1 00:00:00,000 --> 00:00:03,091 [cough] Hi. Welcome back. In this set of lectures we're talking about processes 2 00:00:03,091 --> 00:00:07,098 that are path dependent. And to describe path dependence, what I want to do is use 3 00:00:07,098 --> 00:00:11,080 a model. And this model is going to be based on a set of urns. These are very 4 00:00:11,080 --> 00:00:15,081 simple urn models. And these urn models are going to help us distinguish between 5 00:00:15,081 --> 00:00:19,053 processes that are path-dependent and processes that are what I call 6 00:00:19,053 --> 00:00:23,034 "phat-dependent", where the order doesn't matter but the set of things that matters. 7 00:00:23,034 --> 00:00:27,056 They're also going to help us distinguish between processes that have outcomes in a 8 00:00:27,056 --> 00:00:31,062 given period that are path-dependent, and processes that have equilibriums, that have 9 00:00:33,051 --> 00:00:35,039 long-run distributions over the set of outcomes, that are path-dependent. So the 10 00:00:35,039 --> 00:00:38,092 outcome in a period might depend on what happened in the past, that's fine. But 11 00:00:38,092 --> 00:00:42,068 that's different from saying, the long-run equilibrium, where the system is gonna go, 12 00:00:42,068 --> 00:00:46,044 is gonna depend on the, on the process. So I'm gonna make a distinction between 13 00:00:46,044 --> 00:00:49,097 path-dependent outcomes and path-dependent equilibrium. And I'm gonna do this using 14 00:00:49,097 --> 00:00:53,063 this very simple class of urn models. So what is an urn model? An urn 15 00:00:53,063 --> 00:00:57,037 model is always simple, it's got an urn that contains blue balls and it contains 16 00:00:57,037 --> 00:01:01,015 red balls, and you pick balls out of the urn. So I might reach this urn, and I might 17 00:01:01,015 --> 00:01:04,093 pick out a red ball. That's it. And then it's the probability or picking out balls of 18 00:01:04,093 --> 00:01:08,053 different colors. So the simplest urn model is called the Bernoulli model, and that 19 00:01:08,053 --> 00:01:12,045 works a follows: You just got a fix number of balls in the urn. So this urn has three 20 00:01:12,045 --> 00:01:15,096 red balls and one blue ball. The probability of picking up a red ball is 3/4. 21 00:01:15,096 --> 00:01:19,060 So if you think about this, this model can be used to explain a lot of 22 00:01:19,060 --> 00:01:23,010 casino games like roulette and blackjack: what are the odds of getting a 23 00:01:23,010 --> 00:01:26,096 face card? And you can think that there's, you know, 52 cards in the 24 00:01:26,096 --> 00:01:31,095 deck, and 16 of those cards are face cards. So there's a 16 over 52 chance 25 00:01:31,095 --> 00:01:36,016 that you're going to get a face card. So that's it, so the Bernoulli models are 26 00:01:36,016 --> 00:01:39,089 really simple, probabilities stay fixed. And the probability in each draw in a 27 00:01:39,089 --> 00:01:43,066 Bernoulli model is independent. So if I get a red ball this time, put that ball 28 00:01:43,066 --> 00:01:47,058 back in the urn, that doesn't have any effect on the probability of getting a red 29 00:01:47,058 --> 00:01:51,031 ball in the next period. So when you say something's independent, it means: what 30 00:01:51,031 --> 00:01:55,043 happens now doesn't depend in any way on what happened in the past. What we want to 31 00:01:55,043 --> 00:01:59,056 do is take this simple ideal of an urn model and use it to construct some 32 00:01:59,056 --> 00:02:04,014 path-dependent processes. And the most famous path-dependent process is called the Polya 33 00:02:04,014 --> 00:02:08,033 process. And here's how it works: Start with an urn with two balls, one red, one 34 00:02:08,033 --> 00:02:12,085 blue. You pick a ball. So maybe I pick the red ball out. After you pick that ball, 35 00:02:12,085 --> 00:02:17,030 you look at it, see its color, and then you put in another ball of