 So good morning everyone. I welcome you to lecture number 10 of our course Collective Dynamics of Firms I'm happy to see so many here, so we reached the critical math for all of those who are not here and Just watch the video stream Before I forget it would like to make two announcements the first is regarding this course of course the mTech has a Evaluation of this course and this will take place on the Thing that was the 18th of May About three weeks or the 15th of May something like this and It's important that we have some statistics, you know, so we can certainly deal with small numbers, but Yeah, it has to be a bit sufficient so those of you who are interested to Evaluate this course you're welcome then to join the next two lectures Where we have to pass on these forms and you have to file fill this in I will also give this to Pavlin Though in the exercise you also have the possibility to do so The second announcement is about a talk in the etihad risk center seminar series Some of you probably know that I gave a talk last week On Tuesday about systemic risk some of you were there. I think you were there, right? So and Next week we have an invited speaker Professor Sydney Redner from Boston University Person I invited to Visit it a hat sir and he will talk about statistical physics of citations This came to my mind because Vahan is writing his semester thesis on a related topic actually So for the physicists that certainly of interest now for the others you think about statistics, right? Not statistical physics. You just think about statistics of citation And it's quite surprising to see that of course if you look into publication data You see the same kind of regularities that we see here with the firm data There's not much of a difference and even the the formal models Are more or less the same. This is all multiplicative growth and these kind of things Yeah, you find power laws and block normal distributions and so on it's just another topic But there are similar techniques And so those of you who are interested it's a Tuesday at five o'clock One floor down in D1 to well we if you remind me Alex we can also send out the announcement for this okay, but this I come to the lecture of Today, so this is the outline of the course as far as we came First we talked about empirics and data to learn about the phenomena We would like to describe these were called stylized facts Then we talked about agent-based models that were simple models. We have set up to capture the stylized facts that we have observed in the Data We started with the most simple Models that were stochastic Growth models of non-interacting firms and then step by step in each of the lectures We added a bit more of complexity to capture with some facts that have not been Well, we presented by the models and You remember that we in particular talked about indirect Interactions of firms that was interaction through the market that was this advanced model of Free market capitalism Natalia found a few mistake in the slide so she made an updated version just for your information, right? And So we double-checked that all the equations were okay And during the last lecture we were talking about the redistribution of wealth at another Indirect interaction mechanism so the government collects some Proportion of your revenue in form of a tax and then it's redistributing this evenly And this is another kind of indirect interaction and those of you who followed the hand out and the recording of last year You found then that we are able to reproduce some skew distributions We are not able to do this ourselves, but we tried to Simply believe in what our Japanese colleagues did Today we come to a very different topic Namely direct interaction so far these firms never directly interacted but indirectly interacted and We would like to include in this formal setup now direct interaction and the example that we take is Knowledge transfer Some of you who takes a course on economic networks will See a slide overlap with the economic network Lectures who's taking the course economic network one two, okay This is only a slide overlap and the economic lecture and the economic network course We go much more into the details here. We talk about simple models only and I would like That you understand The kind of argumentation and the phenomena that we observe rather than doing some sort of mathematics That's what we do in the course economic networks So let's come back to our modeling framework. It's the same as we did before so you have a firm characterized by X X is a proxy for the size and later will be a proxy for the knowledge stock of the firm and we asked ourselves what governs the Change of the size of the knowledge stock of the firm and that has to be included in this non-linear function F so Remember the different levels of assumptions that we made for this F function first There were no interaction Then there was the interaction via the mean if you think of examples and please all the think of the model of Egery and Simon there was also an indirect Coupling towards the mean because the growth rates depended on the average growth of the industry or on the average growth of the Economy as a whole right they made these different factors there today we Specified this function in a different way we have now a dependence of their function directly on X i and X j What does this mean? It means that the firms does not grow by itself This was one of the assumptions of the G broad model right we didn't talk about the reason why the firm grows We simply assumed it grows and here We specify the condition it grows because it interacts with another firm J and this firm J may have a patent or access to some Special resources that firm I can use in order to growth. That's the basic idea So let us come to the modeling framework. We called it here cooperation by the way Alex There are two mistakes on the front Page right so if you can remind me on this and we fix it afterwards one issue just for your Information if we are no longer using these email addresses that are printed on the first slide all the time We have disabled these addresses you can communicate with us via Moodle as you did all the time, right? Okay, we call this a modeling framework for cooperation because it basically means that the firms have to cooperate Well in some sense the proxy X is for the knowledge stock as I mentioned before If you want to have an idea then it's a number of patents So you are a specific company that doesn't have a hot-selling product But you have about 15,000 patents that can be used by another company Strengthen the strategic position in the market. Yeah, not to use 15,000 patents not at all But to scare the competitors, right? You know a few recent examples of these kind of Indirections the knowledge growth of which is now for the the X variable is assumed To depend on two terms Which are not really specified here. There are benefits B and there are costs Of course, there has to be a benefit term Why should you interact with other firms if there is no benefit, right? But on the other hand you should not assume that there is no cost involved You know if you think of the famous time like Chrysler case something right then the cost for collaborating Between firms for collaboration between firms was very different internal culture and setup was extremely high and underestimated Right, that's a problem here. So the benefit and the cost depend on The adjacency matrix. What is this? This describes the interaction at some point I have to describe with whom firm I Interacts and we do this in this little setup. I took this graph here from one of our early papers and The corresponding adjacency matrix, so let's look here you see from I Form one exchange knowledge was firm to But with no one else. So then you see here in the row Now in the column, sorry, you see here that from one indirect with firm two, right? There's a one and there are zeros