Game theory challenge: Can you predict human behavior? - Lucas Husted

譯者: Lilian Chiu 審譯者: SF Huang

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

幾個月前，我們在我們的 社群裡下了個戰帖。

We asked everyone: given a range of integers from 0 to 100,

詢問大家： 從 0 到 100 間的整數，

guess the whole number closest to 2/3 of the average of all numbers guessed.

請猜出哪個整數是最接近

所有猜測答案之平均值的 2/3。

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

若所有猜測答案之平均值為 60， 正確猜測應該是 40。

What number do you think was the correct guess at 2/3 of the average?

你認為平均值的 2/3 應該是哪個數字？

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

讓我們看看大家是否能推論出答案。

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

這個遊戲就是在賽局理論家 所熟知的「常識」下所進行。

Not only does every player have the same information —

所有的玩家不僅擁有相同資訊—— 他們也知道別人都有相同資訊，

they also know that everyone else does,

and that everyone else knows that everyone else does, and so on, infinitely.

且其他人也都知道其他所有人 也都知道，以此無限類推。

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

最高的可能平均值會發生在 大家都猜測 100 時。

In that case, 2/3 of the average would be 66.66.

如果這樣的話， 平均值的 2/3 是 66.66。

Since everyone can figure this out,

這點大家都想得出來，

it wouldn’t make sense to guess anything higher than 67.

所以猜測 67 以上的 數字並不合理。

If everyone playing comes to this same conclusion,

如果所有玩家都得到同樣的結論，

no one will guess higher than 67.

就沒有人會猜測 67 以上的數字。

Now 67 is the new highest possible average,

所以 67 是新的最高可能平均值，

so no reasonable guess should be higher than ⅔ of that, which is 44.

所以，合理的猜測 都不會高於 67 的 2/3，

也就是 44。

This logic can be extended further and further.

這個邏輯可以一直延伸下去。

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

每推衍一次，最高可能 平均值就會再變小。

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

所以，合理的做法是去猜 範圍中有可能的最小數字。

And indeed, if everyone chose zero,

的確，如果大家都選 0，

the game would reach what’s known as a Nash Equilibrium.

這個遊戲就會達到所謂的納許均衡。

This is a state where every player has chosen the best possible strategy

這個情況是指：在大家都參與的 前提下，每個玩家都已為自己

for themselves given everyone else playing,

挑選出最佳策略，

and no individual player can benefit by choosing differently.

沒有任何玩家會因 選擇不同策略而從中受惠。

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

但在真實的世界不會發生這種事。

People, as it turns out, either aren’t perfectly rational,

結果發現，人要嘛不是完全的理性，

or don’t expect each other to be perfectly rational.

不然就是不預期彼此是完全的理性。

Or, perhaps, it’s some combination of the two.

或者是上述兩種狀況的組合。

When this game is played in real-world settings,

在真實的世界玩這個遊戲時，

the average tends to be somewhere between 20 and 35.

平均值通常會在 20 到 35 之間。

Danish newspaper Politiken ran the game with over 19,000 readers participating,

丹麥報紙《政治報》舉辦了這個遊戲，

有一萬九千名讀者參與，

resulting in an average of roughly 22, making the correct answer 14.

結果的平均值大約是 22， 因此正確答案為 14。

For our audience, the average was 31.3.

至於我們的觀眾，平均值為 31.3。

So if you guessed 21 as 2/3 of the average, well done.

所以，若你猜 21 是平均值的 2/3，幹得好。

Economic game theorists have a way of modeling this interplay

經濟賽局理論家有種 叫做 K 級推理的方法，

between rationality and practicality called k-level reasoning.

可以針對這種理性和實際 之間的相互影響來建立模型，

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

K 代表的是推理循環重覆的次數。

A person playing at k-level 0 would approach our game naively,

K 級為 0 級裡的玩家， 是天真的玩家，

guessing a number at random without thinking about the other players.

他會隨機猜測，不考慮其他玩家。

At k-level 1, a player would assume everyone else was playing at level 0,

K 級為 1 表示玩家會假設 其他玩家都用 0 級的方式來玩，

resulting in an average of 50, and thus guess 33.

因此平均值會是 50， 他就會猜答案是 33。

At k-level 2, they’d assume that everyone else was playing at level 1,

K 級為 2 表示玩家假設 其他玩家都用 1 級的方式來玩，

leading them to guess 22.

因此他會猜測 22。

It would take 12 k-levels to reach 0.

要 12 級才會達到 0。

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

證據指出，大部分人的 K 級 會停在 1 或 2 級。

And that’s useful to know,

知道這點很有用，

because k-level thinking comes into play in high-stakes situations.

因為在賭注高的情況下 就會用到 K 級思考。

For example, stock traders evaluate stocks not only based on earnings reports,

比如，股票交易員在評估股票時 不僅只是看盈餘報告，

but also on the value that others place on those numbers.

也會考量他人對 這些數據所賦予的評價。

And during penalty kicks in soccer,

足球罰球時，

both the shooter and the goalie decide whether to go right or left

射球員和守門員都要 判斷要向左或向右，

based on what they think the other person is thinking.

他們判斷的根據就是 推測對方會怎麼想。

Goalies often memorize the patterns of their opponents ahead of time,

守門員通常事先就會記住 對手的踢球模式，

but penalty shooters know that and can plan accordingly.

但罰球的射球員知道這一點， 可以依此來因應。

In each case, participants must weigh their own understanding

在每種情況中， 參與者都必須要權衝

of the best course of action against how well they think other participants

他們自己對於最佳做法的理解，

及他們認為其他參與者 對情況的理解程度。

understand the situation.

But 1 or 2 k-levels is by no means a hard and fast rule—

但 K 級為 1 或 2 絕對不是 不能變通的規則——

simply being conscious of this tendency can make people adjust their expectations.

只要能意識到這種趨勢， 就能讓人調整他們的預期。

For instance, what would happen if people played the 2/3 game

比如，重新想想剛才 2/3 的遊戲，

after understanding the difference between the most logical approach

如果玩家知道最合邏輯

和最常見的方法之間的差別後， 會如何猜測呢？

and the most common?

Submit your own guess at what 2/3 of the new average will be

你自己所猜測 新平均值的 2/3 是多少？

by using the form below,

把答案寫在下面的表格中， 我們就能知道了。

and we’ll find out.