They play another twenty-seven hands. Eventually it becomes clear that Stephanie has more than an inkling of how to play this game correctly. Sora wins a little more off Stephanie than the reverse, but their stacks are still close.

"Good," he says, gathering up the deck to signal a break. "So, same question as before. What's your thought process like when you're playing this game in second position?"

"There are six possibilities for me: either you check or you raise, multiplied by three cards. If I have the king I always call or raise, and if I have the jack and you raise first, I always fold. Also, if I have the queen I never raise. Either you have the jack and you'll fold, which doesn't help, or you have the king and I'll lose. Four of the possibilities are nailed down. The other two possibilities are having the queen when you raise, or having the jack when you check."

She takes the queen out of the pile to illustrate her point.

"If I have the queen and you raise, you're either bluffing with the jack or trying to get me to call with the queen versus your king. I can't fold every time, otherwise you can bluff me with impunity, so I have to call at least some of the time. I tried to do it randomly, since I couldn't tell whether you were bluffing by your body language."

She swaps the queen for the jack.

"If I have the jack and you check, you're either holding the king and trying to get me to bluff with the jack or holding the queen and gambling that I have the jack and not the king."

Sora raises his hand. "Good so far. At this stage I'll tell you something you'd have noticed anyways soon enough: I never raise with the queen either, for the same reason. If you have the jack you'll fold and I gain nothing I wouldn't have gained by checking, and if you have the king you'll call and I'll lose more chips than I might have otherwise. Open raising with the queen is a 'weakly dominated strategy' – no matter which card your opponent has, you can always do at least as well or better by checking."

That makes sense.

"Right, so… if I have the jack and you check, you're either holding the king and trying to get me to bluff with the jack, since you know the thing about the queen never raising, or you have the queen and you're doing the queen thing. So in that scenario raising isn't great, since half of the time you're guaranteed to call with the king and the other half of the time you'll have the queen and you might call; but if I check I'm guaranteed to lose. I need to raise and bluff having the king at least some of the time, otherwise my predictability lets you raise with the queen every time."

"Interesting. You're right about the queen and the jack, but can you tell me any more? Is there a specific probability you should use to decide how often you want to raise with the jack or call with the queen?"

"Half… of the time…?"

That doesn't sound right, but she's not sure and there's no penalty for guessing wrong.

"Keep what you just learned in mind. We're going to play with you in the first position now; see if you can spot the pattern for actions that aren't forced. Player one never raises with the queen and will always fold when facing a raise with the jack, so don't worry about those anymore."

Sora gives the truncated deck a wash and a shuffle, then deals each of them a single card face-down. He looks at Stephanie expectantly.

Ante up.

Stephanie's card is the queen. No need to think, this is always a check. If Sora checks back, she has a 50% chance of winning, as pure as it gets.

Sora raises. His face is set in stone.

Well, it was fifty-fifty.

… wait, no it's not. Half of the time he has the king and always raises, but the other half of the time he has the jack and might raise, hoping she thinks he has the king and folds. So, how often does Sora do that?

Stephanie doesn't believe it's half of the time, mostly because Sora gave her the 'are you sure' look when she said that, but it can't be far off. If it's true, or close to true, she wins this hand one time in four if she calls. That's probably a good fraction of the time to call, then.

She decides to pick a number at random, and if the positive residue after dividing by four is zero she'll call. Random number, random number… there are 52 cards in the deck. Sure.

52 ÷ 4 = 0. She calls.

Sora flips up the king and collects the pot.

Not entirely unexpected. It's possible to commit no mistakes and still lose. No use being mad about it.

Trouble is, though, she's pretty sure it was a mistake.

Ante up. Sora deals Stephanie the king this time.

Aha, the hand she's been thinking about. As player one she has the power to check and hope for a raise from Sora with the jack, or to open raise and get a call from Sora's queen, either of which nets her two chips. She's guaranteed one chip at minimum, but if she doesn't fight for two she won't make up for all the hands she's lost with the jack and queen.

Will Sora call a raise? He definitely won't with the jack, and he might with the queen. She probably wants to raise anyways, since she can't extract extra value from jacks and can from queens. Sora might have been doing that in first position, come to think of it. But she can't raise all the time, because if she does Sora will stop calling and she won't be able to farm him for double the chips when he has the queen.

How often is he going to bluff with the jack in this position? Not often. Stephanie is weirdly confident that it's less than half the time, given how fraught bluffing is in poker. So, a quarter of the time. She gets two chips by raising one time in every eight, so that's how often she'll do it.

Random number time! There are thirty pieces in backgammon, divided by eight leaves six. She raises.

Sora calls.

Yes! Stephanie shows the king and rakes in the pot.

Another one.

It's the jack. She's going to check-fold most of the time, but if she never raises Sora can exploit her. Time to make a decision.

… this is just the inverse of the last decision. She'll raise one quarter of the time. Five nations on the continent divided by four leaves one. Check.

He has the queen. Check. Oh, convenient, Stephanie had the jack. The pot's his.

They'll play a few more hands with Stephanie in first position, but not too many. He wants her to catch a glimpse of the underlying math, and for that she'll have to play a lot of hands on both sides.

… she's pausing before she plays the jack or the king but checking immediately with the queen. Sora's trying not to pay attention to the tell, but it's like trying not to pay attention to the beam of a flashlight in the dark. He deliberately calls with the jack once, just to tell himself that he's being fair.

A few rounds later, he decides it's time for the practicum.

"Hopefully you've got a strategy in mind. We're going to play for real now, alternating turns, and we're going to speed it up. We'll stop once it's obvious."

"Once what's obvious?"

"You'll see. It'll be obvious."

They commence, rotating position with each hand. Neither player speaks, they simply tap the table or push chips into the pot in silence. Once they get into a rhythm each hand takes about five seconds to complete. The game starts to noticeably progress.

Ah, it is indeed obvious. Despite the simplicity of the game and the fact that neither of them is monopolizing first position, Stephanie is slowly bleeding chips to Sora. She varies how she plays the hands that call for probabilistic choices, but it's hard to tell whether it's doing any good. This is a loss in every way that matters.

"That should be enough," Sora says, once he has twice as many chips as Stephanie. "Let's talk. How did you decide which probabilities to use in your strategy?"