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

翻译人员: Jiasi Hao 校对人员: 潘 可儿

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

几个月前，我们在自己的社群上 发起了一个挑战。

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

我们问每个人： 从给定 0 到 100 的整数范围内，

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

猜测一个最接近 所有猜测数字平均数 2/3 的整数。

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

即倘若所有猜测数的平均是 60， 那么正确的猜测将会是 40。

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

你认为哪个数字会是 平均数 2/3 的正确猜测呢？

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

让我们看看是否可以尝试并推理出 我们猜测答案的方法。

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

这个博弈是在一先决条件下进行的， 该条件被博弈理论家称为常识。

Not only does every player have the same information —

不仅每一个参与者 都有一样的信息储备——

they also know that everyone else does,

他们也知道其他人都一样，

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

并且其他人也都知道 再其他人也如此，如此无限循环。

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

现在，如果每个人都猜 100， 那最大的可能平均数将会出现。

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

在那个情况下，平均数的 2/3 将会是 66.66。

Since everyone can figure this out,

既然每个人可以明白这个道理，

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

那就没有理由去猜比 67 大的整数。

If everyone playing comes to this same conclusion,

如果每个人都在博弈中 得出同样的结论，

no one will guess higher than 67.

没人会猜比 67 大的整数。

Now 67 is the new highest possible average,

现在 67 是最大的可能平均数，

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

所以合理的猜测 就不应该比 67 的 2/3 大，即 44。

This logic can be extended further and further.

这个逻辑可以不断地被拓展，

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

随着每一步，符合逻辑的 最大可能猜测数会不断变小。

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

因此猜测最小的可能数字 看似非常明智。

And indeed, if everyone chose zero,

确实，如果每个人都选择 0，

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

这个博弈将会达到“纳什均衡”。

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

在这一情况中，

每个玩家在都为自己 选择了最优可能策略，

for themselves given everyone else playing,

and no individual player can benefit by choosing differently.

并且没有单独的玩家 可以通过不同选择受益。

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

但是这在现实世界不会发生。

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

事实证明， 人们要么不是完全理智的，

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

要么不会预期别人能做到完全理智，

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

再或者可能是这两种情况的组合。

When this game is played in real-world settings,

当这个博弈在真实世界中发生时，

the average tends to be somewhere between 20 and 35.

平均数接近于 20 至 35 之间的某个整数。

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

丹麦 Poolitiken 报纸曾开展这个博弈， 有超过 1.9 万读者参与。

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

其平均数结果约为 22， 使得最终正确答案为 14。

For our audience, the average was 31.3.

而我们的观众参与者， 平均数为 31.3。

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

所以如果你的猜测数为 21， 那你猜得漂亮！

Economic game theorists have a way of modeling this interplay

经济博弈理论家有一个 模拟理性和实践相互作用方法，

between rationality and practicality called k-level reasoning.

称为“ k 级推理”。

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

其中 k 代表 一个推理周期的重复次数。

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

一个 k 级为 0 的人 会非常天真地参与我们的博弈，

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

他不会考虑别人的选择 而只是任意地猜一个数字。

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

一个 k 级为 1 的人 会假设别人都在 0 级博弈，

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

进而平均数为 50， 因此猜测数为 33。

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

一个 k 级为 2 的人 会假设其他人都在 1 级博弈，

leading them to guess 22.

导致他们最终猜测数为 22。

It would take 12 k-levels to reach 0.

这将要求 12 的 k 级 来达到猜测数为 0。

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

事实证明大部分人 处于 1 或 2 的 k 级。

And that’s useful to know,

而知道这一点很有用，

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

因为 k 级思维 在高风险情况下时常出现。

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

例如，股票交易员 不仅基于收益报告来评估股票，

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

也基于其他人 在那些数字上摆放的价值。

And during penalty kicks in soccer,

在球赛的点球环节中，

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

射门人和守门员 都凭借他们对彼此想法的预判

based on what they think the other person is thinking.

来决定向右或向左跑。

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

守门员时常提前 记住他们对手的习惯模式，

but penalty shooters know that and can plan accordingly.

但罚球射手知道此事， 并依此做出相应计划。

In each case, participants must weigh their own understanding

每个情况下，参与者必须衡量

自身对最优行为的理解，

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

来对抗他们认为 其他参与者对情况的了解深度。

understand the situation.

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

但是 1 或 2 的 k 级推理 绝不是硬性且速成的规定——

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

仅是人们对这种博弈趋势的意识 使人们调整他们的预期。

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

例如，当大家都了解了最符合逻辑的 与最普遍方法之间的区别，

after understanding the difference between the most logical approach

再来玩这个 2/3 的博弈游戏，

and the most common?

结果又会如何？

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

将你的新平均数 2/3 的猜测整数 填到以下表格并提交，

by using the form below,

and we’ll find out.

我们再来看看。