In this post, I want to talk through a simple mathematical result that forces us to think twice about relying too heavily on averages.
Quick review of expected values
Given an opportunity to play a game with uncertain outcomes, one reasonable way to value to opportunity is to weight each possible reward by its probability of occuring and sum up the results. This quantity is called the expected value, or expectation, of the game. (Note that a simple average is an expected value with equal weight on each outcome.) Another way of saying this is that if you have to pay a fee to play this game, the fee you should be willing to pay is the game’s expected value (or less, of course). Mathematically, if the payoff of the uncertain game is represented by the random variable $X$, the possible outcomes are $x_1,x_2,\dots,x_n$, and the corresponding probabilities of the outcomes are correspondingly $p_1,p_2,\dots, p_n$, then the value of playing the game would be given by
(The notation $E[X]$ is how we represent the expected value of $X$.) For many bets, this approach has intuitive appeal, and there are many scenarios in which it is used directly.
Flipping coins
With this in mind, consider the following game. You flip a coin until seeing a head. If a head falls on the $k$th flip, you win $2^k$ dollars. The question is: how much would you be willing to pay to play this game?
Well, you might begin, there are infinitely many possible outcomes (head after 1, head after 2,…). Each outcome requires flipping $k-1$ tails and then a single head. Using a fair coin, we have
Using the framework of outcome values and probabilities of those outcomes, we can express $x_i$ and $p_i$ for each $i=1,2,\dots$ as
But wait! If this is the case, then for each $i$, $x_ip_i = 2^i/2^i = 1$, so the expected value of the game is actually infinite (the sum of infinitely many 1s)! It seems that, according to this mathematically sound analysis, that we should be willing to participate in this game at any offered price. With this information, how much would you pay to play this game?
What gives?
If you think something is fishy here, you’re right. This problem is so well-known it has a name: the St. Petersburg paradox. It isn’t actually a paradox, but the use of the word refers to the fact that on the one hand, this game has infinite expected value, but on the other, the probability of making a large sum of money is vanishingly small. To put a finer point on it, the probability of winning more than $2^k$ dollars is $1/2^k$! For even moderate values of $k$, this probability is miniscule.
This sort of issue has caused many to reject the exclusive use of expected value as a valuation technique. Some suggest adding a measure of risk (as is customary in any sort of financial application), some advocate for use of the median instead, and still others advocate using the expected value of some utility function applied to the outcomes, rather than the outcomes themselves. There are all interesting areas to explore, but the bottom line is that expectations alone can lead to some head-scratching issues.
Finite resources
One other aspect of the problem to which some attribute the counterintuitive result is the implicit assumption we made that the banker or casino has infinite wealth with which to bankroll the game. Let’s see what happens if the banker only has finite wealth. That is, let’s now suppose that the banker has $W$ (for wealth) dollars with which to fund the game. We’ve introduced a ceiling on the number of rounds before the game ends: $L = \lfloor \log_2 W \rfloor$ (i.e., the number of times you can double the payout before exceeding $W$, rounded down). Having reframed the problem this way, the expected value calculation returns a saner result:
That is, the expected value of the game is now logarithmic in the banker’s wealth. This means, for example, that if the banker has one billion dollars, the game is only worth about \$30. This accords with our intuition about small probabilities of winning anything significant.
Conclusion
Expected value is one of, if not the most important tool/concept in all of probability! Even so, as we’ve shown in this post, it is not a panacea. If you’re not careful, strange (and fascinating) things might happen.