 So good morning everyone. I welcome you to lecture number 11 of our course collective dynamics of firms as I mentioned in the email today. We have the course evaluation. I Assume that there is not a very good statistics from six Students here, but nevertheless we would like to distribute these sheets if some of you Attend the exercise and haven't done this yet, and you can also do it Together with Pavlin. Yeah, maybe you can distribute this here. You return this during the break, right? I think that everyone is familiar with this, right? Or is there a question? The course number. Yes, you are right. What is the course number? I Write it down here the course number is three six three or five Three that's a course number. Yes So you do not need to rush with this You have time until the break And in the meantime, I will entertain you with some slides Okay, so where are we in the course right now? We looked first into data Then we found some Starlights facts about the size distribution of firms and the growth rate distribution of firms and then for the rest of the course We try to Develop agent-based models to reproduce the starlights fact. So the surprising finding was that you were able to reproduce these starlights facts already from very simple assumption that From the outset have not a real economic interpretation, right? If you think about these stochastic growth factors So then it's impressive that you get the Distribution without assuming anything about economics Therefore in the second part of the course we try to enhance these agent-based models step by step the first step that be The first step that we did was Considering indirect competition or indirect interaction. How did we do this? We considered competition in a market in a kind of free market capitalism and then we found that The because of the competition some of the firms Will survive and most of them not This is a nice intermediate result, but it doesn't reflect the state of an economy Well, we have small and big firms co-existing that means in the next step. We try to add assumptions about interaction that results and coexistence of firms and we did this by Two different models one was a redistribution of wealth which had an impact on the inequality distribution and the other Interaction assumption was cooperation by exchanging knowledge. This was done in the last course, right? You learned how firms Depend on the input of other firms There was a baseline model and then to this baseline model we step by step added assumption about utility maximization So the question now is do we cover the economy or the dynamics of firms by just Considering utility maximization. The answer is not as we will see today Because only in rare cases you can really calculate utilities right, of course if you think for of the problem from the perspective of the firm then there is There is a Strategic decision with whom to interact and so on and so on right, but this is usually not really made based on a Calculation of a utility. There are all the social Effects that are included in this decision what we do today is we entirely In the first part focus on the social processes You as a firm you decide to do something not because you are convinced to do this But because you observe that all others are doing it, right? There are hurting effects. There are social influences Yeah, you observe what other big players in the market are doing and Then you try to adopt their behavior because you can observe that these are successful firms You do not know precisely why they are so successful You can just observe they are successful and then you try to copy their business strategy for example their production Layout or whatever right that is the focus of today's lecture So we will talk specifically about hurting effects in firms that firms try to imitate other firms And we will also look into So-called lock-in effects, which means what happens if you copy the decisions of other firms or the Policy of other firms. How does it lead to states that are hard to change? That's behind this past dependence and lock-in effect So with this we we have a number of slides today. I'm not going to read all of these I try to be a bit short This slide should motivate why we are interested in this The best thing is I'm not going to talk about these famous paper by Allison and Glaser From 1997 we have put this in the literature folder if you're interested you can take a look into this I've just addressed the basic problem if you look into the spatial distribution of firms Then you see that there are clusters of firms that Emerge in certain areas the best known example and the most Copied one is the Silicon Valley There were a number of few firm of small firms that have been established either as Spin-offs of Stanford University or because of some local Connection with the industry and then just started to grow and became a very prosperous area So the question is how did it happen? Everyone wants to have a Silicon Valley in his canton or in his country, right? So everyone wants to have so the question is how did it happen? What is behind the success? And people try to understand this because if you go there and look then you see that the firms in these Local clusters have similar business culture, right? There is this saying or rumor that they never had written contracts Everything was agreed on in the pub or in the coffee shop and these kind of things. Yeah, of course That's wrong, right? But it gives you a kind of flavor of what people think was a culture in these early days when Google was founded or Apple or something like this, right? So that's the idea Okay, and We would like to understand how firms influence each other and creating this local culture because essentially the existence of such a local culture is an important ingredient for For the establishment on the survival of something like Silicon Valley I should mention that Silicon Valley is just one example here So we have another example in the Boston area, which is all the wealth studied in the literature Boston 128 that's a route out of Boston in the South where firms locate and aggregate And people want to understand why is it like this? the modeling Has To start with an agent-based Framework that means so we are not looking into density functions of how firms are distributed we look into the problem of aggregation from the perspective of a single agent so It is all the importance that we do not enforce The result namely the emergence of a cluster and a local culture by putting this in somehow in the model Yeah, there are some people that are really smart in Tweaking the model exactly the way they want to get the result of course so here it's important to have not a central authority that Basically governs the interaction behavior of this firm instead it has to come up as a Emerging property as we name it in complex system science and The third important assumption that is listed here is we do not assume That firms get into the state based on utility calculation Why not first of all we do not know the utility We want to do it in a utility based model than for the rest of the lecture We just talk of how should the utility look like and then of course we have to go and have to test these Assumptions with an econometric model and so on and so on you know these from other courses, right? Here our assumption is instead of explaining this from a utility maximization We assume that firms Locally influence each other You see what your neighbor is doing and then you behave in a similar way or you adjust your behavior In a way that you can better collaborate with your neighboring firm and this kind of ideas, right? Here we have listed two classes of modeling The voter model and the bounded confidence model so the voter model is not captured in this Lecture actually, but I should mention it a voter model is a very simple model You assume that an agent is characterized by one of two states Zero or one this can stand for anything, right? Let's assume. It's an opinion or something and then this agent can adopt this state according to The number of other agents in the neighborhood that have the same or a different opinion or state All right, so that means the agent is in state zero and then the agent looks around and sees he's surrounded by all firms in state one and Then the probability that the agent switches towards state one becomes very high or the other way