WEBVTT

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

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

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

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

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

NOTE Paragraph

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

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

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Not only does every player have 
the same information—

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they also know that everyone else does,

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

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

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

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Since everyone can figure this out,

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

NOTE Paragraph

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If everyone playing comes to 
this same conclusion,

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no one will guess higher than 67.

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Now 67 is the new highest 
possible average,

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

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This logic can be extended further 
and further.

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

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

NOTE Paragraph

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And indeed, if everyone chose zero,

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

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

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for themselves given 
everyone else playing,

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and no individual player can benefit 
by choosing differently.

NOTE Paragraph

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

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

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

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

NOTE Paragraph

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When this game is played in 
real-world settings,

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the average tends to be somewhere 
between 20 and 35.

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

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

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For our audience, the average was 31.3.

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

NOTE Paragraph

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Economic game theorists have a 
way of modeling this interplay

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between rationality and practicality 
called k-level reasoning.

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

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

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

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

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

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

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leading them to guess 22.

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It would take 12 k-levels to reach 0.

NOTE Paragraph

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

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And that’s useful to know,

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

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

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

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And during penalty kicks in soccer,

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

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based on what they think the other 
person is thinking.

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

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but penalty shooters know that 
and can plan accordingly.

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In each case, participants must weigh 
their own understanding

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

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understand the situation.

NOTE Paragraph

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

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

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

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after understanding the difference between
the most logical approach

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and the most common?

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

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by using the form below,

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and we’ll find out.