When you look back at your life, do you add, or do you multiply? Consider your life, summarizing the value of it, is this a matter of addition, or of multiplication?

For most people it's addition. You add your income and subtract (add the negative of) your expenses. Every day something new happens, and it adds to your life. That's the general idea most people have.

Wrong!

I mean, it's okay, it's not incorrect, quite, it has its place, it's the Accountant's view.

But what you want to do, to make the most of your life, is to take the Investor's view. You want to multiply: multiply what you are, or have, or do, by (1+r) where r is the return, hopefully a positive number, if your asset, your investment, your accumulated achievement, your total life impact, is growing.

So this little essay is about summarizing a history of events, like your past or imagined-future life, using the concept of a Mean, which means a type of Average. Have a peek, maybe it'll change the way you make decisions in your life.

What is the value of a history of outcomes?

Here is the basic question: you have some outcomes, let's say you measured them, how will you summarize them? Many answers: The sum, the count, the average, the median (the value of the middle one after sorting), the mode (the most common value). Did you ever consider the product as a summary, or the log-sum?

Sum is such a good summary it is the root of the word "summary"; we use it in financial accounting, ‑expenses+income, to see where we are. But sum varies, usually increases, with count or over time, and it hides speed or rate. Count is good but it doesn't give a typical magnitude, just an event count or duration which hides possibly important information with different events counting as the same, one, each. Average is our bread and butter, \( \mu = sum/count \), scaling the sum by the count to try to get at the Typical.

You know all this already, of course!

Now consider three approaches to extracting the Typical from some outcomes (say, a set of N numbers { x_i | 1 ≤ i ≤ N }), using Plus (which includes Minus where x_i < 0) and Times, and Divide. (These are also called the Pythagorean Means.)

Combinator	Common Name	Technical Name	Formula
+	"average"	Arithmetic Mean	$$ \frac{1}{N} \sum_{i=1}^{N} x_{i}$$
*	N/A	Geometric Mean	$$ \sqrt[N]{\prod_{i=1}^{N} x_{i}}$$
/	N/A	Harmonic Mean	$$ \frac{1}{\frac{1}{N}*\sum_{i=1}^{N} \frac{1}{x_{i}}} $$

Each is best for its appropriate context of use, but all help us imagine a kind of summary.

Tourists and Gamblers, Workers and Investors have their preference for Arithmetic and Geometric Means. More on those, presently.

The Harmonic Mean, strange beast, is the inverse of the average of the inverses. Darius inspired this whole essay today by teaching me about the harmonic mean. Thanks, Daryoush! I've heard about the Golden Mean, but this is different.

The Harmonic Mean is best for an electronic circuit with resistors in parallel: the performance of the whole circuit is the harmonic mean of its parts. It's also great for estimating "true" travel durations, where the smallest x_i dominates the results. For example, suppose on a given route, one driver takes 2 minutes, and another takes 100, then the harmonic mean formula is

$$ \frac{1}{ \frac{1}{2} * (\frac{1}{2} + \frac{1}{100}) } = 2 * \frac{1}{ 0.5 + 0.01 } = \frac{2}{ 0.51 } = 3.92 $$

3.92 is a whole lot closer to 2 than it is to 100. In general the Harmonic Mean is appropriate if the deviations are mostly on the delay side and If delays are less informative. Maybe the 2nd driver stopped at McDonald's on the way. Unusually extra-long durations should indeed be discounted. Whereas driving times still don't go to zero even if you drive at 100MPH. The shortest times are on average the truest, in some sense of average. The correct sense is "the Harmonic Mean."

Contrast that with the Arithmetic mean, where neither small nor large dominates, all count equally (including negatives!), just add them up (and divide by N), that's the 'average'.

Or finally the Geometric Mean, where the close-to-zero is what dominates. A product of numbers, where any one is zero, is zero as the entirety.

The Gambler's Mean

Everything has its context. Our context is, we are here in the middle of our lives, plugging along, doing what is needed to survive, watching how it's going, maybe discovering some new things or empowering ourselves to achieve something more, and hopefully making something more of our lives. Am I wrong?

Maybe you need to think about it.

I'll assume not.

Now, does this context actually impose any structure on our thinking? Does it tell us what kind of Mean to use? I say, actually, it does, Yes.

We need to think about this in terms of Gambling, however much we'd like to imagine otherwise. What life actually is, is a random sequence of bets on unknown outcomes. It challenges morality! Responsibility means not wasting resources or opportunities, and minimizing the risks of negative outcomes. But we don't actually control the future completely. Intellectual responsibility means thinking about the unknown as being unknown, not pretending otherwise. Therefore we must consider our lives very much as, no, exactly like, Gambling. Not as in, irresponsibly, but as in, actually betting your life and not actually knowing what the outcome will be. We want to be good gamblers, is what I'm saying, because we are all betting our lives: we ARE gamblers, and confronting that fact will let us think more clearly -- and be more responsible!

