thedashboardeconomist

[thedashboardeconomist]

The St. Petersburg Paradox!

A very nifty thought experiment, first posed by Daniel Bernoulli (cousin of the namesake of Bernoulli's Principle), recently popular for its role in undermining expected utility maximization as part of the behavioral economics movement. The paradox is based off of a game, which is as follows: a casino dealer flips a coin. If the coin comes up heads, the dealer pays out $1; if it comes up tails, the prize pool doubles and the dealer flips again, such that a tails flip and then a heads flip is worth $2, two tails flips and then a heads flip is worth $4, and so on infinitely. Before we get into things, ask yourself: how much would you be willing to pay in order to play this game?

The expected value of this game is calculated as all expected values are: the sum of all possible payouts times the probability of those payouts occurring. In this case, we already know the payouts: (1, 2, 4, 8, 16, 32, ...). The probabilities that match up with them are (0.5, 0.25, 0.125, 0.0625, 0.03125, 0.015625, ...) which is representative of each payout being half as likely to occur as the last, as each requires an additional coin flip. Multiplying them together, we get an infinite string of 0.5s; and, summing up that infinite string of finite values results in infinity. Neat! A game that has an expected payout of infinity! You should have been willing to pay an extremely high amount-- say $100,000,000-- to play; because the expectation is that you would get an infinite amount of money back!

Except that that is patently absurd: a very reasonable person would decline to play that game at a price of $50, let alone $100,000,000. Bernoulli's explanation for this was to consider the utility of each dollar, rather than the dollar values themselves. For instance, the first dollar is much more valuable than the billionth dollar, so we should weight the extremely high payouts lower in our calculation for the expected value of the game. There are compelling reasons to believe that this is never a reasonable way to model human behavior around the game (particularly an argument from Rabin and Thaler which is too abstruse to discuss here, but about which I would happily make another post)-- but as far as the scope of this post is concerned, it'll do for now.

What I was hoping to discuss here was the other oft-discussed statistic of an economic game: the standard deviation. To get to that, we'll start with the variance, whose formula is-- as always-- the expected value squared minus the expected value of the squared payouts. We'll start with the latter object, because it's less weird. Like the original game's payouts, we simply have an infinite series, which looks like this this time: (0.5, 1, 2, 4, 8, 16, ...). Notice that rather than an infinite series of the same finite number, we have an infinite series of an exponentially increasing number of the form 2^n.

As for the expected value squared, it looks a little strange, as we are essentially multiplying two infinite sums together. If we all remember multiplying polynomials together, the rule remains the same here: multiply the first object in the first sum by each object in the second sum, adding the result as you go. What that looks like for us is this: 0.5(0.5 + 0.5 + 0.5 + 0.5 + 0.5 + 0.5 + ... + 0.5), or an infinite string of 0.25s. Next, we repeat using the second object from the first sum, obtaining another infinite string of 0.25s, then the third object, and so on for all of the infinite objects in our sum. Our final result is a sum of infinite strings of infinite sums of 0.25s, which, as mentioned before, is kind of a weird object.

Weirder still is that we now need to subtract the sum from before, the expected value of squared payouts, the (0.5, 1, 2, 4, 8, 16, ...) object, from our other object, in order to find the variance. To accomplish this, we're going to use a little trick. Remember that every object being summed in the expected value squared object is a sum of infinite 0.25s. We know for certain that this object is greater than the first object in the expected value of squared payouts, because that is simply 0.5; and, predictably, infinity is greater than 0.5. We can do this for every single value in the expected value of squared payouts object, because an infinite sum of infinite 0.25s will be greater than any of the finite numbers that comprise the (0.5, 1, 2, 4, 8, 16, ...) series, even as it grows to infinity.

This allows us to represent the variance not only as positive infinity, but it remains as an infinite sum of infinite objects, which means that its standard deviation is also still both of those things-- this is because the standard deviation is always the square root of the variance, and taking the square root of either of those objects does not change what they are. Critically, this means that the infinity that describes the standard deviation of the value of the St. Petersburg game is larger than the infinity describing the expected value!

This is where the actual interpretation of the math gets a little fuzzy. While it's fairly clear what it means for the game to have an infinite expected payout, what does it mean for the standard deviation of that payout to be not only also infinite, but also a greater infinity than the expected value-- especially considering that the value of the game can never be negative? From the definition of a standard deviation, we might feel confident in saying that, in playing the St. Petersburg game many times, we would never reliably pin down the average value of the game: that is, any data collecting effort designed to empirically determine the expected value of the game would never arrive at any one value (or even in the proximity of any one given value) with any certainty or consistency, though that is already implicit in the fact that the expected value of the game is infinite.

That's all of my rambling about this game and paradox for now. Also, I am but a humble economics B.A., so if any serious statisticians, mathematicians, or otherwise knowledgeable individuals happen across this and have a more elegant approach to anything broached here, feel free to add on!