Further and deeper exploration of paradoxes and challenges of intuition and logic can be found in my recently published book, Probability, Choice and Reason.

Blaise Pascal was a 17th century French mathematician and philosopher, who laid some of the main foundations of modern probability theory. He is particularly celebrated for his correspondence with mathematician Pierre Fermat, forever associated with Fermat’s Last Theorem. Schoolchildren learning mathematics are more familiar with him courtesy of Pascal’s Triangle. Increasingly, though, it is Pascal’s Wager, and latterly the Pascal’s Mugging puzzle, that has entertained modern philosophers. Simply stated, Pascal’s Wager can be stated thus: If God exists and you wager that He does not, your penalty relative to betting correctly is enormous. If God does not exist and you wager that He does, your penalty relative to betting correctly is inconsequential. In other words, there’s a lot to gain if it turns out He does and not much lost if He doesn’t. So, unless it can be proved that God does not exist, you should always side with him existing, and act accordingly. Put another way, Pascal points out that if a wager was between the equal chance of gaining two lifetimes of happiness and gaining nothing, then a person would be foolish to bet on the latter. The same would go if it was three lifetimes of happiness versus nothing. He then argues that it is simply unconscionable by comparison to bet against an eternal life of happiness for the possibility of gaining nothing. The wise decision is to wager that God exists, since “If you gain, you gain all; if you lose, you lose nothing”, meaning one can gain eternal life if God exists, but if not, one will be no worse off in death than by not believing. On the other hand, if you bet against God, win or lose, you either gain nothing or lose everything.

It seems intuitively like there’s something wrong with this argument. The problem arises in trying to find out what it is. One good try is known as the ‘many gods’ objection. The argument here is that one can in principle come up about with multiple different characterisations of a god, including a god that punishes people for siding with his existence. But this assumes that all representations of what God is are equally probable. In fact, some representations must be more plausible than others, if the alternatives are properly investigated. A characterisation that has hundreds of million of followers, for example, and a strongly developed set of apologetics is at least a bit more likely to be true than a theory based on an evil teapot.

Once we begin to drop the equal-probability assumption, we severely weaken the ‘many gods’ objection. Basically, if it is more likely that the God of a major established religion is possibly true (however almost vanishingly unlikely any individual might think that to be) relative to the evil teapot religion, the ‘many gods’ objection very quickly begins to crumble to dust. At that point, one needs to take seriously the stratospherically high rewards of siding with belief (at whatever long odds one might set for that) compared to the stakes.

It is true that infinities swamp decisions, but we need not even go as far as positing infinite reward for the decision problem relative to the stakes to become a relatively straightforward one. It’s also true that future rewards tend to be seriously under-weighted by most human decision-makers. In truth, pain suffered in the future will feel just as bad as pain suffered today, but most of us don’t think or behave as if that’s so. The attraction of delaying an unwelcome decision is well documented. In the immortal words of St. Augustine of Hippo in his ‘Confessions’, “Lord make me pure – but not yet!”

A second major objection is the ‘inauthentic beliefs’ criticism, that for those who cannot believe to feign belief to gain eternal reward invalidates the reward. What such critics are pointing to is the unbeliever who says to Pascal that he cannot make himself believe. Pascal’s response is that if the principle of the wager is valid, then the inability to believe is irrational. “Your inability to believe, because reason compels you to and yet you cannot, [comes] from your passions.” This inability, therefore, can be overcome by diminishing these irrational sentiments: “Learn from those who were bound like you. . . . Follow the way by which they began; by acting as if they believed.”
Even some modern atheist philosophers admit to struggling with the problem set by Blaise Pascal. One attempt to square the circle is by saying that in the world where God, as conventionally conceived, exists with a non-zero probability, there is a case for pushing a hypothetical button to make them believe if offered just one chance, and that chance was now or never. Given the chance of delaying the decision as long as possible, however, it seems they would side with St. Augustine’s approach to the matter of his purity.

Pascal’s Wager has taken on new life in the last couple of decades as it has come to be applied to the problems of existential threats like Climate Change. This issue bears a similarity to Pascal’s Wager on the existence of God. Let’s say, for example, there is only a one per cent chance that the planet is on course for catastrophic climatic disaster and that delay means passing a point of no return where we would be powerless to stop it. In that case, not acting now would seem a kind of crazy. It certainly breaches the terms of Pascal’s Wager. This has fittingly been termed Noah’s Law: If an ark may be essential for survival, get building, however sunny a day it is overhead. Yes, when the cost of getting it wrong is just too high, it probably pays to hedge your bets.

Pascal’s Mugging is a new twist on the problem, which can if wrongly interpreted give comfort to the naysayers. It can be put this way. You are offered a proposal by someone who turns up on your doorstep. Give me £10, the door-stepper says, and I will return tomorrow and give you £100. I desperately need the money today, for reasons I’m not at liberty to divulge. I can easily pay you anything you like tomorrow, though. You turn down the deal because you don’t believe he will follow through on his promise. So he asks you how likely you think it is that he will honour any deal you are offered. You say 100 to 1. In that case, I will bring you £1100 tomorrow in return for the £10. You work out the expected value of this proposal to be 1/100 times £1100 or £11, and hand over the tenner. He never comes back and you have, in a way, been intellectually mugged. But was handing over the note irrational? The mugger won the argument that for any low probability of being able to pay back a large amount of money there exists a finite amount that makes it rational to take the bet. In particular, a rational person must admit there is at least some non-zero chance that such a deal would be possible. However low the probability you assign to being paid out, you can be assigned a potential reward, which need not be monetary, which would outweigh it.

Pascal’s mugging has more generally been used to consider the appropriate course of action when confronted more systemically by low-probability, high-stakes events such as existential risk or charitable interventions with a low probability of success but extremely high rewards. Common sense might seem to suggest that spending money and effort on extremely unlikely scenarios is irrational, but since when can we trust common sense? And there’s no reason to believe that it serves us well here either.

Blaise Pascal was a very clever guy and those who over the centuries have too quickly dismissed his ideas have paid the intellectual (and perhaps a much bigger) price. Today, in an age when global existential risk is for obvious reasons (nuclear annihilation not least) a whole lot higher up the agenda than it was in Pascal’s day, it is time that we revisit (atheists, agnostics and believers alike) the lessons to be learned from ‘The Wager’, and that we do so with renewed urgency. The future of the planet just might depend on it.

Exercise

In the Pascal’s Mugging Problem you are offered £3,000 tomorrow if you pay the stranger £25 today. You believe that there is a 1 in 100 chance that the stranger will return to pay you.

Is handing over the £25 rational from an economic point of view? Would you hand over the £25? What if the stranger offered to pay you £10,000 tomorrow, and you believe there is a 1 in 125 chance that he will return to pay you?

Would your answer be different if any of the sums involved were different?