of about from two indirects with firm three and with form four So you see three and four. They are this year one. That means the adjacency matrix Contains all the information With whom from I interact that is the important message. That means to be later on can tweet this as a set of dynamic equations and the adjacency matrix simply describes the interaction Benefits in terms of knowledge transfer are called knowledge spillovers in Economic models, so that means there is something more that you get by interacting with the firms You get access to new design features. You get access to new product features for example, you get access to a different kind of Workforce for example at this firm specialized on particular issues and has very skilled workers on a particular setup for example to develop some Drugs for the pharmaceutical industry then by interacting with this firm to also get access to the Skills of the workforce right and the cost of collaborations. We will list a few later on But it's important to have this in mind From the dynamic equations you immediately see that firm I would only interact with any other firm If the benefits are larger than the cost right otherwise it would face a decrease in the Important variable which is a knowledge stock here. That means it invests more in the collaborations and it gains It's a very clear thing and this is called strategic interaction So that means you as an agent Calculate what are my benefits? What are my costs and then you make a strategic decision I interact with this guy, but not with that guy. That's the idea behind this Okay, look into this Bit more in detail So we have in mind applications towards innovation networks of course firm can Invent themselves and can create innovation themselves without Collaborating with other firms that happened in the past and still happens But on the other hand, you know from other course in particular the m-tech students that this is not the end of the story To have an innovation is one thing to establish this innovation somewhere on the market complete difference That means if you have the pattern for whatever end to interface in Communication, let's assume you you are the inventor of Blu-ray standard or something of the 8 HD no, so then Of course You have to make sure that most other firms share the standard Right. Otherwise, you will not really gain a lot of market share if the other firms follow other Standard so what you do in such a situation you vary early on Make strategic interaction with these firms in order to make sure that they are involved in The development of this standard and the related patents and use them afterwards That's the idea behind it, right and You and in most firms are not a single firm has all the knowledge as I as I mentioned before that Is needed to develop a particular pattern and it's not although a good idea to have everything in house It's in most cases much better to Collaborate with other firms and when once a product is developed or once you change the focus of your production Then you have the chance to simply cut this Interaction with the other firm you do not lead to lay off 1,000 people that you have hired Two years ago to develop a particular thing, right? So you see there is an idea behind this That's a innovation network as it was plotted here in the Nemo project Well, I give and hint here so the different colors Identify different types of Actors in this innovation network. There are firms. There are universities There are other government agencies So now let's come to the model in particular I Said that now the Interaction function depends on x J x k and so we could also assume it depends on x I If it depends on x I then we are back in the square where we talk about the multiplicative growth for example, right? So I want to separate these two effects therefore, I today only talk about The dependence on the other firms I could as I said Although assume it depends on my own knowledge stock Yeah, but for the modeling purpose to show how much is the growth due to interaction and not to my self you know Because I want to separate these effects. I assume that f only depends on other firms and Then we can have this equation here as the most simple assumption So I wanted to use this feature today. So do you like this or not? I have seen that even in the video you can see the red laser point, right? But only if you know that I use that Very interesting and and this can be surely seen. I assume right so So the first and most simple assumption is It depends on other firms x j This is these are the entries of the adjacency matrix that describe with whom I interact and then instead of having costs directly Involved in the interaction, I assume there are some general costs They are expressed in this term with the phi That means there is a constant Now cost proportional to my own growth if I interact with you, so then I have to increase my Administration in the company I have to Have an office that's related to Capture all the contracts that describe the interaction with you and to checks is and to Archive this and so on and so on you can assume that we can capture the cost by this Drostrum here. It's not a cost specifically related to my interaction with you It's a general cost related to the fact that I interact with other people and therefore grow All right, and the more I grow that means the more I interact with other people or with other firms The higher is this cost so you understand what the meaning of this In a fizz yeah, please Yeah No, it simply means that the growth of my knowledge stock Depends on your knowledge stock. It's proportional to your knowledge stock if you are the firm with the 1000 patterns and I interact with you then I grow more when If you are the firm with a one pattern, right, that's the idea here So but the important assumption here is it does not depend on my knowledge stock and I explained why I have chosen This assumption to simplify the problem and to really look into what's the meaning of the interaction here so For the adjacency matrix we can in the simplest case assume a random network a random network is If you have the nodes which represents the firms and you draw random lines with a certain probability you choose To firms and then with a probability P which is maybe 0.2 you draw a line between these firms right and then you get a random network and If you know the P That's maybe the M or the P is important Variable here if you know the P that means the probabilities that you and I have a link then I can Calculate the number of firms on average Because there are n-minus other firms in the system and with each of them I probe a law I probe a link which is accepted with a probability P Then the M is the average number of links okay We can assume that this costs or Delusion flux can be set to zero. I mean the phi is an arbitrary constant So we can for simplicity set this to zero what we want so and then We see that this is nothing but exponential growth Where here the a is the adjacency matrix and the x0 is The knowledge stock of all these firms that have been connected to me initially That means there right this is a vector here and then we can calculate how the knowledge stock of all the firms Develop over time But what we do is we transfer this equation into a market share equation by simply using relative variables then instead of having this simple Expression we get a more complicated expression here But this is due to the fact that I go to for relative variables And that the market share is constrained to one the sum is constrained to one right you can do this little exercise at home To transfer equation one to this equation if you want no But it's a very simple way of getting this Okay, now We do some computer simulations to better understand how the Firms evolve and we Makes the following important assumptions There are two time scales in the system. There is one time scale at which the firm Increases