revaraver / 1

for git bash

WEBVTT

1 00:00:00.012 --> 00:00:01.758 Hi again. It's Matt,

2 00:00:01.758 --> 00:00:05.403 and now we're talking more about strategic reasoning.

3 00:00:05.403 --> 00:00:10.896 And in particular let's go through and and analyze the Keynes beauty contest

4 00:00:10.896 --> 00:00:14.651 game now and talk about the Nash equilibria of this game.

5 00:00:14.651 --> 00:00:17.541 So, remember what the structure of this game was.

6 00:00:17.541 --> 00:00:22.099 Each player named an integer between 1 and 100, so you've got a population of

7 00:00:22.099 --> 00:00:24.452 players, they're all naming integers.

8 00:00:24.452 --> 00:00:30.125 the person who names the integer closest to 2/3 of the average integer named by

9 00:00:30.125 --> 00:00:33.634 people wins, other people don't get anything.

10 00:00:33.634 --> 00:00:36.758 ties are broken uniformly at random. Okay.

11 00:00:36.758 --> 00:00:42.681 So again, what are other players going to do? You have to reason through that and

12 00:00:42.681 --> 00:00:47.012 then what should I do in response? So these are the key ingredients of a Nash

13 00:00:47.012 --> 00:00:51.445 equilibrium and the Nash equilibrium is everybody's choosing their optimal

14 00:00:51.445 --> 00:00:55.789 response, the one that's going to give them the maximum chance of winning in

15 00:00:55.789 --> 00:00:58.521 this game to what the other players are doing,

16 00:00:58.521 --> 00:01:02.323 that's going to be a Nash equilibrium. Okay. So let's take a look.

17 00:01:02.323 --> 00:01:06.880 so, how are we going to reason about this? Suppose that I think that the

18 00:01:06.880 --> 00:01:12.020 average play the averaged integer named in this game is going to be some number

19 00:01:12.020 --> 00:01:14.740 X. so, I, you know, including my own

20 00:01:14.740 --> 00:01:17.561 integer, I think this is going to be the average.

21 00:01:17.561 --> 00:01:22.081 Well, what has to be true about my reply to that, my reply should be 2/3 of X,

22 00:01:22.081 --> 00:01:26.769 right? I should be naming the integer closest to 2/3 of whatever I believe the

23 00:01:26.769 --> 00:01:30.578 average is going to be. So my optimal strategy should be naming

24 00:01:30.578 --> 00:01:34.546 an integer closest to 2/3 of X. So here, we're just working through

25 00:01:34.546 --> 00:01:39.344 heuristically, we'll, we'll get to formal definitions and analysis in a little bit,

26 00:01:39.344 --> 00:01:42.282 but let's just go through the basic reasoning now.

27 00:01:42.282 --> 00:01:47.134 Okay, so I should be trying to name 2/3 of what I think the average is going to

28 00:01:47.134 --> 00:01:49.596 be. Well, X has to be less than a 100, right?

29 00:01:49.596 --> 00:01:53.096 There's no way that the average guess can be more then 100.

30 00:01:53.096 --> 00:01:57.578 So the optimal strategy for any player should be no more then 67 right? So if I

31 00:01:57.578 --> 00:02:03.299 think that everybody's rational I, so, if I believe that's true, then I think that

32 00:02:03.299 --> 00:02:06.867 nobody should be naming an integer bigger than 67.

33 00:02:06.867 --> 00:02:12.477 Okay, so what does that mean? Well, that means that I can't think the average is

34 00:02:12.477 --> 00:02:17.590 any higher than 67, right? So, if, if the average X is no bigger than

35 00:02:17.590 --> 00:02:20.980 67, then I should be naming no more than 2/3 of 67.

36 00:02:20.980 --> 00:02:26.495 Right? Now, you can begin to see where this is going, so that means that if I

37 00:02:26.495 --> 00:02:31.917 think everybody else understands the game and understands that nobody should be

38 00:02:31.917 --> 00:02:36.757 naming a number bigger than 67 and nobody should be naming numbers bigger than 2/3

39 00:02:36.757 --> 00:02:39.507 of 67. we keep going on this, so nobody should

40 00:02:39.507 --> 00:02:42.352 be naming anything more than 2/3 of this, of 2/3 of 67.

41 00:02:42.352 --> 00:02:46.677 Now, obviously, when you, if you just keep looking, everybody's going to want

42 00:02:46.677 --> 00:02:49.592 to be a little bit lower than everybody else's guess.

43 00:02:49.592 --> 00:02:53.172 So wherever the average is you should be lower than that.

44 00:02:53.172 --> 00:02:58.108 What's the only number which, everybody can be naming, and consistently choosing

45 00:02:58.108 --> 00:03:01.369 the best response they have to what the average guess is.

46 00:03:01.369 --> 00:03:05.889 the unique Nash equilibrium of this game is for every player to announce one.

47 00:03:05.889 --> 00:03:09.678 Okay? Well that's, yeah, so, so we're driven all the way down to,

48 00:03:09.678 --> 00:03:14.236 to announcing one and that's a unique Nash equilibrium, and what happens now,

49 00:03:14.236 --> 00:03:17.978 we all announce one we all tie, and somebody wins at random.

