"I tried to copy you towards the end," Stephanie admits. "I think you have to call with the queen or bluff with the jack less than half the time, and in first position you raise with the king the same number of times you bluff with the jack. I don't know – is there an exact number at all?"

Sora smiles. "There is, in a sense! I'm not surprised you didn't find it, but I wanted to let you look. You can't succeed if you don't try. Thank you for trying! Let's talk about second position with the queen first."

He slides the queen into the center of the table, then pauses.

"Whose picture is this on the card? The queen of diamonds I'm familiar with has a portrait of a woman named Rachel on it."

"Kainath, goddess of forests. The king of clubs is Ocain and the jack of hearts is Alram. Can you tell me the numbers now?"

"Yes, sorry. So, you're player two and you have the queen. You want to avoid losing money to me when I have the king, so one thing you could do is always fold when I raise. Always doing the same thing in a particular situation is called a 'pure strategy', and there are a lot of those in this game. But, if you use the pure strategy of always folding your queens, I will notice. That would let me use the pure strategy of always bluffing with the jack, letting me beat you every time with two of the three cards. You can't let that happen. But you also can't call every time – I'll notice that too, much faster, and then I'll use the pure strategy of never bluffing with the jack. One of the 'mixed strategies' somewhere between 0% folding and 100% folding is a「Nash equilibrium」that leaves me indifferent to—"

"A 'Nash equilibrium'," Stephanie repeats slowly, sounding out the words. The phrase "næʃ ekwɪˈlɪbriəm" has failed to translate. It involves math, somehow.

"… okay! On the off chance that we've just struck paydirt I need you to lose a game of Rock Paper Scissors. You're wagering away your right to share this knowledge with anyone until I say otherwise."

Stephanie's indignance at this treatment is vastly outweighed by her curiosity about the forbidden gambling knowledge. She plays scissors to Sora's rock, wagering her silence on the matter, and commits to paying attention to what he has to say.

"Thank you," Sora says. "If that comes in handy later you'll be glad I did it, and if it doesn't it won't matter much, I promise. Moving on! In a— it's a single idea named after a person, that might be the problem. We'll call it a 'stable equilibrium', does that translate?"

It does.

"A stable equilibrium is one where your opponent is indifferent between all of their options. It's not the best strategy unless both players are using the stable equilibrium – remember, if you always fold the queen I can do much better by bluffing the jack every time – but if you use the stable equilibrium as player two then the best I can do is use player one's stable equilibrium. You've made all of my other options equal or worse. If I have other equally good options we'd say I have multiple stable equilibria; otherwise I have just one."

Is it a waste of his phone's battery life to write this down using the whiteboard app? Maybe. Is it worth it? It will be, if Stephanie is picking up what he's putting down.

"There are different ways of finding mixed strategy stable equilibria, but here's an intuitive way to think about it. If I have the jack and check, my 'expected value' is losing one chip no matter what. If I raise, it's losing two chips if you call with either card or winning one if you fold, which you'd only do with the queen. So you need to call my bluff with the exact probability that make the 'expected value' of the bluff equal to losing a chip, which is minus one. Since the value of bluffing into a king is minus two, the expected value you want when you're holding the queen is zero, and your equilibrium call probability is one third."

sora → J

check EV = −1

bluff | Q → ⅓ ⋅ (−2) + ⅔ ⋅ (1) = 0

bluff | K → (−2)

bluff EV = ½ ⋅ (−2) + ½ ⋅ (0) = −1

check = bluff :D

"… and that last symbol is a smiley face, don't worry about it."

Stephanie can't read Sora's native language any more than he can read hers, but he's talking out loud as he goes. She can remember what the text means.

