"You still haven't told me why I should bluff a third of the time, or what you should do if you have the queen. You always check to start, but what if I raise?"

"I've given you everything you need to answer those questions yourself."

Stephanie has not heard of Socratic questioning by that name, but she is familiar with the concept and is growing to despise it. She was promised this answer, dang it!

She starts to write out the probabilities on the cell phone when she realizes that Sora wasn't exaggerating when he said he'd given her everything she needed.

"I have to raise with the king every time, and my king-and-jack logic is the same as yours: I raise with the king three times as often. That means the probability of bluffing the jack has to be one third!"

"Yes! Excellent! That's exactly right, Stephanie. Only one more position to look at: what's my stable equilibrium strategy if I have the queen in first position and I'm facing a raise from you?"

Sora loses one chip if he folds, gains two if he calls a bluff, and loses two if he calls and she has the king. Conditional probability: Sora holds the queen half the time, he checks some fraction of the time with the king and more often with the jack but every time with the queen… except her payout doesn't vary with the rate that he plays either of the other cards, they just proved that, meaning his call probability is the same as hers when holding the queen.

"You call with the queens a third of the time, just like me."

Well done, grasshopper. Sora takes the phone back and starts drawing.

"Here's my perspective as player one. The expected value of a king is a little more than one chip, the expected value of a queen is losing a third of a chip, and the expected value of a jack is losing a chip."

(7⁄6 −⅓ −1) ÷ 3 = −1⁄18

"For player two it's the opposite – every round your expected value is gaining one eighteenth of a chip. And that's all there is to it. Any questions?"

"How did you find these numbers in the first place, just by guessing? And how do you change them when your opponent isn't playing perfectly?"

"With 「linear programming」."

Shiro may or may not have been napping. She rolls over, staring balefully at Stephanie from beneath a curtain of greasy hair.

"There are sixty-four possible strategies. Define a square matrix of that size for the expected values of each strategy versus each other strategy, then multiply it by negative one to get the expected values for the other player. It's a two player zero-sum game, so you restrict your answers to positive reals that sum to one and take the minimax of both matrices to get the Nash equilibrium."

"Thank you, Jane von Neumann, but regular mortals don't prove basic results using matricies with four thousand elements."

"If you strip out dominated and weakly dominated strategies it's under a hundred elements, which incidentally is also how to exploit your opponent when they aren't playing optimally. You don't even need a calculator to take the transpose of— why are you wasting the battery on that?! There are no electrical sockets in this castle!"

"Some people are visual learners!" he defends himself. Then he looks askance at Stephanie. "Princess, there are two reasons I wanted to play this game. The first one was to show you the game theory. This whole business of minimizing your opponent's expected value isn't just how to play this game, it's how to play all games. With one bit of hidden information, limit betting, two players, zero sum, the solution is small enough to show you. If you add more cards, more players, change the bet sizes, the algebra gets more complicated but the ultimate strategy is the same. This is why the cell phone is a better chess player than you: it has an algorithm for looking at the expected value of every move and plays to minimize your gains over the long run. Or maximize its own gains, those are equivalent."

He switches the cell phone off.

"The other reason was to show you why you aren't going to play poker, even if we figured out how to get around your wager. You suck. Empirically. If Shiro and I doubled down and taught you as much as we could until the sun rose, you wouldn't retain half of it and wouldn't have the experience to apply the other half correctly. You're in a supporting role tomorrow. The end."

Stephanie is disinclined to shy away from the truth in her heart. Facing the fear you cannot name is one of the ancient arts passed down by humans through the centuries, and an essential one to master if you hope to be queen. There is no room for stupid failure when your decisions affect the lives of your subjects. Still, it would be hard to avoid this conclusion even if she wanted to, not when Sora has spent the last hour hammering it home. The stack of chips on his side of the table is taller than hers. 'The end' indeed.

She understands why Sora swore her to secrecy, now. This information is not generally known in Elkia, nor in the Eastern Union, nor Rapture, and it's unlikely to be widespread anywhere else on the continent (Elkia has few non-human residents but it's hardly isolationist). The secret of 'minimizing expected value' is key to creating perfect strategies for any number of commonly-played games. It's a superweapon.

"Thank you," she says shortly. "What is your plan, then? Surely you have one."

"My plan is to wash off, find some clothes for tomorrow, eat dinner, practice poker with Shiro, and get eight hours of sleep."

"I don't— yes, all of those things, but why did you want me to join the tournament?"

There is no particular reason to answer this question. Stephanie's assistance is both superfluous and completely assured; her active participation is barely required.

However, there's something to be said for keeping your teammates appraised of the plan. He tells her.

"Have you never heard the expression 'only break one rule at a time'?" Stephanie asks incredulously.

"Only break one law at a time, maybe. Game rules are different. The more rules are involved, the harder it will be to pin down exactly what's going on. Besides, I'm not breaking the rules – I'm interpreting them creatively."

"The original plan was to take your chips and use them to augment our starting stack," Shiro informs her. "In real poker, the ratio between your stack and the blinds^† determines how aggressively you can play. It's possible to push stronger players out of tournaments by extracting chips from weaker players and using your resources to go set mining – we wanted to avoid that, at least until we got past the first few tables and skill started to dominate luck."

^†Upon hearing this, Stephanie is enlightened as to what the M-ratio is.

"And you didn't think anyone would notice you sitting at the table with twice as many chips as you were supposed to have?"

"You'd be surprised. It's easy to develop tunnel vision when you're sitting at the poker table. No, the reason we can't do that is because it's probably cheating. Chip dumping is explicitly against the rules; we weren't planning chip dumping per se, since you aren't playing in the tournament, but receiving chips from someone directly rather than at the table via subterfuge sounds like a presumed extension of the chip dumping rule, rather than an unwritten rule."

"And your real plan isn't cheating because…?"

"Because it complies with both the spirit and the wording of the rules?"

"We'll need more information on how cheating works to be sure," Sora admits. "Otherwise, what she said."

Stephanie is just manifestly unqualified to judge whether this is true. Sora and Shiro are undeniably better at games than her – well, Sora is, Shiro could conceivably be drafting off Sora's experience – but this plan does not have the ring of sound reasoning to it. Ordinarily that would be enough for her to call it quits. Her grandfather will be remembered forever for not giving up well beyond every indication that he should, and Stephanie is determined to avoid having the same legacy.

Unfortunately, she's not calling the shots anymore. She might have to suck it up and hope everything pans out.

"Do you want me to show you to the baths?" she asks. Washing off usually helps her cool down, both figuratively and physically.

Shiro is wrapped in a film of dried sweat and a dress that hasn't seen the inside of a washing machine in months. Her long, oleaginous hair is unkempt enough to warrant shears. The soles of her feet are stained soot-black from walking barefoot across the earth and sleeping rough, complete with dried bloodstains and blisters from the same. You would need neither eyes nor ears to detect her approach.

"No thank you," she tells Stephanie.