The Infinite Money Paradox

Vsalsa! Here Kevin and I have a simple coin toss game that requires no skill, has no trap or gimmick, and can generate

infinite

wealth. The thing is...no one really wants to play it. Because? How is it possible that an incredibly easy game with endless advantages makes practically everyone react with a huge yawn? To play this game, we will turn to the most rational and calculating man in history: Vsauce2's old friend, Dwight Schrute. You approach the table to flip a coin. Your prize starts at $2. If the result of the coin toss is FALSE, the game ends and you win $2.

If it appears DONE, you play another round and your prize is doubled. Every time you get a FACT, you keep playing and the prize keeps doubling: from $2 to $4 to $8 $16 $32 $64. $128, and so on… forever. But as soon as you get a FALSE, you are done and can collect your winnings. So, if you hit a FALSE in the third round, then your prize will be $8. If your first FAKE comes in Round 14, you would take home $16,384. No matter how unlucky you are, you will never earn less than $2. If things go very well... then things could go very well. Now that you know the possible benefits, how much would you be willing to pay to play this game? $3?

How about $20, $100? The profits could be

infinite

, so the question is: how much is the possibility of infinite wealth worth to you? We can determine the precise answer, but first we need to know the expected value of the game, which is the sum of all its possible outcomes relative to their probability. That determines the point at which we choose to play a game or, in the real world, the point at which we decide to purchase insurance for our home or a life insurance policy. If our risk is less than our possible reward, we should play. If we pay too much relative to what we are likely to get from playing, then we shouldn't play.

Here is Schrute's expected value. You have a 50/50 chance of losing on your first toss and returning to the beet farm with $2. With probability ½ and a payoff of $2, its expected value in the first round is $1. The probability of winning two rounds is ½ * ½, or ¼, and your prize there would be $4. That's another dollar in expected value. For three successful throws, it is ½ ^3, or ⅛, multiplied by $8. Another dollar. 1/16 * 16… 1/32 * 32… 1/64 * 64… For n rounds, the expected value is the probability (½)^n * the payoff of 2^n, so the value of n does not matter, the result will be 1. The expected value of the game is 1 + 1 + 1 + 1 + 1… forever.

Because each round adds $1 of value no matter how rare the event is. The expected value is infinite. And therein lies our

paradox

. Because you would think that a rational person would pay all the

money

he has to play this game. Mathematically, it makes sense to pay any amount of

money

less than infinity to play. No matter how much money you risk, in theory you're getting the deal of a lifetime: the reward justifies the risk. But nobody wants to do that. Who would empty their bank account to play a game where they know there is a 75% chance of coming away with $4 or less?

It's confusing because expected value is, mathematically, how you determine whether you'll play a game. Look, if I offered you a coin toss game in which you would win $5 if it came up heads and lose $1 if it came up tails, your expected value of each round would be the sum of those possible outcomes: (50% probability * +$5) + (50% probability * -$1). Half the time you will win $5 and the other half you will lose $1. In the long term, you will earn an average of +$2 for each round you play. So paying less than 2 dollars to play that game would be a great deal. When the price to play is less than the expected value, it's a no-brainer.

And since the expected value of the Schrute game is infinite, paying anything less than infinite money to play it should also be a no-brainer. But is not. Because? What's interesting about this game is how mathematics conflicts with... real humans. Enter: Prospect theory. An element of cognitive psychology in which people make decisions based on the value of wins and losses rather than just theoretical results. The reason people don't want to empty their pockets to play this game despite its infinite payoffs is that the expected marginal utility (its real value to them) decreases as those mathematical payoffs increase forever.

This solution was discovered a few years ago. A few hundred years ago. In 1738, Daniel Bernoulli published his “Exposition of a New Theory of the Measurement of Risk” in the Commentaries of the Imperial Academy of Sciences in St. Petersburg, and what we now call the St. Petersburg Paradox was born. Bernoulli did not question the expected value of the St. Petersburg match; Those are cold, hard numbers. He just realized there was so much more. Bernoulli introduced the concept of expected utility of a game, which until the 20th century was called moral expectation to differentiate it from mathematical expectation.

The main point of Bernoulli's resolution was that utility, or how much you care about a thing, is relative to an individual's wealth and that each unit tends to be worth a little less to you as you accumulate it. So, as an example, not only would winning $1,000 mean a lot more to someone who is broke than, say, Tony Stark, but even winning $1 million wouldn't hurt Stark Industries' research and development. And there is also a limit to a player's comfort with risk: John Maynard Keynes argues that high relative risk is enough to prevent a player from playing a game even with infinite expected value.

Iron Man can afford to lose a few billion. You probably can't. And the value itself is subjective. If I won 1,000 peanut butter and jelly sandwiches, I would be THRILLED. If someone allergic to peanuts won them, I'd be...less excited. So. Well well. Taking all this into account, how much can you afford to lose in the St. Petersburg match? How much do you want to play? Bernoulli used the logarithmic function to calculate prices that took into account not only the expected value of the game, but also the wealth of the player and his expected utility. A millionaire should feel comfortable paying up to $20.88 to trade Schrutes, while someone with just $1,000 would pay a maximum of $10.95.

Someone with a total of $2 of wealth would, according to the logarithmic function, borrow $1.35 from a friend to pay $3.35. Ultimately, everyone has their own price that influences their wealth, their desires, their comfort with risk, their preferences, how they want to spend their time, what else they could be doing with their money, their own happiness... And The thing is... this game can't even exist. Economist Paul Samuelson points out that a game of potentially infinite gains requires the other party to be comfortable with a potentially infinite loss. And no one is okay with that. So if the important elements are variable and the game cannot exist, what's the point?

The St. Petersburg

paradox

reminds us that we are all more than mathematics. The raw numbers might convince a robot that it's a good idea to bet his robot house on a series of coin tosses, but deep down you know it's a very bad idea. Because you are not a calculation of expected value. You are not a logarithmic function. Numbers are part of you and help you live your life. But in the end, it's... you. Made. And as always, thanks for watching.