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

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Showing Revision 4 created 11/14/2019 by lauren mcalpine .

  1. A few months ago we posed a challenge
    to our community.
  2. We asked everyone: given a range of
    integers from 0 to 100,
  3. guess the whole number closest to 2/3
    of the average of all numbers guessed.
  4. So if the average of all guesses is 60,
    the correct guess will be 40.
  5. What number do you think was the
    correct guess at 2/3 of the average?
  6. Let’s see if we can try and reason
    our way to the answer.

  7. This game is played under conditions known
    to game theorists as common knowledge.
  8. Not only does every player have
    the same information —
  9. they also know that everyone else does,
  10. and that everyone else knows that
    everyone else does, and so on, infinitely.
  11. Now, the highest possible average would
    occur if every person guessed 100.
  12. In that case, 2/3 of the average
    would be 66.66.
  13. Since everyone can figure this out,
  14. it wouldn’t make sense to guess
    anything higher than 67.
  15. If everyone playing comes to
    this same conclusion,

  16. no one will guess higher than 67.
  17. Now 67 is the new highest
    possible average,
  18. so no reasonable guess should be
    higher than ⅔ of that, which is 44.
  19. This logic can be extended further
    and further.
  20. With each step, the highest possible
    logical answer keeps getting smaller.
  21. So it would seem sensible to guess the
    lowest number possible.
  22. And indeed, if everyone chose zero,

  23. the game would reach what’s known
    as a Nash Equilibrium.
  24. This is a state where every player has
    chosen the best possible strategy
  25. for themselves given
    everyone else playing,
  26. and no individual player can benefit
    by choosing differently.
  27. But, that’s not what happens
    in the real world.

  28. People, as it turns out, either aren’t
    perfectly rational,
  29. or don’t expect each other
    to be perfectly rational.
  30. Or, perhaps, it’s some combination
    of the two.
  31. When this game is played in
    real-world settings,

  32. the average tends to be somewhere
    between 20 and 35.
  33. Danish newspaper Politiken ran the game
    with over 19,000 readers participating,
  34. resulting in an average of roughly 22,
    making the correct answer 14.
  35. For our audience, the average was 31.3.
  36. So if you guessed 21 as 2/3 of
    the average, well done.
  37. Economic game theorists have a
    way of modeling this interplay

  38. between rationality and practicality
    called k-level reasoning.
  39. K stands for the number of times a
    cycle of reasoning is repeated.
  40. A person playing at k-level 0 would
    approach our game naively,
  41. guessing a number at random without
    thinking about the other players.
  42. At k-level 1, a player would assume
    everyone else was playing at level 0,
  43. resulting in an average of 50,
    and thus guess 33.
  44. At k-level 2, they’d assume that everyone
    else was playing at level 1,
  45. leading them to guess 22.
  46. It would take 12 k-levels to reach 0.
  47. The evidence suggests that most
    people stop at 1 or 2 k-levels.

  48. And that’s useful to know,
  49. because k-level thinking comes into
    play in high-stakes situations.
  50. For example, stock traders evaluate stocks
    not only based on earnings reports,
  51. but also on the value that others
    place on those numbers.
  52. And during penalty kicks in soccer,
  53. both the shooter and the goalie decide
    whether to go right or left
  54. based on what they think the other
    person is thinking.
  55. Goalies often memorize the patterns of
    their opponents ahead of time,
  56. but penalty shooters know that
    and can plan accordingly.
  57. In each case, participants must weigh
    their own understanding
  58. of the best course of action against how
    well they think other participants
  59. understand the situation.
  60. But 1 or 2 k-levels is by no means
    a hard and fast rule—

  61. simply being conscious of this tendency
    can make people adjust their expectations.
  62. For instance, what would happen
    if people played the 2/3 game
  63. after understanding the difference between
    the most logical approach
  64. and the most common?
  65. Submit your own guess at what 2/3
    of the new average will be
  66. by using the form below,
  67. and we’ll find out.