that same color. Then 36 00:02:17,030 --> 00:02:21,030 you do it again. So maybe the next time I pick another ball out, and again it's 37 00:02:21,030 --> 00:02:25,040 red, so I add another red ball. Well, then maybe then next time, I pick a blue ball out, 38 00:02:25,040 --> 00:02:29,065 and then add in a ball that's blue. Now if we think about this Polya process, what's 39 00:02:29,065 --> 00:02:34,026 gonna happen is that over time, these probabilities are gonna change. So if we 40 00:02:34,026 --> 00:02:39,008 think about the Polya process in action, what was, what's gonna happen is, we're 41 00:02:39,008 --> 00:02:44,043 gonna get lots and lots of balls in here. Lots of red balls, lots of blue balls, 42 00:02:44,043 --> 00:02:48,049 depending on what I pick out. And the probability of balls, of the different 43 00:02:48,049 --> 00:02:52,009 colors in the balls can be changed; it's going to start out that the probability of 44 00:02:52,009 --> 00:02:55,082 red balls is the half. Then it might go to two-thirds as I pick another one. Then it 45 00:02:55,082 --> 00:02:59,018 might go to three-fourths if I pick another red ball, but then if I pick a 46 00:02:59,018 --> 00:03:02,068 blue ball it's gonna go to three-fifths. So what we see is, the probability of 47 00:03:02,068 --> 00:03:06,050 picking a red ball is going to change over time, and it's going to be path-dependent. 48 00:03:06,050 --> 00:03:10,001 It's going to depend on the path of previous choices. What I'd like to do now 49 00:03:10,001 --> 00:03:13,080 is state two results about the Polya process. I'm not going to prove them. In 50 00:03:13,080 --> 00:03:17,058 the next lecture I'll prove why they're true. In between this lecture and the next 51 00:03:17,058 --> 00:03:21,037 lecture what I'd like you to do is see if you can figure out why they're the case. 52 00:03:21,037 --> 00:03:24,097 So do some examples, with, you know, just with picking three balls from the urn and 53 00:03:24,097 --> 00:03:28,043 four balls from the urn, you know, four sequences along the path, and see if you 54 00:03:28,043 --> 00:03:32,052 can see why these are true. So here's the first result, and it's sort of surprising. 55 00:03:32,052 --> 00:03:36,091 The first result is, any probability of red balls and blue balls is an equilibrium on 56 00:03:36,091 --> 00:03:40,059 the long run. So you can end up with 60 percent red balls, you can end up with 57 00:03:40,059 --> 00:03:43,074 4 percent red balls, you can end up with 99 percent red balls. Any one of 58 00:03:43,074 --> 00:03:47,098 those things could happen, and they're all equally likely. So you're equally likely 59 00:03:47,098 --> 00:03:51,009 to get 4 percent red balls or 85 percent red balls. That's amazing. And it 60 00:03:51,009 --> 00:03:55,027 turns out, it's not very hard to prove. So I want you to see if you can figure out 61 00:03:55,027 --> 00:03:59,026 why it's the case. Let me talk about why this is important though, in terms of the 62 00:03:59,026 --> 00:04:03,023 Polya process, and also just sort of why the Polya process is important. So let's 63 00:04:03,023 --> 00:04:06,092 think of the Polya process as a model of fashion, let's say. So suppose that 64 00:04:06,092 --> 00:04:10,080 there's different two colored shirts you might buy. You might buy a red shirt or a 65 00:04:10,080 --> 00:04:14,067 blue shirt. And so you look in the store window, and you see one red, one blue. And 66 00:04:14,067 --> 00:04:18,054 you're not sure what to do, so maybe you choose red. Well, now your friend's going 67 00:04:18,054 --> 00:04:22,059 to come buy a shirt. And she looks and she thinks well, I see one red and one blue in 68 00:04:22,059 --> 00:04:26,059 the store window, but I see that my friend bought a red. So she sees two red and one 69 00:04:26,059 --> 00:04:30,035 blue. And so maybe her probability of buying a red shirt is two-thirds because 70 00:04:30,035 --> 