Chapter 426: Guessing Expectations

Zheng Jie was somewhat curious, "Huh? The game is just starting now?

"But hasn't the most fundamental principle and logic of this game already been revealed?"

Clearly, this 'Wallet Game' or 'Two Envelope Paradox' that Yang Yuting mentioned wasn't complex in principle.

Simply put, two slips had different amounts written on them, with one amount being ten times the other. The question was whether to switch to the other slip after drawing one.

If the method for calculating expected return was wrong, one would reach the erroneous conclusion that 'no matter how you switch, it's a positive return.'

But after thoroughly investigating, you'd actually just return to the starting point, which was that it didn't matter whether you switched or not.

It was simply a 50% chance of switching from a small envelope to a big envelope, or a 50% chance of switching from a big envelope to a small envelope.

So as long as one knew that 'from the perspective of probability and expected return, switching makes no difference,' this couldn't really be called a puzzle.

So what else could be made into a puzzle?

Yang Yuting smiled and continued, "Just keep listening.

"Now, I'm still a God’s Imitator.

"But this time I'm in a 'Self-Selected Game,' and I must follow the basic rules in the 'Self-Selected Game' hall.

"The specific gameplay is still choosing one of two slips, then deciding whether to switch to the other one. Once you decide, you can't go back or change.

"What's written on the slips are 'chip' values. In 'Self-Selected Games,' chips can be exchanged for visa time but are affected by inflation.

"However, every time this game is played, the two numbers will change. That means you can't use past experience to judge the next values.

"Now, draw."

Yang Yuting placed two written slips in her left and right hands respectively.

Zheng Jie volunteered again, "Let me!"

He pretended to consider carefully, then took the slip from Yang Yuting's left hand.

Opening it, the number '10,000' was written on it.

"Do you want to switch?" Yang Yuting asked.

Zheng Jie hesitated, "This..."

Clearly, although he already understood the specific principle, actually putting himself in the specific game scenario still made him hesitate.

Because the principle of this strategy would be affected by the specific game rules.

Dai Yifan was somewhat puzzled, "Since we already know the principle is that switching or not switching makes no difference, then it just depends on mood.

"If it were me, I'd switch. Since they're the same anyway, switching gives more information."

Yang Yuting asked, "What if after switching, it becomes 1,000? In 'Self-Selected Games,' these two chip amounts would have a huge difference and might put you at a disadvantage in subsequent games."

Dai Yifan was stumped. He fell into silence again.

Zheng Jie seemed to think of something. He said quite seriously, "If it's in the early game, don't switch. If it's in the mid-to-late game, you can try observing other players first, then decide whether to switch."

Yang Yuting nodded and asked, "Why make such a decision?"

Zheng Jie was somewhat excited, "Because this problem is no longer the original 'Two Envelope Paradox' problem. It has become a problem of 'guessing the God’s Imitator's psychological expectations'!

"In the original problem, if you drew '10,000,' there was no way to judge whether this was a relatively small or relatively large number.

"The other number could be 1,000 or 100,000.

"But when this problem has a specific scenario, it's different because there's a 'range of expectations'!

"So if encountered in a 'Self-Selected Game,' first you should try to guess how many chips the God’s Imitator is willing to put into this game at most, then use this expectation to judge whether the number you currently drew is large or small.

"Considering the 'Self-Selected Game' situation, although God’s Imitators can issue unlimited chips, they can't really issue them infinitely.

"If I drew the number '10,000' right at the start of the game, and the other slip was '100,000,' that would mean in the early game, players could easily obtain 100,000-level chips in this room.

"This would cause very serious inflation in this room, with chips able to exchange for drastically reduced visa time.

"This would directly discourage players entering the room later, causing the entire room's price system to collapse.

"So if a God’s Imitator really designed such gameplay in a 'Self-Selected Game,' they would definitely strictly limit chip output in the early stage.

"These two numbers could only be '1,000' and '10,000.'

"That would be equivalent to... if a player drew '10,000,' they'd have found a guaranteed win method in this room.

"But as the game progresses, to attract more players to enter, the God’s Imitator might also change the amounts of these two numbers midway.

"For example, changing '1,000' and '10,000' to '10,000' and '100,000.'

"Because there are no restrictions in the rules, God’s Imitators can change at any time.

"After changing, perhaps some players drew '100,000' chips. If other players saved chips from this room, their wealth would be diluted, equivalent to reshuffling.

"Although this is a method of killing the goose that lays golden eggs, for players joining later, it gives them reasons to play a few more rounds.

"So considering the God’s Imitator's thinking, in the early game, if you draw 10,000, stop. In the mid-to-late game, you can observe other players' situations, or try switching yourself. Perhaps there's a possibility of switching to 100,000."

Yang Yuting appreciatively clapped her hands, "Very well said! That's exactly it!"

Qin Yao said with emotion, "So that's how it is. Amazing, Zheng Jie!"

Zheng Jie was somewhat embarrassed, "Not at all, not at all. Mainly this problem was guided well, progressing from shallow to deep.

"If principle analysis hadn't been done at the start, if I suddenly encountered this in a game, I probably wouldn't have thought of it."

Yang Yuting continued, "So the first variation of this problem is 'guessing expectations.'

"If I use real-life examples, everyone should understand more easily.

"Suppose now it's your company or organization's annual party, and the company's boss or your superior wants to play this game with you. The amount drawn is your year-end bonus or award.

"Also two slips. The slip you drew says '10,000.' At this point, the boss asks if you want to switch.

"Would everyone switch?

"Auntie Zhou, you speak first?"

Zhou Guifen was suddenly asked and was somewhat surprised. But after thinking, she still said, "Don't switch!"

Yang Yuting nodded, "Mm, why?"

Zhou Guifen said matter-of-factly, "We should thank heavens if our school doesn't delay wages. How could they possibly give a 100,000 year-end bonus! Absolutely impossible. Even 10,000 would be incredibly lucky."