its knowledge stock Dx after dt and there is a second time skill at which the firm decides With whom to interact We have a random Link together will be interact. I calculate my knowledge stock then I find out This is not sufficient My costs are higher than my benefits for example here There are no cost direct cost as I said, but that's irrational. Well, and then I Basically cut the link and try with someone else. That's the idea Well, that means there are two time skills one time skill at which the DX after dt relaxes to a Stationary state and then another time skill at which this stationary state is perturbed by Changing the interaction pattern No, this is a basic idea. Did you understand? What I just said that's important So we have this coupled set of equations X I Dx I after dt then we calculate given the net fixed network structure What is the growth of the knowledge stock then we end up in stationary values, right? That's relaxes as I've shown on the last slide so and then Something is changed That's the interaction pattern instead of interacting with you in the next round or interact with you That means some of the links are revived That's the idea. We have a specific assumption about the reviring mechanism. We rank the firms So, you know already from the last lecture about inequality and the Lawrence corporate ranking of firms means so we start with the best performing firm which has the highest X I and then Order them towards the least performing firms There's always the best firm and there is the second best and so on and then we decide that the least performing firm That's removed from the system. That's our idea. So we remove this firm because it's not well performing and we replace this firm by a A new firm that means in this model there's entry and exit dynamics The exit is for only for one firm That's the least performing firm and in order to keep the number of firms constant We have a new firm added to the system and how does this new firm interact with the existing firms? By a random wiring to existing firms With the probability P the new firm gets links assigned to the existing firms. That's the idea of this model There are any questions about the model. It's a very simple model and now We can ask the following question What's the likelihood that I am hit By the remover that I am the least performing firm in this simulation That's an important question or you can put it the other way around. What's the likelihood that I'm always One of the good firms well performing firms And the second question is that the whole pattern converges to some Yeah favorite network topology Remember the network topology represents the interaction patterns you can ask is there an optimal interaction of the firms? That's a question No Okay, so we do it very simple here. We look into some Computer simulations we start at t equals zero We have I think 100 firms these firms are Randomly connected to other firms. We calculate their Xi We look which one is the least performing firms. That's removed a New firm is put into the system that gets a number of this previous firm and this Randomly we wired to the existing set of firms and Then we Recalculate What is the knowledge stock in the next film if there was nothing about you? You have not been Included in the reviring mechanism then of course your knowledge stock doesn't change in the next step, right? So only for those where some changes Occured the knowledge stock would change and this is an arbitrary snapshot So what do we see? First of all we see a number of firms that are completely isolated here, right? They do not interact with anyone so That means Their knowledge stock is probably very low, right? Why are they still in the system? I mean Right, right. These are the least performing firms, but thanks God They are not the only ones, right? So that means every time where someone is chosen for removal There is plenty of choice, right? Okay, that's the first thing then you see these clusters here that emerge And then you also see that there are long change. So 84 depends on the growth or 84 Not depends but Supports the growth of knowledge stock of 50 and so on there are these chains Let's go to another arbitrary time step. So I have marked a number of firms in red Why did I do this in order to show you that there is a cycle actually there are two cycles in this picture? 21 37 or so I cannot read this here So, okay, you I do not read need to read some numbers, but you see them, right? So they form a cycle That is a very important issue If you are the firm 21 Why should you support firm number 17? 37 sorry in Growing its knowledge stock, right? Because there is a random link, but okay, why should you do this, right? As if you think of a rational decision We can also go back to this Little pictures that we had before here. So you see there are only indirect Interactions firm one supports the growth of firm two, but from two does not support the growth of from one, right? So why should firm one? Support the growth of from two. What's the reason for this? Is there any incentive to do this? That's a question here, right? So and this is answered in This picture here these red firms form a cycle. You see step by step though This support this firm that support that firm that firm and so on so and 49 supports 21 So that means even if 21 has nothing to get from 37 It still contributes to the knowledge growth of 37 because we are all part of a cycle, right? and therefore It gets from another firm which it's not directly supporting that's the most important thing Do you understand this so and then there is a second cycle which you hopefully see it's between 49 and what's this eight? Yeah, so They mutually benefit each other and now you see that These firms always get or gain in knowledge growth Because they are part of a cycle that means they support others But at the same time they get through the cycle support from others, right? And if we now look for the least performing firms The red ones are certainly not the least performing firms, right? There's always someone in the system that is much worse in Performance than the red firms. Do you understand this very important? We call this indirect cooperation Because there was no direct Interaction between all these firms or in indirect reciprocity Direct reciprocity is a trivial thing. Yeah, I benefit you because you benefit me, right? That's direct reciprocity Indirect reciprocity is much much more difficult to explain. I support you Despite the fact that I never get anything big from you. Why do I do this? That's a important question in economics. It's a question in game theory, but also in social sciences Why should I support you and Here you see the answer. I support you because I get indirectly From us being part of a no society group Cycle here in this network terms. That's all. That's the reason Okay, and now these guys are quite safe whenever they Least performing agent is chosen No, they are fine. They are not selected. So you see This is another snapshot. What do you? Recognize that the cycle is growing Right, the red the number of red firms connected among each other is growing Where are they growing? Because there is this random reviving after every time step. That means a new firm by chance has a Link to the cycle or another cycle Appears just by chance, right? And if this cycle ever appears All the firms that are part of the cycle benefit a lot from this indirect reciprocity You understand what I mean though and therefore They will no longer be removed from the system because they are the well-performing agents, right? So and as long as there is our white firms in the system There's plenty of choice to remove these right Because they are always less performing than the red ones and therefore of course The cycles are growing all the time Because the white firms are removed New firms come into the system are randomly rewired to the system connections to other firms and