around and so on Right, so that's a voter model. First of all the binary variable minus one plus one something like this and secondly It's a frequency Frequency-based model you calculate the frequency the percentage of agents in your system your neighborhood and then you adjust your decision according to what all the others are doing or A different from all the other what all the others are doing. That's all the possible Yeah, if you see everyone is now in state one then you think it could be a beneficial to be in state zero Right and have a different business model for example. That's the idea these models these voter models are well studied in various Scientific disciplines notably in mathematics, but also in physics I'm not going to talk about this. I could because I did some work myself in this direction But that's not important today. Maybe the important thing that you should consider the voter is not voting, right? Yeah, that's maybe important for you to remember, right? No one votes here. The voter simply adapts to what the voter sees in the environment We talk about the second class of models today. That's called the bounded confidence model The difference is that the agents are characterized by a state X But this can be a continuous variable. Let's assume it's something between zero and one, but it can be any value and Whether or not you interact with other agents depends on whether you have a similar X Whatever X is. Yeah, let's assume X is your is your business model the way you Handle contracts or something like this, right? And then of course if you have a firm on the other end that has a complete different culture Complete different way of handling whatever administrative process is then this will not work out, right? That means the assumption is the more Similar the firm is in the way it runs its business and more easy for you is it is to interact with us Let us describe the baseline model first This is a model So we should improve Maybe it's a little to section a bit here That's a model that was first in reduced in a social context Where people thought about who talks to whom right if you go for a party and Then you talk the first place may be to people that are similar to you because then you have something to share with them, you know you have something to talk about and If there are people that are not of your Social level Completely different than you have problems to interact with these people. That's the assumption here, right? that means each of these agents is characterized by a value X and Then for each possible Interaction so we choose two agents we compare the X of i and the X of j and We assume that the only interact if the difference is less than D in this model here, right? So that's a threshold of course if the D is very large Then we end up with the so-called mean field limit because then everyone can really interact with everyone else But if the D is very small Then he may find a different phenomenon who can imagine what we will see if the D is very small Or rather small we have an intuition what we would see Yeah, how would the structure of the system look like? If the D is moderately small If the D is large, what would be the result? Everyone interacts with everyone then because of the interaction we become very similar, right? So that means we end up with a mean culture, right? So that's a kind of mean fields limit Whereas if the D is very small So our interaction is much restricted. Yes Yes, we have some kind of communities of class there So there are all these guys that have a very large X Then there are all these guys that have a mean intermediate X and then there are all these guys have a small X, right? So and those guys do not talk to each other if they belong to different groups. That's the idea The important here in the is written in the lower part This is an adaptation process because you interact with these other agent you get to know the agent and therefore your Local variable your local culture whatever the X stands for becomes more similar to what the other is The other is characterized by so you see here So xj for example is the xj of the previous of the previous Time step we could add the time step there, right Alex so and So the change here depends then on The difference of the two and the mu is a kind of convergence parameter So the if you choose point five here, then it's a symmetric process That means I change if I'm I I change my X the same way as you change But if it's different, then of course someone changes small so Here is Some Result about it. There are a few physicists in this room What do you guess from this kind of dynamics? Let us try to understand this if you drop This part here in the beginning and you just look into the second derivative here, then it reminds you of what Well, have you seen this kind of equation before with a second derivative in space? Hmm in all diffusion equations for example, you know Every diffusion equation has a second derivative in space, right? So you know what a diffusion process is, right? Diffusion processes you start with a delta function and then in the causal time it gets broader broader and broader, right? So and then it's very broad at the end That's the underlying process as well here, but there is a difference the mu because of this or the axis bound to The axis bound to 1 the mu can not be larger than 1 so and look into this factor here Then it means that the factor in front of this is negative You want to talk about the difference to the diffusion process or No, okay So the factor here in front is of course there is a row square there and so on but this doesn't matter Right, so we just talk about the second derivative and then you see there's a pre-factor and this pre-factor is negative So what do we see if we have? Negative diffusion you do not see this we see this, right? That's what you can already guess without any further knowledge about mathematics from this equation, right? In the cause of time this will not this distribution will not broaden but get narrow That's the that is the Basic meaning of this or as we have put it here if you have some initial fluctuations where some of the X are a bit more prominently represented then this will be amplified and other cultures that are less prominent in the Initial state will die out in the cause of time. So that's what you what you Can already guess from this so here are computer simulations that show us exactly this so here, please have a look you have a Very broad distribution in the beginning and then in the course of time. That's time it becomes narrow and it converges to just one culture Why is it like this? It's like this because The mu and the D are Appropriated to you remember with the divorce the D decides Who's talking to whom right if I make the D smaller as you can see in this one Then not everyone randomly chosen can talk to everyone else And then you see instead of one culture or one cluster emerging You see these two class right? That's the difference Excuse me, I don't know no idea But that's something you can get with Different numbers of agents as well. So and although you remember what the mu is doing the mu gives him Somehow described the impact of this interaction, right? If the mu is very small then we interact it will be converged but only very very little That means my x variable changes on a very slow timescale Which on the other hand means that there are ample possibilities in between? That allow me to change my direction, right? That's something we talk about when we talk about past dependence afterwards So this is a plot that Maybe I should go here Shows you Where in this where at the end these initial distribution converges here You see it converges to one state or one opinion here It converges to two opinions and then this example it converges to all the discrete opinions and you see Two things here so we assume that all of these x are possible in the beginning So and then you see the final distribution is characterized by these levels, right? So you have four different cultures Because you have a local convergence of these guys here They converge to that and those guys converge to that kind of culture, right? So that's obvious So you have four different levels, but what else do you see you see that there is some overlap in these levels Do you see this here, right? So that means if you are an agent, let's say with point five Yeah, initially x equals