We asked everyone: given a range of integers from 0 to 100, guess the whole number closest to ⅔ of 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 at ⅔ of 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 than ⅔ of 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 as ⅔ of 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 rule— simply 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 the ⅔ game 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 what ⅔ of 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！

Want more game theory? How about this?

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。