Chapter 77
We decided to build our own decks out of the overflowing Pookimon cards and play in earnest.
The basic rules of this game are as follows.
First, each deck has 60 cards.
You can include up to four copies of the same card.
When the game starts, both sides draw seven cards, then draw one card each turn after that.
And each deck has Pookimon. You evolve them step by step and beat the stuffing out of your opponent to win.
End of explanation.
A-rin, first of all, stuffed her deck full of cards that looked powerful.
A deck made up only of strong Pookimon and the energy needed to use their moves.
By contrast, after skimming the rules, I quickly calculated the probabilities and expected values in my head.
Then I put in a certain proportion of draw cards and search cards that could find the cards I wanted.
As a result, the number of Pookimon cards actually in the deck went down.
"Sis, is this deck really okay? There are way too few Pookimon. This game is about beating your opponent up with Pookimon, you know?"
A-rin, who hadn't even played a single game yet, was already acting like she knew everything.
How ridiculous.
"Yeah. It's fine."
"Hmph... older sister, I filled mine with only strong Pookimon! Don't go waah-waah crying after you lose!"
A-rin taunted me.
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And so the game began.
A-rin must have drawn her deck's main Pookimon well, because she put it on the field right from turn one.
I, on the other hand, hadn't drawn my main Pookimon, so I had no choice but to put a comparatively low-value Pookimon on the field.
"Hee-hee, I'm already ahead, aren't I?"
"We'll see."
A-rin taunted me, all excited.
A few turns passed after that.
A-rin kept evolving her Pookimon smoothly, one after another.
But it seemed she just couldn't get the energy cards she needed to attack me.
"Hmph-."
A-rin made a displeased face.
She kept playing with a lighthearted expression, as if thinking she'd just draw it next turn.
But even after the next turn, it looked like she kept drawing nothing but useless cards.
A-rin's face gradually turned more and more tearful.
She had run into what you'd call a completely dried-up hand.
"Ugh..."
Meanwhile, even though I hadn't drawn my key cards at first,
I made good use of my draw and search cards, drew cards like crazy, and steadily built up my board.
"Older sister, why are you drawing so many cards...!"
"Nihi-heh."
"Moo-."
A-rin puckered her lips and showed her displeasure.
And then, on turn three.
I evolved my Pookimon and attached plenty of energy, fully ready to beat A-rin up.
All the while, A-rin still wasn't in a position to launch even a single proper attack at me.
"Go! Blow away A-rin's Pookimon!"
I ordered the Pookimon on my field to attack.
Kaboooom-!
"Heeeek-!"
A-rin was helpless against it.
Then she clutched her chest, staggered, and collapsed face-first onto the floor with a thud.
"Guh... o, older sister... you're too strong...!"
Even if A-rin was still inexperienced, she was more serious about the game itself than anyone else.
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After the game ended,
A-rin started carefully reviewing why she'd lost.
"Hmm... the cards I needed just wouldn't come into my hand... Did I lose because I was unlucky?"
It seemed A-rin thought the reason she lost was luck.
Reviewing your plays is good, but you only grow if you do it in the right direction.
I decided to properly explain to A-rin why she'd lost.
"A-rin. It's true luck was involved, but there's a more fundamental reason than that."
"Huh? What's that?"
"From the start, A-rin's deck was at a disadvantage. Let me explain."
I spread out A-rin's deck and said,
"A-rin, you filled your deck to the brim with only strong Pookimon and energy, right? Then the only card you see each turn is the one you draw at the start of the turn."
"Yeah. I drew one each turn."
"That's the problem. Look. For A-rin to evolve and attack on turn two, all of these cards have to be in your hand at the same time."
I pointed them out one by one with my finger.
"One basic Pookimon. One evolution Pookimon. And two energy cards for the attack. Four cards total."
"Yeah. I know."
"But without draw or search cards, you have to gather all of that purely within the nine cards you draw over two turns. Then do you know what the probability is?"
I wrote the numbers on the whiteboard, swish-swish-.
"A-rin's probability of drawing the basic Pookimon and its matching evolution Pookimon and successfully evolving is about 38 percent. And the probability of drawing at least two energy cards of the right type to attack is 14 percent. Since those two probabilities are independent, if you multiply them?"
"I know how to multiply! 0.38 times 0.14 is..."
A-rin started calculating the probability I'd mentioned at lightning speed.
"...0.0532!"
"Ding-dong-ding- correct. So with A-rin's deck, the probability of being ready to attack by turn two is only 5.32 percent."
"Gasp. Seriously? Why is it that low?"
A-rin's eyes went wide.
"Yeah. A-rin's deck is the kind that only wins if that tiny probability comes up. You got lucky and drew the Pookimon you needed to evolve, so the evolution went smoothly—but you didn't also draw the energy at the same time. That's why you lost and had no choice but to get beaten by me without being able to do anything."