point five then you can end up in this cluster But you can also end up in that cluster and this really depends with whom you interacted initially Right, that means for those where you are part of two levels here Initially here there and there It really depends on the sequence of interaction with whom you talk first who influenced your x into a particular Direction and then on the question whether or not this was amplified. I Mentioned this because afterwards we talk about path dependence And this is already something where it depends on the path of your interaction That's a meaning of path dependence whether you end up in this local culture on that local So now we extend the baseline model I hope that everyone understood what's the meaning of the bounded confidence in their values And you understood that the D is a very important parameter that Somehow defines the number of levels that you can get and you also have seen in this last slide this ambiguity for agents that can choose between that and that final state This is the message and now we try to understand this To extents us. Why do we want to extend that there's an important issue missing in this model namely that if you Got to know these other guy and have interacted with this other guy Then you should basically assume that in the next time step You do not ignore that you had a previous interaction with us. No That's important Let us assume you are from a and you will have talked to firm B in the beginning, right? About sharing some patterns or something like this, right? So and then you talk to firm C But talking to firm C after talking to firm B is not completely independent because you as a firm already know what you talk to firm B and You are influenced by this Right and we try to capture this in a particular assumption here Which we call the in group The assumption here is the following Interaction is a costly act So you invest something you invest your time Maybe although your money or you disclose some of your whatever firm knowledge Business secrets. I don't know right so therefore there's some costs associated with this and you therefore in the next time You will not ignore the costs you have already invested into this which in terms means that those other firms have Basically an influence on you because you talked to them before and Have in as we assumed in this model. You have somehow adjusted towards that direction What we try now is that the sequence of Interaction that we assume leads to a social network I got to know you because we interacted my X became a bit more similar to you So therefore you somehow influences me and I will keep this in mind in my in group and Because I now adds a sequence of my Interaction partners to this kind of in group. I'm no longer just dependent on my own X I'm also dependent on the influence of those who I interacted with before, right? You see it makes sense to think about this like this you know Is it clear what we have changed and why we now assume that The X does not only change According to the bounded confidence model where we become closer after interaction also changes because there is an influence of the Past interaction partners which are summarized in an in group. That's the idea Okay, so here we formalize what I just said so we Consider this in group influence Here in this Effective value of X. So you see The effective value of X is composed of two parts. One is my X and The other one is the average X of the partners. I have interacted with and I can wait this influence with a parameter alpha if I choose alpha equal one then I only follow the Influence of the in group partners So that means my own opinion doesn't count anymore And if I choose alpha equals zero, then it's the other way around. It's the original bounded confidence model I have a sequence of random interaction and I do not care with whom I talked before Yeah, so these are the two different Assumption for the alpha. We have a very mild assumption only We say, okay It the group influence Increases with the in group size which makes a lot of sense, right? That's the only assumption you can choose other assumptions here. So this is alpha is more or less proportional to I So it's a one with a little bias No, it's a it's below one with a little bias, yeah Okay, so that's the only assumption here and then We change the rule of the bounded confidence model a bit. We assume now That you only interact. It's the effective value of the X is similar Before you looked into the X only and now you look into the effective access means your X plus the in group influence, please Yeah No, we don't assume this we can do it in the very first step So in the very first step we choose each other, right? So we interact if the D If our X is less than D So Then we change our X according to the bounded confidence in a while We become a bit more similar and we add each other to each other's in group That's what we do you consider that you have interactive with me before and I consider that I have interactive with you Before that means in the second round. I cannot no longer start with a naked X I instead start with my X effective, right? So which keeps the previous interaction in mind So the next slide in a very complicated way just summarizes what I said here That's a mathematical formulation of After we interact it I add you to my in group And if we didn't interact I don't add you to my in group But it also says I remove you from my in group If you were previously in the in group, that's also important, right? So let's assume after number of interactions with other partners I Just by chance have another interaction with you as a firm But our X because we had different other experience in The meantime our X have diverged So then why should we keep each other in the in group not at all, right? So that's what This also means I hope you understand this so this is a removal and that's the inclusion Now we have a kind of group effect So we just took this example of the first interaction I have added you to my in group now, right? So and then I don't interact with anyone anymore, right? Let's assume. Well, but you interact with all these other people, right? What does it mean? It means you change your X constantly And I even without interacting I'm affected by this Because this interaction with the in group has another Has another effect on me you understand I do not need to interact with the other people to be affected by the group because your In group is growing and because you are part of my in group So therefore if your X is changing Although because of your interaction with your in group also my X is changing You got this point, right? That means we have a social network that is growing here and this social network influences us Even if we are not active Because the network changes. Yeah, the network adapts So now let's come to the results here. So that's already the result But we have a computer simulation and I prefer to show you this Let me see That's exactly what we what we have just described by the way this video together with the paper is On our website you can download it and you can watch it again We just failed to include this into the notes, right? I'm sorry for this Though this is what we have described the beginning. We have a number of agents these agents Two of these agents are randomly shown Chosen and then their X is compared, right? And if their X difference is less than D They interact and otherwise not so let me So you see the second time steps nothing happens Why not because there were two agents chosen that don't interact, right? So therefore nothing happens third time step Nothing happened if the If the agents move in this space This has no meaning Well, it You just look into the color of the agent. So the grayscale basically tells us about the X Right, it's from black to white and So the link means the two of us have interacted And at that time the X of the two of us was within this boundary. That's the idea So now let us run and then you see there are Networks Emerging from this kind of interaction. So I let this run a bit. So then you yeah, as I said you should not watch the Movement in space. You should better watch How the network evolves So let us stop here at this point. So what you see Is there are two components in the system. There are all these firms interacting so Which is indicated by the links. You also see that these firms have a similar gray