then may either form a new cycle or Become part of a cycle. That's the idea So we see a core of cooperating firms and then we also see this here. We see this parasitic Periphery, so why are these guys there? Let's look here at 97 52 and 94 why are they there right we got they benefit from the fact that 13 is doing so well, right? That's the idea. They never contribute anything to 13 But thanks to the fact that 13 benefits 97 and 97 has a lot of growth from this benefit 42 and 94 all the benefit But to a less and less and less degree that means there is a point in time with the whole thing Well, these firms will be removed and now let's assume without looking into into Further snapshots what will happen in the next time step? Or not the next time thought in some time step in the future Who can who of you can imagine how the whole game will end? Mm-hmm. Yeah, there are only firms in one or two courts It doesn't necessarily have to be one core, but at the end we have a situation where we have all the red firms only right and That's the most important day right because then we look for the least-performing firm And the least-performing firm now for the first time is a red Not a white because the whites are all gone We pick one of the red firms and we remove it from the system. What do we do by this? We destroy the cycle right? Because the red firm had a connection Yeah, incoming connection benefited from another firm and gave something to another firm And that firm is removed so that means the whole cycle collapses, right? You understand why right? That means they let's put it like this the worst case scenario is not that we feed a parasitic periphery, right? We can afford this If we are the red firm, you know, we can afford feeding the parasitic periphery The worst case scenario is if there is no parasitic periphery anymore because then it's on us Yeah, to be removed. That's the problem Right, and that's the same as you if you think of this competition model of indirect competition model of free capitalism No, as long as there was a least-performing firm in the system, right? This was fine your Fitness was above the average because there were guys in the system that were not performing well Right the problem just occurs if these guys are no longer there because then you are the least-performing firm, right? Therefore, you understand that the example plays a huge role here It's not bad to have other least-performing firms in the system because this improves your ranking No We see here in this Graph is indeed what we have just described. We see that The networks or m remember is the average number of links, you know It's scaled by the total number of firms So that's network time here and then you see that a network is growing Let's look for the first for the blue curve here So the network is growing here and then at some certain point you see crashes These crashes occur because the cycle was hit by the selection process. There were no Parasitic firm anymore no white firm There were only red firm and they were taken out and therefore the whole thing collapses The shocks are more severe as you can see if the link density is not very high That's very understandable if you have a sparse network if you have only a few interactions Then of course, you have not many cores You have maybe one core or two core and if these are destroyed and you see a real drop down, right? So whereas here the shocks are not very severe because there are because of the high link densities a high number of interactions You have always some firms to interact with right? You I hope you understand the meaning of this This is our baseline model The baseline model was taken from prebiotic evolution We were actually not talking about firms, but about catalytic cycles and prebiotic evolution That's where the model was developed for but it describes some things that we can well understand for the rest of the lecture we do nothing to then making this physics or Biologic model an economic model I mean we go now and add step-by-step things that are important to understand to become an economic model We extend now the basic model as I said so The first and most important thing is if you compare this model that I just described in detail with an economic model is This is assumption of random links, right? That's the first thing that the economist tells you okay There is no random linkage to other firms a firm Carefully decides with whom to interact and that's the first thing we have to put into the model, right? It's very clear so The second thing is if we do this we should not assume that the firm knows all other firms in the system, right? That's very important. So the firms have maybe some Finded information only so that's called bounded rationale Let's see that we get this into this model and the third assumption is We should not avoid the cost right if firms decide to interact with someone then there are costs involved It's not like on Facebook where you have two thousand friends Yeah, why do you have two thousand friends because it doesn't cost you anything to have two thousand friends, right? If you really had to communicate with them, yeah in a pair-wise interaction You would very soon realize that the cost outperform the benefits, right? But because this is not implemented therefore It's not bad, but in a real economy. It's important that You consider the cost of interaction with a particular firm Okay, so now let us go for this. There's a bit of mass, but you should not be scared Let us drop first For a moment the assumption of the cost then the equation looks like this We already discussed this and from this bearer equation Or we have no cost involved we can Calculate a gross return That's simply dividing X dot which is a growth rate by the size of the firm and We can prove That for this gross return We can estimate this by the largest Eigenvalue of the adjacency matrix that's something we capture in detail in The economic network lecture, but not here You simply assume this. How can I know about how can I know about the? Gross return I can proxy it by the largest Eigenvalue of the adjacency matrix. That's also called for benios per wrong Eigenvalue, right? Therefore the pf there so we continue Right after the break of ten minutes. So what is cool about? this equation if We can estimate the Gross return by the largest Eigenvalue of the adjacency matrix It's cool that we do not need to solve dynamic equations, right? So we have a good proxy and that's sufficient We are in economics here and no longer in I don't know non-linear dynamics So that means people are not really interested in solving this equation They're interested in proving something about that's a different thing so Okay, then This is the approximation for the costs if we go This I'm sorry. This is the approximation for the benefits Yeah, the gross return now. Let us introduce the costs to the to the model here so we can introduce the Net return which is a small R and that's a benefit minus the cost What are the costs the costs are what I have to invest in order to keep the links active? Each link has the same cost. That's the assumption here and K. I Is the number of links that I have to maintain? and if I Because I do not want to solve this equation here in time I use the approximation for the asymptotic limit namely that the our eye is estimated by The largest eigenvalue here and these are the costs This holds on the long run For asymptotic times, right, but that's enough for what we want to do with this We are not as I said interested in these intermediate steps So we now define a performance measure which is called pi here and Pi is the social welfare. We come to the Definition of the social welfare here. So the social welfare is basically the sum over all Benefits minus all of the costs That's the idea. So and we call in Network efficient if the social welfare that means This difference between cost and benefit is maximized This is