[This is the third in a series that started here]
You have been selected for the jury in the Sam Bankman-Fried (SBF) trial. You try to forget the rather uncharitable definition of a jury: Twelve people not smart enough to get out of jury duty. In opening arguments one of the attorneys describes the defendant as a brilliant mathematical genius. Early in the trial you learn about something called “Effective Altruism” or “EA.” Not long thereafter you find out that both the EA movement and SBF are driven by a calculation known as “Expected Value.” You wonder what that is.
Understanding Expected Value
Suppose I tell you I am going to flip a “fair” coin where “fair” means that it is evenly weighted on each side. In other words, the coin has an equal likelihood it will land on heads or tails. I am going to flip this coin 1000 times. You are asked what proportion of the tosses you expect will be “heads”. You are on a jury and your understanding of this question will determine whether a man spends the rest of his life in prison. What is your answer?
If your answer is “50%” you already have a good intuitive understanding of Expected Value. (Be prepared to be the only one in the jury room who does). But to get to the level of judging SBF you need a little more information. Formally, Expected Value is the product of the probability of an event, times the payoff generated by that event averaged over a large number of trials. Imagine a game in which you are paid one dollar for each Heads and nothing for Tails. You must pay a fee to play the game. You wish to pay no more than the minimum value you expect to gain. How much should you be willing to pay? Your answer is the expected value. In tabular form it looks like this:
Since, in the long run (meaning many repeated trials), you can expect to end up with 50 cents, this is the most you should pay to enter the game.
Let’s take a look at a series of flips in a plot. Look at the first five tosses below. The vertical y-axis records the accumulated proportion of Heads which occur as the number of tosses (horizontal x axis) increases. Note the first five tosses result in T-H-T-H-H because the first five accumulated recordings of Heads, as a proportion of all flips, are 0, 0.5, 0.33, 0.5 and 0.6. Prove it to yourself. At each step, count the number of Heads appearing so far and divide that by the total flips so far.
Round Two
This continues, although the fractions become a bit more irregular, as the series grows larger. Note what happens…HEY!!! HEY!!! OK, LISTEN UP!!!!! I HEARD SOMEONE SNORING!!!! THERE IS A MAN ON TRIAL HERE AND HIS LIFE IS IN YOUR HANDS!!
Pay attention as the number of tosses reaches 75 (the alert reader may be concerned that, despite being beyond the halfway point on the x-axis, 75 is far less than half of the total 1,000 tosses. Fear not, the x-axis scale is not linear; it is logarithmic to enhance the visual, the conclusion is unaffected). At 75 tosses, the proportion of Heads “settles down” to about the 50% your intuition suggested. After that the deviation from 0.5 varies little. This is called “convergence” in that the early flips, when aggregated with more and more flips, converge to the mathematical expectation, in this case, 50%. This is one version of the Law of Large Numbers you may have heard about.
Suppose you believe we just got lucky in this first experiment of 1,000 tosses. Fine, let’s test it with a second round of another 1,000 tosses. The new plot is added to our earlier graph (check the first five tosses, H-H-H-H-T, as you did before). This time, despite the first four tosses being Head, again at about 75 tosses, the series begins to converge to 50%.
Although the graph becomes very messy, if we run 20 tests of 1,000 tosses each, they all exhibit the same convergence behavior beginning at about the same point in the series.
You have just encountered the simplest version of the concept of expected value. Here is a slightly more involved example.
Keep in mind that SBF can make these computations, typically involving many more rows, in his head.
Next week: A jury of his peers?