The surprise quiz paradox

In high school, a teacher announced he would give a pop quiz during the next week. I just laughed. Tricks are for rabbits, silly teacher—you can’t surprise a rational student.

To explain why, I’ll use the technique of “backwards induction,” which I introduced in the ultimatum post.

So let’s start from the end and work backwards.

Could the quiz ever be a surprise on Friday, the last day of the week?

No, it cannot. If I happened to make it until Thursday night without taking a quiz, I would think, “Hey, the professor has not given the quiz yet. Since tomorrow is the last class of the week, the quiz has to occur on Friday. I won’t be surprised when he pulls out the quiz.”

So Friday is ruled out. I might then ask: could the quiz ever be a surprise on Thursday?

Again, it cannot. If I made it to Wednesday night without being quizzed, I would think, “The quiz can either be on Thursday or Friday. But Friday is ruled out. Therefore, the quiz has to be on Thursday. I won’t be surprised.”

And so Thursday is ruled out. Repeating this logic, I could then rule out Wednesday, Tuesday, and Monday as possible days when the quiz would be a surprise.

It is logically impossible for the teacher to give a surprise quiz.

And this is why the teacher can randomly give the quiz any day of the week, and I will not be able to predict it.

What gives?

The surprise quiz paradox is a special case of the unexpected hanging paradox. There are two explanations for the apparent paradox.

The first is that not everyone is a computing machine. It took five steps of reasoning to eliminate the days of the week. Most of us get bored after thinking one or two steps, which is why most of us can only anticipate a couple moves ahead in chess.

The second is finer, epistemic point. Although teachers cannot surprise me with a quiz, that doesn’t mean I can predict when the quiz will be. This is because the phrases “not surprised” and “able to predict” are two different concepts. For instance, when I toss a fair coin, I am not surprised if it turns up heads. But that doesn’t mean I am able to predict the outcome of any particular flip.

The technical way to say it is that events you could not predict are the realizations of random processes. A head or a tail is a realization of a coin flip. A true surprise is a realization that’s “vastly different” from what you expected. Like if a coin landed directly on its side, ten times in a row.

Similarly, I am not at all surprised when the market swings up or down. Fluctuations are a normal part of the market, and it’s very difficult if not impossible to consistently predict financial markets.

Surprise is a powerful force in the market, and that’s why the Fed keeps investors guessing about interest rates, as I discussed before.

Randomness is a critical factor in decisions where you have imperfect information, such as whether to trust a car salesperson, or whether to invest pre-tax or post-tax. I’ll be writing about these topics some time this year. If you’re rational, I know that you won’t be surprised when I do

Two jokes

I’ll end with two jokes about surprises.

From Snopes

When Albert Einstein was making the rounds of the speaker’s circuit, he usually found himself eagerly longing to get back to his laboratory work. One night as they were driving to yet another rubber-chicken dinner, Einstein mentioned to his chauffeur (a man who somewhat resembled Einstein in looks & manner) that he was tired of speechmaking.

“I have an idea, boss,” his chauffeur said. “I’ve heard you give this speech so many times. I’ll bet I could give it for you.” Einstein laughed loudly and said, “Why not? Let’s do it!”

When they arrived at the dinner, Einstein donned the chauffeur’s cap and jacket and sat in the back of the room. The chauffeur gave a beautiful rendition of Einstein’s speech and even answered a few questions expertly.

Then a supremely pompous professor asked an extremely esoteric question about anti-matter formation, digressing here and there to let everyone in the audience know that he was nobody’s fool. Without missing a beat, the chauffeur fixed the professor with a steely stare and said, “Sir, the answer to that question is so simple that I will let my chauffeur, who is sitting in the back, answer it for me.”

From K.B. at Scientific American:

I heard this tale in India. A hat seller, on waking from a nap under a tree, found that a group of monkeys had taken all his hats to the top of the tree. In exasperation he took off his own hat and flung it to the ground. The monkeys, known for their imitative urge, hurled down the hats, which the hat seller promptly collected.

Half a century later his grandson, also a hat seller, set down his wares under the same tree for a nap. On waking, he was dismayed to discover that monkeys had taken all his hats to the treetop. Then he remembered his grandfather’s story, so he threw his own hat to the ground. But, mysteriously, none of the monkeys threw any hats, and only one monkey came down. It took the hat on the ground firmly in hand, walked up to the hat seller, gave him a slap and said, “You think only you have a grandfather?”