00:04:34,021 she sees more red than blue. So then she buys a red. So now somebody else comes in and 71 00:04:34,021 --> 00:04:38,017 they buy a shirt. And they're thinking, "okay, what's more popular, red or blue?" And they 72 00:04:38,017 --> 00:04:42,019 see three people buying red and one person buying blue. And so they are three times as 73 00:04:42,019 --> 00:04:45,075 likely to buy red or blue. What you're getting is a Polya-type process. Now, 74 00:04:45,075 --> 00:04:50,026 what's interesting about this is that, if people make decisions in that way, what 75 00:04:50,026 --> 00:04:54,060 this result tells us is: anything could happen, and anything is equally likely. 76 00:04:54,060 --> 00:04:57,081 We're as likely to end up with 4 percent red shirts as 99 percent red 77 00:04:57,081 --> 00:05:02,004 shirts. We could use this same model to think about people buying Macintosh 78 00:05:02,004 --> 00:05:06,049 computers or IBM machines, right? Whether you use DOS or Mac? And the things that 79 00:05:06,049 --> 00:05:11,006 you think about using, the probability that you buy a Mac machine is proportional 80 00:05:11,006 --> 00:05:15,063 to the number of people who bought Mac machines in the past, you're going to get 81 00:05:15,063 --> 00:05:20,057 a Polya-type process. So, what plays out, what happens in reality is gonna be very 82 00:05:20,057 --> 00:05:25,087 contingent on the history, and anything could happen. So, that's result one. 83 00:05:25,087 --> 00:05:31,023 See if you can figure out why it's true. Here's result two: Result two is that any 84 00:05:31,023 --> 00:05:36,065 sequence of events that has R red balls and B blue balls is equally likely. So, if 85 00:05:36,065 --> 00:05:41,081 I get red, red, red, blue, that has the same probability of happening as drawing 86 00:05:41,081 --> 00:05:45,084 up blue, red, red, red. Why is that the case? Again, to use some examples, see if 87 00:05:45,084 --> 00:05:49,070 you can figure it out. I'll prove it in the next lecture. So what this means, and 88 00:05:49,070 --> 00:05:53,051 the reason this is interesting: It basically says, if you know the frequency 89 00:05:53,051 --> 00:05:58,007 of red and blue balls. That, that doesn't tell you really anything about the order. 90 00:05:58,007 --> 00:06:02,070 So if you just see the set of things that happened, you can't infer anything about 91 00:06:02,070 --> 00:06:06,064 the order, because any order is equally likely. So, again, that's sort of 92 00:06:06,064 --> 00:06:10,088 surprising. Let me move on to another process. This next process is called the 93 00:06:10,088 --> 00:06:14,069 balancing process, and it's sort of the inverse of the Polya process. In the 94 00:06:14,069 --> 00:06:18,082 balancing progress, again we have an urn. There's a red ball and a blue ball. But the 95 00:06:18,082 --> 00:06:22,090 difference here is, when I pick a ball out, let's suppose I pick a red ball, what 96 00:06:22,090 --> 00:06:26,082 I do is, I add a ball of the opposite color. So I pick out a red ball, I add a 97 00:06:26,082 --> 00:06:30,079 blue ball. Well, think about what this process is going to do. Suppose I pick a 98 00:06:30,079 --> 00:06:34,081 red ball. I add a blue ball. Suppose I pick another red ball. I add another blue 99 00:06:34,081 --> 00:06:38,074 ball. Well now it becomes incredibly likely I'm going to pick a blue ball. So 100 00:06:38,074 --> 00:06:43,060 if I pick a blue ball, then I add a red ball. We can see, is this process is gonna 101 00:06:43,060 --> 00:06:49,008 balance itself out. So the balancing process in the long run is gonna end up 102 00:06:49,008 --> 00:06:53,008 with 50 percent red balls and 50 percent blue balls. Now [inaudible] okay, this is 103 00:06:53,008 --> 00:06:56,073 balls, where is this gonna apply to the real world? What it applies to, is situations 104 00:06:56,073 --> 00:07:00,019 where you want to keep different constituencies