Posted June 11, 2013 By Presh Talwalkar. Read about me , or email me .

The animated show Bob’s Burgers details the adventures of a family of five trying to run a restaurant. Probably the funniest part of the show is the hand-written daily burger specials which include puns like:

—We’re Here We’re Gruyere, Get Used To It Burger

—Poutine – On The Ritz Burger – comes with poutine fries

—The Final Kraut Down Burger – comes with sauerkraut

—It’s Fun to Eat at the rYeMCA Burger – comes on Rye, w/Mustard, Cheese & Avocado.

(more of them here).

Another running gag is the scheming nature of the youngest daughter, Louise. In the episode, “The Kids Run the Restaurant,” from Season 3, Louise finds a perfect chance to execute one of her plans. While her parents are away at the hospital, Louise creates a secret underground casino.

The plot unfolds with an interesting game of rock-paper-scissors. I describe the scene from the show.

(If you’re not interested in the show, you can skip ahead to “Analyzing the Game: No Scissors” with virtually no loss of continuity).

WARNING: Major spoilers ahead

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"All will be well if you use your mind for your decisions, and mind only your decisions." Since 2007, I have devoted my life to sharing the joy of game theory and mathematics. MindYourDecisions now has over 1,000 free articles with no ads thanks to community support! Help out and get early access to posts with a pledge on Patreon. .

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The background to the game

Louise’s casino is running smoothly until the landlord, Mr. Fischoeder, makes a surprise visit and starts gambling too. At first he loses several hundred dollars, but then he tries his hand at the game of rock-paper-scissors. Mr. Fischoeder is somehow extremely good and keeps winning, with each game being worth $100. Louise ends up losing all the money the casino has, plus she ends up in the hole trying to recover her losses.

Ultimately, Mr. Fischoeder ends up $5,000 ahead.

It is at this moment that the parents return. The father Bob has come back from the emergency room with his left hand wrapped in a bandage. He had cut himself during food prep and required stitches in the skin between his index and middle finger.

Bob is stunned and tries to ask Mr. Fischoeder to cancel the debt. But Mr. Fischoeder insists that he must pay the $5,000, either now, or in future rent payments.

It is at this moment that Louise hatches up another scheme. She suggests Bob should play rock-paper-scissors with Mr. Fischoeder. Why? With his bandaged hand stitched up, Mr. Fischoeder won’t expect him to play scissors.

Here is the dialogue.

LOUISE: We could play him double or nothing. BOB: What no, we’re already in the hole. I’m not going to owe him $10,000. LOUISE: Dad, I can’t beat him but you can. With that (points to Bob’s wrapped up hand). BOB: This? Why would I play him with this? My hand is stiched. Plus, I’m not even left-handed. LOUISE: Exactly! If you use that hand, he’ll think you can’t throw scissors. And that’s why you are going to throw scissors. BOB: But I really can’t. My fingers won’t separate. LOUISE: Yeah keep saying that. We need him to think you think that. BOB: No, Louise, I can’t actually do it. LOUISE: Great, so you know what are going to have to do. BOB: Oh my God. LINDA (the mother): That’s my girl, that’s my little mind-gamer. My little “Amorosa.” TINA (older sister): But what if Mr. Fischoder knows that we think he knows that Dad can’t throw scissors? Or what if he thinks that we know that he thinks that? Or what if he thinks…? GENE (older brother): Any outcome is possible, Tina, life is chaos!

The conversation highlights many interesting topics of game theory.

–Louise’s strategy depends on surprise

–Linda wonders about the effect of common knowledge (“I know that you know that I know…”)

–Gene is being comical, but he sort of speaks the truth. With randomized strategies, one cannot predict the exact realization, so life in a way is a mystery.

The situation also raises a specific game theory problem. A standard game of rock-paper-scissors is written out in matrix form as follows.

The +1 represents a win and a -1 a loss. The best strategy for each player is to randomly throw each option with a 1/3 chance.

That’s if each player can play all options.

But what happens when your opponent can’t play a choice? For instance, what happens if you know your opponent cannot play scissors?

What is the best thing for you to do?

Analyzing the game: no scissors

Suppose for a moment you know your opponent cannot make the scissors gesture. How should you best play?

Here is how the game will look after crossing out the scissors strategy.

Logically, here is how to think. If your opponent doesn’t play scissors, then you should realize there is no reason for you to play rock. Why? It’s because rock only wins against scissors, but you know your opponent can never play scissors. Therefore, rock can at best tie (against rock) or it will lose (against paper).

Therefore, if your opponent can’t play scissors, you should never play rock. (In game theory jargon, it is said that rock is a dominated strategy).

Of course, your opponent can reason this out as well, and that means your opponent is well aware you will never play rock. Both of you realize that your opponent never plays scissors, and you never play rock.

This means the game is reduced even one step more to the following diagram.

(Incidentally, this process for removing bad strategies has a technical name. It is known as iterated deletion of strictly dominated strategies and it goes by the acronym IESDS. I wrote about this previously in my article How my professor almost lost $250 playing game theory with his class.)

In this reduced game, you can play either paper or scissors, and your opponent can play either rock or paper. The interesting part is this game is asymmetrical: you can win in two different ways (by winning paper against rock or scissors against paper, but your opponent can only win in one way: by playing rock against scissors. The other case of paper-paper is a tie.

Evidently you have an advantage here. What is the best thing for you to do?

We can go through the motions of calculating the mixed strategies (as explained in detail here and here).

It turns out that you should play paper with a 2/3 chance, scissors with a 1/3 chance, and your opponent will play rock with a 1/3 chance and paper with a 2/3 chance.

Your expected payout ends up being 1/3, so the game is actually in your favor.

(This makes sense: your opponent has put himself at a disadvantage by not being able to play scissors).

Returning to the show, Mr. Fischoeder actually faces a positive expected value to play the game. And he will want to play paper with a 2/3 chance and scissors with 1/3 chance.

How does this play out?

What happens in the show

In the episode, Mr. Fischoeder plays paper. In a way, this is a necessary plot device. The dramatic scene is going to happen when Bob plays scissors, so it has to be the case that Mr. Fischoeder loses with paper. It would be much less interesting if he played scissors and they merely tied–of course with Bob’s Burgers, such a deflating ending is entirely possible.

That said, it is a fine coincidence that playing paper was mathematically correct in “playing the odds” and that’s exactly what Mr. Fischoeder did.

Bob very humorouly sacrifices his own health to win at the game. He painfully makes a scissors gesture with his fingers, which rips apart the stitches that he just received in the emergency room.

Mr Fischoeder graciuosly accepts his loss, explaining he misread the situation and thought Bob was too much of a “wussy” to play scissors.

Bob ends up in the emergency room again to receive stitches, but he does win the game and erase his $5,000 debt.

Perhaps the next day Bob, after he’s recovered, can celebrate by eating his “Mission A-Corn-Plished Burger.”

(images are lo-res captures from episode of Fox Bob’s Burgers)