 Thanks everyone for being here. This is joint work with several friends and co-authors from Toulouse, Doshin, Yassine, Yassine Lee. And we started on this several years ago when I was visiting Toulouse. Again, I'm sad to say it's still preliminary, but I hope that with kids' vaccines over the horizon, I am soon in France finishing this up with them. So to motivate this paper, let me start with an observation. I think you can find many definitions and lots of sort of public interest in the idea of product ecosystems. Many of the big platform sponsors have them, right? Whether it's Apple's collection of iStar devices knit together by their software or the Android ecosystem, sometimes the definitions you find on the web are kind of built around user groups the way we often think of platforms, other times they're sort of built around devices. But I think it's fair to say that most of the multi-sided platform intuition that we bring to thinking about these ecosystems is based on models of platforms with two sides. So here's a collection of sort of seminal papers from the early to mid-2000s that build out that intuition. A second observation, sort of a related but slightly different starting point for this paper, is that the literature on platforms has a great deal to say about the cases of monopoly and competition, but a little less about provision of complementary inputs to a multi-sided platform. And so those two observations sort of beg a series of questions that you might think of as questions about complementary ecosystems. Things like how does the position of a device within an ecosystem, here you think of meaning a product network, how does it influence equilibrium prices and demand? Or what happens to prices and demand when complementary multi-sided platforms serve user groups that completely or partially overlap with one another? And those are going to be the starting points for our paper. We're also going to motivate it with an application and in some sense this is where we really began the process of writing the paper. We're thinking about overlapping complementary platforms and we're thinking about it in the context of licensing and specifically licensing as it applies to the the internet of things. So here's a way to build up the application that we're going to take with us all the way through the presentation today. Here's maybe a very stylized view of the 2G world. There's a monopoly input provider, think of it as Qualcomm. They own a bunch of patents that you need to use for CDNA. They license two downstream goods. So they provide an input into handsets and into network infrastructure and there are demand externalities between these two goods. The better the network, the more demand for devices that can use it, the more devices, the more the incentive to roll out a network that uses them. So this is a really familiar kind of notion of a platform. Now imagine we take this hypothetical monopolist into the world of the internet of things. In the internet of things there's still handsets and infrastructure but these wavy lines represent both the limitations of my ability with PowerPoint and the idea that there's network externalities amongst all of the downstream devices which extend from cars to electric meters to appliances in your kitchen and whatnot. To the extent that they all use the same radio interface, the owner of patents that read on the radio interface can charge a price to the handset provider and another price to the infrastructure provider and another to the cars and the appliances and so forth. We might be interested in how those prices internalize all of the downstream network effects amongst devices. And then lastly, because we're interested in complementary platforms, you might complicate this model in a somewhat realistic way and say, well, what if there are multiple owners of essential patents? So not only do we have Qualcomm but we also have say Ericsson licensing patents that are necessary to use the radio interface for all of these IoT devices. This is the setup that we're going to study in this paper. A set of providers of essential inputs to perfectly competitive set of downstream industries whose products have network externalities amongst them. I'll come back and talk about why this is or is not a great way to think about the real world of standard essential patent licensing but for now let's take the setup as kind of the motivation for the paper. So with that in mind, what do we do? We build first sort of a model of pricing in that kind of a world and the trick behind the paper is we linearize demand to keep the whole thing tractable. And then the first application of this is going to be what I think of as ecosystem pricing. We'll study the case where there's only one upstream input provider who's a monopolist and we show how prices and quantities are related to measures of network centrality. So this is a model of how do you set the prices for your device ecosystem. Then we'll add sort of more platforms and in that world we can study how sort of basically the Cournot problem. So we'll think about Cournot compliments for multi-sided platforms and in particular we'll sort of show that at least for a single device within that network you can kind of overturn the Cournot intuition. You can get prices falling as the number of complementary input providers increases. And then lastly kind of at the edges of what we've gotten finished for this paper we'll talk a bit about partially overlapping platforms. So a world where say you have two upstream providers that overlap on one part of the network but have monopolies on other parts of the the device network and there we'll show how they have incentives to sort of overextract rents from the shared device and that distorts us away from what we think of as the baseline prices in this model. So that's that's where we're going. So let me talk about the setup first the model and then we'll do the monopoly case. The competing or actually not competing complementary platforms case and then overlapping and then conclude. So the setup I think I hope is pretty simple. There's end devices or sides produced by perfectly competitive downstream industries with zero marginal cost. Upstream we have the platforms or the licensors in the general