Jeopardy!, The Princess Bride, and Game Theory

I like Jeopardy! It’s cerebral and I usually get home from work around the time it comes on. For the most part, the trivia aspect is fun, but there is some level of contestant interaction that gets exciting too.

The winner of Friday’s show (Jan 20, 2017) employed an absolutely brilliant strategy in final Jeopardy! It’s something that regular watchers of the show probably caught, but I thought it was cool and wanted to talk about it, so here goes.

Background:

Going into Final Jeopardy! the three contestants scores were:

Neil (returning champ): $9,200
Cathy:  $1,200
Hardy: $12,200

Final Jeopardy! category: WOMEN SINGERS

Final Jeopardy! clue: What she calls her “Love of Many Colors album”, a 2016 release by this singer is her first No. 1 country album in 25 years.

Assuming you’re familiar with how Final Jeopardy! works… you probably know that sometimes the leader ought to assume that their closest competitor will bet all of their money.

In the case above, it means that Hardy assumes Neil bet $9,200. If Neil is correct, then he will end up with $18,400.

Hardy ended up betting $6,201. We can surmise that he did, in fact make two assumptions:

H-1. Neil would bet to 2x his score
H-2. Neil would respond correctly

Why? Because if assumptions H-1 and H-2 above hold, then Neil will end up with $18,400. The only way Hardy can win at that point is if he is correct and bets at least $6,201. (How much above $6,201 may depend on how confident he is about the category).

This is where it gets interesting. In game theory, there is something called level-k thinking. The best way to describe level-k thinking is by example.

Imagine you’re playing Rock-Paper-Scissor against Bob.

You like Rock. It’s strong, easy to throw… you know it’s the best. So, you throw rock. That’s you’re level 0 strategy. It means that you did not consider AT ALL what you’re opponent might do.

But now you think, “what if Bob knows that I like to throw rock?” Now, you’re at level-1. “If Bob knows I throw rock, then I should throw paper.”

Level-2: Bob has figured level-1 out, and conclude he’ll throw rock, so you throw paper

Then it can just go on and on ad finitum.

That’s roughly k-level thinking, although some of the details may be a little wrong. The moral is that when you have a strategic game, you are apt to consider what your opponent thinks,

Back to Jeopardy! So, Hardy bets $6,201. Of course, Neil doesn’t know what Hardy bet; but he’s clever. What does Neil do?

He bases his strategy on two assumptions:

N-1. Hardy would bet based on assumptions H-1 and H-2
N-2. Hardy would respond incorrectly

What makes this so cool is that N-2 is really a corollary to N-1. Think about it, if Hardy bets $6,201, then the ONLY way Neil can win is if Hardy responds incorrectly. So, there’s no reason to operate under any other assumption. If Neil assumes Hardy will answer correctly, then Neil loses regardless of his bet.

If N-1 and N-2 hold, then Hardy ends up with $5,999. What then should Neil bet?

No more than $3,200. And, indeed, that’s exactly what he did. He was wrong, but he ended up with one more dollar than Hardy. Final scores:

Neil: $6,000 – Winner!
Hardy: $5,999
Cathy: $2,400

Now, nothing stopped Hardy from employing a higher k-level strategy. He could have bet $6,199 assuming that Neil would have bet assuming N-1. But he would have looked really silly losing by one dollar if Neil had just decided to bet his whole score.

Plus, at some point, you just end up looking like Wallace Shawn from the Princess Bride.

PS: The correct Final Jeopardy! response was: Who is Dolly Parton?

 

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