Further On The Wealth Of Stations

Recall that my fellow students and I have resolved to investigate the role of chance in the spread of wealth amongst the populace by creating a series of games with which we might approximate its ebb and flow. Our first such game[1] was of so simplistic a construction that it offered no meaningful insights into the matter, but at least shed some light upon the manner in which we might answer such questions as what are the chances that, in a perfectly fair game, a few players might fare significantly better than the many or what are they that a player having had a run of poor luck might conclude the game ahead of a fellow who had had better?
We have since spent some mental effort improving the rules of our game and it is upon these changes that I shall now report.

A notable deficiency of our first game is that it involved no trade, but simply assigned to each player a randomly chosen profit or loss of up to five percent at each turn. As students of wager, we immediately recognised that the players' funds at the conclusion of the game should be very nearly governed by the log normal distribution since most every product of many identically distributed, but otherwise unrelated, random quantities must be.
That some players should profit well by such a rule whilst many should suffer by it was consequently of no surprise whatsoever, as was the unlikelihood that a player having had poor fortune might turn the table upon a more fortunate fellow.

A Model Of Trade

To model the implications of a perfectly fair system of trade we changed the rules of the game so that each player entered into a trade with a randomly chosen counter party at each turn. The consequence of this trade was that some randomly chosen percentage of the funds of the more impoverished player would be transferred from one of them to the other with
\begin{align*} u &\sim U(-c, c)\\ x^\prime_1 &= x_1 - u \times \min\left(x_1, x_2\right)\\ x^\prime_2 &= x_2 + u \times \min\left(x_i, x_2\right) \end{align*}
Here, $$U(-c, c)$$ stands for a uniformly distributed random variable taking values between $$-c$$ and $$c$$, $$x_1$$ and $$x_2$$ the players' funds before the trade and $$x^\prime_1$$ and $$x^\prime_2$$ those after it.

Deck 1 demonstrates one thousand turns of this game for one hundred players with a trade of up to five percent of the lesser of the players' funds.

Deck 1: Our Trading Game

Just as we did for our original game, we have marked the players' initial funds with a red line.

Evidently, some players fare a good deal better than others despite the rules applying equally to all, a fact that is highlighted by arranging them from the wealthiest to the poorest at each turn, as is done by deck 2.

Deck 2: Sorted By Decreasing Funds

The Distribution Of Outcomes

Unfortunately, it is not at all obvious how the players' outcomes should be distributed at the conclusion of this game. What we can do, however, is construct a histogram of the players' final funds, as is done by deck 3.

Deck 3: The Histogram Of Outcomes

On first inspection this doesn't look so very different to a histogram that we might expect from a log normal distribution but, as demonstrated by deck 4, if we construct a histogram of the logarithms of the outcomes we find that it is rather more biased toward the left than we should expect from a log normal distribution.

Deck 4: The Histogram Of Log Outcomes

In retrospect this shouldn't have come as any particular surprise since the logarithm of each player's funds at the conclusion of the game must be finitely bounded above and infinitely unbounded below. This follows from the fact that whilst a player's funds may fall ever closer to zero, and hence their logarithms ever more negative, they cannot exceed the total initial funds held by all of the players.

Unfortunately, to our very great frustration, the distribution that governs the players' outcomes has proven to be beyond the reckoning of my fellow students and I. To answer questions regarding their fortunes we have consequently had to construct decks to play the game a great many times, much as we did to produce these histograms.

Figuring The Players' Chances

The first thing that we wished to know were the chances that a player might finish the game with greater than some many times their original funds or with no more than some fraction of them and we have put together deck 5 to answer that question.

Deck 5: The Chances Of Losses And Gains

As might well have been expected given the logarithmic histogram of the players' eventual funds, it proves far more likely that a player will suffer a relative loss than gain an equivalent profit but that a lucky few will prosper well above their fellows.
Once again we have found that in a perfectly fair game, serendipity alone might favour the few at the expense of the many!

The next question that my fellow students and I have concerned ourselves with is what are the chances that a player who had had some poor fortune at the outset might eventually prevail upon another who had fared rather better.
That the players contract trade with each other at random means that we cannot consider pairs of them in isolation whilst figuring such chances; instead we gave halved funds to half of the players and doubled funds to the others and counted how many times the former concluded the game with greater funds than their counterparts amongst the latter after five hundred turns of the game, as demonstrated by deck 6.

Deck 6: The Chance Of A Comeback

Evidently there is precious little chance that an unfortunate player might prosper above their more fortunate fellows, again suggesting that a relatively good start to the game is likely to result in a relatively good end.

An Appetite For Trade

It might reasonably be presumed that the wealthier one is, the more willing one might be to trade. To investigate the consequences of this presumption we changed the rules of our game so that each player should contract a number of trades at each turn equal to the smallest whole number that his funds were no greater than at its commencement, as has been done in deck 7.

Deck 7: Varying Appetites For Trade

On first inspection it certainly appears that this has exacerbated the degree of inequality at the conclusion of the game. To quantify this, deck 8 calculates the chances of proportional losses and gains given our new rule.

Deck 8: The Revised Chances Of Losses And Gains

Clearly our suspicions that the revised game had led to greater inequality were correct; the chances that any given player should increase their funds at all have decreased whilst those that they should do so significantly have increased.
Naturally, we also wished to know the effect upon the chance that a player might have a reversal of fortune and so we added the revised rule to the deck that we had used to figure it, yielding deck 9.

Deck 9: The Revised Chance Of A Comeback

That the new rule increases both the inequality of outcomes and the chance that an unlucky player should eventually prosper above a lucky one might seem rather contrary to common sense. However, if one considers the fact that a more prosperous player has an equal chance of increasing both their profits and their losses beyond those of a less prosperous player at each turn it is perhaps unsurprising, since the fortunes of the former must consequently swing back and forth that much more vigorously than those of the latter.

In Conclusion

Whilst my fellow students and I found these games to be rather more satisfying than our first, given that the players were compelled to interact with each other at each turn, they still fall somewhat short of the mark, lacking rules to reflect the fact that wealth might be created through labour and lost through consumption.
We have therefore resolved to contrive new rules at such time that our studies permit!
$$\Box$$

References

[1] On The Wealth Of Stations, www.thusspakeak.com, 2016.