"Ugh... so that's what it was..."
After running the numbers a few times, A-rin pointed at my deck.
"Then what about you, older sister? You couldn't even draw your main Pookimon at the start. So how did you win?"
"Good question, A-rin."
I smiled with satisfaction and spread out my deck.
"I boldly cut back on the strong Pookimon. Instead, I stuffed in lots of draw cards that let me draw cards and search cards that find the cards I want."
"Yeah. That's why you had so few Pookimon."
"Right. But look. Even if I didn't have my main Pookimon in my opening hand, I had this search card instead. This card brings a basic Pookimon out of the deck with a 100 percent guarantee and puts it on the field."
"Gasp, 100 percent?"
"Yeah. It's not leaving it to luck; it's bringing it in for sure. And this draw card lets you throw away the cards in your hand and draw seven brand-new cards."
I fanned the cards out with a shrrk-.
"I chained search and draw cards together and looked through more than 18 cards in my deck. That's more than twice as many as the nine cards A-rin saw over two turns. By steadily clearing out cards I didn't need right away, I greatly increased the chance of getting the cards I wanted."
"Wow..."
A-rin gaped.
"So that's how you play card games..."
She had a look on her face that said she'd realized something.
"So the conclusion is this, A-rin."
I said, holding up my index finger.
"A-rin's deck had a very low expected value for executing the play you wanted. By contrast, my deck was built to raise that expected value as much as possible. That's why I beat you so easily."
"Then... does this deck always lose?"
A-rin made a sulky expression.
"No, that's not it."
I shook my head.
"If you're lucky and the perfect cards come in at just the right time, A-rin could still win. For one or two games, at least."
"Oh! Then that's a relief!"
"But do you remember what your older sister said last time?"
"Huh? What was it again?"
"In an imperfect-information game, you can win or lose a single match right away. What's important is building a strategy that gives you an advantage in the long run."
"Aha...!"
A-rin looked as if she'd realized something.
She must have remembered what I'd said when we dissected the game's structure mathematically last time.
"Right. Even if you can win one or two games through luck, over a hundred or a thousand games, the side with the higher expected value eventually wins. Your deck is extremely unfavorable in that regard."
"I see..."
A-rin looked back and forth between her deck and mine.
Then she clenched both fists.
"I learned something good from you, older sister, so I'll try building a deck that raises my expected value for winning, too-!"
"That's the spirit, A-rin."
I patted A-rin on the head.
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From that day on, A-rin underwent a surprising change.
Whenever I tried to teach her math, A-rin would normally complain, "Ugh, too much trouble," and act so bothered.
Now she was begging to study math on her own.
She especially wanted to practice probability calculations hard.
"Older sister! I want to study probability again!"
"Really?"
"Yeah!"
If my little sister wanted to study on her own, what older sister in the world could refuse!
I got excited and made problems for A-rin myself.
Most of them were probability problems, just as she requested.
"A-rin, but why do you suddenly want to study probability?"
"If I'm good at probability calculations, I think I can make the optimal strategy! That Nash equilibrium you talked about, right? I need that!"
A surprisingly plausible answer.
"Nice."
So my hard work teaching her math wasn't for nothing.
To want to study on her own like this just because she wanted to get better at games.
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[Problem: In a 60-card deck, there are 3 Rajamong ex cards and 4 Rare Candy cards. If you draw 7 cards for your opening hand, what is the probability that both Rare Candy and Rajamong ex are in that opening hand?]
A-rin was lying face-down on the blanket, kicking her feet in the air as she solved the problem.
With her tiny hands, she held a pencil and scribbled away.
"Hmm... since this is an independent-trials case, a binomial distribution..."
A-rin muttered.
"The probability of no Rare Candy is 61.21 percent... the probability of no Rajamong is 69.86 percent... the probability of neither is 41.58 percent..."
Numbers spun around in A-rin's tiny head, ping-ping-.
"Since it's the complement, if I subtract it from the total probability... the probability of drawing both is... 10.51 percent!"
"Ooh-. That's correct."
Her mental arithmetic really was insanely fast.
She was a genius who outpaced me in sheer processing speed.
She could even grind out complex probability calculations to the second decimal place as if it were nothing.
I was planning to give her credit for roughly 11 percent after rounding, though.
I peeked at the paper over A-rin's shoulder.
It seemed she'd done most of the work in her head, because the paper had only a neat answer written on it.
...At this point, isn't she basically a human computer?
"Heh. Older sister, I solved them all! Give me more, more!"
A-rin shook the worksheet with sparkling eyes.
"Okay. Then this time I'll give you something a little harder."
In this way, A-rin's math studies progressed steadily.