Pascal’s Mugging

The Concept of Pascal’s Mugging — Pascal’s Mugging is like Pascal’s Wager. If you don’t know what Pascal’s wager is, here’s a refresher. Pascal’s Wager is that if (his version) of Christianity is correct, then the reward for believing in God is infinite, and the punishment for lack of believe is also infinite. If Christianity isn’t correct, then the reward or punishment for belief or lack of belief is finite. Thus, no matter how improbable God’s existence is, any non-zero probability of his existence justifies a belief in God.

Pascal’s Mugging is similar. Imagine that a mugger stops Pascal, and pulls him into a dark alleyway. The mugger, however, forgot to bring a weapon. The mugger, being clever, decides to try to make a deal with Pascal. (This isn’t an exact retelling.)

“If you give me your wallet, tomorrow I will return here at this time with twice the amount of money in the wallet.”

Pascal laughs off the proposal, but the mugger tries again.

“If you give me your wallet, tomorrow I will return with 100 times the amount of money you gave me. If you think there is even a 5% chance that I am an honorable person, the expected payoff is much greater than what you are giving me today.”

Pascal again refuses, but the mugger is nothing if not persistent. (This is where things become absurd, and the argument breaks down).

“There is some probability p that I am honorable. If I promise that, tomorrow, I will return with more than 1/p times the amount of money in the wallet, the expected payoff for you is positive.”

In the version of the story I first read, the mugger succeeds by promising 1,000 quadrillion happy days of life to Pascal in exchange for the wallet. That’s quite a lot to promise for a wallet, but apparently either Pascal thought the mugger being honorable was exceedingly unlikely, or the contents of the wallet was extremely valuable.

Why it fails — I think it’s pretty clear that there’s something problematic with the mugger’s logic. You can’t just promise someone increasingly large quantities of something good in order to convince them to give you something. Then again… I hear the lottery is pretty successful.

But here’s the difference between Pascal’s mugging and the lottery: in the case of pascal’s mugging, the probability that the person making the proposition will be able to provide what they’re promising approaches zero far faster than what they’re promising grows. If I were giving the mugger ten dollars, I’d estimate about a 1% probability that they’d be able to procure $100 the next day (expected loss: $9). But if the mugger promised me ten billion dollars, I’d put the probability that they’d be able to procure those 10 billion dollars at far, far less than 1 in 10 billion. The more they promised, the greater the expected loss, because the lower the probability that they’d be able to give me the money. This is the obvious problem with Pascal’s mugging.

There’s a problem with it that’s far less obvious, however, and also more less applicable to daily life. Imagine you’re planning on buying a $50 stock, and you estimate that each day over the next 10 years, there’s a fifty-fifty chance of the stock going up or down by 5% of it’s value at that time. What’s the expected value of the stock after 10 years? $50, same as when you started. The average of a 5% increase and a 5% decrease is a change of 0%. But that’s just the average. What’s most likely going to happen is that the stock will be worth around $0.52, or 52 cents, by the end of that 10 year period. Why the discrepancy? Imagine the stock alternated between going up 5% and going down by 5%. Every two-day cycle (up the first day, down the next) would take off 0.25% from the stock’s value (1.05*0.95=0.9975). Those 0.25%’s add up, and over time the stock will most likely lose value. The only reason the expected value comes out to $50 is because there are a few astronomically unlikely cases where the stock becomes enormously valuable, and when those are factored in, the average comes out to $50. The average person can’t live life maximizing expected gains. Unless someone is fabulously wealthy, a loss has greater impact on that person’s life than an equivalent gain in wealth. Only the rich can afford to have the extremely diverse investment portfolios necessary to ensure they receive the expected value rather than the most likely value. Only the rich can afford to safely place the bets you need to become rich.

This is definitely something to think about.

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