God's imitator

Chapter 427 Guessing Expectations

Zheng Jie asked curiously, "Huh, has the game just started?"

"But haven't the game's most fundamental principles and logic already been revealed?"

Clearly, the principle behind Yang Yuting's "wallet game" or "double envelope paradox" is not complicated.

Simply put, there are two slips of paper with different amounts written on them, one of which is ten times the amount of the other. The question is whether or not to exchange one for the other after drawing one.

If the method for calculating expected returns is incorrect, it will lead to the erroneous conclusion that "no matter how you change it, the returns will be positive".

But after a thorough investigation, it turns out that we've simply returned to square one, meaning that it doesn't matter whether we change it or not.

It's simply a 50% chance of getting a small red envelope in exchange for a large one, or a 50% chance of getting a large red envelope in exchange for a small one.

Therefore, once you know that "in terms of probability and expected returns, it doesn't matter which way you switch," it's not a mystery at all.

What other questions could be set?

Yang Yuting smiled and continued, "Let's continue listening."

"Right now, I am still a copycat."

"But at this point, in the 'Selection Game' section, I must follow the basic rules in the 'Selection Game' lobby."

"The specific way to play is to choose one of the two slips of paper, and then decide whether to switch to the other one. Once you decide, you can't go back on your word or change it."

"The note says the value of 'chips.' In the 'Choose Your Own Game,' chips can be used to exchange for visa time, but they are affected by inflation."

"However, the two numbers change every time you play this game. In other words, you can't use your previous experience to judge the value next time."

"Now, let's draw."

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

Zheng Jie volunteered again: "I'll do it!"

He pretended to consider it for a moment, then took the note from Yang Yuting's left hand.

Upon opening it, I saw the number '1' written on it.

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

Zheng Jie hesitated: "This..."

Obviously, even though they understand the specific principles, they still hesitate when they actually put themselves in the specific scenario of the game.

Because the principle behind this strategy is influenced by the specific rules of the game.

Dai Yifan was somewhat puzzled: "Since we already know that the principle is that it doesn't matter whether we change it or not, then let's just go with the flow."

"If it were me, I would switch. Since they're all the same, switching will give me more information."

Yang Yuting asked, "What if it changes to 1000? In the 'Choose Your Own Game,' there can be a big difference between these two chip amounts, which could put you at a disadvantage in the later games."

Dai Yifan was stumped by the question and fell silent again.

Zheng Jie seemed to have thought of something, and he said quite seriously, "If it's in the early stages of the game, don't switch. If it's in the mid-to-late stages of the game, you can try observing other players before deciding whether to switch."

Yang Yuting nodded and asked, "Why was this decision made?"

Zheng Jie was somewhat excited: "Because this problem is no longer the original problem of the 'two-envelope paradox,' it has become a problem of 'guessing the psychological expectations of imitators'! In the original problem, if you draw '1,' there is no way to judge whether it is a relatively small number or a relatively large number."

"The other number could be 1000, or it could be 10."

"However, when this problem has a specific scenario, it's different, because there will be an 'expected range'!"
"So, if you encounter this in the 'Game of Choices', you should first try to guess how many chips the copycat is willing to put into this game, and then use this expectation to judge whether the number you draw is big or small."

“Considering the ‘selection game’ scenario, although copycats can issue unlimited chips, they can’t actually issue unlimited chips.”

"If I draw the number '1' at the very beginning of the game, and the other slip of paper is '10', then it means that in the early stages of the game, players may easily obtain chips in the range of 10 in this room."

"This will cause very serious inflation in this room, and the visa time that chips can be exchanged for will be greatly reduced."

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

"Therefore, if copycats really designed such gameplay in the 'selection game,' they would definitely have to strictly limit the output of chips in the early stages."

"These two numbers can only be '1000' and '1'."

"That means... if a player draws '1,' they've found a sure-win strategy in this room."

"However, as the game progresses, in order to attract more players, copycats may change the amounts of these two numbers midway through the game."

"For example, change '1000' and '1' to '1' and '10'."

"Because there are no restrictions in the rules, copycats can change their rules at any time."

"After the change, some players may draw '10' chips. If other players keep the chips in this room, then their wealth will be diluted, which is equivalent to a reshuffling."

"Although it is a short-sighted approach, it gives new players a reason to play a few more rounds."

"So, if we consider the copycat's idea, then stop when you get 1 in the early stages of the game. In the mid-to-late stages, you can observe other players' situations, or try to exchange for something else. There might be a chance to get 10."

Yang Yuting clapped her hands approvingly: "Well said, that's exactly it!"

Qin Yao exclaimed, "So that's how it is! Zheng Jie is amazing!"

Zheng Jie was a little embarrassed: "No, no, it's mainly because the question was guided well, from simple to complex."

"If the underlying principles hadn't been analyzed beforehand, I probably wouldn't have been able to figure it out even if I suddenly encountered it in the game."

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

"If we give real-world examples, it should be easier for everyone to understand."

"Imagine this is your company's or organization's annual meeting, and your boss, or your superior, wants to play this game with you. The amount you draw will be your year-end bonus or welfare."

"There are two slips of paper. The slip you draw says '1'. At this point, the shopkeeper asks you if you want to change it."

"Should everyone change?"
"Aunt Zhou, why don't you tell us first?"

Zhou Guifen was a little surprised to be asked this question, but after thinking for a moment, she still said, "I won't change it!"

Yang Yuting nodded: "Yes, why?"

Zhou Guifen said matter-of-factly, "We're just grateful that our school doesn't delay paying salaries, how could we possibly give out a 10 yuan bonus at the end of the year? Absolutely not, we'd be lucky to get even 1 yuan."

(End of this chapter)