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


  • 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?


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


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


  • 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.


  • 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.


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


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


  • 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.


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


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


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


  • 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.


  • 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.


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


  • 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.


  • 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.


  • Want more game theory? How about this?


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


  • Find out the answer in this video.


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


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