I started this post with the title “Death and Taxes.” (I know– I will move on to lighter subject matter soon.) But my references already both have this same new title above, and it sounds more interesting to me anyway, so I continue the theft.
This is a bit of a hodge-podge of ideas stemming from a convoluted train of thought this past weekend, but there is at least some common ground here. It began with the common saying about death and taxes, and led from there in two interesting directions. The first direction was taxation, specifically an argument about a flat tax, and an interesting counter-argument for progressive tax (i.e., higher tax rates for higher incomes) based on the idea of logarithmic utility of wealth.
The second direction was death (!), where I was reminded of an interesting article in an interesting book, that also involved the useful application of logarithms, in this case to a study of humans’ capacity to kill each other. This is the source of the title of the post; we’ll get to this in just a moment, but first let’s tackle progressive tax and log utility.
The idea is pretty straightforward: if you have $1,000,000 and your neighbor has $10,000, then adding– or taking away– $1,000 means a lot more to your neighbor than it does to you. In other words, the marginal usefulness of income decreases as the absolute income increases. This is often made mathematically precise by using the logarithm of some numeric representation of wealth as an indicator of the perceived utility of that wealth. As applied to a discussion of taxation, the argument against a flat tax is that with a flat (i.e., constant percentage) income tax, the perceived loss in relative utility from income tax is less for the wealthier than for the poorer.
I am not sure I buy this as a defense of progressive tax. But it’s an interesting idea. And this is only the lesser, motivational half of the story anyway. The more interesting half, to me, is an essay by Brian Hayes titled “Statistics of Deadly Quarrels,” discussing same-titled research by one Lewis Richardson on the history of “deaths… caused by a deliberate act of another person.” This includes not just wars, but individual murders, covering the period from about 1820 to 1950.
As Hayes points out, this aggregation of small and large scale, of “abominable selfish crime” and “heroic and patriotic adventure,” was intentionally provocative: “One can find cases of homicide which one large group of people condemned as murder, while another large group condoned or praised them as legitimate war. Such things went on in Ireland in 1921 and are going on now in Palestine.” Note that this was written in 1960.
Ok, enough heat, now for some light. Richardson took the very useful, and necessary, approach of classifying the various conflicts according to not their absolute number of deaths involved, but to the (base 10) logarithm of the number of deaths. For example, there were (and still are) only two wars with a “magnitude” of 7, meaning that 10-100 million people died, namely the two World Wars.
This logarithmic scale is handy, since it allows for the difficulty in obtaining very precise data, particularly for small-scale conflicts such as murders involving only a few people, but even for the larger-scale wars where it is not always clear exactly who died and when, and whether they should be “counted,” perhaps dying months or years later from complications resulting from wounds received in battle. Using a logarithmic scale allows comparison of sizes of conflicts without needing to be as precise about the exact size of any one conflict.
The essay is just one of several interesting reads in Hayes’ book referenced below. I recommend checking it out; to tease without spoiling the fun, another interesting observation from the study was the list of magnitude-6 wars. These are the wars involving between about 500,000 and 2 million deaths, the largest conflicts other than the two World Wars. It turns out that there are “only” seven such wars, and presumably we should all know what they were. I didn’t do so well, only being able to name three, and one of those was an uneducated guess. Can you name all seven?
Reference: Hayes, Brian. Group Theory in the Bedroom, and Other Mathematical Diversions. New York: Hill and Wang, 2008. This is a great collection of essays, one of which is titled as below. The “group theory in the bedroom” refers to mattress flipping; it is also an interesting read, and I think would make a great introduction of students to some simple group theory.
Reference: Richardson, Lewis Fry. Statistics of Deadly Quarrels. Pittsburgh: Boxwood Press, 1960.