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Sunday 31 May 2009

Pascal’s Wager – For Richer, For Poorer

My Grandmother's Lottery

My grandmother, on being told by a friend that if they won the lottery they wouldn’t know what to do with the money, was affronted: “Don't be stupid”, she said, “I’d spend half going down one side of Main Street and the other half going up the other side”. Unfortunately, in the real world, we need to keep an eye on the both sides of the equation – is it worth taking the chance of becoming rich if the downside means being poor?

Blaise Pascal, in another context, tried to answer this question five hundred years ago. Admittedly, in a sign of our changing times, he was less concerned about his terrestrial wealth than his immortal soul. Nonetheless, his argument – forever encapsulated as “Pascal’s Wager” – can just as effectively be applied to the arguments for and against a balanced investment approach.

Divination Without Hindsight

I often read – for research purposes only, you understand – opinions on investment by the various analysts, professional and amateur, who populate the proliferation of media channels that we’re faced with these days. Although there are honourable exceptions most of the ‘information’ that spews out is really nothing but an updated version of the Oracle of Delphi – various attempts to divine the future in order to identify the precise way one should allocate one’s investment monies to maximise one’s returns.

Often this divination is developed in such a way to suggest that the author has an inside line on the future and that alternative ways of considering it are absurd. That such analysis – if we can dignify it with that term – is generally based on selective sifting of the evidence is almost inevitable. As I can testify, giving a fully balanced view of any complex process is pretty well impossible in any report less than a few hundred pages long. Still, the tendency is to promote a view that there is a right way and wrong way to invest at any one time.

Looking Forward, Looking Backwards

That view, of course, is correct – with hindsight. In the rear view mirror we can always see exactly what we should have done to become rich. We should have bought commodities in 2003, we should have bought government bonds in 2006, we should have shorted the markets in 2007. Unfortunately, most of us didn’t because, frankly, trying to understand the future in a world of vast complexity – trying to pick out the correct signal from the overwhelming noise – is just about impossible.

This analysis, though – if you’ll excuse me by dignifying it in that way – is really no better than that of the mass market pundits. Anyone can assert a view in the absence of evidence but the mere fact the world gives you a pulpit to do it from doesn’t mean that it’s right. The question is, whether we should take risks in advance of time in an effort to become wealthy when, if we select wrongly, the alternative is to end up poor. Pascal has an answer for us.

Blaise Pascal

Blaise Pascal was one of a group of fifteenth century French mathematicians who helped to develop many of the statistical analysis techniques that we apply to a wide range of problems today. Together with Pierre de Fermat (who was taking time off from posing bloody stupid theorems to waste years of future mathematicians' time) he figured out how to calculate probabilities as applied to games of chance – the beginning of the development of probability theory and, far in the future, risk management.

Pascal, however, oscillated throughout his life between the twin callings of mathematics and religion, eventually succumbing to the latter and, in so doing, accidentally moved the application of probability beyond its original use for gamblers. In the first known use of statistics to measure future risk and aid decision making he created the proposition known as Pascal’s Wager.

Pascal’s Wager

In Pascal’s world God was an everpresent. For the most part atheism was unknown, it was simply an argument of which God you believed in. Of course Martin Luther's habit of nailing wordy proclamations to cathedral doors was in the process of causing the schism of the Established Church across Europe, leading eventually to the development of all sorts of new world views. In the middle of this muddle Pascal found himself torn between religion and science and decided to analyse the problem, rationally, through the application of the newly invented probability theory.

At one end of Pascal’s spectrum was the possibility that God exists and that by living an irreligious life you condemn yourself to eternal damnation. At the other end was that God doesn’t exist and by living an abstemious life you deny yourself the pleasures of a worldly life. In all we have four conditions:

God ExistsGod Doesn't Exist
Behave WellEternity in HeavenBoring Life
Behave BadlyEternity in HellFun Life

Pascal assumed that all four conditions were equally likely and then multiplied the probability of each condition by his own personal weighting. As, in his world view, the downside of Eternal Damnation massively outweighed the other conditions the weighting he applied to this meant the "Behave Well" line won easily. Note two things here - firstly the weightings were Pascal's personal ones and secondly that the outcome wasn't an abstract probability but a real world decision - which Pascal appears to have acted upon.

Modern philosophers who are, by and large, a nit-picking bunch of academics have pointed out a number of possible weaknesses in Pascal’s Wager. Stuff like – which God are you talking about and why do you think there’s an afterlife anyway? Just wait till Judgement Day, we’ll see who’s nit-picking then.

