My Talk In Istanbul On Suspense And Surprise

1) We can come up with a lot of explanations for why this might be, but the short answer is that it’s not in our model. We take as a primitive that we’re looking at a situation where you get entertainment — as opposed to pain or boredom — from learning the outcome over time. One relevant factor is that if you need to learn information to make a decision, then you probably want to know as soon as possible and entertainment isn’t important. That’s why we focus on noninstrumental information in the paper. Second, there seems to be an interaction with the “stakes.” If you don’t care about something at all — if you have no investment in the outcome — you probably get no entertainment from it. So you want to watch sports games for a team you follow, not for random teams you’ve never heard of. And if you’re too invested in the outcome — the doctor is telling you whether you have cancer — then suspense might be painful instead of pleasurable. This raises interesting issues in settings like gambling, where you might choose your own stakes.

2) Let me first ignore your question and tell you a bit of what we’ve done. Suppose we *know* p in advance. If we want to maximize suspense or surprise in a game that’s first to N points, or a playoff series that’s the first to win N games, what N should we choose? Well, the bigger is N, the slower we reveal information, and slow information revelation gives more surprise. But there’s a tradeoff — we also get rid of the aggregate uncertainty, which reduces surprise. If you beat me in 70% of games, then I still have a chance of winning a best of 1 or a best of 3 series. I have no chance of winning a best of 99 series. It turns out that the closer the winning probability is to 50%, the greater N should be. For p=80%, we play a single game. For p=70%, we do a best of 5. If there are no other costs limiting the number of games, then for p=50% we’d approach infinitely many games.

OK, now back to your question. Maximizing the likelihood of picking the better team just means taking N to be as large as possible. It turns out that when we aren’t sure about p but the support contains 50%, surprise and suspense are also maximized by taking N to be as large as possible (although it grows very, very slowly when there’s not much probability about 50%). Of course, surprise and suspense aren’t necessarily monotonic in N when there’s uncertainty over p — when we’re almost sure p is about 70%, we have a local maximum at a best of 5 series. Then 7 is worse, 9 is even worse, but maybe we catch back up at 9999 (or whatever).

In terms of your specific proposal of how to measure the tradeoff of game length vs. test power, and how to compare that to the tradeoff of length vs. suspense, I have to think more about that.

3) By your definitions the suspense is equal to the expected surprise, correct? So the problem of choosing an information policy to maximize expected surprise would be the same as choosing one to maximize expected suspense. This gives us a single policy to consider. In fact, we have worked this out (it’s not in the paper), and qualitatively we get the same result as in our Suspense-maximizing policy.

The key is that K-L divergence and variance both share the mathematical property that the expected sum from *any* fully revealing sequence of signals is equal to a constant. So to maximize (our suspense / your proposed measures), we want to dole this out evenly over time — 1/T of this in each of T periods. This is the property we use to get all of the other features of optimal suspense. (Numerically, using K-L divergence instead of variance gives somewhat faster information revelation — we move towards the edges more quickly).

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Thanks for your interest in the paper, and let one of us know if you have any followup questions!

First, I wouldn’t toss aside our result that surprising games have jumpy belief paths that look approximately Brownian. Think about, say, basketball — it’s a very popular sport! There you’d get belief paths that really do look Brownian — small changes up or down every 30 seconds, no possibility of any individual big jumps until the very last few seconds. Tennis belief paths also look more like our suspense solution than our surprise one (aside from a few big ones, most points are small and don’t affect the game very much… but they add up). Loosely speaking, by our definitions soccer is a very suspenseful game (rare big belief jumps) and basketball is a surprising one (frequent small jumps).

We’ve played around with the idea of surprise=(belief change / standard deviation), but it doesn’t give anything interesting. The problem is that you can scale all belief changes down by a factor of a million, and it doesn’t change the new measure of surprise! Revealing essentially no information still gives you lots of surprise…