1 00:00:06,646 --> 00:00:10,302 A few months ago we posed a challenge to our community. 2 00:00:10,302 --> 00:00:15,192 We asked everyone: given a range of integers from 0 to 100, 3 00:00:15,192 --> 00:00:22,056 guess the whole number closest to 2/3 of the average of all numbers guessed. 4 00:00:22,056 --> 00:00:26,776 So if the average of all guesses is 60, the correct guess will be 40. 5 00:00:26,776 --> 00:00:31,414 What number do you think was the correct guess at 2/3 of the average? 6 00:00:32,733 --> 00:00:36,107 Let’s see if we can try and reason our way to the answer. 7 00:00:36,107 --> 00:00:41,406 This game is played under conditions known to game theorists as common knowledge. 8 00:00:41,406 --> 00:00:44,499 Not only does every player have the same information — 9 00:00:44,499 --> 00:00:46,706 they also know that everyone else does, 10 00:00:46,706 --> 00:00:52,618 and that everyone else knows that everyone else does, and so on, infinitely. 11 00:00:52,618 --> 00:00:58,538 Now, the highest possible average would occur if every person guessed 100. 12 00:00:58,538 --> 00:01:03,268 In that case, 2/3 of the average would be 66.66. 13 00:01:03,268 --> 00:01:05,205 Since everyone can figure this out, 14 00:01:05,205 --> 00:01:09,625 it wouldn’t make sense to guess anything higher than 67. 15 00:01:09,625 --> 00:01:12,748 If everyone playing comes to this same conclusion, 16 00:01:12,748 --> 00:01:15,517 no one will guess higher than 67. 17 00:01:15,517 --> 00:01:19,659 Now 67 is the new highest possible average, 18 00:01:19,659 --> 00:01:25,439 so no reasonable guess should be higher than ⅔ of that, which is 44. 19 00:01:25,439 --> 00:01:28,980 This logic can be extended further and further. 20 00:01:28,980 --> 00:01:33,710 With each step, the highest possible logical answer keeps getting smaller. 21 00:01:33,710 --> 00:01:38,275 So it would seem sensible to guess the lowest number possible. 22 00:01:38,275 --> 00:01:41,133 And indeed, if everyone chose zero, 23 00:01:41,133 --> 00:01:45,065 the game would reach what’s known as a Nash Equilibrium. 24 00:01:45,065 --> 00:01:49,419 This is a state where every player has chosen the best possible strategy 25 00:01:49,419 --> 00:01:52,524 for themselves given everyone else playing, 26 00:01:52,524 --> 00:01:57,334 and no individual player can benefit by choosing differently. 27 00:01:57,334 --> 00:02:01,514 But, that’s not what happens in the real world. 28 00:02:01,514 --> 00:02:05,479 People, as it turns out, either aren’t perfectly rational, 29 00:02:05,479 --> 00:02:09,038 or don’t expect each other to be perfectly rational. 30 00:02:09,038 --> 00:02:12,369 Or, perhaps, it’s some combination of the two. 31 00:02:12,369 --> 00:02:15,219 When this game is played in real-world settings, 32 00:02:15,219 --> 00:02:20,219 the average tends to be somewhere between 20 and 35. 33 00:02:20,219 --> 00:02:26,076 Danish newspaper Politiken ran the game with over 19,000 readers participating, 34 00:02:26,076 --> 00:02:32,056 resulting in an average of roughly 22, making the correct answer 14. 35 00:02:32,056 --> 00:02:35,758 For our audience, the average was 31.3. 36 00:02:35,758 --> 00:02:41,018 So if you guessed 21 as 2/3 of the average, well done. 37 00:02:41,018 --> 00:02:44,681 Economic game theorists have a way of modeling this interplay 38 00:02:44,681 --> 00:02:49,802 between rationality and practicality called k-level reasoning. 39 00:02:49,802 --> 00:02:54,642 K stands for the number of times a cycle of reasoning is repeated. 40 00:02:54,642 --> 00:02:58,949 A person playing at k-level 0 would approach our game naively, 41 00:02:58,949 --> 00:03:02,676 guessing a number at random without thinking about the other players. 42 00:03:02,676 --> 00:03:07,876 At k-level 1, a player would assume everyone else was playing at level 0, 43 00:03:07,876 --> 00:03:12,416 resulting in an average of 50, and thus guess 33. 44 00:03:12,416 --> 00:03:17,192 At k-level 2, they’d assume that everyone else was playing at level 1, 45 00:03:17,192 --> 00:03:19,492 leading them to guess 22. 46 00:03:19,492 --> 00:03:23,096 It would take 12 k-levels to reach 0. 47 00:03:23,096 --> 00:03:27,916 The evidence suggests that most people stop at 1 or 2 k-levels. 48 00:03:27,916 --> 00:03:29,395 And that’s useful to know, 49 00:03:29,395 --> 00:03:34,005 because k-level thinking comes into play in high-stakes situations. 50 00:03:34,005 --> 00:03:39,379 For example, stock traders evaluate stocks not only based on earnings reports, 51 00:03:39,379 --> 00:03:43,112 but also on the value that others place on those numbers. 52 00:03:43,112 --> 00:03:45,402 And during penalty kicks in soccer, 53 00:03:45,402 --> 00:03:49,543 both the shooter and the goalie decide whether to go right or left 54 00:03:49,543 --> 00:03:52,735 based on what they think the other person is thinking. 55 00:03:52,735 --> 00:03:56,691 Goalies often memorize the patterns of their opponents ahead of time, 56 00:03:56,691 --> 00:04:00,288 but penalty shooters know that and can plan accordingly. 57 00:04:00,288 --> 00:04:03,551 In each case, participants must weigh their own understanding 58 00:04:03,551 --> 00:04:07,743 of the best course of action against how well they think other participants 59 00:04:07,743 --> 00:04:10,144 understand the situation. 60 00:04:10,144 --> 00:04:14,924 But 1 or 2 k-levels is by no means a hard and fast rule— 61 00:04:14,924 --> 00:04:20,345 simply being conscious of this tendency can make people adjust their expectations. 62 00:04:20,345 --> 00:04:24,357 For instance, what would happen if people played the 2/3 game 63 00:04:24,357 --> 00:04:28,250 after understanding the difference between the most logical approach 64 00:04:28,250 --> 00:04:29,850 and the most common? 65 00:04:29,850 --> 00:04:34,291 Submit your own guess at what 2/3 of the new average will be 66 00:04:34,291 --> 00:04:36,233 by using the form below, 67 00:04:36,233 --> 00:04:37,813 and we’ll find out.