case there's M we'll start with K equals one. Each platform licensor holds essential patents for every device and charges P sub I super K to downstream sector I. So this is the device charged by platform K for inputs to produce device I. All right aggregate demand the total demand for each device is given by this linear demand system. There's an intercept device specific less the sum of the input price charged by all of the platforms that are licensing inputs to that device plus a set of demand externalities from device J to device I. Okay. You can express this in matrix form. Right so if you write it this way the vector of demands equals the device specific intercepts less the prices right here prices is an end device by M platform matrix was multiplied by ones plus the externalities. Okay. And so to make this thing stable so that there's no sort of profit pump kicking around in our demand system we have to assume that the largest eigenvalue of this entire matrix of downstream externalities is less than one and in that case you can write the demand system this way right it's the bring this the externalities over here and then invert this matrix pre multiply by the the linear demand system and there you have it. Okay. And then there's one more thing to to the set up before we get to actual results let me I'm sort of going to pre define the initial notion of centrality and this is something that we borrow from the literature on social networks. All right so Katz Banasic centrality here will denote it by C centrality for Katz Banasic an asset I'm not sure how to pronounce it is equivalent to the identity matrix less this thing a plus a prime over two inverted post multiplied by the intercepts so what is this thing one way to think of it is it is a fixed point or an eigenvector of the average undirected externality matrix so the average undirected externality matrix is just a plus a prime over two it's the inbound externalities from devices j to i plus the outbound externalities from device i to all the j's divided by two so you can think of it as you know sort of as though we don't care what direction the externalities are going in you know sort of we want the average strength of them as between devices i and j right that's what this matrix gives us okay that's a way and the dimensionality of this thing is n by one so every device has a Katz Banasic centrality a different way of thinking about this that you can pull from the literature on networks is that if we take this average undirected externality matrix call it b then a device's Katz Banasic centrality is its own demand intercept plus the value of all k step links to each device weighted by a what that means is you can think of a step as one multiplication by the matrix b right so we say the steps value is the value of the externalities inbound and outbound okay and so you have a one you can get to from one device to any other device by sort of a one step link or by taking two steps on the network or by taking three steps on the network or four out to infinity okay so weighting those steps by the demand the kind of baseline demand for each device that gives you this geometric sequence okay and this geometric sequence will converge if the biggest eigenvector of b is less than one and that's Katz Banasic centrality okay so this is something that shows up in the literature on social networks the literature on pricing on networks it's going we'll use it immediately all right and I think that's all we need for setup let me stop many clarifying questions about the setup okay then first result okay in this world what's the prices charged by an ecosystem monopolist right an ecosystem monopolist the upstream think about the the slide with you know one license or licensing many devices it sets prices equal to this expression one half a plus one quarter a minus a prime post multiplied by the Katz Banasic centrality vector and the equilibrium demand at those prices is one half the Katz Banasic centrality vector okay a bunch of remarks about this then how do we think about it what does it mean there's three parts to each price there's the baseline okay one half a that's exactly what you would get in the simplest you know monopoly linear demand no network effects kind of model okay this vector of prices then there's two pieces that are really familiar going on in the deviations or distortions away from the baseline price one is value extraction okay so one quarter times this a matrix which you can think of as an upward shift in the demand for each device because of the network effects from the other devices in the ecosystem feeding back into it right post multiplied by centrality and then there's an externality internalization that's a mouthful an internalization effect which is minus one quarter a prime right so you subsidize devices if they produce a lot of outbound network externalities that increase demands for other goods in the downstream ecosystem and again post multiplied by the centrality measure okay so that's basically the decomposition of prices first result I mean this shows up in the the various papers on two-sided platforms or you know kind of a next next comment rather than first result is that symmetry means we just get the baseline prices right so if the externality matrix of the ecosystem that characterizes the ecosystem is symmetric equilibrium prices are the same as you'd get in a model of independent goods linear demand monopoly prices second we go into some detail about this in the paper there's a link between what we do and Armstrong's model in 2006 right so Armstrong characterizes these distortions here in terms of these externality weights multiplied by demands or in fact utilities if you go all the way back to kind of principles in his paper you can see right away that our paper if demand is the same thing as one half of centrality okay substitute this into here this is the expression in our paper for the distortions away from the baseline price okay so our model of ecosystem pricing is essentially Armstrong we just express everything in terms of primitives here as opposed to endogenous care you know features of the environment like the demand for the other goods okay I think I've already said this but you know there are there are links that