color here Which means that the X value is has has become rather similar That's no surprise. That's because they could interact and they converge to a similar one So and now you see these green links here actually, right? This means we are at a point in the simulation where we have chosen these two guys If we just compare their X X i and x j there would not be any interaction possible By just comparing the X. But what matters here is not the X What matters is the X effective that means the X plus the influence of the of the group, right? And because they have these groups. So basically The group influence somehow forces them to also interact or allows them to interact If you think of a business model though, these guys would probably not talk to each other But thanks to the groups they belong to they have the ability to still talk to each other, right? Just by themselves the CEO of company a would not talk to the CEO of company b But because both of them belong to consortium that usually talk to each other These guys also talk to each other. That's the idea here, right? okay If the green link would not exist then these two Cultures would evolve completely independently. I hope you see this here It's only thanks to the fact that there is the in-group Otherwise, we are already at a point where this completely separates so Now let me just continue with it. There's also then a point In time where a red link appears. I hope I see this somewhere It's here. So this means If I would choose these two guys at this particular moment No, they would not interact anymore. That means they have been Evolved in a direction that they even with the x effective are not able to interact, right? So that means the group Would not help them to interact Yeah, you understand this So they don't interact But you see there are other links, right? And the in-group that was the example that I tried to explain before the in-group evolves by itself You understand? So even if these two guys do not interact Those guys interact and other guys and these two groups interact And this means that the in-group has its own dynamics and indirectly changes the Effective acts of these two agents in a way that they Can't converge to each other even that they never ever interacted again after this point. You understand this I hope yeah, so it descends to the dynamics of the in-group That we see this. Okay. Let me just finish with the simulation So the nice result or maybe we skip it here because the nice result is already on this page At a larger time step we see that these two groups converge to each other Which is a very nice Result because it tells us what is the role of the social network in Finding a common culture in these two in these simulations here, right? Without the social network of these firms We would not see the emergence of a local culture We would see instead as you have seen in the video the emergence of two cultures That means the separation of the set of firms into groups that can no longer interact with each other That's the important message here Let me just look into one particular example here Well, we try to estimate Under what circumstances these Talk to each other. There's another remark Alex. So because this was copied from another talk of mine This you threshold is somehow related to This is The epsilon is related to the D. I think this is never said on one of these slides, right? So Now you say well Thanks to the in-group. We all talk to each other. We will always find a Local culture that we wouldn't find Without the social network, right? But this is essentially not really true If you look here Into the convergence To what's the local culture that means the frequency of reaching consensus Consensus means we are all part of one Of one culture. So then you see that for small d That's the epsilon of course the Probabilities that we end up in a local culture increases But if the d gets larger this probability decreases. This sounds a bit miraculous, right? Because didn't we say that the big d would Allow us To interact with more people That's true. If you look into the blue curve the blue curve is for the Model without any social network and without any memory effect. So here you see that this is Indeed increasing with the d But here's another effect Now that because we interact it and because our in-group is working well We convert to a local culture very fast Much faster than in the case without a memory Thanks to the in-group input and this means that The likelihood that we can later on still talk to these guys that are on on on the fringe of these Variable, yeah, namely by at very small at very large x. This probability that we can still talk to them This probability that we can still talk to them decreases very fast Because we have converged in the initial state very far We find a local culture But at the same time we lose Our ability to talk to firms which has have extreme Opinions or extreme x Therefore this Decreases afterwards. So that's a message of it I hope you understood this, so Of course, it's a benefit if The ability to talk to each other is very low Now then you see there is a benefit here the red curve is above the blue curve So for small d, but if the d is larger then you see The facts that we speed up our convergence process and our in-groups are working well And we all talk to each other we converge very fast and those guys who have extreme opinions are left out of this process That's an interesting finding, right? Okay, so this slide gives us a bit of a motivation. I can be very fast of this About this So what do we mean by this local culture? Local culture means that we somehow follow a social norm in our business practice. There are Of course legal contracts and written Norms and rules, but there are also Non-written rules. That is simply the way We trust each other the way we behave, right? I as company One in the silicon valley will not send out emails to your Employees of company b and ask them whether they want to join our company for a slightly better salary or these kind of things You don't do this, right? So there's no legal statements that you shouldn't do this This is what we call a local culture here, right? So And this is uh, of course beneficial if you have People or firms accepting this kind of unspoken rules or social norms It eases the process a lot In interacting between firms. That's something you understand immediately. Therefore, there is a measurable Value associated with converting to a similar business practice for example Yeah, that's a message of the slide basically with this right in time We stop for the break of 10 minutes and then we continue with past dependence You can please return the sheets here put them here on my table. Just continue, please I hope you understood from the first half of the lecture the following first of all There is not always utility Calculation of firms. There's also social interaction Of course in this particular case of subordinate confidence model, you could assume an underlying utility, right? And your utility is maximized if you interact with firms that are more similar to you And you have less utility if you interact with firms that are more different from you, right? This model can be mapped onto a utility maximization model, but in general I want to emphasize in particular this last part where We talk existing studies focus on a game theoretical analysis. What does it mean? It means this usual Prisoner's dilemma that you have in order to establish cooperation, right? You'll probably know this from other Lectures I cooperate with you if you cooperate with me, but how can I know from the outset that you will cooperate with me, right? So that's a dilemma situation And therefore So I usually choose something where I know that my my payoff is maximized and that would be not to cooperate because otherwise I lose my Cooperation investment That's the way this was this issue is usually discussed In economics, we try to choose another approach here where we talked about these benefits in a way of influencing and changing the Yeah business practice in a way that you are better aligned and therefore get a kind of indirect utilities that we are not calculating So but there is already one thing that I mentioned several times in the beginning It for some agent it really depends With whom they interact it first because this drives them either to these or to the other direction