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

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

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

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

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

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

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

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Not only does every player have [br]the same information —

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they also know that everyone else does,

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and that everyone else knows that [br]everyone else does, and so on, infinitely.

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

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In that case, 2/3 of the average [br]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 [br]anything higher than 67.

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If everyone playing comes to [br]this same conclusion,

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no one will guess higher than 67.

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Now 67 is the new highest [br]possible average,

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

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This logic can be extended further [br]and further.

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

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

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And indeed, if everyone chose zero,

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

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

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for themselves given [br]everyone else playing,

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and no individual player can benefit [br]by choosing differently.

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

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

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

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

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When this game is played in [br]real-world settings,

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the average tends to be somewhere [br]between 20 and 35.

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

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resulting in an average of roughly 22, [br]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 [br]the average, well done.

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Economic game theorists have a [br]way of modeling this interplay

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between rationality and practicality [br]called k-level reasoning.

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

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

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

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

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

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At k-level 2, they’d assume that everyone [br]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.

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The evidence suggests that most [br]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 [br]play in high-stakes situations.

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

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but also on the value that others [br]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 [br]whether to go right or left

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based on what they think the other [br]person is thinking.

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Goalies often memorize the patterns of [br]their opponents ahead of time,

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but penalty shooters know that [br]and can plan accordingly.

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In each case, participants must weigh [br]their own understanding

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

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understand the situation.

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

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

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

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after understanding the difference between[br]the most logical approach

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and the most common?

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

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by using the form below,

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and we’ll find out.