Reading Difficulty: 1/5

Prerequisites: Have a brain

My friends and I were watching some pretty sus anime last winter. As part of that, we watched about episodes of No Game No Life, which is not an anime you should watch. Like, the premise is interesting but execution-wise it's so unwatchable. Uh, anyway, let's roast Episode 2.

Background

Some guy named Sora, the protagonist, made some girl named Steph really mad, and now they're playing Rock Paper Scissors to settle their dispute. The twist: Sora "guarantees" that he will play paper by adding the rule that if he plays anything other than paper, he loses.

For example, if Sora plays rock and Steph plays paper, then they both lose so it's a tie.

Here is the outcome table provided by the anime, which is accurate:

(The blue icons on the left indicate Sora's choice, and this selects a row of the table. The pink icons on top indicate Steph's choice, and this selects a column of the table. The intersection is the outcome: Sora's face (the only masculine face in the table) is a win for Sora, Steph's face is a win for Steph, and the Japanese character is a draw.)

Click this if you need more clarification on the rules

Click this if you need more clarification on the table

Obviously, Sora has a disadvantage here (and this is how he lures Steph into playing in the first place). However, because of anime shenanigans, Sora actually wants a draw (or more). To make this mathematical, we can assign payoffs to the possible outcomes. The stakes can be roughly approximated as follows:

• Win for Sora =
• Draw =
• Win for Steph =

Positive values indicate benefits for Steph, whereas negative values indicate benefits for Sora. (This is an oversimplification.)

None of this actually matters much for determining optimal play correctly, but it helps give context for...

The "Logic"

(Feel free to skip this.)

Steph's Thought Process

• Sora wants a draw, and this has a chance of happening. I won't let Sora get his draw, I want to win.
• If I play Rock or Scissors, then I have a of winning, whereas if I play Paper, I only have a chance of winning. So playing Paper is out of the question - My only options are Rock or Scissors.
• But Sora "guaranteed" that he will play Paper, so playing Rock would be quite risky. So I should play Scissors.
• ...but this is the obvious thought process, so Sora is just expecting me to play Scissors so that he can respond with Rock and obtain a draw. To curtail this, I can just win by playing Rock.
• ...but if I play Rock, there is actually a chance that I will lose!
• In fact, Sora will likely play Paper. It is the only option where he can win. Moreover, his chances of losing are only if he plays Paper, whereas he loses with probability if he plays anything else.
• From these probabilities, it is obvious that Sora's only logical option is Paper, so I will play Scissors.

Steph plays Scissors. Sora plays Rock. This results in a draw.

Sora explains that he predicted all of Steph's thought process, which is why he knew she would play Scissors. Sora says that Steph's correct choice was Paper, and she would have chosen Paper if she was smart enough to figure out that Sora figured out what Steph was thinking.

Questions for you:

1. Both Steph and Sora are being stupid. What is the main error?
2. What should Steph have done?

Give this some thought before reading on!

Game Theory

There are many egregious fallacies being made, but the overarching one is that the probabilities don't make sense. For instance, while it is true that Steph wins in 2 out of 3 possible outcomes if she plays Scissors, but that does not mean her actual "probability" of winning will be . This sort of "local" thinking really misses the greater context.

In fact, I put "probability" in quotes because it is not even clear that probability has any significant role whatsoever! Obviously, both players of this game have a brain - they're not going to just flip a three-sided coin to make their decision. So why should we be assigning probabilities at all? Sure, a logical player could still have a "personal probability" as to how likely they think certain events are, from their point of view. (This goes into a field called epistemology that is out of scope of this post.) But such "probabilities" are ultimately subjective and don't make for rigorous argument.

So what is the proper way to reason about this game?

It turns out that all we need is a very basic trick from Game Theory. Let's pull up that outcome table again.

Observe that Sora, if he is playing optimally, has no reason to play Scissors. No matter what Steph plays, Sora would always be better off playing Paper than playing Scissors. So, we say that for Sora, Paper dominates Scissors.

What this means is that we can completely ignore the possibility that Sora plays Scissors! It simply makes no logical sense, and we can literally delete that row from the chart. There is no need to consider it.

Now the game is simpler to reason about! And with this simplification, it is now Steph's turn to make a realization: She has no reason to play Rock, because no matter what Sora plays (either Rock or Paper), Steph would always be better off playing Paper than playing Rock. For Steph, Paper dominates Rock.

So, we can completely ignore the possibility that Steph plays Rock. Thus we may delete that column from the table.

No more simplifications can be made. We have reduced this to a game of Matching Pennies. Due to the symmetry in choices, there is not much more theorizing that can be done by either Sora or Steph. Both Sora and Steph ought to essentially flip a coin now. If they do so, then the expected payoff would be . That is, Steph has a slight advantage in the expected winnings from playing this game.

In sum: Steph's only reasonable options are Scissors and Paper. Sora's only reasonable options are Rock and Paper. Under optimal play, neither of each player's two choices is better than the other, and Steph has a slight edge.