we build out in the paper to a literature on pricing on a network or social networks and games and lastly to sort of try and think about what in the real world this may give us some intuition for I think it's interesting to think about what might be central devices in a in a product ecosystem right maybe it's phones or things you know I'm interested in echo or nest these things that Amazon and Apple Google may be selling us to sort of try and sit in the midst of all the other devices and connect them together okay so let me now sort of provide three examples so we get a feel for how network structure relates to pricing all three examples are going to have the following kind of common features the externalities are going to just take one of three between any pair of devices will take one of three values new data or zero general think of new bigger than data we'll set all of the device specific intercepts to one and we'll define this term D is equal to the diff the positive difference between new and data okay so first example will be a star right what does a star network look like sort of graphically you can think of it like this there's one device in the middle that produces the big dark arrow externalities to n peripherals and each of the n peripherals produces small externalities back onto the star device i equals one and then there's no relationship amongst the star these two themselves okay in terms of the matrix there's a column of news here and there's a row of it eight is here and everything else is a block of zeros so that's a star network what are the prices you get from a star so for the star good you can see as long as D is bigger than zero you subsidize it and you subsidize it more the more peripherals you have and for the peripherals you can see we charge above the baseline monopoly price of one half and the extent of the markup you know kind of the rent extraction that you take on each of the peripherals depends on both D which is the this difference between our two parameters new and and the cat's banash its centrality of the star device team uh team before you move on uh david has a question uh it's in the chat with baby david can unmute and just ask about the central devices yeah it seems at least in the telecom application the central device might be the the network access like yeah there's any iot and to this example with the subsidy the hand it might be a model that's sort of another model explaining why people give away the handsets to that and an interpretation because the handsets provide a lot of create a lot of application usage on the network yeah i that sounds good to me um i i'll be you know to be honest i find it not a priori easy to know what this the central device is which is something we should think about a bit but i you know i kind of i think that interpretation sounds plausible um okay let me let me return to a couple more examples just to sort of give intuition here's a here's a different structure okay so in this structure there's n devices and they exist in sort of a hierarchy right so device number one produces the most outbound externalities okay device number two produces the next most outbound device number you know all the way down to n which produces the weakest outbound externalities and gets therefore the most inbound in matrix form it looks like this right so the first row the outbound externalities from good one to goods two through n um is a row of eight us then the second row has a mu and then some eight us and the third row has a couple news and then some eight us okay i think i said it backwards um but the math is right okay in this example one thing to notice is that um a a plus a prime is symmetric so that the cat's been asked its centrality of every device is equal right asymmetry and pricing comes only through the difference between a and a prime and not through centrality okay and what do you get out of it so you can derive the expression for each goods its price is one half minus d over four times this expression times centrality and this expression is um negative if n plus one is less than two so we subsidize the devices that produce a lot of outbound externalities and we exploit or raise the prices of those that produce few outbound externalities relative to inbound okay and uh every device gets a different price and then a last example this one i don't i can't think of a real world network that would look like this but i think it's useful for intuition imagine a ring imagine you know sort of a set of devices that are that are linked and sort of a ring structure where each one has one inbound um externality of size eta and one outbound externality of size mu and then you see these guys in the corner here make it a circle okay that brings us right back to sort of the baseline prices right so if the externalities are you know sort of if if there is a symmetry here turns out this is a ring type of a symmetry as opposed to a reflective type of a symmetry we often just get the baseline prices okay so let me you know what do we take away in general from these examples i think um we see this familiar trade-off between external extraction of rents and internalization of externalities that are that are there and two-sided pricing kind of models but we see the structure of the ecosystem in terms of externalities between devices matters through centrality um nevertheless we still seem to sort of the general intuition that you want to subsidize devices that produce a lot of positive externalities under other devices that that intuition still seems to hold and then in general symmetry sense pushes us back towards these the kind of baseline world okay so that's compliment you know so that's the baseline set up that's a monopolist now let's think about complementary platforms right remember this world this was our motivating example right so now we're going to we're in a world where there's just one upstream provider licensing or setting prices for end sides that were interrelated now let's add this second upstream provider okay and i promised i would say a few things about actual whether this actually applies to licensing um i think there's a tradition of papers that take a Cournot compliments type view of standard essential patent licensing there's good reasons