"Then, suppose I have the king. If I check I either win a single chip if you check back or two chips if you try to bluff me with the jack, and if I open raise I either win a single chip if you fold or two chips if you call with the queen. You need your calling ratio with the queen and your bluffing ratio with the jack to set those expected values equal to one another. Calling one third of the time with the queen is optimal if you bluff with the jack into my king one third of the time as well. That bluff ratio is also optimal, I'll show you that in a minute, but for now just take my word for it. Here's the math."

sora → K

check | Q → (1)

check | J → ⅔ ⋅ (1) + ⅓ ⋅ (2) = 4⁄3 EXPLANATION SOON

check EV = (4⁄3 + 1) ÷ 2 = 7⁄6

raise | J → (1)

raise | Q → ⅔ ⋅ (1) + ⅓ ⋅ (2) = 4⁄3

raise EV = (4⁄3 + 1) ÷ 2 = 7⁄6

check = raise !!!

"As you can see, if I'm not lying about bluffing with the jack one third of the time being part of your stable equilibrium, this strategy makes my ability to choose worse than useless. I lose an average of one chip per jack and gain just over one chip per king. My only hope to do better is to deal myself more kings than jacks, which won't happen over lots of hands unless I'm cheating."

"Yes?"

"… I need to digest this," Stephanie says diplomatically. She's picking up what Sora is putting down, but perhaps not as fast as she'd like. "Tell me why bluffing with the jack one third of the time is part of my stable equilibrium?"

"No. First we consider player one. This is a little more complicated. I need to decide how often to value bet the king and bluff the jack. You have a pure strategy for playing the jack if I raise and a pure strategy for playing the king no matter what I do, so we need to look at what I'm going to do against your queen range. If you fold to my raise you always lose a single chip, and if you call your expected value is winning two chips against the jack or losing two chips against the king. To be in equilibrium you need to be indifferent between calling and folding, so I need to set the expected value of your call to losing a chip. What should I do?"

He almost continues explaining, but decides that Stephanie seems appropriately invested in the lesson. She can try again. As a treat.

"Feel free to use the cell phone if you want."

Stephanie's enthusiasm for the thing has diminished somewhat now that she's paid the price. She'll use it to write down her reasoning though; thinking about what Sora thinks Stephanie thinks Sora thinks when he open raises is almost enough recursion to give her a headache.

There are two unknowns in this question. She decides to use the suit of clubs to represent the odds that Sora has the king, followed by the suit of hearts to represent the odds of the jack (this is not the convention she learned from her tutors, but she wants to cut down on the number of symbols she has to explain and it lets her use the nifty colored quill tool a little more).

stephanie → Q

fold EV = −1

call | K → ♣ ⋅ (−2)

call | J → ♥ ⋅ (2)

call EV = −2♣ ÷ (♣ + ♥)  +  2♥ ÷ (♣ + ♥)

The expected value of a fold is always losing one chip, but the expected value of a call apparently forms a surface. She frowns. "There are infinite possibilities. How do I find the right solution?"

"Yup! Clubs and hearts could be lots of things, if this were just an algebra problem. But it's also a gambling problem, which means there are some tools we can use to search through a smaller space. Do you want a hint?"

Stephanie was an unusually bright child and received a stellar education growing up, which is why she has been following along up until this point. However, pedagogy for future queens does not cover measure theory mathematics. She would like a hint, please.

"The first thing we'll do is say that clubs and hearts have to be positive real numbers, to satisfy the first axiom of probability. Complex and negative probabilities aren't meaningless if you're doing「quantum mechanics」, but they don't come up in games."

Yet.

"Multiply both sides by clubs-plus-hearts to reduce it to real solutions only." That should be enough of a hint, he thinks.

She does this.

−2♣ ÷ (♣ + ♥) + 2♥ ÷ (♣ + ♥) = −1

−2♣ + 2♥ = −♥ −♣

3♥ = ♣

"I still don't have a solution," she complains.

"That looks like a solution to me! I'm in a stable equilibrium if I open raise the king three times as often as I open raise the jack. I don't just have one stable equilibrium mixed strategy, I have an infinite number of them."

Well, an infinite number of them as long as the odds of him bluffing are between zero and one third. All outcomes must add up to one, unless you're in a really existential thought experiment where the law of total probability doesn't apply.

Oh, this one is going to be fun.