happy, so let's suppose, put a 105 00:07:00,019 --> 00:07:03,088 convention in the United States, you might be decide to put it in the northern 106 00:07:03,088 --> 00:07:07,067 state or southern state. If you put it in the northern state, what you're doing is 107 00:07:07,067 --> 00:07:11,055 increasing the probability that you're gonna put it in the southern state four years later. If you put it in 108 00:07:11,055 --> 00:07:15,029 the northern state again then you're really increasing the probability that you'll put it in the southern 109 00:07:15,029 --> 00:07:18,089 state, so it's like you're putting in balls of the other color. Or, think of the 110 00:07:18,089 --> 00:07:22,045 International Olympic Committee deciding where they should put the Olympic 111 00:07:22,045 --> 00:07:26,021 Games. Should they put it in Asia, should they put it in North America? Do they put it 112 00:07:26,021 --> 00:07:29,084 in Europe? Do they put it in South America? Now you can think of the balls as being four 113 00:07:29,084 --> 00:07:34,044 different colors. But as you pick balls that say Europe, as you put more, put the 114 00:07:34,044 --> 00:07:39,026 Olympics in Europe twice in a row, in effect, you're putting balls of the other 115 00:07:39,026 --> 00:07:43,075 continents into the urn, making it more likely you'll pick those. And what this 116 00:07:43,075 --> 00:07:48,024 tells us is that you're gonna end up with equal probabilities of the different 117 00:07:48,024 --> 00:07:52,073 continents. Or in the case of the northern and southern states, equal likelihood in 118 00:07:52,073 --> 00:07:57,039 putting the convention in the north or in the south. So here's what we get, and this is sort of interesting: In 119 00:07:57,039 --> 00:08:01,077 the Polya process, we could put in balls of the same color, what we got is that 120 00:08:01,077 --> 00:08:05,098 anything could happen. We could get any probability between zero and one, and 121 00:08:05,098 --> 00:08:10,013 they're all equally likely. So just as likely to get 99%, as 4%, as 60%, as 122 00:08:10,013 --> 00:08:14,048 33%. In the balancing process, the only thing you can get is 55, 50%. You get an even 123 00:08:14,048 --> 00:08:18,051 mix. So the Polya Process has, the equilibrium is incredibly path-dependent. 124 00:08:18,051 --> 00:08:22,070 In the balancing process, it's not at all. So that's the distinction I want to 125 00:08:22,070 --> 00:08:26,090 highlight. I want to make a distinction between the period outcomes. What happens in a 126 00:08:26,090 --> 00:08:31,042 particular period, versus: in the long run, what does the distribution of balls in the 127 00:08:31,042 --> 00:08:35,004 urn look like. So when we think about path dependence, we can think of it either in 128 00:08:35,004 --> 00:08:38,048 terms of these outcomes, or we can think of it in terms of the equilibria that are 129 00:08:38,048 --> 00:08:42,062 generated. So that's gonna be an important distinction. Path-dependent outcomes just 130 00:08:42,062 --> 00:08:46,035 means that what happens in a period depends on what happens in the past. 131 00:08:46,035 --> 00:08:50,044 That's gonna be true of both the Polya process, and the balancing process. 132 00:08:50,044 --> 00:08:55,033 Path-dependent equilibrium means, what happens in the long run depends on the process 133 00:08:55,033 --> 00:08:59,079 along the way. So if I think of the Polya process and the balancing process, the 134 00:08:59,079 --> 00:09:04,013 Polya process has both path-dependent equilibria and path-dependent outcomes. 