Pareto Principle and Gini Coefficient explained in Terms of Trading Cards
The Pareto principle states that for many outcomes roughly 80% of consequences come from 20% of the causes. It was named after the Italian economist Vilfredo Pareto who observed that approximately 80% of Italy’s land was owned by 20% of the population. He then discovered that for other countries a similar allocation was true. Surprisingly, this distribution approximately applies to numerous other examples in economics, engineering, computer science, sports, health and safety as well as my trading card collection. I discovered that about 25% of my cards make up about 80% of my collection’s total value. Funnily, to a little lesser degree, the principle was also true eight years ago when my collection contained half as many cards and was worth about 30% of today’s value. Back then, 25% of my cards made up about 70% of the total value.
The Pareto Principle is a special case of the more general phenomenon of Pareto distributions which are described in terms of power laws. In mathematics, a power law describes a functional relationship between two quantities, where the relative change in one quantity results in a proportional relative change in the other quantity. In other words, one quantity varies as a power of the other:
One can see in fig. 1 that the power law relation indeed nicely describes the distribution of what proportion of value is given by the x-most valuable cards. On the other hand, if you plot the cumulative proportion of value on the y-axis you obtain a nice saturation curve.
My current collection (blue curve) obviously features a steeper slope than in 2013 (orange curve), meaning that fewer cards make up a bigger proportion of the total value. In a way, one could say that my collection became more unjust in terms of the distribution of card values.
One commonly used parameter to quantify the inequality among values of a frequency is the so-called Gini coefficient. It was introduced by the Italian statistician Corrado Gini who proposed it as a measure of inequality of income or wealth for different countries. A Gini coefficient is a single number between zero and one whereas zero expresses perfect equality, e.g. all cards are of equal value, and one expresses maximal inequality among values, e.g. the total value is concentrated on only one card withall other cards being worthless.
As depicted in fig. 2, the Gini coefficient can be derived from the Lorentz curve which is in our example the plot of the proportion of the total value of the cards (vertical axis) that is cumulatively given by the bottom x of the cards. Perfect equality would be given by a straight line at 45 degrees. Now the Gini coefficient is geometrically defined as the ratio of the area between the straight line representing perfect equality and the given Lorentz curve (A) divided by the area of the triangle below the straight line (A+B):
G=A/(A+B).
To illustrate how the x- and y-axis for the Lorentz plot can be determined, I chose ten cards as an example in fig. 3. The formulae for calculating the x-axis are given in column D and those of the y-axis in column E, whereas the first couple of cells are not evaluated due to illustrative purposes.
As determining the area of the Lorentz curve can be cumbersome, especially when the curve cannot be approximated by a nice function which can easily be integrated, there is also an analytic formula to calculate the Gini coefficient:
In the formula above, x_i represents the i-th element in non-descending order, i.e. you have to sort the table of cards according to their value in an ascending way. The parameter i represents the rank of each card in the ordered list and the number n stands for the total amount of cards.
Again, I tried to illustrate the formula above in terms of the Excel example given above. Column E is an auxiliary function to conveniently obtain the first sum:
The calculation of the Gini-Coefficient is given in cell G3 with the unevaluated formula depicted in G4.
For my total collection, the Gini coefficient was at 0.59 in 2013 and increased to 0.70 as of now (see fig. 5). The reason for the increased inequality of the value distribution could be that I just put a larger emphasis on buying more valuable cards in the last few years. But the fact that precisely the more valuable cards have shown the greatest increase in value should also be a significant contributor, as well. Almost as in actual society, the rich are becoming richer.