a specific notion of the term efficient efficient is not what you and I think efficient would mean efficient is a firmly defined term and economics the network is called efficient if it maximizes the social welfare All right, it's something you would Maybe dub with another term. Therefore, it's important that you understand that that's the meaning of efficiency And now we would like to find out what we can learn about the social welfare So now there is the difference between let's say a corner physics and economics In a corner physics, you probably would have started by simulating these equations for a number of firms and looking into how they Everett link density evolves and all these sort of things right exactly as you have seen in the different snapshots But this is not what we do in economics and not what we're interested in What we do now is we prove a number of things about this equation And we are in particular interested in the fact what's the role of the cost here That is the difference to the baseline model if you recall what so is there a cost That is critical for forming a network for example Do we see that no network evolves if you reach a critical cost for example days these are questions So and here as I said, I'm not interested in Doing the proofs with you if you want you can go to the papers and can look them up You also find some in the literature folder here or we what I present here are just the results and Remember that we are interested in the cost and we didn't do any simulation We just proved and this is also the argument why we have chosen linear our cost Because with linear cost you can prove everything We first started with non linear cost and we could not prove everything and Then the paper was also not accepted the physician and the economics Jonas What was Pardon me at the moment. It's linear Yeah, of course, of course. Yeah, so we started with non linear functions and we come later to quadratic cost as well But that was not accepted because you could not prove about it, so You see method first, right? So we just better use the not so realistic assumption But to prove everything then using the realistic assumption and not proving everything, right? That's a trade-off now. I mentioned this in particular for the three physicists, right? So there's that's a difference basically Okay, this is what we have proven The most important Thing is that you understand that there are critical values of the sea you can see this in The same way as we talk about phase transitions or bifurcations in non linear dynamics, right? so If the sea is this is a normalized cost if the sea is larger than one in a normalized way then The efficient network is the empty graph remember what the efficient network is that is the network That maximizes the social welfare and you'll be also know precisely how the network looks like there is no network, right? That's the first answer then the second Important proof of is the sea is less than one the network is connected Okay, this means there is one component It's not the case that we have three different components for example that are separate. There's one component and then The third important threshold involved here if the sea is less than point five The efficient network is a fully connected graph Everyone understands what the fully connected graph is everyone has a link to any everyone else right and then there are a few specific proofs for What happens between one half and one below one half we know fully connected network What is between one half and one so there are specific? mixtures of hops and Clicks so these are structures in the network and we are able to provide a formal algorithm That constructs the efficient network. That's the meaning. I think we captured this in the lecture economic network So I skip it here. What this tells you is we know a an algorithm that constructs the Efficient network for us if the sea is in between Which is also very nice. That means we do not really really need to do computer simulation We can write down in a formal way how the network will be constructed and looks like so Okay Though the first general insight is very obvious the higher the cost the sparse as a network Right below one half fully connected above one nothing That's very clear The second nice thing also from the viewpoint of the economist is There is always an efficient network Even if it doesn't look like what we assumed particularly at the empty network, right? In particular, this is important here between one half and one. We know how the efficient network looks like Okay These are the proofs and now let us go to the computer simulation and let's see if we find these structures that we have just Proven to exist This is a different thing. I want you to really understand the difference here What we did is we have proven the existence of a state But we have not proven that we reach this state. These are two different things, right? So think about the paradise, right? So we know that the paradise exists Or we cannot find any way from where we are now to the paradise. This is a complete different thing, right? And We are addressing this question now How does it look like if we try to get to the efficient network where we already have proven how the network looks like? so And we do this here with the simulation we start with an empty graph so and then We have the same assumption of the separation of the time scales as before we take two we take two two agents and Then we change something about the link structure We let the whole system relax into a quasi stationary state then we change the network Let it relax again and again. So that's the idea so and For the perturbation of the network we have now a few more refined assumption Remember what the economists have criticized about the baseline model the baseline model We did everything with random choice. Yeah, that was a random link and then someone was taken from the system and Firm was replacing this lease performing agent and was randomly reviled to the rest of the system and so on This was a criticism therefore. We start now with the empty network. So that means the number of links are growing all the time We do not change the number of nodes as you see, right? So we just we have a hundred nodes. Let's assume and then we choose two agents and we think about Whether these agents link or not and we have two different mechanisms. So which looks like a Roman to here means II incremental improvement No, I I mean and best response. What do we mean by incremental improvement? So we take two agents a and j every of these agents Calculates the net return Remember what the net return was. I think it was written on the previous slide Benefit minus cost right for himself and then if We find out that you and I are both better off by taking by keeping the links and we accept to have the link If we find out that I am better off, but you are not We are not accepting the link. I would probably accept the link, but you not right because why should you do it? That's the idea of incremental improvement, right? You understand this So then the second just to contrast this procedure is called best response I and J are taken and Look into the possibility to have a link But before I and J do any make any decision They calculate what will happen if instead of J I take a link to K or to M or to N or to any other agent in the system So that means the agent before committing himself to this link makes a huge Calculation by checking every other possibility in the system. That's called best response and only after this very Strong procedure it turns out that this is the best choice I could have then I take the link if it turns out that this is not the best choice. I Would not accept the link, right? I cannot link it to someone else, but I would not accept this You understand the difference, right? One is I just check the link between you and me and the other one is I check every single link in the system So the criticism is really obvious You have to know in best response. What are the other who are the other agents in the system? You have to do a very computationally Costly procedure to find out what would be the case if I were to link to you and so on, right? Okay, so and then we stop if we do not find any better improvement in the system Which means for incremental improvement bilaterally stable network That means for for any two links. We do not find a better solution and For best response, it's a Nash Equilibre. I'm just mentioning this to tell you that's a different There's a difference in this. So now let's look into the average return That's a performance criterion Over the time time measured in networks update and then you see so this is the cost here That in best response you reach about 45 here. So what's the meaning of 45? 