## 字幕列表 影片播放

• A few months ago we posed a challenge to our community.

幾個月前，我們在社群上發佈了一則挑戰。

• We asked everyone: given a range of integers from 0 to 100, guess the whole number closest toof the average of all numbers guessed.

內容是：「 0 到 100 的整數，猜猜全部人所猜數字平均的三分之二會是多少？」

• So if the average of all guesses is 60, the correct guess will be 40.

如果所有人猜的數字平均為 60，那麼答案就會是 40。

• What number do you think was the correct guess atof the average?

你認為平均的三分之二會是多少呢？

• Let's see if we can try and reason our way to the answer.

來看看我們能否推理出答案。

• This game is played under conditions known to game theorists as common knowledge.

這個挑戰是在賽局理論學家熟知的「共同知識」下進行的。

• Not only does every player have the same information, they also know that everyone else does, and that everyone else knows that everyone else does, and so on, infinitely.

意思是，每名玩家都有相同的資訊，他們也知道彼此有這樣的資訊，不斷以此類推。

• Now, the highest possible average would occur if every person guessed 100.

現在，如果全部的人都猜 100，就會出現最高的平均數。

• In that case, ⅔ of the average would be 66.66.

如此，平均的三分之二就會是 66.66。

• Since everyone can figure this out, it wouldn't make sense to guess anything higher than 67.

既然所有人都想得到，那猜任何高於 67 的數字就不合理了。

• If everyone playing comes to this same conclusion, no one will guess higher than 67.

如果每個人都這麼想，表示沒人會猜高於 67 的數字。

• Now 67 is the new highest possible average, so no reasonable guess should be higher thanof that, which is 44.

現在 67 成了可能最高的平均數，所以沒理由猜高於 67 的三分之二的任何數字，也就是 44。

• This logic can be extended further and further.

這邏輯可以無限延伸。

• With each step, the highest possible logical answer keeps getting smaller.

每經過一次，可能的答案都會越來越小。

• So it would seem sensible to guess the lowest number possible.

所以，最合理的就是去猜最低的數字。

• And indeed, if everyone chose zero, the game would reach what's known as a Nash Equilibrium.

而如果全部的人都猜 0，這挑戰就會達到所謂的納許均衡。

• This is a state where every player has chosen the best possible strategy for themselves given everyone else playing, and no individual player can benefit by choosing differently.

這是指每個玩家都選擇了對自己最有利的答案，而任何人選了不同的答案就會對自己不利。

• But, that's not what happens in the real world.

但，現實並不是如此。

• People, as it turns out, either aren't perfectly rational, or don't expect each other to be perfectly rational.

結果，人類並不完全理性，或不期待他人能夠完全理性。

• Or, perhaps, it's some combination of the two.

又或者，是兩者的結合。

• When this game is played in real-world settings, the average tends to be somewhere between 20 and 35.

全世界在這挑戰得到的平均，通常會落於 20 至 35 之間。

• Danish newspaper Politiken ran the game with over 19,000 readers participating, resulting in an average of roughly 22, making the correct answer 14.

丹麥報紙 Politiken 進行了這項挑戰，有 19,000 名讀者參與，得到的結果平均數約為 22，這使得正確答案落在 14。

• For our audience, the average was 31.3.

TEDEd 的觀眾，得出的平均數則為 31.3。

• So if you guessed 21 asof the average, well done.

所以如果你猜 21，做得好！

• Economic game theorists have a way of modeling this interplay between rationality and practicality called k-level reasoning.

賽局理論學家有個推測理性與實際性的方法，稱為 K 等級推理。

• K stands for the number of times a cycle of reasoning is repeated.

K 代表推理過程重複的次數。

• A person playing at k-level 0 would approach our game naively, guessing a number at random without thinking about the other players.

一名 K0 等級的參與者會天真地玩這場挑戰，不考慮其他人，而隨便猜一個數字。

• At k-level 1, a player would assume everyone else was playing at level 0, resulting in an average of 50, and thus guess 33.

K1 等級的人會認為所有人都是 K0 等級，所以平均數為 50，因而猜測答案為 33。

• At k-level 2, they'd assume that everyone else was playing at level 1, leading them to guess 22.

K2 等級認為所有人皆為 K1，而猜 22。

• It would take 12 k-levels to reach 0.

重複 12 次後，答案會變成 0。

• The evidence suggests that most people stop at 1 or 2 k-levels.

證據顯示大多數人皆停在 K1 或 K2 等級。

• And that's useful to know, because k-level thinking comes into play in high-stakes situations.

這很有用，因為 K 等級推理能在處於高風險的狀況下派上用場。

• For example, stock traders evaluate stocks not only based on earnings reports, but also on the value that others place on those numbers.

舉例來說，股票交易者不只會用收益報告來評估一支股票，他們也會看其他人對這些數字的評價。

• And during penalty kicks in soccer, both the shooter and the goalie decide whether to go right or left based on what they think the other person is thinking.

而足球的罰球，罰球者與守門人都以猜測對方當時的想法，來決定自己該向左或向右。

• Goalies often memorize the patterns of their opponents ahead of time, but penalty shooters know that and can plan accordingly.

守門員通常會記憶對手的習慣，但罰球者也知道，所以他也可以反其道而行。

• In each case, participants must weigh their own understanding of the best course of action against how well they think other participants understand the situation.

每個情況下，參與者得依據敵手對情況的了解程度，來衡量自己的決定。

• But 1 or 2 k-levels is by no means a hard and fast rulesimply being conscious of this tendency can make people adjust their expectations.

但 K1 K2 等級沒有什麼明確的規定，只要理解規律，就能讓人們調整答案。

• For instance, what would happen if people played thegame after understanding the difference between the most logical approach and the most common?

舉例來說，如果人們都理解了，最有邏輯與最普遍的方法得出答案的差異，再來玩這 ⅔ 遊戲，結果會如何呢？

• Submit your own guess at whatof the new average will be by using the form below, and we'll find out.

用下面的表單填下你對於三分之二遊戲的新答案，我們就能得知了！

想要看更多賽局理論？這個如何？

• Why are so many gas stations built across the street from each other?

為什麼那麼多加油站都會蓋在彼此對街？

• Find out the answer in this video.

答案就在這則影片裡。

A few months ago we posed a challenge to our community.

# 【TED-Ed】賽局理論挑戰！你能預測其他人的行動嗎？ (Game theory challenge: can you predict human behavior? - Lucas Husted)

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Mackenzie 發佈於 2019 年 11 月 25 日