for this one is that you know sort of if these firms own patents that cover sort of a radio which is sort of an ip that reads on many devices and they have to be used and all of the devices are going to communicate with each other then in some ex post sense all of those patents are known to be that they are complementary okay but there's also some reasons to be kind of cautious about pushing this line of thinking too hard in the real world most of these standard essential patents are subject to commitments to license at fair and reasonable rates which many people think is less than what could be charged by an unconstrained monopolist and they also say that the rates will be non-discriminatory which means that there might be sort of constraints on your ability to set the price for one device differently than the price for another device and you know sort of even if we did think that there were the patent provided like an unconstrained monopoly i think there's legitimate questions about whether the price itself is something that's negotiated subject to threat of an injunction on downstream devices which might give us towards this sort of unconstrained world versus reasonable rates that would be set by a judge and then who knows where these prices come from so all of this is to say we're going to contribute to a literature that views licensing of essential patents as a corno compliments problem but with being sort of cautious about taking our model to literally for a way that these kinds of prices are set in the real world all right so to get to what the model says we need to adapt our definition of cat's benesity centrality right so it turns out that the the network used to compute centrality when you have multiple upstream platforms is this you know so instead of a plus a prime over two right we have this expression that puts more weight on the outbound externalities associated with each device okay and here's how it factors into the symmetric equilibrium prices right so define lambda here is one over m plus one right in a symmetric equilibrium every platform is going to charge this expression lambda times a small a plus a second lambda times this difference which is exactly the same as in the monopoly expression times centrality defined according to this different matrix which puts more weight on outbound externalities than inbound externalities okay a couple of remarks about this right so these baseline prices lambda a that's the that's the solution to corno from 1838 okay that's the baseline that you get without any of this downstream externalities just a corno compliments okay oops the weight right in this expression you can think of these as weights m over m plus one and on this matrix one over m plus one the weight on outbound externalities here is in the centrality matrix is going to increase as you add more upstream monopolists right so the more severe the corno problem the more weight we put on outbound externalities I think the intuition for that is that the many monopolies problem leads to dissipation of inbound externalities right what does that mean inbound externalities you can think of them as sort of raising the intercept of the demand curve for any individual monopolist right so the output of other goods in the ecosystem is translated into more demand for the focal good okay but the more monopolist you put into the system right they're charging a rent themselves on each of the devices in the ecosystem and that's a downward shift in the intercept of the demand system right so the many margins problem um is going to dissipate you know sort of the inbound externalities on here right these are the ones that cause you to raise your price right and so our notion of centrality is going to put more and more weight on outbound externalities as in uh in the way we compute centrality okay but but for that you know so you can see that this is exactly the same expression we had for the monopolist but for two pieces right one is this corno term and one is the different notion of centrality all right extending the the comparison to corno a little further right we can say but the aggregate baseline prices right this is kind of the corno result m times lambda times a right this is what we would have absent these are the prices absent any downstream network externalities this expression increases in m right this is basically this part is m over n plus one which is increasing in m okay then an interesting question is for a particular device right somewhere in the downstream ecosystem could you flip that result right could you have some you know some device that gets cheaper uh as we add more upstream monopolists turns out the answer is yes okay and to to show that we use an example of what we call an augmented star network okay so this is it's a star network because we're going to have one device that sits in the middle and generates outbound and inbound externalities is in the prior star example but we're going to augment it by we're going to sort of increase demand for the star relative to everything else right we're going to set its intercept to beta bigger than one okay so theorem three in the paper core knot is that for the augmented star network when beta exceeds this value five thirds times root n minus one the total price of a peripheral device is smaller when there are two upstream monopolist than when there is one the intuition for this is the following right with this kind of augmented star network what the monopolist does is subsidize the star and extract rents on all of the peripherals we saw that earlier when we talked about the star network for a monopolist right they raised the price of all of the peripherals above one half and they dropped the price on the star to sort of generate aggregate externalities right when we go to two upstream monopolies you can think of both of those things moving towards the middle okay right so it's going to reduce its subsidy to the star device and it's going to reduce its rent extraction on the peripherals and for these parameter values that reduction in rent extraction will end up being larger than the increase in price associated with the basic multi margins problem right the corner effect okay and then again to link this back to applications i think it