Infinity poses a different sort of challenge to expected utility. In this chapter we’ll meet two famous examples.
Suppose I’m going to flip a fair coin, and I’m going to keep flipping it until it lands heads. When it does, the game is over.
How much would you be willing to pay to play this game? Most people aren’t willing to pay very much. After all, you probably won’t win more than a few dollars. Most likely you’ll only win
And yet, bizarrely, the expected value of this game is infinite! There’s a
Visually, the St. Petersburg game looks something like Figure 14.1. We can’t show every possible outcome, because the payoffs get larger and larger without limit. So we have to cut things off at some point. But we can get the general idea by displaying just the first few possible outcomes. (The rest are shown faded and only partially.)
As the number of potential flips grows, the rectangles get narrower because the outcomes become less probable. But they also get taller. The payoff doubles every time the probability is cut in half. The result is an infinite sequence of rectangles, each with the same area, namely
Most people, when they first encounter this puzzle, try to take the easy way out. “The game is impossible,” they say. “Nobody could actually fund this game in real life. No casino has unlimited money, not even the government does. And nobody can guarantee they’ll live long enough to finish the game either, since it could potentially go on for any number of flips.”
But there are two problems with this answer.
First, notice that we can make the game instantaneous. Instead of flipping a coin over and over, we throw a dart at a square board just once. If it lands in the left half, you win
We still need an infinite bankroll, since there’s no finite cap on the possible amount you could win. But consider whether you’d be willing to risk your life to play the game if the casino did have enough cash on hand. Imagine if God descended from heaven and offered to run the game. Would you gamble your life to play then? It’s a fanciful scenario, but if our decision theory really is correct then it should still work even in fanciful scenarios.
Second, even if the game had a finite limit—let’s suppose it was
Figure 14.2: Daniel Bernoulli (1700–1782)
The St. Petersburg game was invented by the mathematician Nicolaus Bernoulli in the
His solution: replace monetary value with real value, utility in other words. Recall that the more money you have, the less value additional money brings. So even though the payoffs in the St. Petersburg game double from
Figure 14.3: Bernoulli’s logarithmic utility function
What is the real value of money then? How much utility does a gain of
Notice how, for example,
The difference this makes to the St. Petersburg game is displayed in Figure 14.4. The rectangles don’t all have the same area of
So if Bernoulli is right about the utility of money, the fair price for the St. Petersburg game is only about
Unfortunately, although Bernoulli was probably right that money decreases in value the more of it you have, that doesn’t actually solve the paradox. Because we can just modify the game so that the monetary payoffs grow even faster.
The decimal value of
Instead of the payoffs increasing like this:
Figure 14.5: St. Petersburg’s revenge
Now the utilities are the same as the dollar payoffs were in the original version of the game:
What’s the right solution to the St. Petersburg paradox then? Nobody knows, really. Once infinities get involved, the whole expected value framework seems to go off the rails.
Some decision theorists respond by insisting that there’s a finite limit on utility. There’s only so good an outcome can be, they say.
But others don’t find this response plausible. There may be a limit on how much good you can get out of money, because there’s only so much money can buy. But money is only a means to an end, a medium we can exchange for the things we really want—things of intrinsic value like pleasure, happiness, beauty, and love. Is there really a finite limit on how much value these things can bring into the world? If so, what is that limit, and why?
Figure 14.6: Blaise Pascal (1623–1662)
The modern theory of probability has a curious origin. It started with Blaise Pascal, a French mathematician and philosopher living in the
Pascal was a devout Catholic. So, once he developed the tools of decision theory, he applied them to religious questions. Like: is it rational to believe in God?
Pascal realized he could think of this question as a decision problem. If God exists, then believing gets you into heaven, which is very good. Whereas not believing gets you a one-way ticket to hell, which is terribly bad. So believing in God looks like the better option.
But Pascal also realized that probabilities matter as much as the potential payoffs when making decisions. Playing the lottery might win you millions, but the odds are very poor. So spending your money on a cup of coffee might be the smarter choice.
Likewise, believing in God might be such a long shot that it’s not worth it, even if the potential payoff of heaven is fantastic. Of course, Pascal himself already believed in God. But he wanted to convince others to do the same, even if they thought it very unlikely that God exists.