45 means So we have 50 agents you know 50 agents and we have normalized the whole thing in such a way that So the maximum return would be 50 or the 50 agents If we are linked to all other agents, of course, we have some costs, right? 50 times 0.01 is Five right now that means it's somewhere around 45 That's the best response that sounds good So that means we find what supposed to be from a rough calculation the optimal the optimal Revenue, right? So and what is with the best response with the incremental improvement? It's less than this Aha, so why is it less because these other agents they have checked all possibilities, right? Well, these agents just took the first offer and Took it after they found out that that's not worse than what they have right now, right? So but now comes the interesting thing here. This is what you would expect, right? So you think more information is always better Yeah, I go to the media mark and I inform myself about a camera So but then before I buy this in the media mark I go to all these other shops and compare the price and then I go to the online shops and compare the price And then I find the best offer right so you think more information is better So and this seems to confirm this picture But only if the costs are very low the cost increase you see there's a picture in words So the best response agent Do not Get higher revenue. That's what I wrote here above a critical C Incremental improvement is better and that's something we would like to understand now So we do this by looking into these graphs so remember What we have proven before I just gave you the results if the C was less than point five What do we get? So the fully connected network, right? That was the proof And it pretty much looks like a fully connected network. It's a cost is very low Remember best response even finds this network, right? But what happens if we increase the cost? So let's take point two What did we prove? There should be a fully connected network, right? And what do the stupid agent do? So they do not find the fully connected network, right? Instead what they find as an equilibrium network this one That means they do not find the efficient network for interaction They find another network and that's even an equilibrium network, but it is not efficient Efficient means it maximizes the social welfare And this does not maximize them social welfare This is simply the best they could get under the current condition, right? The maximum is somewhere else probably here, right? That is what you have to have in mind to interpret this result. So why don't these agents find it? That's the question now We have proven how it should look like and these agents by making strategic decisions do not find it What is the reason? The reason is in the past dependence of the way they create their interaction networks Remember I'm not sure how much you're familiar with Evolutionary optimization or these kind of things probably a bit, right? Okay for For the given cost the structure of the Landscape looks a bit like this, you know Okay, that's the structure of the landscape. So we have proven that this is the best, yeah fully connected network That's what we know. Okay, well, we start somewhere here Right, so and now you are agent I and you are agent K You accept This because you think okay Now I have nothing then I have something so let's accept this link, right and then By accepting this decision you end up in this state That is an equilibrium state, but it is not the best equilibrium that exists, right? So but from this state you can never leave Right, that's the problem and even worse for the best response case Because in the best response case after you got into one of these you calculate all the other possibilities And then you find out well These are all worse than what I have now This is understandable because you see it takes a long long way to get out of this hole And then into this here, right? So that means you have to accept in order to find the paradise you have to accept a huge detour of States that are worse than what you have now, right? Understand what I mean. So that is a very important Insult and because you are the best response agent and you have checked all possibilities. You will never ever accept any Detouration yeah, anything that is Worth than what you have now That means the best response agent is even more trapped than the incremental improvement agent The incremental improvement agent by chance may find a way out because by putting you into my network and by accepting a link I of course change the global structure for me It's beneficial, but I could probably make a jump from here to there, right? That's possible for incremental improvement, but it's not possible for best response Then you end up here where you are maybe very close to to the optimal that's the idea So I wanted to understand this So that means a historic evolution What we have observed here like we start with an empty network and we let the network grow until it finds a Equilibrium state so if you consider this evolution we create a path dependent and that means that we get stuck in Equilibrium state that are not optimal or they are called here No, it's written here inefficient equilibrium Or suboptimal states The evolutionary optimization techniques of course deal a lot with how you can get off these kind of states, right? So but here because we are thinking of rational agents agents or bounded rational agents agents that Carefully decide about having a link or not No, you never leave to be irrational. Yes, of course this This is very important So and you see here we have still proven the fully connected network is the efficient one, right? And now you see what they did, please Mm-hmm. Mm-hmm Yes This is true. This would be let's call the second order rationality, right? So But Rational means in the sense it's used in economics means not that you have Higher knowledge about global states. This is not the assumption Rational agent is an assumption. It's an agent that has full knowledge about His decisions and the consequences and the decisions of the other and their consequences So that means I could calculate if I think about our interaction I could assume a mindset where I think like you and then argue about your consequences This is the assumption. That means I could Take your mind and see the system and I could take my mind and see the system But it does not mean rationality mindset that I'm Godfather looking into the system and see how these stupid guys by their full rationality Have a global state emerging, right? That's something you cannot really predict No, because that's a collective effect This is something that that is not not part of your of your strategic thinking Understand what I mean. So even if you calculate your own utility for a longer time horizon No, you may not see this because that is something that is not It's not the property that associates your other agents the property that is associated to the whole system, right? But that's a it's a tricky point actually, yeah There is another there is an emerging state