And this is a very important scenario The scenario where we all converge to one common culture is a very rare scenario Basically, it only refers to mean field interaction That means if everyone can talk to everyone else in the system then we have this one cluster But the more realistic case is that we have a number of clusters And this means that the way to how we got there is very crucial And this is denoted as path dependence in general. So what does it mean? It means that we have small events random events in the beginning that During the evolution of the system, which is now the interaction scenario of the firm become Get or get amplified in a way that at the end the macroscopic state That means the state of the system as a whole depends on these Random interactions that have been amplified over the car That is a very important Effect If mr. Yulet has never met mr. Packard while they were students, right? So they would not have founded a company called Yulet Packard or these kind of things. There's a random event that led to a Amplification which was then Reinforced over time Up to the fact that we see a macroscopic state in which we have a very big company that's dominating the market, right? This is the kind of things you should have in mind, but there are other important economic Examples one of them is about competing standards, right? You think about here. This is an old example beta max versus vhs Probably do not know what this refers to it refers to the to this video cassette that You had when we had by the time you were born or something, right? Now we're talking about blu ray versus. We already forgot hd what the other standard probably, right? That's a similar situation. You have two competing standards in the market And then tiny fluctuations in the beginning decide about Which one will dominate you would assume the better one, right? That's natural. That's not true, right? If always the better one would dominate and the word would look like completely different That's a very important message actually It is not that the better one is Amplified all the time This we talk about amplification at a point where it's not clear which one is the better one, right? Okay, let's take one example from my own work here, which just shows you results of computer simulations Let's think about ants Searching for food. So that's this example and what we simulated here. So in the middle, you have the nest and then you have five food sources So here are two and there are two and there's a fifth one so these Two this one and this one have approximately the same distance from the nest No, there's no advantage of either going here or going there So the ants by chance found this food source earlier And then they started to exploit this food source so The black line that you see here, which actually is a gray line Showed the path that they use for exploiting this food source. You can think of a market You can think of a patent or whatever, right? Okay, so this patent was discovered first and then everyone went Into this business and tried to exploit this patent as much as possible creating whatever gorilla glasses for Smartphones or I don't know right something like this But you see there is a sharp turn in this path So actually you would have naively expected that they simply march on and detect this Right, so that was a random accident at that point that they have chosen this way Once they have chosen this way, then you see They exploit although the next food source that is in the same direction Which is smart enough, right because So once you're made it to here, so why shouldn't you take the benefit from exploiting the next market again? Which is very close to this one This all happens at a time where they already know these sources actually they know all the sources But the fact that they have started to build a route into this direction This fact forced them somehow because it was re-amplified to first exploit this area And only after they found out that they have exploited everything here, then they change as you see here Their business and then go into the other area As you can see here that see then they exploit this other area And then at the very end they are smart enough to leave the whole Southern part of the system and then exploit this food source in the north This was done by agents that have no representation of their environment at all, right But what I want to say here is the fact that there is an early path by one end or two ends that Let them into this area. This was amplified over the course of evolution And therefore they first spend their time here before they went to this area. That's the message here, right? You cannot calculate the advantage of being here first. There is no advantage That was a random event and this was amplified and then of course once you had this Highway built into this area then of course you you use the highway, right? So you do not discard your investment I mean aunts are not talking like this, but we could talk like this in a very anthropomorphic way, right? Okay, you do not discard your investment that you made by building this route No, you first take what you can get from this area, right? So it's quite logic and natural Okay This is as I said called path dependence and we try to understand now how this happens So the first Important feature is initially there is no advantage of either of these that's important If there isn't advantage then of course in particular agents that have a bit smarter representation of the environment Will certainly choose the better the better Alternatives then of course So that means equally everything is this initially everything is almost of equal value So but then we have a small perturbation So you two randomly choose to drive on the right side, right? And then this is amplified Why is it amplified? We got all those people who then choose to drive on the left side. So they are Eliminated from the system, right? So therefore you have a strong amplification effect and at the end everyone drives to the right side There is no not the slightest advantage of doing it. It was simply an initial symmetry break That was re-amplified and the same The opposite happened in in in countries like UK, right so and then because these were the early uh business partners of Japan when you go to Japan and we want to drive them on the left side, right so Like that you're because they first met the people from the UK That was basically the way how this was decided Was not really decided was simply amplified. Yeah, okay The second important ingredient of this is once everyone drives on the right side Or once everyone has bought a Equipment that can read blu-ray disc The system gets locked into this Because of your initial investment you cannot simply change the whole system and say okay from tomorrow on Everyone is using a different standard, right? You cannot do this Because everyone gets used to this and it is prohibitively costly to change the state No one is saying you cannot change the state, right? But it costs a fortune, right? It costs a fortune And this is very important because you Have reached a state at which you cannot afford to change the state anymore This state is again again and again re-amplified, right? That's the consequence of it So and that's called lock-in effect No So in the previous example you have seen already that we somehow understand how social norms emerge based on this lock-in effect There are a few interactions in the beginning these interactions are amplified Other agents are involved the system of or the social network is growing There are more and more links I added to the system and then you see That eventually some social norm emerges That was not decided initially it is simply the result of the whole process And once this norm has emerged so there is a very very difficult situation to change this again, you know Okay, that is the Rational of what we are talking about now So Once the system is locked in this has further consequences Right, so we are just talking about one dynamic process that leads us into a state that we cannot really leave anymore But the fact that we are in that state now Has consequences and here we took examples also from the technical area So once we all decided to have a DOS system on our computers Or once we had computers with a DOS system then people started to write software for these systems, right? So and we got