So what do gamblers do to figure out the averages?

To calculate the "average" casino day, try this: Drive two busloads to the casino, 100 tourists, each with $100 to bet, and look at their wallets at the end of the day. A few will win, most will lose some, and some will lose everything and have zero left. Add up the total, divide by the number of people, that's the average return on going to the casino with $100. Maybe the average is $75, or $50, but it's surely less than $100 unless the casino had a bad day, and it's surely more than $0 unless there was a robbery. Let's call that the one-day, Holiday, or Tourist's average, it's also just the "average", technically speaking. (Context will help determine what "average" is intended to mean in different situations; it's not always this one.) The Tourist's Average is history-oblivious. It is path-independent, because every path is identical, has length one, is a single event.

Indeed the Tourist's Average is very different from the Investor's or Gambler's Average, which is path-dependent and history-bound. The gambler goes to the casino 100 days in a row, and takes all his or her winnings from one day and plays them the next day and then again the next day. At some point s/he is going to have a bad day and hit zero. Then the rest of their casino adventures are in the bar and the restaurant, because the betting wallet is empty. The gambler's average return is the geometric mean: MULTIPLY all the returns, and take the Nth root of the product, that's the Gambler's Average. Because if there is a single zero in that list, then the whole result is zero.

Thus with the geometric mean, similar to the harmonic mean, the smallest also dominates the results. Suppose you take the Gambler's Average of a 50% loss and a 50% gain.

$$G = \sqrt{(0.5)*(1.5)} = 0.866$$ Even though x₁=0.5 and x₂=1.5 are both 50% away from the neutral, no-change return, the loser which returns 0.5 of the investment drags the whole result down from neutral even though the winner is identically 50% away from neutral. Indeed if any instance hits zero, then the whole thing goes to zero. \( \sqrt[N]{0 * \prod_1^N{x_{i}}} = 0 \)

In exactly this way, arithmetic vs geometric mean differ for investment gains. It turns out 50% loss is not equal to 50% gain, so you can't just add them up like a tourist, your "average" of 100% return of capital might be an ACTUAL loss, because your two successive bets returned +50% and -50% with the result being a final cumulative return of 0.866 - 1 = -13.4%. Failure to lose ought to mean 100% return of principal; but if you had a brain fart and a 50% loss, now you need a 100% gain, Mister Gambler.

Okay this has been fun, and yes now you are better armed to not be taken advantage of by hedge funds and investment advisers that advertise their average returns, (or by Snopes when they misrepresent pharmaceutical results).

Right? Let's make up a nice clear case. Suppose your hedge fund made one bet in January, and lost 50%, then another bet in February and March together, and gained 75%. Then on April 1 your million dollars invested just became \( 1M * 0.5 * 1.75 = 0.875M \), which is a 1/8th or 12.5% cumulative LOSS, but they can advertise to their new customers, Hey, we had some losses, some gains, but our "average" return was (-50% +75%)/2 = +12.5% GAIN. April Fools!

So no, when you are dealing with cumulative results, use the Geometric mean. Unfortunately humans prefer to think in terms of the additive or Arithmetic mean, which is what gives the meaning to April Fool's Day.

Don't be a fool. We (you too) are in cumulative mode, like the gambler who bets his wallet day after day. So use the Geometric Mean, since we are all betting our whole wallet, our whole life, day after day. If we have a zero on any given day, we are completely done, so avoiding the zero event is important, and thinking about life Geometrically is how we actually can do our best.

Postscript

This little essay on means was actually the catalyst for my longer paper generalizing economics from two-way financially-valued transactions to one-way flows with multiple kinds of value. This "Benefax Flows Model" enables you to assess the value of your decisions or choices according to your own particular valuation of the larger, multi-dimensional value space including but not limited to financial value or Benefax, while considering the downstream ripple effects as value multipliers, just like the Investor's Mean.

If you can think of your choices having value in the long term, then the knock-on effects that your choice triggered in other people's choices influenced by what you chose, in all the different dimensions personally valued by you, ought to be part of your decision. If a choice gives you a legacy that multiplies over time, then that would be a good choice, and multiplicative thinking would help you to be clear about it, so as to maybe make better choices in this uncertain world, and without restricting you to financial payoffs only.

Perhaps parenthood, entrepreneurship, and knowledge creation are the most obvious, enormous, rippling Benefax generators, while everyone benefits from rippling benefactual flows in the form of Connection and Gifts as both recipient and giver, producing much of what is meaningful and long-term satisfying in life. Therefore, while expanding your thinking from just money, also think multiplicatively: your choice has a downstream knock-on effect, which multiplies the value of that choice, and might help you to most wisely choose.

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