Chapter 71
After summarizing what we'd learned about perfect information games, Arin watched intently.
“Heehee, this is fun!”
“Okay, but Arin, do you remember how Sis divided games into two types earlier?”
“Yeah! Perfect information games and… imperfect information games!”
“That's right. But everything we've looked at so far has been a perfect information game. Chess, Go, and the stone-removal game. Games where you can see all of your opponent's cards.”
“Now that you mention it, yeah.”
“But imperfect information games are completely different. It'd be hard to explain with words alone, so—”
I got up and took a deck of playing cards out of the drawer.
“Let's try it ourselves.”
Riffle-riffle-.
I picked out exactly three cards from the deck of fifty-four.
………
J Q K
………
“If the simplest example of a perfect information game was the stone-removal game, then this is the simplest example of an imperfect information game. It's called Kuhn Poker.”
“Kuhn Poker?”
“Yep. Mathematicians stripped poker down to its essence and removed everything else, turning it into a simple game. Here are the rules.”
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[Kuhn Poker Rules]
- Only three cards, J, Q, and K, are used. (Strength ranking: K > Q > J)
- The two players are each dealt one card. The remaining card is left unseen.
- Both players bet one chip at the start.
- The first player acts first: check (pass) or bet (put in one more chip).
- If the first player bets → the second player can call (match the bet) or fold (give up).
- If either player folds, the other player takes all the chips in the pot.
- Once both players have finished betting one or two chips → reveal the cards.
- The player with the higher card takes all the chips in the pot!
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After reading all the rules, Arin wiggled her fingers.
“Oh, it's simple! I wanna play!”
“All right. I'll go first. Okay, let's begin.”
✒️✒️✒️
Tap. Tap.
We each took some chips.
Then, before dealing the cards, we each put one chip in the middle.
The first round.
Peek-.
I took a quick look at my card.
My card was a K.
A guaranteed winner.
“Bet.”
Tap-.
“Umm… call!”
Arin matched my bet.
When we revealed our cards, mine was a K and Arin's was a Q.
“Sis wins.”
Roll-roll-.
I swept all the chips in the middle over to my side.
The second round.
This time, my card was a J.
The weakest card, guaranteed to lose.
Even so, I calmly pushed my chips forward.
“Bet.”
“Gasp… you're betting again?”
Arin hesitated as she looked at her card.
“Ugh… Sis, don't you have a K again…? Fold!”
“Heh-heh, so easy…!”
Arin covered her card.
Though it was easy to infer that it was a Q.
I smiled and pulled the chips toward me.
Of course, I never showed my J.
We played several dozen more rounds like that.
Little by little-.
A pile of chips grew higher and higher in front of me-.
while the chips in front of Arin gradually dwindled until her side looked bare-.
“Huh… why do I keep losing?”
Arin puffed out her cheeks as she looked at her dwindling chips.
“Muuuu… Sis, aren't you getting awfully lucky? You keep getting good cards!”
I shook my head.
“No. The cards were randomized the same way for both of us. This isn't a difference in luck; it's a difference in strategy.”
“Strategy…?”
“Imperfect information games have to be approached differently from perfect information games. The important thing is guessing what cards the other player has, right?”
“Right…?”
“We can each have one of J, Q, or K. If Arin has a K, then it's a guaranteed winning hand, so it'd be best to bet, right?”
“Hmm… yeah. Since it wins for sure, it's better to put in more chips.”
Arin nodded.
“That's the K strategy. Since it's a winning hand anyway, you bet to squeeze as many chips out of your opponent as possible. If you check, you take two chips, but if you bet and your opponent calls, you get four chips.”
“Oh, I see! K always bets!”
“Then what about the weakest card, J?”
“Hmm… J always loses, so… check?”
“That's the trap.”
I raised a finger.
“If you reveal a J, you lose for sure. So if you meekly check, your chance of winning is zero. But—”
I brought up the hand where I'd bet with a J earlier.
“What if you bet while holding a weak hand?”
“Wha… you mean bet with a weak hand…?”
“Then your opponent might get scared and think, ‘Oh no, she must have a good card,’ and fold. Then you win a game you were originally guaranteed to lose.”
“…Ah! Was that what happened earlier? You had a J then, Sis?!”
“Heh-heh. That's right. That's called bluffing.”
Arin's mouth fell wide open.
“That's so unfair, Sis! You had a weak card but pretended it was strong!”
“What did you expect? Were you under the impression you were playing against a saint?”
Ruuuumble-.
Arin looked shocked, as if she'd just learned the truth about the world.
I raised one more finger.
“But you can't keep using this strategy forever. What would happen if Sis bluffed every time she had a J?”
Arin thought for a moment, then her eyes lit up.
“Ah! Then I can just call, thinking, ‘Sis is lying again!’”
“Correct. If you bluff too often, you'll get caught and lose, but if you don't bluff enough, it has no effect. So—”
I scribbled on the whiteboard.
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[Kuhn Poker Optimal Strategy]
- K (strongest) → Always bet. Squeeze out as much as possible.
- J (weakest) → Bet only occasionally. Pure bluffing.
- Q (middling) → Usually check.
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“You should bluff with a J exactly one time in three. Only 1/3 of the time.”
“Why exactly 1/3?”
Good question.
“Because that's the frequency that confuses Arin just enough. When Sis bets, whether Arin calls or folds, she breaks even. The ratio that makes either choice equally good is 1/3. Then, no matter what Arin does, she can't beat Sis.”
I explained with a smug expression.