The potential payoff of believing in God is special though, Pascal realized: it’s not just very good, it’s infinitely good. If you believe in God and you’re right, you go to heaven for eternity, a neverending existence of pure ecstacy. Whereas if you don’t believe in God and you’re wrong, the payoff is infinitely bad: an eternity in hell.
So Pascal figured the decision problem looks something like this:
Table 14.1: Pascal’s Wager
God Exists | God Doesn’t Exist | |
---|---|---|
Believe | ||
Don’t Believe |
The
It doesn’t matter, as it turns out. Whether we use
Why do these finite values not matter in the end? How do they get drowned out?
Well, said Pascal, even atheists must admit that there’s some small chance God exists. Nobody can be
Also,
What about not believing? It has infinitely negative expected value:
Some people criticized Pascal’s argument on the grounds that belief is not a decision. Whether you believe in something isn’t voluntary, like deciding what shirt to wear in the morning. You can’t just decide to believe in God, you can only believe what seems plausible to you based on what you know.
But, Pascal famously replied, you can decide how you spend your days. And you can decide to spend them with religious people, reading religious books, and going to a house of worship. So you can decide to take steps that will, eventually, make you a believer. And since believing is so much better than not, that’s how you should spend your days.
A more serious problem with Pascal’s argument is known as the many gods problem.
In Pascal’s day, in France, Catholicism dominatd the religious landscape. So for him, believing in God just meant believing in Catholicism’s conception of God. But there are many possible gods besides the Catholic god, like the god of Islam, the god of Judaism, the gods of Hinduism, and so on.
What happens if you choose the wrong God? You might go to hell! The god of Mormonism might send you to hell for believing in Catholicism, for example. There might even be an anti-Catholic god, who sends all Catholics to hell and everyone else to heaven!
So the correct decision table looks more like this:
Table 14.2: Pascal’s Wager with many gods. The … stands in for the many different gods that might exist, each of which has its own column. There is also a row for each of these possible gods, since we have the option to believe in that god.
Catholic God Exists | Anti-Catholic God Exists | … | No God Exists | |
---|---|---|---|---|
Believe Catholic | … | |||
Believe Anti-Catholic | … | |||
Don’t Believe | … |
What’s the expected utility of believing in the Catholic god now? It turns out there’s no answer! The calculation comes out undefined:
Imagine we start with an infinite list, a list of all the counting numbers for example:
But not so fast! Suppose we start again with all the counting numbers:
The moral: there are many ways to take an infinite quantity away from an infinite quantity. Some of these leave an infinite quantity remaining. Others leave nothing remaining. There are still others that would leave just one, two, or three items remaining. (Can you think of your own examples here?)
So
The expected value framework doesn’t seem to work well when infinities show up. The St. Petersburg problem gave us similar trouble. Researchers are still trying to figure out how to make decisions that involve infinite quantities.
In the St. Petersburg game, what is the probability of winning
Suppose we modify the St. Petersburg game by capping the number of flips at
According to Daniel Bernoulli’s solution to the St. Petersburg paradox, the utility of the coin landing heads on the
According to Daniel Bernoulli, the utility of
In the text we discussed one reason this doesn’t resolve the paradox. What was the reason?
Some people respond to the St. Petersburg paradox by arguing that there’s a limit on how good an outcome can be. Utilities have an upper bound, they say.
Bernoulli’s logarithmic utility function does not have an upper bound. In other words: for any real number
Consider the first form of Pascal’s Wager, displayed in Table 14.1. What is the expected utility of believing in God if the probability God exists is
Consider the first form of Pascal’s Wager, displayed in Table 14.1. Suppose the probability God exists is almost zero, but not quite: one in ten trillion. What is the expected utility of believing in God?
According to the many gods objection, Pascal’s Wager argument fails to account for the possibility of other gods besides the Catholic conception. What happens to Pascal’s decision table when other gods are included? Choose one.
Consider the “many gods” version of Pascal’s Wager, displayed in Table 14.2. What is the expected utility of believing in Catholicism?
Suppose an atheist responds to Pascal’s Wager as follows: “Belief isn’t something we choose. I can’t just decide to believe in God any more than you can decide to believe unicorns exist.”
How would Pascal reply?
Explain why the expected utility of believing in God is undefined in Pascal’s Wager, if we include other possible gods like the “anti-Catholic” god.
Write your answer using complete sentences. You can include equations, tables, or diagrams, but you must explain what they mean in words. Your answer should include an explanation why