Sora will assist with removing Stephanie's ball gown. Shiro's torso expands noticeably when he unlaces the back.

Ah, much better. She pulls her arms out of the sleeves and starts shucking the rest off as fast as possible.

"So, assuming this tournament is mostly rookies by Earth standards, let's start with our six-max ranges. I want to know what you were thinking when you raised six-five suited under the gun that one time."

"That the table was full of nits and I could fold on a board with face cards?"

All of Sora's opponents folded that hand, in reality, but your chain of reasoning cannot count on being lucky. Poor play combined with bad luck can destroy your chances of victory, whereas if you play well enough bad luck can only slow you down.

"You gave them money."

She is speaking in terms of expected value, of course.

"My raise was huge and our range in that game was polarized, they were only going to call with jacks or higher."

"I'm going to need you to walk me through this one, big brother."

"Gladly…"

Stephanie is looking forward to—

There are still preparations to be made – cellars to inspect, furniture to arrange, luncheons to plan, seating arrangements to come up with for important guests once they bust out, speeches to rehearse – and Stephanie was supposed to complete her final reviews that afternoon. Instead, her scheduled pre-tournament stud match with Chloe Zell seems to have mysteriously taken up nearly her entire day. She'd better get on with all that.

Fine.

The bare minimum amount of work will take until late in the evening.

Late in the evening, Stephanie changes into her nightclothes, lights some of the candles in her bedroom, and sits down in her rocking chair with a quill and some parchment. She doesn't have enough energy for her original plan, but she's going to muscle through the game theory homework before bed so she can put it behind her.

Shiro's explanation for how to find Nash equilibrium strategies with a matrix didn't have enough functional details for her to pick it up. She'll ask for clarification later. In the meantime, she's going to use Sora's technique. She doesn't have a way to find the expected value of strategies in the real game yet, so she'll also reuse the Idiot Poker method and solve a simpler problem first.

The Game of Pure Strategy played with only the ace and the two has exactly two strategies: bid the same rank or use the other card. No way to mix it up – which ironically makes it a true game of pure strategy, unlike its fully-realized version. The Nash equilibrium strategy is therefore remarkably boring: both players bid their ace for the ace and their two for the two, resulting in a draw no matter the order. Using the other strategy will result in a loss unless your opponent foolishly does the same thing. Impossible to improve upon.

How about the ace, two and three… make it the two, three and four instead, to avoid the possibility of a draw. Stephanie is going to use 'probability of victory' rather than 'number of chips won' as her metric for finding expected value, and she's not sure how to fit tied games into that scheme so she's kicking them into the long grass. The Game of Pure Strategy also has a fixed number of rounds in addition to a binary win condition, so maybe the Idiot Poker method isn't applicable after all? She'll give it a shot.

There are six orders the prize cards can land in. Because any two of the three cards will win, the game is mostly the same as Rock Paper Scissors: four beats three, three beats two, two loses to four but gives a three-in-four chance of winning the subsequent two cards. It's not completely identical – if the bids for the third card are equal then the game is unfortunately still a draw – but it's close enough that the true mixed strategy equilibrium is probably close to 'play each card with equal probability', biased a little towards bidding the same rank as the prize.

Stephanie is in the process of writing this out when she realizes the answer to Sora's question.

See, even though she does not have a working "get your opponent's expected value" function to use in her solution, she can guess that it'll be a slightly uneven distribution over all of your remaining cards. If the first card to land on the board is the king, you need to have at least some probability of bidding any of your cards to keep your opponent from exploiting you. Sora bid his ace for the king, presumably because he knew Juno was liable to bid king for it and wanted to get some value out of the deal. So the first round strategy has thirteen probabilities.

But that's just the first round for one possible card – those thirteen probabilities change depending on which prize card is first. Worse, the number of permutations in the game is the factorial of the number of prizes: two for two cards, six for three, 24 for four, and 120 for five. Stephanie calculates the factorial of thirteen and gets a number greater than six billion. Six billion probabilities, multiplied by which cards you've already bid and which cards your opponent has already bid…

Mystery solved: Sora didn't use the Nash equilibrium strategy because he hasn't spent millennia memorizing numbers. Stephanie lights the parchment with a nearby candle and tosses it into the empty fireplace to keep anyone from fishing it out of her wastebasket. The whole idea of a Nash equilibrium doesn't feel quite as formidable as it did a few short hours ago, but it does still seem like the ideal basis for creating mixed strategies: narrow down your possibility space to only the good options, then select a mixture of those good options to stop your opponent from knowing your plan with certainty.

Alternatively, maybe Shiro's「linear programming」trick is faster than Sora's guess-and-check method for very large spaces, which might mean that poker has a set of approximately-stable-equilibria that are small enough to find and memorize but strong enough to win a tournament against a lineup of opponents who've never heard of game theory before.

Drat. She should've stayed with them and learned more. She goes to bed, feeling unsure and alone.

What a busy day it is. Fayette can't remember the last time the castle was prepared to host so many people – not during her tenure, that's for sure – but it's exhausting. Her work is normally completed before sunrise.

Her final task of the morning is to assist the young miss presently staying in the Rapture suite. Princess Stephanie had their new outfits brought to the wardrobe overnight by the laundry service, and Fayette is positive that one of them is another highly impractical dress made from rare silks and expensive filigree that takes a second person's help to don. She's not going to let pass an opportunity to reuse her old court apparel.

She knocks on the door.

No answer.

She knocks again, more stridently.

Silence.

There's no time for dallying. Ignoring etiquette, she pushes the door open and goes inside.

The young woman is alone, asleep under a mound of blankets. Her older brother is gone. The curtains are pulled shut, casting everything in gloom. The ball gown is draped over the back of a chair – carefully, in a way that will only cause the slightest twinge of chest pain for whoever has to clean it up later.

It is probably against protocol to call royal guests sluggards. Emphatic cheerfulness it shall be.

"Good morning!" she announces at full volume, making her way for the curtains.

Shiro sits up, wiping a strand of drool from the corner of her mouth.

"Morning," she says languidly, and flinches when the chambermaid throws open the curtains and lets the daylight in.

Disboard's sun is slightly bluer and more intense than Earth's, probably an F-type main-sequence star if the mechanics of converting hydrogen into helium are the same. Shiro is trying to hold off on speculating whether the existence of magic and elves implies radically different astrophysics until she reads at least one (1) book on the subject. Regardless, it's spectacularly uncomfortable until your eyes adjust.