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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.

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

• In that case, ⅔ of the average would be 66.66.

• Since everyone can figure this out, it wouldn't make sense to guess anything higher than 67.

• If everyone playing comes to this same conclusion, no one will guess higher than 67.

• Now 67 is the new highest possible average, so no reasonable guess should be higher thanof that, which is 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.

• 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.

• Danish newspaper Politiken ran the game with over 19,000 readers participating, resulting in an average of roughly 22, making the correct answer 14.

• For our audience, the average was 31.3.

• So if you guessed 21 asof the average, well done.

• Economic game theorists have a way of modeling this interplay between rationality and practicality called k-level reasoning.

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

• A person playing at k-level 0 would approach our game naively, guessing a number at random without thinking about the other players.

• 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.

• At k-level 2, they'd assume that everyone else was playing at level 1, leading them to guess 22.

• It would take 12 k-levels to reach 0.

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

• And that's useful to know, because k-level thinking comes into play in high-stakes situations.

• 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.

• For instance, what would happen if people played thegame after understanding the difference between the most logical approach and the most common?

• 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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A2 US TED-Ed average guess level player guessed

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

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Mackenzie posted on 2019/11/25
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