135 00:09:04,013 --> 00:09:08,047 But the balancing process only has path- dependent outcomes. The equilibrium is 136 00:09:08,047 --> 00:09:13,009 always one half. It doesn't depend on what happened along the way. It's always going 137 00:09:13,009 --> 00:09:17,077 to end up with equal amounts. Now, this is an incredibly important insight, because it 138 00:09:17,077 --> 00:09:22,011 affects how we think about identifying path dependence in the real world. I'll 139 00:09:22,011 --> 00:09:26,026 give you an example. In the balancing process, history matters at each step in 140 00:09:26,026 --> 00:09:30,047 time. So when you think about, what's the odds of getting blue balls in period seven, well that 141 00:09:30,047 --> 00:09:34,053 hap, depends on how many blue balls I picked in periods one through six. So each 142 00:09:34,053 --> 00:09:38,064 period's outcome depends on the past. So if I'm writing a narrative, I'd say, "oh boy, 143 00:09:38,064 --> 00:09:42,054 this really was contingent on what happened previously". However, that doesn't 144 00:09:42,054 --> 00:09:46,081 mean that what happens in the long run depends on the path. What happens in the long run 145 00:09:46,081 --> 00:09:50,076 could be completely independent of the path, like in the balancing process. So 146 00:09:50,076 --> 00:09:54,098 history can matter at each moment in time, but it can't matter in the long run. And 147 00:09:54,098 --> 00:09:59,003 so if I just tell a story that says, you know, period seven dependent on six, eight 148 00:09:59,003 --> 00:10:02,056 dependent on seven, nine dependent on eight, seven, and six, that in no way 149 00:10:02,056 --> 00:10:06,033 means that history was contingent. It could very well be that the equilibrium 150 00:10:06,033 --> 00:10:10,020 was set in stone beforehand, that the process was naturally going to go to one 151 00:10:10,020 --> 00:10:14,012 place. Now, what are the examples of that? Well, two examples from history might be 152 00:10:14,012 --> 00:10:18,023 what we call in America "Manifest Destiny", the idea that the United States was likely 153 00:10:18,023 --> 00:10:22,030 to be a continent that stretched from sea to shining sea. So, the history played out 154 00:10:22,030 --> 00:10:26,002 in particular ways, but there would be some people that argue that it didn't 155 00:10:26,002 --> 00:10:29,065 matter what that path was, we were destined to be a country that stretched 156 00:10:29,065 --> 00:10:33,044 "from sea to shining sea". So, even though, the Louisiana Purchase, particular wars, 157 00:10:33,044 --> 00:10:37,016 the gold rush, all those things happened in a particular sequence, some people 158 00:10:37,016 --> 00:10:40,093 argued it didn't matter, eventually we were gonna become one nation, stretching 159 00:10:40,093 --> 00:10:44,032 from sea to sea. Another example is the railroads. People argue, once the 160 00:10:44,032 --> 00:10:48,000 railroads are invented, that they sort of built themselves. So sure, there were 161 00:10:48,000 --> 00:10:51,083 these, you know, people like the Carnegies and the Stanfords that laid the tracks. 162 00:10:51,083 --> 00:10:55,051 But the thing is, it didn't matter who those people were, and it didn't matter 163 00:10:55,051 --> 00:10:59,043 what order they were laid. The tracks were gonna be laid connecting the cities one 164 00:10:59,043 --> 00:11:02,098 way or another. And it really doesn't, you know, particular sequences, the fact 165 00:11:02,098 --> 00:11:06,067 that this track update, you know, track such-and-nineteen is going to be connected 166 00:11:06,067 --> 00:11:10,026 to track such-and-eighteen. So yes, each lay of each track was path-dependent, 167 00:11:10,026 --> 00:11:13,095 but the long-run outcome may not have been. The tracks are gonna be what they're 168 00:11:13,095 --> 00:11:17,055 gonna be, because that's where it's economically efficient for those tracks to 169 00:11:17,055 --> 00:11:22,004 be laid. So again, at each moment in time, the event depends on past history. But the 170 00:11:22,004 --> 00:11:26,095 long-run equilibrium doesn't depend on past history. Now remember when we talked 171 00:11:26,095 --> 00:11:31,024 about path dependence, we were saying the outcome probabilities depend on the 172 00:11:31,024 --> 