the investment rules vs. trading rules. For example, in the yard-sale model, the exchange amount at risk is bounded by the net assets of the "poorer" of the two traders. This limits the potential gain (or loss) of the "richer" trader. In this model, the maximum gain or loss is that of the investor. Consequently, while the likelihood of winning or losing doesn't change for each participant, the expected value of the amount at risk changes after the first few rounds if "Random" or "Double or Nothing" is selected.
|Basic Flow||First, select the number of investors, which may range from 2 to 200.
Second, select the initial distribution of wealth. It may be constant (every investor has 100 units), uniform (investors have random initial assets uniformly distributed between 0 and 100 units), triangle (roughly equivalent to uniform, but sorted), or exponential (each investor has initial assets that are 95% of the next wealthier investor).
Third, determine the amount invested. "Random" means that the value at risk is a uniformly distributed fraction of the investor's net worth. "Double or Nothing" is as one would expect: if the investor loses, their net worth will be zero, if they win, their net worth will double. Other choices include fixed amounts such as .1, 1, 5, 10, 20, etc., units of value.
Fourth, run any number of Monte Carlo trials, where each trial entails a random selection of an investor, and the determination of an amount won or lost. The amount won or lost depends on the rule selected, but in no case can be more than the investor's current assets. You can run 1, 10, 100, 1,000 or 10,000 trials at the push of a button, and keeping the [ENTER] key on your keyboard depressed will, in most browsers, repeatedly run that many trials. If the amounts at risk are large and the number of investors is low, very few trials are required to achieve substantial concentration. Conversely, if the lowest amount is selected (.1) and the maximum number of investors (200) is operative in the market, even after a large number of trials, there will be a normal distribution of wealth and the wealthiest trader will have only gained 25 - 30% of their initial asset value.
Fifth, at any time, the results can be sorted to better visualize the nature of the distribution of wealth. If the "Keep Sorted" checkbox is checked, the evolution of the distribution of wealth may be easily viewed. On the other hand, not continuously sorting can help illustrate how an early leader may be overtaken.
Finally, the percentage of total wealth of the leader may be viewed. While this tends to increase to 100%, this does not necessarily happen monotonically. For example, three investors may have, respectively, 80%, 10%, and 10% of the total wealth. If the wealthiest goes broke in a single trial, the new "leader" only has 50%.