used to these DOS To these microsoft products, right and then it's even harder to to leave it We do not just discard the computer. We also discard everything that It relates to this computer our whole way we process our Our administrative work and these kind of things. That's a very important thing Or the other way once you have decided initially that you bet on the auto mobile And then you create this huge highway system surrounding los angeles, right and after 50 years and you recognize that you probably Had a wrong assessment of the future But there is no way of changing this anymore, right? You can of course you can pay for wrecking all these Highways and building whatever railroads or something like that, of course in principle you could do But you would never get a majority for this or people would feel affected a lot about These kind of changes the whole suburb area suburban area is Created according to the existing transportation system and so on and so on and so on you understand that this has consequences And this means that the initial decision that you made Becomes more and more reinforced on higher levels, right? You can take if you want you can take the whole Euro crisis and talk about this in a similar manner, right? So there were wrong decisions made in the very beginning, right? You had the time to change this You even could afford to change it Whatever reasons to not do it Because you didn't change it. It was we amplified and now it's amplified to a state where you can no longer change it All right, that's the important message here Okay, we would like to understand this in a more formal way of course This reminds on my Core course on systems dynamics and complexity that I teach in the fall who attended this for a few Okay, most of you I can be very good. Did you hear this before about the polio process? Yes, and we were talking about this I thought because the How do we split material between these different courses? I can explain this to you in systems dynamics and complexity. We basically look into non-linear dynamics What we do here in this course collective dynamics of firms and we look into agent based models From a is interacting with from b from a makes a decision from b Grows and so on so these kind of things we look into agents not into the system as a whole macroeconomics would be that The related part of systems dynamics So here we look into agent based dynamics and therefore we have removed everything from past Of past dependence from this course systems dynamics because this refers to agents making decisions Right, so therefore it's now part of this course and no longer part of the other one But it's good to know then I can be a bit more shorter. So Okay, this is a typical way of how to illustrate a positive feedback process to Yeah, people That do not see the same thing from an equation. It's a graphical way of explaining it If you have a stable process then of course if you have a Small fluctuation here. This small fluctuation will not change the position of this ball, right? Because there is a force that brings the ball always to this state why Because it's a stable state The situation is completely different if you have an initial unstable state Then whatever you do The tiniest fluctuation drives you either to the left or to the right side And then you cannot stop the system from evolving either to this or the other side, right? That's basically the equivalent This is a positive feedback process here Yeah, so that means every single deviation from the initial state Is amplified gets larger larger and larger. That is the idea here So we try to model this with the so-called polier process It's named after the mathematician George polier who first thought about this. So everyone is aware of this, right? Who's not aware You are not aware It wasn't okay. Yeah, that's what I said Originally it was taught there, but then we removed it because it's basically an agent based model So that's the idea of Mr. Polier now. Well, you have an urn The polier urn is just an urn and initially You have a black and a white ball in so And then you add constantly balls to the system And you choose between the black and the white balls with a Probability that is proportional to what's already in the system So let's take this initial situation one black and one white ball. What is the probability to choose a black ball? 50 percent. Yeah, one half because there is One over two Now Because 50 percent is a random choice you choose a black ball. So what would be this situation in the next step? You have two black balls and one white ball, right? So that means it's the probability to choose the next black ball is proportional to The frequency of the black balls and the probability that the next ball will be a black ball is Yeah, it's two third Where's this one third right because you have two black balls divided by three Where this one black ball divided by three So okay, and what you see is that this initial tiny fluctuation namely to choose a black instead of white ball Is amplified over the evolution of the system and then you end up in a situation where you can no longer handle Changes the situation in the system. So We refer to this slide because that is your self-study task of this week. You should simulate this polio process So Using our hopefully so the idea here is To use a random realization, you already know how to generate random events and are You choose a random number between zero and one and You have the following rule that's described here You Calculate at every instance in times a fraction of black and white balls in your own that is a small x And then You add a black stone if The random number you have chosen is less than x and the white stone if it's larger than x You can invert this and can do it the other way around if you want. So it doesn't change the result No, but what you have to remember here is that so initially The x is 0.5 So to choose a random number that is above 0.5 is the same has the same probabilities and To choose the random number that's below 0.5, right? so But then as we described before you add a black or white stone and then the x is moving along this axis Because you have either added a black or white stone and that means you decrease or increase the area from which you can Choose a random number, right? Let's assume so the axis evolving Into this direction then of course the probability of choosing A random number that is smaller than x is decreasing all the time Why is it decreasing because the x is decreasing and the x represents the fraction of the Black balls in the system right So and then of course there's nothing left and the probability to choose a white ball is increasing all the time because This is much larger Yeah, you got the point. So that's the result here So from a simulation of mr. Sturman, but you hopefully get the same thing. So you start here with at 0.5 there are two things that you should Recognize from this picture The first one is that these Curves there are 10 simulations here Somehow lock well, so they become straight lines And because this refers to the x it means that the x does not change Right, that's the first thing that you have to recognize. Do you see this? Right, so here initially there are fluctuations, but then it'd be this dumps out and then the x is more or less stable Why is this the case? It's so if time is measured in number of balls added So then let's assume we are here, right proportion of black stones is about whatever 0.005 so it means so if we have a hundred If we have a hundred stones here then five of these Five of these stones are only black and the other 95 are white, right? And now we have 200 then 10 are black and 190 are white and now We add a new stone okay, so then it's Not a hundred ninety divided or no the other way around 10 divided by 200 it's 11 divided by 201 right, so that's about the same right 11 divided by 201 It's basically 10 divided by 10 right that is in the whatever fifth digit after the comma There is a little chain and that's why this gets stationary because the system size has increased to a number Where small changes like adding another ball do not matter anymore and that's the lock in effect That's the first thing you have to recognize the second thing you have to recognize is what who guesses What's the second or what could you recognize? That's true. Yes, the first steps are most important and then there