that is not Likely foreseen by the agents because the agents only argue on the level of the agents They know about your decisions and their Consequences or they can assume this so but they they only argue about your utility function Not the utility function of the system. You understand that's a difference. This is not additive Think about public good games are these kind of things, right? So public good games. So we all make the wrong decision. Yeah Right being free riders and not cooperators, right? That's the situation of the public good game Whereas if you would so that's a rational decision We have our two choices and we then we see okay. This choice cost me more than another one So therefore I take the other one right which is free riding that means I'm rational in my decision But I'm not counting in that the whole system including myself in the next on the upper next time set becomes Is in a worse state than it is now? That's a different thing You know rationality always refers to the decisions that are made on the agent level Based on the utility function on the agent level No, that's important, but it's a good point. Okay. Now. Let's move on a bit to incorporate other Things so we have talked about costs now But we have not talked about breaking up links That's something we didn't discuss yet or did I include this? Let me just check No so Now let us assume that we want to break up our Link because we find out that we should better invest into something else and then We have a cost involved so here we see incremental improvement is for the Where we are creating links again both of us benefit, but now if after some time if We evaluate our personal network if you find out there is a Link that is no longer beneficial for me Then I am allowed to cancel this link and I do this decision myself here Both of us in incremental improvement have to agree to create the link So that means both of us have to benefit from it, but here it's a different way It's sufficient that one of us has Less Benefit from this link think of relationships. Yeah You can break up your marriage if you want so you do not need the agreement of other your wife for example, right? So but the assumption here is that of course you do not get your full investment back You have invested time you have invested Knowledge for example in the case of knowledge transfer that you have transferred to the agent and so on so Only a part of this and we measure this by a So-called severance cost and here you just have to keep in mind Alpha equal norm Zero means nothing we get back and alpha equal one means We get back our full investment then you can assume how the network would look like Does someone have a guess how the network would look like for these two extreme cases? Who has a clue? What about the upper rows there? What would the network look like if you get nothing back by breaking up And of course, it's not rational to break up, right? true Because you lose a huge investment that you made over the years, right? instead of you have if you have a That was this case a full loss of them if you have no loss of investment Then you can all the time change your relationship to other companies with no cost and you will reach a more efficient state Much more easier because all the time you evaluate your links, right? Okay, here we look into this This is the case where you do not get back anything, right? And then you see that you have this very sparse structure of the network and So they're not very many links But the agents also do not really collaborate. I mean if you see this here, then you can imagine that from these Cycle structures in particular look at this cluster. You get a lot of benefit because these agents fully They collaborate in a very nice way, right? So but there are other structures like these is per peripheric parasites for example That are not Cut off Why are they not cut off? Well, because of course you you lose your investment though then at some point It's much cheaper to just let them run and Stay in the network. So and now we increase the alpha to 1 Right, so then you see that the network becomes much denser Why does it become much denser because the agents have more possibilities to rewire the link and to find better possibilities, right? These other agents get stuck very early in the previous picture in this picture. They become much much more Yeah flu flu and in reviring the links and Here you see the most optimal structure That they can get remember what we have discussed before in Efficient network and these kind of models should include a fully connected network Remember this? So if you look here, then you see these are all fully connected networks That means they they are close to be efficient network the real efficient network for this case would be which one Yeah, but we have one globally connected where everyone is globally connected This is no longer possible because of this incremental improvement, right? So these agents because they are already connected to all their neighbors Have no reason to break one link up and to revire it to someone else, right? So because it's always beneficial for them to keep these links No, so therefore we see these clusters of fully connected networks here if there are no severance cost so If there are again if there are no costs it means the agents can try more Because there is no consequence in first probing this partner then probing that partner then probing that partner and so on and so on There's no cost involved. Therefore the agent can try better And find the more optimal structure We still understand that this is not the most optimal structure. It's a global optimal, but it's very close to it Right at least locally we see these patterns. You understand if I instead increase This severance cost so and we remember that in this model alpha equal zero was the Loss of the full investment then you have this picture, right a very sparse network, which is completely suboptimal But the agents Do not change it because they feel the loss of what they have invested Okay Now let us move in the last five minutes to what's non linear costs As we already discussed the non-linear case is the more realistic one But on the other hand the one where you can Not prove everything in the nice way as this was possible for the previous two cases What we did now is we included The dynamics looks a bit differently. We have a dissipation term Here that is That already appeared in the baseline model if you remember, right? There was some dissipation the more you grow the more you all the lose because of your Organizational cost for example, then you have this benefit Term which is the same as we have discussed before and you recognize that it only depends on the other agents Knowledge stock if you gain Otherwise not and then you have this additional term of Costs that goes with x square What's the difference between these costs and these costs though these are costs that? appear for interaction Right, so it's proportional to a IG if you do not interact and you do not have this cost this cost You always have this is let's assume that's a tax you have to pay or something like this It's proportional to your knowledge Yeah, and this only appears for the interaction and we do now the following we have two different strategies for the link creation and For the link deletion There is the unilateral link deletion and creation that means only I decide about this I I Evaluate you as a potential partner, and then I create a link If it's beneficial, and if I find out that's not beneficial I cut the link, right? Or the other one is I evaluate this possibility But you all the evaluators possibility and only if the two of us gain as in the case of incremental improvement We create a link and only if the two of us lose we break up. That's the idea. Yeah, okay and So the assumption then is that of course we do not have all the information about all the other agents in the system This