“Ugh… that's so annoying…”
“And with the middling Q, betting doesn't gain you much. A K will call and beat you, while a J will fold immediately. So it's better to mostly check quietly.”
Arin stared at the strategy chart.
“But, Sis. Then is there… no way to guarantee a win like there was in the stone-removal game?”
Oh.
Arin had asked about the crucial point.
“That's right. That's the decisive difference between the two games.”
I slowly wrote on the whiteboard.
………………………………………………
Stone-removal game → Calculate the current position, and you get the answer.
Kuhn Poker → You have to calculate what your opponent knows and doesn't know, too. So the answer comes in the form of probabilities, not a single answer.
………………………………………………
“In perfect information games, you can see all the cards, so if you calculate everything to the end, one move emerges as the answer! You're definitely going to win or definitely going to lose.”
“Yeah.”
“But in imperfect information games, you can't see your opponent's cards, right? So there are no guaranteed wins or guaranteed losses. Instead, mixed strategies like ‘Bluff once every three times in this situation’ are the closest thing to a winning strategy.”
Arin made a bewildered expression.
“So the answer is to mix your choices according to probabilities?”
“Yep. You can still lose any given round because of luck. There is no single best move; only an optimal strategy.”
“Heeey…”
Arin looked at me with fascinated eyes.
“If you stick to this strategy, then over the long run, on average, you'll gain the most. When both players stick to their optimal strategies and neither can gain more by changing their strategy alone, that's called a Nash equilibrium.”
“Nash equilibrium…”
“Of course, I'm not talking about a monster that lives in a lake.”
“That's Nessie! I know that much!”
“And when Sis bets, Arin thinks, ‘Does she have a K, or is she bluffing with a J?’ You look at your opponent's actions and keep updating your guess about which card they're most likely to have. That's called Bayesian inference.”
“Bayesian inference…”
“Of course, I'm not talking about a color that looks like an even mix of brown and yellow.”
“Sheesh-! That's beige! Don't tease me!”
I chuckled at Arin's reaction.
“It may seem a little simple, but rock-paper-scissors is another example of an imperfect information game. Arin, what would you do if I always threw rock?”
“Huh…? Then of course I'd throw paper.”
Arin looked at me as if she couldn't imagine what kind of question that was.
“Right? If you know what move your opponent is going to make, you can take advantage of it, right? Just like knowing someone keeps betting with a J lets you exploit them by calling with a Q.”
“Aha.”
“That's why the optimal strategy in rock-paper-scissors is to throw rock, paper, and scissors completely randomly, each with a 1/3 probability.”
“I get it. Then at least my opponent can't read my pattern and gain an advantage, right? I understand it more or less.”
Arin picked up the cards again.
Her eyes were completely different now.
“Sis. Let's play again. I won't fall for it this time.”
“Oh, are you up for it?”
✒️✒️✒️
A few rounds later.
My card was a J.
I calmly pushed my chips forward again.
“Bet.”
Before, Arin would have gotten scared and folded.
But this time was different.
Hmm-.
Arin stared at her card, a Q, and thought hard.
“Sis bet. If she has a K, that's the obvious play, and if she has a J, it's a bluff. But seeing how Sis bets every time… I get the feeling her bluffing frequency is a little high.”
Arin pushed a chip forward with a tap-.
“Call!”
When we revealed the cards—.
Revealed were my J and Arin's Q.
“Yoo Arin wins!”
Roll-roll-.
The chips rolled toward Arin.
“Woohoo-! I won-! I caught Sis bluffing for the first time!”
Arin threw both arms into the air in triumph.
“Heh-heh. Well done. You read my bluff perfectly.”
As I gathered up the chips, I added,
“But Arin, there's one thing you need to remember. This game involves luck, so even if you play perfectly, you won't win every round. In fact—”
I wrote Kuhn Poker's expected payoffs on the whiteboard.
………………………………………
First player's average expected payoff ≈ −1/18 chip
Second player's average expected payoff ≈ +1/18 chip
………………………………………
“When both players play Kuhn Poker perfectly, strangely enough, the first player suffers a very slight loss on average.”
“Huh…? Why?”
Arin wore a bewildered expression.
Maybe she'd used her brain too much today, because steam began to puff out of her head-.
“Because the person who acts first gives away a hint about their hand first. The second player sees that before deciding, so they have more information.”
“Ah… so moving first is a disadvantage?”
“In imperfect information games, the order in which you act is information in itself.”
Arin looked back and forth between the cards and chips. Then she muttered quietly,
“That's fascinating. Chess is a game where you calculate everything to the end with your mind, but card games like this… are games where you read your opponent's mind and fight using probabilities and strategy.”
Oh, that's my little sister.
She summarized what I'd taught her today perfectly in one sentence.
“Exactly. Mathematically speaking, there are two types of games. And now Arin is someone who knows about both. There is definitely a difference between someone who knows that and someone who doesn't.”
Pat-pat-.
I gently patted Arin on the head.
“Woohoo-. Games can be viewed through math too! This is fun!”
“Then I'll summarize what we learned today.”
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[Imperfect Information Game]
- Kuhn Poker is a representative example of an imperfect information game that has already been completely analyzed.
- Optimal strategies can be calculated using Nash equilibrium.
- An optimal strategy is given not as a single action, but as a probability distribution.
- Even in real poker, top-level players use approximate versions of these strategies.
- Negotiations, auctions, and investments in the real world can also be viewed as imperfect information games.
- The fields that study these are probability theory and game theory.
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