is the third thing to recognize The initial condition is 0.5. So that's true. So that's all the what warhund said. So what do you also recognize? You think of these 10 simulations here? Okay, so we shouldn't We shouldn't They're very different right there. There is not much of a statistics here. That's true If you would do the four thousand simulation, everything would be black here. So that means this process has no preferred stationary value, right So in fact, you can prove mathematically that every single value of these x's is possible You can get any number That's the second important thing. So you lock in to something No But it's completely unclear and Completely unpredictable for the linear process. Which x you choose at the end. That is the important method You can be locked in into any state Right Is it good or bad? What do you think that's correct? Yes That's bad if we if we want to predict something then we better know What the outcome should be and so that's true Right, so that means there is basically an open evolution. All right, so this Gives us our number of possibilities, right We cannot predict the system what we have ample of choices Well, you can see it from both ways And we have just one value to converge right then we are just talking about the time at which we lock in Okay, these are the two important messages I can skip this slide because we just talked about this right It converges to a path dependent equilibrium What value we have at the very end depends on the sequence of how we got there and every new addition to the urn Generates a smaller smaller and smaller impact That is the message of this slide here Okay, we can also skip this slide because we already discussed this In the beginning you have a chance to invert a process Right, that's the first step after the initial conditions the initial condition was one black one white, right? So now we have chosen accidentally a black though then we have two black one white So that means the blacks are dominating What is the probability that we can at this this is step number two, right That we can invert Starting from step number two that we can invert the situation where the blacks are the majority We want to have the white as a majority we can think about this so The first we have to choose more white, right so in the beginning So the probabilities that I choose one white is one third because the blacks are dominating No, two over three versus one over three one third So then I have two whites to black the probability to choose one white is 50 50 to white to black so then I have the whites dominate And then the probability to choose again one white is then three over five so and if you Multiplies this because we are talking about independent effects and the probability to invert a state like this Which is right one step after starting with the game Is only one tenths. That's very small. That means 90 percent of The probability Reinforce the given initial fluctuation and only 10 percent Is the probability to invert this No, so you see it's almost impossible But if you then go for a larger number so then you see it's literally Not happening anymore. Okay so now We come to the non-linear of polio process here the assumption in the linear process if you have to describe this in the exam Then the assumption is that the probability to choose a black or a white stone is proportional To the percentage or the frequency of the black or the white stones in the url, right? That means this is there was a linear dependence p Equals or proportional to x and x is the fraction of these boards Right So what you can think if you want to do an application you can think of all other possibilities here so You can say okay the proportion of choosing a black at the probability of choosing a black stone is just Decreasing with the number of black stones It's proportional to the number of white stones for example, then you would get something like this Which is here called minority voting. No, or you could say, okay I choose a black stone if the proportion is Below 0.5 and if it is above 0.5 I choose whites right So that's you have ample possibilities of setting this relation Between what you see in the system and what is the probability of doing the next step? So and that's called non-linear polio process so and Choosing any form of this proportionality Results in a Considerable difference the linear polio process. What is this? Yeah, the probability of adding a black stone is directly proportional to the existence number of Frequency of black stones And then we remember from this picture that we have discussed that every possible Every outcome is possible, you know Yeah, that means the ball can be here, but although there there was no preferred state, but if I choose any of the linear functions Then this linear Any of these dependencies in a non-linear way. I'm talking about this kappa here, you know Which is by the way the eta there. Oh, alexa. So, okay Then this picture changes So if we have something that's more than proportional for example, so then we have a banded shape like this If it is more than 50 percent then the probability to changes is increasing if it is less than 50 percent the probability of changing of choosing the black stone is Yeah, more than proportionately decreasing. So if we have something like this as a dependency then We end up with only two equilibrium states here Namely zero or one So that means instead of having a million possible outcomes. We only have two possible outcomes So if you want to predict the system as far hand said, then You are much better in tuning the mechanism at which the process is amplified Right, that's what you basically do. You choose a non-linearity. That's in favor of you You are the company a and you have invented The whatever phone, you know the a phone Well, and now you want the market to lock in your phone. You want to be here, right? Then you choose An amplification that is not just proportional to the number of people using the a phone Yeah, but gives you additional incentive. So for example You reduce the price Proportional to the number of people using the phone already, right? That's a non-linear feedback that increases it. Yeah, or you say for every tense Customer, you know buying the a phone. Yeah, I give a set of the a tablet free for free or something like this You know for every fifties or something So you think about ways of twisting these Probability in favor of you and then the non-linear vote. I'm all would predict it You end up all with that, right? This is of course not realistic, but this is what companies are doing. So they try to Choose in non-linearity that allows a super linear more than linear Amplification of the choice of the customer. That's by the way the same as whatever Microsoft does and they say well, we have this nice browser and if you buy the whatever suit Sweet for for office works and you get the browser for free, right? So that's the idea. Okay Where the conflict with the european commission came from and these kind of things you understand that you as a company have a possibility To change this outcome in favor of you But you should be aware that all the other competing companies try to do the same, right? We talk about the limitation afterwards So what you get then Is a much much stronger lock-in effect as you can see here Why do I say it's much stronger because they do not only lock in they also lock in in either of the states So instead of having any possible outcomes. They have only two possible outcomes. That's why I say it's a stronger lock-in effect I hope you got the point, right? Other examples of lock-in effects. So the famous example discussed in the literature. I will be very short. It's a Keyboard I Assume everyone knows what the quality keyboard So people found out that this is a suboptimal way of writing Well, there are other keyboards now keyboards like microsoft natural keyboard of kind of this, you know Just never materialized Even the small iphone now. Yeah, so where you don't have a keyboard anymore this It mutates the quality keyboard the question is why is this like this? That's an early amplification The quality keyboard was chosen at a time Where you had a typewriter. Do you still know what the typewriter is? So okay typewriter is something that you press a and then a little Um, how do you call it? Yeah Yeah, yeah something. How do you call it? Lever a lever