50 00:03:17.978 --> 00:03:22.429 If, if I try to deviate form that, if I try to announce a higher integer, I'd

51 00:03:22.429 --> 00:03:26.971 just be higher than the average guess, so I wouldn't be at 2/3 of the mean.

52 00:03:26.971 --> 00:03:29.581 So this is going to be a stable point. Okay?

53 00:03:29.581 --> 00:03:33.532 So, let's see what, what actually happens when people play this.

54 00:03:33.532 --> 00:03:38.392 So part of this reasoning is you're trying to form expectations of what other

55 00:03:38.392 --> 00:03:43.227 players are doing and you need to make sure that those expectations actually

56 00:03:43.227 --> 00:03:46.425 match reality. So let's have a peek at some plays of

57 00:03:46.425 --> 00:03:50.024 this game. So this, this is a plot here where we're

58 00:03:50.024 --> 00:03:55.553 actually giving you the results of the online course of when it was taught last

59 00:03:55.553 --> 00:03:59.879 year, we had players play this game, and so these are the results.

60 00:03:59.879 --> 00:04:04.445 And here from 2012, we had more than 10,000 people actually participate in

61 00:04:04.445 --> 00:04:09.085 this particular game. What do we see? So, down here on this, we

62 00:04:09.085 --> 00:04:15.142 have integers going from 0 to 100 and then over here, we have the frequency.

63 00:04:15.142 --> 00:04:21.088 So, how many people named the given integer? So the, the 50 right here is

64 00:04:21.088 --> 00:04:23.693 the, is the mode, so we get the mode of 50.

65 00:04:23.693 --> 00:04:27.742 The most often named integer was 50, 1,600 people named 50.

66 00:04:27.742 --> 00:04:33.310 Well, obviously, they hadn't gone through all the reasoning and it takes a while to

67 00:04:33.310 --> 00:04:37.106 sort of figure out what the equilibrium of this game is.

68 00:04:37.106 --> 00:04:42.172 what's the mean here? So the mean was 34, so actually there's some interesting

69 00:04:42.172 --> 00:04:44.930 things. Some people naming 100, a number that

70 00:04:44.930 --> 00:04:49.704 could never really win, right? So it's not clear exactly what what, it

71 00:04:49.704 --> 00:04:54.130 could, it could end up winning if everybody named 100 then you could end up

72 00:04:54.130 --> 00:04:58.465 in a tie there, but then you would be better off naming 67 instead.

73 00:04:58.465 --> 00:05:03.227 So so when we, when we end up looking through this, what we end up with is some

74 00:05:03.227 --> 00:05:06.634 people naming high numbers, but very few people,

75 00:05:06.634 --> 00:05:12.432 then we end up with some interesting spikes a bunch of people just named 50.

76 00:05:12.432 --> 00:05:16.012 Not clear exactly what the reasoning is on, on 50.

77 00:05:16.012 --> 00:05:22.192 interestingly if you think that a bunch of people are going to do that you might

78 00:05:22.192 --> 00:05:26.512 want to name 2/3 of 50. Okay, well, there's a big spike here at

79 00:05:26.512 --> 00:05:32.162 33 where a bunch of people believed that other people were going to name 50.

80 00:05:32.162 --> 00:05:37.070 if we keep going, so down here. If we keep going and looking at this,

81 00:05:37.070 --> 00:05:40.413 what we see, then we see another spike at 2/3 of 33.

82 00:05:40.413 --> 00:05:44.657 So some people said, okay, well, maybe a bunch of people are going to think that

83 00:05:44.657 --> 00:05:48.317 the average is going to be 50, they are going to name 33.

84 00:05:48.317 --> 00:05:52.830 I'm going to go one better than that. I am going to name something around 22,

85 00:05:52.830 --> 00:05:55.980 23. you know what the winner in this game

86 00:05:55.980 --> 00:06:01.168 was? The winner was actually 23. So 2/3 of the average guess here was

87 00:06:01.168 --> 00:06:06.532 about 23 because the mean was, was 34 and so one of these people randomly would end

88 00:06:06.532 --> 00:06:12.099 up being the winner of this game. Okay? there's actually a spike of people who

89 00:06:12.099 --> 00:06:18.104 went all the way to the Nash equilibrium and it's interesting here, because the

90 00:06:18.104 --> 00:06:22.230 Nash equilibrium works if you believe that everyone else is going to name the

91 00:06:22.230 --> 00:06:24.579 integer one, then that's your best response.