00:11:35,088 sequence of past ev events, past outcomes. I want to make a distinction between that, 173 00:11:35,088 --> 00:11:39,094 remember we had that path dependence and phat dependence where outcome 174 00:11:39,094 --> 00:11:44,041 probabilities depend on past outcomes, but not their order. So I call this "phat" 175 00:11:44,041 --> 00:11:48,093 because remember I can think of path and rearrange the letters to get phat, I'm 176 00:11:48,093 --> 00:11:53,051 just switching the h over here. Now I also do this because "phat" is a little bit 177 00:11:53,051 --> 00:11:57,052 of, you know, joke about thick description. Right? So fat means thick. 178 00:11:57,052 --> 00:12:02,015 And thick descriptive accounts take into account the full sequence. The Polya 179 00:12:02,015 --> 00:12:06,095 process is phat. It's not path. And the reason why is, all that matters is the set 180 00:12:06,095 --> 00:12:11,070 of outcomes. If I had red, red, blue, that means I've added two red balls and one 181 00:12:11,070 --> 00:12:16,045 blue ball. And if I had blue, red, red, that also means I've added one blue ball and 182 00:12:16,045 --> 00:12:21,028 two red balls. So all that matters is the set of outcomes, not the sequence. Why is 183 00:12:21,028 --> 00:12:24,074 this so important? Why do we care about it being path-dependent versus 184 00:12:24,074 --> 00:12:28,039 phat-dependent? Well, let's do just a tiny bit of math and see why. How many different 185 00:12:28,039 --> 00:12:31,090 paths are there? Let's suppose I pick, I have 30 periods, and I wanna know, how 186 00:12:31,090 --> 00:12:35,055 many different paths are there? Well, there's two things I could have picked the 187 00:12:35,055 --> 00:12:39,016 first period, red or blue. Two I could have picked the next period. Two I could 188 00:12:39,016 --> 00:12:43,026 have picked the third period, and so on. So there's two times two times two ... 30 189 00:12:43,026 --> 00:12:47,001 times, different histories. That creates over a billion different histories, 190 00:12:47,001 --> 00:12:51,011 different paths. If I wanna know how many sets there are, well, I could have zero, I 191 00:12:51,011 --> 00:12:55,021 could have added zero blue balls. I could have added one, I could have added two, I 192 00:12:55,021 --> 00:12:59,004 could have added three, up to 30. So there's 31 sets. So we got a billion 193 00:12:59,004 --> 00:13:03,055 paths, and only 31 sets. So if something's set dependent, that means that, in round 194 00:13:03,055 --> 00:13:08,001 31, that means there's only 31 different possibilities that matter. If its path 195 00:13:08,001 --> 00:13:12,060 dependent, that means there's a billion possibilities that matter. And this raises 196 00:13:12,060 --> 00:13:17,013 two questions. First up, could something actually be path-dependent, depend on all this going 197 00:13:17,013 --> 00:13:21,083 different paths? And in second, if there is one, can I construct it using my simple 198 00:13:21,083 --> 00:13:25,085 urn model? Well, the answer to both questions is: yes, you can. So here's the 199 00:13:25,085 --> 00:13:30,026 simple process using the urn model: It's called the Sway process, and it's gonna 200 00:13:30,026 --> 00:13:34,073 give us some really interesting insight. So the Sway process works as follows: I start 201 00:13:34,073 --> 00:13:39,010 with one blue and one red ball. When I pick a ball out, I add a ball of that same 202 00:13:39,010 --> 00:13:43,056 color, just like the Polya process. But I also add 2^(t-s) minus 2^(t-s-1) balls 203 00:13:43,056 --> 00:13:48,002 of the ball chosen in each period s less than t. What does 204 00:13:48,002 --> 00:13:52,053 that mean? That's complicated. Let me explain it with a picture, that makes a lot 205 00:13:52,053 --> 00:13:57,004 more sense. So in period one, I pick a blue ball, I add a blue ball. In period 206 00:13:57,004 --> 00:14:01,061 two, I pick a red ball, so I add a