Let's say you're on 'Jeopardy!' and you're absolutely routing your two opponents. You have $40,000 going into the final round, while one of your opponents has, let's say, $15,000. You're guaranteed to move onto the next day, but the final category comes up and it has something to do with baseball, which is your favorite sport. How much — if anything — do you risk?The discussions of this article go along two paths. One is to compare how many of the ten Final Jeopardy! answers everyone got right (most participants have gotten at least seven of the ten - both sites have a fair number of people who are very knowledgeable about baseball). The other is to loudly decry the article's premise that, since you should be fairly confident that you will respond correctly (the article, on the Yahoo! Sports site, assumes you're a baseball fan), you should bet your entire $40,000 haul, rather than the $10,000 that will guarantee a win. Since the Preschool is all about the math these days, let's look at this in a little more detail. Specifically, let's look at the following situation: you have $40,000, the second-place competitor has $15,000, and the Final Jeopardy! category is something squarely in your wheelhouse. How much do you bet?
This (like the Scat decisions I'm supposedly writing about) is a problem of expected value. Suppose that you estimate the probability you will respond correctly as p; for example, if p is 0.8, it means you think you're 80% likely to be correct. If you respond correctly (as you will do 100p% of the time), you gain whatever you've bet; call this number b. If you respond incorrectly (which will happen 100(1-p)% of the time), you will lose b dollars. Your expectation for this decision is then 40,000 + bp - b(1-p), which simplifies to 40,000 - b + 2bp. If your only goal is to maximize your expectation for this decision, then the answer is simple: if p is at least 0.5, you should bet everything; if it's less than 0.5, you should bet nothing.
However, this is not your only goal, because if you win the game, you get to come back tomorrow, and try to win more money. Thus, we must also factor in your expectation of future winnings. (Incidentally, the primary argument I'm seeing against betting everything is that "you're risking a lot in future winnings.") Let's assume, for the sake of argument, that the second-place competitor will bet his entire $15,000, and will respond correctly. Thus, if you bet $10,000 (or less), you will win (or tie), and come back tomorrow, regardless of your response. If you bet more than $10,000, and respond incorrectly, you will lose.At this time, I should point out that we could make this a more complicated model, estimating the probabilities that the other competitors will respond correctly. Also, you could bet something other than $10,000 or $40,000. Even if you factor in the chance of the other competitors missing, betting something like $20,000 seems a doubly bad idea; you leave $20,000 "on the table" if you are right, and risk losing if you are wrong. You could bet slightly less than $40,000, thus improving the probability that you will win if everyone misses the final answer. In fact, I think $39,998 might be the best bet if you're "going for broke." That said, complicating the model takes time, and might be better suited to a paper than a blog post—any undergrads looking for a thesis, this might be a starting point! Once we decide to stick with a simple model, betting anything other than $40,000 or $10,000 makes the math work out less nicely, and a couple of dollars here and there isn't going to make much difference, nor will the tiny probability that you'll win with $2. Let's assume, then, that this is a binary decision: bet $10,000 and always win, or bet $40,000 and lose if you're wrong.