# Finally On The Wealth Of Stations

In our recent investigations we have found that games comprising of random returns upon funds[1], of random trades between players[2] and of random outcomes of labour, trade and sustenance[3], with the latter subject to some bare minimum of expenditure, invariably rewarded a fortunate few at the expense of an unfortunate many, despite having rules that applied perfectly equitably to all.
For our final analysis, my fellow students and I have sought to develop a rule by which we might cuff the hands of providence!

### Rules Of Labour, Trade And Sustenance

Recall that each turn in our last game was comprised of three rounds during which each player was, one by one, subjected to its given rule.

Firstly, a round of labour gave to each player a random positive return upon their funds in recognition of the fact that by one's efforts, one might add some value to those assets in one's possession; by farming or mining one's land, for example. Specifically, by the rule
\begin{align*} u &\sim U(0, b)\\ x &\leftarrow x + u \times x \end{align*}
where $$U(0, b)$$ stands for a uniformly distributed random variable with lower and upper bounds of zero and $$b$$, and $$x$$ the player's funds.

Secondly, a round of trade committed each player to a number of trades equal to the largest whole number less than or equal to their funds at the start of the round, with each trade contracted with a randomly chosen counterparty and transferring some random share of the lesser of their funds from the one to the other according to the rule
\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*}
where $$x_1$$ and $$x_2$$ represent the players' funds before the trade and $$x^\prime_1$$ and $$x^\prime_2$$ those after it.

Finally, a round of sustenance took from each player a quantity of funds given by the rule
\begin{align*} u &\sim U(0, d)\\ x &\leftarrow x - \left(u \times x + d_0\right) \end{align*}
where $$d$$ is the upper limit of the proportion of their funds that they might expend above and beyond those absolutely required to survive, given by $$d_0$$.

These rules are played out by deck 1, to the aforementioned benefit of the few at the expense of the many.

Deck 1: Rules Of Labour, Trade And Sustenance

Once again, for perspective, the red line records the players' initial funds.

### A Rule Of Simple Taxation

For our first attempt to reduce the impact of chance upon the players' outcomes, my fellow students and I turned to a simple redistributive tax. Specifically, we took from each player a share of their profit, if any, after the rounds of labour and trade according to the rule
\begin{align*} t_i &= e \times \max\left(x^\prime_i - x_i, 0\right)\\ x^\prime_i &\leftarrow x^\prime_i - t_i \end{align*}
where $$x_i$$ are a player's funds at the start of the turn and $$x^\prime_i$$ are those after the rounds of labour and trade. We then shared the total gathered funds equally between all players with
$x^\prime_i \leftarrow x^\prime_i + \frac{1}{n}\sum_{i=1}^n t_i$
where $$\sum$$ is the summation sign.

To measure the consequences of this rule for the players' outcomes, we put together deck 2 in which ten percent of their profits are redistributed at each turn.

Deck 2: A Rule Of Simple Taxation

Evidently, the introduction of taxation has had a dramatically levelling effect upon the players' fortunes! This is emphasised by deck 3 in which we plot a histogram of their final funds after some several hundred games.

Deck 3: The Histogram Of Outcomes

That, unlike those of every other game that we have proposed thus far, its highest bar is not the leftmost is conclusive evidence that the rule of taxation is an effective guard against the cruellest ravages of fate.

To quantify this effect, we put together deck 4 to count the number of times that the players' final funds fell below some fraction, or above some multiple, of their initial funds during these games, from which we were able to approximate the probabilities of such outcomes.

Deck 4: The Chances Of Losses And Gains

Finally, we once again sought to figure the chances that players having had an unfortunate start might reverse their fortunes by concluding the game with greater funds than their more fortunate fellows. We did this with deck 5 in which we gave to half of their number halved funds and to the other half doubled funds and counted how frequently the former had fared better than the latter after five hundred turns.

Deck 5: The Chance Of A Comeback

Giving just shy of a one in three chance of turning the tables, it is clear that the rule of taxation has gone a fair way toward mitigating the advantages and disadvantages brought about by contingency!

### A Rule Of Progressive Taxation

Encouraged by these results my fellow students and I determined to investigate the consequences of dividing the burden of taxation according to the capacity of each player to shoulder it. To this end we changed the tax that we gathered from each player after the rounds of labour and trade to
$t_i = e \times \left(\max\left(x^\prime_i - x_i - l_1, 0\right) + \max\left(x^\prime_i - x_i - l_2, 0\right) + \max\left(x^\prime_i - x_i - l_3, 0\right)\right)$
where
$0 \leqslant l_1 \leqslant l_2 \leqslant l_3$
The upshot of this is that, of a player's profits, that part beneath $$l_1$$ garners no tax, that part between $$l_1$$ and $$l_2$$ a share given by $$e$$, that between $$l_2$$ and $$l_3$$ one of twice $$e$$ and that above $$l_3$$ one of thrice $$e$$.
Note that, crucially, every player pays in tax the exact same share of their profits from within each of these ranges, in that any finite proportion of no funds is trivially equal to nothing.

Deck 6 demonstrates the impact of this progressive taxation upon the players' fortunes, with a rate of ten percent applied to profits at levels of one, two and four percent of their initial funds.

Deck 6: A Rule Of Progressive Taxation

This certainly appears to have had an even greater levelling effect for the players, but to be sure we put together deck 7 which redraws the histogram of the outcomes of hundreds of games with the new rule of taxation.

Deck 7: The Histogram Of Outcomes

That the peak of the histogram has moved further to the right and it falls away to zero much more quickly than before are clear evidences that the rule of progressive taxation has made for a much fairer game!
Deck 8 quantifies this by calculating the probabilities that a player might conclude the game with few than some fraction, or greater than some multiple, of their initial funds.

Deck 8: The Chances Of Losses And Gains

Finally, we put together deck 9 to figure how this rule affected the chances that a player who began the game poorly might conclude with greater funds than one who had begin well, by again assigning halved funds to half of the players and doubled funds to the others.

Deck 9: The Chance Of A Comeback

Evidently our rule of progressive taxation has very nearly erased the advantage to be had from a fortunate outset; a result that should be most welcomed by any player who has suffered at the hands of providence!

### In Conclusion

Time and again we have found that rules inspired by the ebb and flow of wealth amongst the populace have lead to games in which a fortunate minority of players profit greater above an unfortunate majority, despite their being applied with perfect fairness. Finally, however, with progressive taxation we have reckoned upon an effective countermeasure to the whim of chance.

Now I have no doubt that the Baron and his kin should object most strongly to the suggestion that providence, rather than worth, might account for their fortunes, nor that they should reject the claim that they should rightly pay a greater share of taxation.
But, given the outcomes of our games, I believe that it is now for them to show that their prosperity is not simply a consequence of their having been born into wealth and spared a fair level of taxation!
$$\Box$$

### References

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

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

[3] Further Still On The Wealth Of Stations, www.thusspakeak.com, 2016.