Embarrassing update: as pointed out by Vladimir Nesov in the comments, all of my quantitative points below are incorrect. To maximize expected winnings, you should bet on whichever alternative you judge to be most likely. If you have a so-called logarithmic utility function — which already has the property of growing faster for small amounts than large — you should bet proportional to your odds on each answer. In fact, it’s exactly arguments like these that lead many to conclude that the logarithmic utility function is in some sense “correct”. So, in order to be led to betting more on the low-probability choices, one needs a utiltity that changes even faster for small amounts and slower for large amounts. But I disagree that this is “implausible” — if I think that is the best strategy to use, I should adjust my utility function, not change my strategy to match one that has been externally imposed. Just like probabilities, utility functions encode our preferences. Of course, I should endeavor to be consistent, to always use the same utility function, at least in the same circumstances, taking into account what economists call “externalities”.Anyway, all of this goes to show that I shouldn’t write long, technical posts after the office Christmas party….
The original post follows, mistakes included.
An even more unlikely place to find Bayesian inspiration was Channel 4’s otherwise insipid game show, “The Million Pound Drop”. In the version I saw, B-list celebs start out with a million pounds (sterling), and are asked a series of multiple-choice questions. For each one, they can bet any fraction of their remaining money on any set of answers; any money bet on wrong answers is lost (we’ll ignore the one caveat, that the contestants must wager no money on at least one answer, which means there’s always the chance that they will lose the entire stake).
Is there a best strategy for this game? Obviously, the overall goal is to maximize the actual winnings at the end of the series of questions. In the simplest example, let’s say a question is “What year did England last win the football world cup?” with possible answers “1912”, “1949”, “1966”, and “never”. In this case (assuming you know the answer), the only sensible course is to bet everything on “1966”.
Now, let’s say that the question is “When did the Chicago Bulls last win an NBA title?” with possible answers, “1953”, “1997”, “1998”, “2009”. The contestants, being fans of Michael Jordan, know that it’s either 1997 or 1998, but aren’t sure which — it’s a complete toss-up between the two. Again in this case, the strategy is clear: bet the same amount on each of the two — the expected winning is half of your stake no matter what. (The answer is 1998.)
But now let’s make it a bit more complicated: the question is “Who was the last American to win a gold medal in Olympic Decathlon?” with answers “Bruce Jenner”, “Brian Clay”, “Jim Thorpe”, and “Jess Owens”. Well, I remember that Jenner won in the 70s, and that Thorpe and Owens predate that by decades, so the only possibilities are Jenner and Clay, whom I’ve never heard of. So I’m pretty sure the answer is Jenner, but I’m by no means certain: let’s say that I’m 99:1 in favor of Jenner over Clay.
In order to maximize my expected winnings, I should bet 99 times as much on Jenner as Clay. But there’s a problem here: if it’s Clay, I end up with only one percent of my initial stake, and that one percent — which I have to go on and play more rounds with — is almost too small to be useful. This means that I don’t really want to maximize my expected winnings, but rather something that economists and statisticians call the “utility function”, or conversely, to minimize the loss function, functions which describes how useful some amount of winnings are to me: a thousand dollars is more than a thousand times useful than one dollar, but a million dollars is less than twice as useful as half a million dollars, at least in this context.
So in this case, a small amount of winnings is less useful than one might naively expect, and the utility function should reflect that by growing faster for small amounts and slower for larger amounts — I should perhaps bet ten percent on Clay. If it’s Jenner, I still get 90% of my stake, but if it’s Clay, I end up with a more-useful 10%. (The answer is Clay, by the way.)
This is the branch of statistics and mathematics called decision theory: how we go from probabilities to actions. It comes into play when we don’t want to just report probabilities, but actually act on them: whether to actually prescribe a drug, perform a surgical procedure, or build a sea-wall against a possible flood. In each of these cases, in addition to knowing the efficacy of the action, we need to understand its utility: if a flood is 1% likely over the next century and would cost one million pounds, but would save one billion in property damage and 100 lives if the flood occurred, we need to compare spending a million now versus saving a billion later (taking the “nonlinear” effects above into account) and complicate that with the loss from even more tragic possibilities. One hundred fewer deaths has the same utility as some amount of money saved, but I am glad I’m not on the panel that has to make that assignment. It is important to point out, however, that whatever decision is made, by whatever means, it is equivalent to some particularly set of utilities, so we may as well be explicit about it.
Happily, these sorts of questions tend to arise less in the physical sciences where probabilistic results are allowed, although the same considerations arise at a higher level: when making funding decisions…