calms and makes hits the paper, right? So okay, and then there is a a and the keyboard was chosen in a way that there is no conflict between these Letters moving forward and backwards, right? So people thought about what is a minimal conflict? yeah, so They thought about what is a natural sequence of Of letters in a word and then of course they made sure that these Letters that naturally appear one after the other are not on the same place of the keyboard because then you Had a limited speed of of typing because you have to wait for the letter to move forwards may click Move backwards and then the next comes, right? So that was the optimization criteria. It made perfect sense at that time But in the age of a computer you could have chosen anything, right? But that's a very strong re-amplification effect so and because all these people were trained like this and there were Were education books and courses to to train people like this as it was amplified amplified amplified again without any Rational other than this early one and there are thousands of examples for this No, okay. Now. Let us come to the last five minutes To discuss a small model, which is a bit more realistic, but The message is not Different from what we just discussed Now we have two Different kind of agents here the assumption before was that every agent is similar No, and then gets randomly assigned black and white What so with a certain frequency? But there was no personal preference of the agent the agent was not asked Do you want to do you prefer to be black or white? This was not the case, but now we have Exactly this assumption here. We have the r agent and the s agent you think of black and white again and the r agent has a preference for uh for The product a And the preference for another product b and for the s agents these preferences are differently So precisely, I think it's written here the r agents prefers to have product a and the s agent prefers to have product b All right for natural Reasons you know you understand that is so that means if I let the system run and Have no other effects and I would find a distribution at the end which shows me A product and b products according to the frequency of a r agents and s agents You understand because the r agent prefers the a product and the s agent prefers the b product That would be the outcome of this that means I could from the outset completely predict How the distribution of the two products? A and b would look like So that's described here But now we have another assumption say in addition to Having a personal preference. Let's for the just for the moment. Yeah, so let's talk about the apple iphone and the Samsung iphone right? That's something we could choose. Yeah, just to give you an idea These are the apple fans and these are the samsung fans to share the body Now we have an additional feedback the agent Has not just his own preference His choice also depends on how many other agents in his environment have chosen the apple product or the samsung product Until you see so this agent has a preference for the apple product But you can see if very many people in his environment have chosen product b, which is the samsung phone Then of course the utility he gets from all the choosing a samsung phone Is larger at some point than the utility from choosing an apple phone Why is this not because of his preference? But because so many others did this choice and then he can no longer talk about the The iphone store. He has to talk about the google app store or something like this, right? There is a point of choosing what the other people have chosen because it eases communication And you have always something to talk about right even in our group. So then we talk about our android products, right? So It's very clear if you do not have any preference and you see If if you do not have any Enhancements and you see a distribution according to the initial preferences how you are born and enter the system But the important thing so that's then what you get here Yeah, the equilibrium is point five and then any randomly chosen sequence of either apple agents or samsung agents Changes this equilibrium towards this random number, but it converges basically to point five so now the difference is I do not just look into my preference I also look into what did the other people choose So and then I have to compare these utilities so for agent For the r agent the r agent wanted to choose the apple product right here. So this ar is larger Than the br that was written two slides before That's the samsung product But because more people have chosen the samsung product this utility is larger than this utility And then you can calculate a difference here and a minus nb And if this difference is below a Certain threshold here, which we call Delta r here Then the r agent which the one who prefers the apple product Would choose the samsung product even that he has a natural preference for the apple product You can do the other way around you can calculate a difference at which the preference The number of people using apple product is so large that an agent who naturally prefers to buy a samsung product And has a larger utility from buying a samsung product because of his preference will eventually choose Let me just finish with this. Um, will eventually choose Something Will they choose the apple product, right? So and this is in the this is the slide at which I want to stop actually You see in the beginning We have this kind of random walk process between r and s dominance r means apple product b means B means Samsung product so But you see when this Delta s or delta r are hit these are the two boundaries. That means there is a critical number of agents choosing apple or critical number of agents choosing samsung then the others log in What they locked in There's no way of leaving the state anymore because then you can only after this point You can only amplify the samsung or the apple state, right? so That means in the beginning you have a bias because of the preferences But then it really depends on the sequence how agents enter the market And how many other agents are there and then you see either something ending up here or something ending up Right, so that's again a polio process as we have seen so It's a linear polio process because we have This proportionality to with the number of agents But at the same time it's not a perfectly linear process because we have this preferences added The agents start with the bias, right? So you can also say it's a non-linear process But in a rather complicated manner, right? So and therefore we have this log in into either of these states This is because of the preferences if we drop the preferences You can test this when you do your little exercise at home We drop the preferences and we are back at the linear polio process Where everything is just proportional to the number of choosing apple and something right That means here we have a combination There is this amplification effect, but there is an initial bias. So essentially the equal The possibility of equal States at the end You could choose every say this is broken down in favor of log in either in a and b Of course, this process is not really realistic. Therefore on this slide We discuss a few limitations of this. I want you to read the slide at home It is important to understand that even this picture does not capture a full model of Technology adoption Because there can be other companies entering the markets, yeah? There can be ample ways of influencing your preferences. Yeah, I just lower the price Maybe you change your preference or these kind of things. Yeah, or I have a mobile Vendor behind this, yeah Like sunrise versus swiss com and then you also choose according to what sunrise or swiss com are offering in terms of Hardware right and all these kind of things. So it's important to understand that This is a caricature of an adoption process But at the same time, it's not a complete model of Of Describing Interaction in the market. That's important to understand So with this we stop next week. We don't meet because there is a holiday and we meet in two weeks again You know, so I would send you an email in the meantime because this week The week when we meet again is the last week as far as I understood, right or No, it's a 20. No, we have the 24th and Yeah We have two two lectures though. Okay. Okay. Thank you very much