Vsauce! Kevin here, and I have a simple coin-flipping
game, that requires no skill, has no catch
or trick, and can lead to infinite wealth. The thing is… nobody really wants to play
it.
Why?
How is it possible that an incredibly easy game with infinite upside causes virtually
everyone to react with a massive yawn?
To play this game, we’ll turn to the most rational, calculating man in history: long-time
friend of Vsauce2, Dwight Schrute. You walk up to the table to flip a coin. Your prize
starts at $2. If the coin flip results in FALSE, the game is over and you win $2. If
it lands on FACT, you play another round and your prize doubles. Every time you get a FACT,
you keep playing and the prize keeps doubling -- from $2 to $4 to $8 $16 $32 $64. $128,
so on and so on… forever.
But as soon as you get a FALSE, you are done and you collect your winnings. So if you hit
a FALSE in the third round, then your prize is $8. If your first FALSE comes in Round
14, you’d walk away with $16,384. No matter how unlucky you are, you’ll never win less
than $2. If things go really well… then things could go really well.
Now that you know the potential payoffs, how much would you be willing to pay to play this
game? $3? What about $20, $100? The winnings could be infinite so the question is: how
much is a chance at infinite wealth worth to you?
We can determine the precise answer, but first we need to know the game’s expected value,
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 take out insurance on our house or a life insurance policy. If our risk is
less than our likely reward, we should play. If we’re paying too much relative to what
we’re likely to get out of playing, then we should not play.
Here’s the expected value of Schrute.
You’ve got a 50/50 chance of losing on your first flip and heading back to the beet farm
with $2. With a probability of ½ and a payoff of $2, your 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 $1 in expected value. For three successful flips, it’s ½ ^3 -- or
⅛ -- times $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 no matter
the value of n, 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 occurrence
might be. The expected value is infinite.
And there’s our paradox. Because, you’d think a rational person would pay all the
money they have to play this game. Mathematically, it makes sense to pay any amount of money
less than infinity to play. No matter what amount of money you risk, you’re theoretically
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’s a 75%
chance they walk 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-flipping game where you won $5 on heads
and lost $1 on tails, your expected value of each round would be the sum of those possible
outcomes: (50% chance * +$5) + (50% chance * -$1). Half the time you’ll win $5, half
the time you’ll lose $1. In the long run, you’ll average +$2 for every round you play.
So paying anything under $2 to play that game would be a great deal. When the price to play
is less than your 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 it’s not. Why?
The thing that’s so interesting about this game is how the math conflicts with.... actual
humans. Enter: Prospect Theory. An element of cognitive psychology in which people make
choices based on the value of wins and losses instead of just theoretical outcomes.
The reason people don’t want to empty their pockets to play this game despite its infinite
gains is that the expected marginal utility -- its actual value to them -- goes down as
those mathematical gains 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 on the Measurement of Risk"
in the Commentaries of the Imperial Academy of Science of Saint Petersburg -- and what
we now call the St. Petersburg Paradox was born. Bernoulli didn’t dispute the expected
value of the St. Petersburg game; those are cold, hard numbers. He just realized there
was a lot more to it. Bernoulli introduced the concept of the expected utility of a game
-- what was, until the 20th century, called moral expectation to differentiate it from
mathematical expectation.
The main point of Bernoulli’s resolution was that utility, or how much a thing matters
to you, 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’s
broke than it would to, say, Tony Stark, but even winning $1 million wouldn’t affect
the research and development at Stark Industries.
And there’s also a limit on a player’s comfort with risk, with John Maynard Keynes
arguing that a high relative risk is enough to keep a player from engaging in a game even
with infinite expected value. Iron Man can afford to lose a few billion. You probably
can’t.
And 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, they’d be… less thrilled.
So. Okay, okay. Given all this, how much can YOU afford to lose in the St. Petersburg game?
How badly do you want to play? Bernoulli used the logarithmic function to come up with price
points that factored in not only the expected value of the game, but also the wealth of
the player and its expected utility. A millionaire should be comfortable paying as much as $20.88
to flip Schrutes, while someone with only $1,000 would top out at $10.95. Someone with
a total of $2 of wealth should, according to the logarithmic function, borrow $1.35
from a friend to pay $3.35.
Ultimately, everyone has their own price that factors in 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 gain requires the other party to be comfortable with potentially
infinite loss. And no one is cool with that.
So if the important elements are variable and the game can’t exist, what’s the point?
The St. Petersburg Paradox reminds us that we’re all more than math. The raw numbers
might convince a robot that it’s a good idea to wager its robohouse on a series of
coin flips, but you know deep down that’s a really bad idea.
Because you aren't an expected value calculation. You aren't a logarithmic function. The numbers
are a part of you and help you live your life. But in the end, you are… you.
Fact.
And as always, thanks for watching.