Pascal's Investment Wager

We can apply Pascal’s Wager to modern investment approaches. We can choose to guess which way the markets are going and go all-in, or we can choose to adopt a longer-term, balanced approach. We know the latter will, almost certainly, mean we end up reasonably wealthy. We know that the former is, at best, a 50-50 gamble - worse, probably, but let's be generous. So we end up with a table that looks like this:

Guess CorrectlyGuess Wrongly
Gamble on FutureVery RichVery Poor
Don't Gamble on FutureComfortably WealthyComfortably Wealthy

If we assume that each condition is equally likely (it isn't, the Very Rich result is very unlikely) we then need to apply personal weightings to get to a decision point. Depending on how old you are, and how many dependents you have and myriad other possibilities your weightings may change. You may also be an inveterate risk taker or someone who has an overwhelming urge to be very rich or just very, very stupid. There will always be a group of people for whom the possible downside of being very poor is outweighed by the possible upside of being very rich.

Risk Management and Decision Making

Personally I reckon the gamble of poverty in old age isn't worth the risk, and I suspect that the majority of investors would feel the same, most of the time. In the middle of some periodic burst of stockmarket mania, however, it isn't always easy to think rationally - we use shortcuts, follow the herd and, all too often, follow the lemmings off a cliff. Pascal's Wager is one of a number of rationality tools that we can use to ground ourselves in the reality of what our decisions really mean.

The great breakthrough that Pascal made is that Pascal’s Wager is not just about probability, it’s about rational decision taking. It was the start of humankind teasing the signal out from the underlying noise and beginning to plan for the future.

That I think those people who are prepared to risk everything on the stockmarket lottery when they can guarantee themselves a safe and comfortable future are slightly mad is neither here nor there. That I suspect that they aren’t taking any kind of rational view of the Faustian pact they’re making is more to the point. If you’re going to deal with the devil it helps to know who’s dealing the cards.

A Piece of the Mountain:The Story of Blaise PascalPascals WagerAgainst the Gods: The Remarkable Story of Risk

Related Articles: Bulletin Boards Are Bad For Your Wealth, Overconfidence and Over Optimism, Cardano's Gambit


  1. I have no interest whatsoever in "The Market"/"God"/"The Leader" or whichever altar you might happen to be talking about, but you are making the same mistakes Pascal made. And I used to be a big fan of Pascal myself until I realized this.

    The point is that it is very presumptuous and obsessive to focus your life on some random exceedingly unlikely potential like becoming wealthy or going to heaven. (Especially calling this "planning for the future" when it is really just throwing it all away for absolutely nothing.) This is the very hit of the drug, addiction. A temptation of spoiling our lives away while missing the point that a "boring" life without the drug is not necessarily bad but may actually be more rewarding, especially since we are consciously aware of it.

    In the end, when you are at your last day and you think about your life spent accumulating wealth/"treasures in heaven" and nothing "else" of merit, your life is really over, and like your "Forrest Gump" analogy, you didn't even realize you completely wasted it. You won. Won what?

    This same nature results in the kind of hatreds that worshippers always exude. If you are not on your knees at the altar, you are a fool or an evil deceiver of fools. The hatreds that investors have for non-investors is par with the hatred that Christians have of atheists, or Muslims for Jews.

    The impression that I get is that it will be very bad for me in the future if I don't buy stocks/real estate/gold right now. As if other people aren't saying the same things about accepting Jesus.

    So go lie in bed with a bible, wad of cash, and bar of gold in your over-priced McMansion for all the comfort they give you.

  2. Pascal's wager, when used as an argument for a particular kind of behaviour rather than a particular kind of belief, is simply an application of what in probability theory is called expectation value.

    There are four parameters for each possibility: the proportion you invest in it; the probability of it happening; the outcome in public units and your personal valuation of those public units.

    Multiply them together and then add them all up and you'll get you overall expected returns: customized to your preferences. Of course, in Pascal's case, you can't really plan to invest half your time in pious living and half in sinful debauchery, they don't mix, you have to choose. But in money matters, you can do precisely that. More interestingly, the point of investment is not to maximize retirement income, it's to maximize personal objectives. Some classes of investors may expect to be quite well off whatever they do and value security above income: the loss, to them, of all their income is far greater than twice the loss of half of it. Conversely, others may be looking forwards to a bleak old age due to a financial disaster and for them, losing a few extra quid a week isn't going to make a lot of difference whereas a punt on a high-risk investment might, just might, lift them out of poverty. The key thing is to weight the outcomes for your own situation, not just go by the financial bottom line.