red ball, and I also add a blue ball for the blue 207 00:14:01,061 --> 00:14:06,012 ball that I pick in period one. Period three, let's suppose I pick a blue ball, 208 00:14:06,012 --> 00:14:11,004 so I add a red ball again for the red ball I picked in period two. And now I add two 209 00:14:11,004 --> 00:14:15,061 blue balls, so I'm multiplying times two the blue ball that I picked in period one. 210 00:14:15,061 --> 00:14:20,065 Period four, suppose I get a red ball, now I add a red ball. I add a blue ball 211 00:14:20,065 --> 00:14:28,035 for period three. I add two red balls for period two. And I add four, or two times 212 00:14:28,035 --> 00:14:34,039 two, blue balls for period one. So what happens is, for each ball that I pick, I'm 213 00:14:34,039 --> 00:14:38,065 going, I'm adding one, then two, then four, then eight, then sixteen, then 32, as I 214 00:14:38,065 --> 00:14:43,030 move through time. The reason that's why I call this "the sway", because as decisions 215 00:14:43,030 --> 00:14:48,001 go, as you go back in time, decisions take on more and more weight. This is really 216 00:14:48,001 --> 00:14:52,046 interesting, because what's creating this full path dependence? It's the path taking 217 00:14:52,046 --> 00:14:56,025 on more and more influence. So, when people talk about path dependence, when 218 00:14:56,025 --> 00:15:00,025 they give examples of the law, when they talk about institutional choices, when 219 00:15:00,025 --> 00:15:04,050 they talk about technological adoptions, they all think about early movers having a 220 00:15:04,050 --> 00:15:08,070 bigger effect and the past really sort of having increasing weight. What turns out, 221 00:15:08,070 --> 00:15:12,095 one way you can get full path dependence, is by having exactly that sort of process. 222 00:15:12,095 --> 00:15:17,002 By having the past take on exponentially more weight over time. So that's really 223 00:15:17,002 --> 00:15:21,005 interesting. Very simple urn model has told us how something that we had some loose 224 00:15:21,005 --> 00:15:24,098 intuition about, for something really to depend on full history, not just on the 225 00:15:24,098 --> 00:15:28,096 set, you've got to have the early path take on more and more weight over time. Or 226 00:15:28,096 --> 00:15:32,072 at least that's one way in which you can get full path dependence. I realize 227 00:15:32,072 --> 00:15:36,058 there's a lot going on here, but here's what's really interesting and it's really 228 00:15:36,058 --> 00:15:40,035 useful. If you take a really simple class of models based on urns, just these really 229 00:15:40,035 --> 00:15:44,006 simple urn models, from those urn models we learned some interesting things. We 230 00:15:44,006 --> 00:15:47,083 learned how to construct a very simple path-dependent process, called the Polya 231 00:15:47,083 --> 00:15:51,035 process; we constructed a balancing process, which showed us you can have 232 00:15:51,035 --> 00:15:55,012 path-dependent outcomes in each period, but not have path-dependent equilibrium. 233 00:15:55,012 --> 00:15:58,079 You can go to a particular thing each time. And we even constructed something 234 00:15:58,079 --> 00:16:02,060 called the sway process, that gives us full path dependence and that showed us 235 00:16:02,060 --> 00:16:07,018 one way you can get full path dependence and not just phat dependence is by having 236 00:16:07,018 --> 00:16:11,044 the weight of history increase over time, which is something intuitively I think 237 00:16:11,044 --> 00:16:15,037 that a lot of historians and a lot of scholars of technology have felt was true. 238 00:16:15,037 --> 00:16:19,011 And this sort of shows us why, in fact, that may be the case. Why, when we see 239 00:16:19,011 --> 00:16:23,034 some of these things really contingent, it might be the case that what happened early 240 00:16:23,034 --> 00:16:26,090 on had a larger effect. Okay, thank you.