92 00:06:24.579 --> 00:06:28.907 But, in situations where a bunch of people don't necessarily understand the

93 00:06:28.907 --> 00:06:32.974 game and haven't reasoned through it, then you actually would be better off

94 00:06:32.974 --> 00:06:36.510 naming a higher number. So Nash equilibrium is a stable point if

95 00:06:36.510 --> 00:06:41.007 everybody figures it out and everybody abides by it, then it's the best thing

96 00:06:41.007 --> 00:06:45.367 you can do but it might be that some of the players aren't necessarily figuring

97 00:06:45.367 --> 00:06:48.452 out exactly what goes on. Okay. Now suppose you, you start with

98 00:06:48.452 --> 00:06:52.847 this game and they're not necessarily playing the Nash equilibrium, but now we

99 00:06:52.847 --> 00:06:56.217 have them play it again. Right? So, they get to do this, play it

100 00:06:56.217 --> 00:06:59.812 again, and then see what happens. Well, now, these people should realize

101 00:06:59.812 --> 00:07:03.817 that they overestimated, right? There's a bunch of people here who are naming

102 00:07:03.817 --> 00:07:08.012 numbers too high, they should be moving their announcements to, to lower numbers,

103 00:07:08.012 --> 00:07:12.054 right? They should be moving down. And if, if, if I anticipate that

104 00:07:12.054 --> 00:07:16.709 everybody's going to adjust and move downwards I should move my announcement

105 00:07:16.709 --> 00:07:20.208 downwards as well. So let's have a peek at what happens.

106 00:07:20.208 --> 00:07:25.063 So here is, is a subset of players actually from, from one of the classes I,

107 00:07:25.063 --> 00:07:28.736 I did on campus, where they got, this is the second play

108 00:07:28.736 --> 00:07:31.301 of the game. So after the first play, then we have

109 00:07:31.301 --> 00:07:34.283 them play again. Now you can begin to see that things, you

110 00:07:34.283 --> 00:07:38.105 know, the, the 50s have disappeared, all the numbers up here have disappeared,

111 00:07:38.105 --> 00:07:41.504 people have moved down, and in fact, a lot more people have are

112 00:07:41.504 --> 00:07:45.677 moving towards the equilibrium once you get to the second part, the second

113 00:07:45.677 --> 00:07:48.306 chance. So if you've played this game, you begin

114 00:07:48.306 --> 00:07:51.842 to see the logic of it. You played again and now we get closer to

115 00:07:51.842 --> 00:07:54.715 Nash equilibrium. So, Nash equilibrium does is a better

116 00:07:54.715 --> 00:07:58.254 predictor here. if from experienced players who have

117 00:07:58.254 --> 00:08:03.219 played this game understood it and, and interacting with the same population, you

118 00:08:03.219 --> 00:08:06.948 can begin to see things unraveling and moving back towards

119 00:08:06.948 --> 00:08:10.418 all announcing one. Okay. So Nash equilibrium, basic ideas, a

120 00:08:10.418 --> 00:08:16.040 consistent list of actions, so each player is maximizing his or her payouts

121 00:08:16.040 --> 00:08:21.680 given the actions of the other players. Should be self-consistent and stable.

122 00:08:21.680 --> 00:08:27.613 the nice parts about this, each players action is maximizing what they can get

123 00:08:27.613 --> 00:08:32.867 given the other players. nobody has an incentive to deviate from

124 00:08:32.867 --> 00:08:36.242 their action if an equilibrium profile is, is played.

125 00:08:36.242 --> 00:08:41.492 someone does have an incentive to deviate from a profile of actions that do not

126 00:08:41.492 --> 00:08:45.742 form an equilibrium. So these are the basic ideas and we'll be

127 00:08:45.742 --> 00:08:49.112 looking at, at Nash equilibrium in much more detail.

128 00:08:49.112 --> 00:08:52.975 So, in terms of of, of making predictions, you know, why, should we

129 00:08:52.975 --> 00:08:57.615 expect Nash equilibrium to be played? Well, I, I think there is sort of

130 00:08:57.615 --> 00:09:01.272 interesting logic here. in this logic, actually goes back to, to

131 00:09:01.272 --> 00:09:06.380 some of the original discussion by Nash. when we want to make a prediction of

132 00:09:06.380 --> 00:09:11.488 what's going on a game we want something which if players really understood

133 00:09:11.488 --> 00:09:16.812 things, it would be consistent. And the interesting thing is we should

134 00:09:16.812 --> 00:09:21.607 expect non-equilibria not to be stable, in the sense that, if players understood

135 00:09:21.607 --> 00:09:25.572 it and see what happens in a non-equilibrium, they should move away

136 00:09:25.572 --> 00:09:28.522 from that. And we saw exactly that in the, in the,

137 00:09:28.522 --> 00:09:32.927 the second round of the, the beauty contest game, then people start moving

138 00:09:32.927 --> 00:09:36.937 down toward the Nash equilibrium. So it's not necessarily true that we

139 00:09:36.937 --> 00:09:40.272 always expect equilibrium to be played, [COUGH] but we should expect

140 00:09:40.272 --> 00:09:44.292 non-equilibrium to vanish over time. And the, there'll be various dynamics and

141 00:09:44.292 --> 00:09:48.817 other kinds of settings where there will be strong pushes towards equilibrium over

142 00:09:48.817 --> 00:09:52.897 time, but they might have to be learned and they might have to evolve and, and so

143 00:09:52.897 --> 00:09:55.332 forth. So, as this course goes on, we'll talk

144 00:09:55.332 --> 00:09:59.222 more and more about some of the dynamics and, and things to push towards Nash

145 00:09:59.222 --> 00:09:59.895 equilibrium.