is also important So let us now look into some computer simulation this computer simulation you have already seen Here that's the baseline case This is the result of the baseline model that we have assumed in the baseline model. Remember we had a core of core operating agents There can be more than one course if possible Yeah, and then we have this periphery of agents that simply are Parasitic well that benefit from the existence of the core and we use this what was the outcome of the baseline model as the starting point for our quadratic cost model So and this is what you get here so so here was the assumption that we have a What's called here random unilateral link creation? That means I randomly create a link to you but I Then check all my links and if the link is not optimal I cut it again You understand so that means I'm have potentially everyone in the system as a cooperation partner But I also check whether this cooperation is useful or beneficial for me and if not I cut the link That's the idea So we start here and that's what we get at the end so Yeah Yes, this is again with the two times scales so I Calculate my acts right everyone calculates his knowledge stock and then After this we change the network and then we pick you as an agent and you evaluate your link structure, right? No, do you do it? You do it the other way around so you create a link the system runs you check was So and then you are picked for checking your links, right? And then you find out okay some of the links didn't gain give me that much what I was expecting Right and then you delete one of your links. That's the idea. So that means the link creation and the link deletion is Basically in between there is the knowledge growth because you need to get the chance to find out Whether this was a beneficial or not, right? You can assume that these agents are all Capable of doing this in their mind, but the assumption is a different one, right? There's a state where we randomly revire links to other agents Then we let the knowledge stock growth and then we check okay Who actually contributed to my knowledge stock and who not and then we make a link deletion So and this is the result of it so result is not surprising, right? because We see that only those links survive, but we have a bilateral connection If you do not benefit to my growth and I I will cut the link But the other way around it's also true if I do not benefit for your growth to your growth You will also cut the link so at the end only those links survive which by chance are bilateral links All right, so in a in this random creation of our a few events that Created bilateral links and they will survive right so in the other case You think about the cycle right so the cycle was also beneficial for us But I cannot know that the cycle is beneficial for us I just look at you and then I see you do not contribute to my growth right so and therefore I cut the link I'm stupid idiot. Yeah, so because by this cutting the link I also killed the cycle right but that's something I cannot know Because I don't have a global picture. That is what I said on the previous slide Right and therefore these these cycles do not exist anymore, but only bilateral links And free riders of course are completely isolated here, right because everyone checks that he doesn't get anything from these Okay, now let us move to the second Example just speed up a bit so Now we assume that we have random bilateral link creation and optimal bilateral link deletion bilateral different from unilateral means I And you have to agree on this not just me and Then you see we start with the previous picture from the previous slide the end state Where we had this bilateral interactions all the time and then we see that what is the result of this assumption? Only a fully connected cluster appear Yeah, so that means so those guys who were free riders in the first instance of they don't count anymore, right? so and those guys Who have some connections to neighbors? Merge together in a cluster where they all connect Because this is from here to there you understand that this is more beneficial right Where as in these other Interactions, there is no chance of being more beneficial. That's a rational decision that they then form one cluster Okay, and now the last case is the one where we wait the links So we say okay, if there is this additional term now with benefits That are Associated to shorter paths. I can have access to your patents through ten other firms But maybe I have access to you directly or only through one other firm then we give these Shorter access to knowledge the higher weight That's the assumption here, right? So that's the first Assumption and the second assumption is that if we recognize if we recognize, let me just finish with this if we recognize that these New links Contribute to a cycle If we can foresee this then we give it a higher weight because we can have this indirect benefit as we learned already And then you see from this starting point. We end up of course in something that has cycles, right? It's not very much surprising because we were able to foresee this possibility and therefore gave more Attention and more weight to these decisions, right? That was the idea Okay, so with these Simulations I want to close so we have Seen a different viewpoint here Today we took the picture of firms directing in Directing interaction interacting with each other that was a viewpoint of today instead of indirect interaction We started with a very simple model wall where this Interaction was randomly assigned that was our baseline model But then the increase the complexity of the model by assuming why is a firm interacting with another firm and there were these different Benefits and the different costs involved that is what we did today. So this is called strategic decision No or strategic network formation because the firm has a certain strategy to follow These are that alternative So we talked about then this Optimal structure and why the firms do not find this optimal structure even that we have proven that this optimal structure should exist Then there are a few challenges involved I mean we are we're talking about only small networks today Are we able by using these models to explain for example large R&D networks? That's one question. I mean you understood that the firms do local decisions But if I look into a macroscopic picture, am I able to use these very simple Individual models when agent makes regular decision to get a macroscopic structure. That's one challenge So and then of course we would like to see more realistic structures and we would like to link this to observations So the two slides that follow here are just copied from from a talk of mine But I talked about financial networks and economic networks. So they are just as appetizers for your Interest in economic networks. Oh remember that we use this that we Teacher's course and parallel so okay So this is for you you have to create this baseline model to understand how it works What it's very easy and these are the questions you should be able to answer the questions I'm not interested in any proof of something like this, but you should at least understand these kind of things for example Why does the network breaks down in the baseline case? No, why is it that from a random network with cycles? I end up only in this bilateral Interactions or why is it that from the bilateral interactions? I get a fully connected cluster. What has changed in the incentives? Remember that the benefits they define an incentive structure for the firm and dependent on the And dependent on the net benefit that means benefits minus cost I can somehow structure the network right as you have seen I get different types of networks by tuning the incentives and You should understand what has changed in the incentive structure that we get from this picture to the other picture So thank you very much for your attention