creates an interesting question think about internet of things patent pools right now we're we're starting to see patent pools forming up to license say cars and the next thing will be you know appliances or electric you know whatever right if you take the phones as already said new pools are forming to license different kinds of peripherals i think it's an interesting question for policy whether we should think about you know sort of platform pricing incentives in antitrust review of those pools just a clarifying question a couple of times you you mentioned when we have two monopolies so so just to clarify when you say two monopolies you mean like chrono with two firms right i mean chrono i mean input they each have them their monopolies in the input market not downstream throughout the the ecosystem throughout this paper is provided by perfectly competitive zero marginal cost producers of each device okay yeah because two monopolies is just something that doesn't often come up yeah i it sounds like i'm using terms that are oxymoronic throughout i hear you okay last this is sort of the last example kind of question that we deal with in the paper a bit thus far symmetry sort of both you know mainly symmetry on the network effects but also sort of symmetry amongst the upstream monopolists has tended to push us towards baseline pricing right so the bait where baseline is what you would get in either independent monopoly pricing of each downstream device or in a world which is equivalent to corno you know so the simplest linear demand corno what happens if we break the symmetry by allowing the upstream platforms to have partial overlap that is you know sort of they each have an input that they monopolize and you know sort of two inputs are required for some of the devices but only one input for others okay so consider a model where there's two upstream input monopolists and three downstream devices okay each of the upstream input monopolists is a true monopolist downstream on one device but they overlap on the other right i'm reaching for an example that i think is getting used in too many papers these days but think about apple and epic they both charge for gains but they seem to have exclusivity in other things right so epic has this graphics engine that apple doesn't sell and it licenses it to developers apple makes a phone or app store or whatever piece of the ecosystem you want to say you know pick and and they don't overlap in those areas right so that sets up the following kind of a problem right so there's a star device both firms charge for the star right um then each has its monopoly over one peripheral okay um i don't know if andre is on here but um another motivating example for this is a hbs case he wrote about a company called gree which i teach and i find pretty interesting i think gaming is a setting where you might find these kinds of partial overlap um right do you want to comment i see you unmuted i mean i can only say yes that is true um so in this example right just to simplify things let's let's actually make this symmetric even let's let's set mu equal to eta okay and what are the equilibrium prices it turns out for the star device the equilibrium prices are one-third which is the corno baseline price plus an additional markup right so that on on device number one the star we charge something above the corno baseline and then on these non-overlapping goods right we each charge what we would you know sort of the monopoly baseline um for a you know a good without a corno problem we charge a half okay so what's the intuition for this what's going on neither platform fully internalizes the externalities from subsidizing the star device okay we don't subsidize in some sense device number one sufficiently because when platform one lowers the price on the star device it's producing some externalities for platform two and conversely when platform two subsidizes the star device it's producing some externalities for platform one that don't get internalized okay so in some sense platform this is a in this example we see how partial overlap kind of exacerbates the double marginalization problem here right the aggregate price for the the star device where we overlap is even larger than it would be in a corno world without these network effects downstream okay and with that may conclude i think i got one minute left what we try to do in this paper is build a tractable model of complementary ecosystems with multiple upstream input monopolists and multiple downstream devices that are interrelated to each other through network effects and ask about how prices relate to not only the magnitude but sort of the structure of the downstream network externalities and out of that comes this sort of prominence of cats been asked such centrality and then we use it to to ask questions about how platform pricing interacts with double marginalization problems a lot of the tradeoffs that you see in the baseline monopoly case are familiar from the existing literature on multi-sided platforms i think that's good but again sort of i think our contribution is to create this link to centrality and i think when we get to a world with complementary platforms we have some interesting things to say about how you can get sort of perverse examples if you're thinking you know sort of that the corno problem always predominates right in particular we show that you can get prices for one side in any case declining with the number of upstream monopolist liars of inputs or you can show we can show how partial overlapped leads to partial internalization of externalities even when you know kind of the network effects are symmetric and so you can kind of get exacerbation of the corno problem as we saw in that last example and that's it i think i'm more or less on time and looking forward to the discussion thank you very much team this is this is excellently right on time so thanks for for measuring time so exactly now we'll have a alex white discussing before we open it up for general discussion alex okay uh can you hear me yes thank you very much are you are you sharing slides or not no slides okay no slides thank you very much to the organizers and tim when tim and i first discussed this talk about a month ago we had a little conversation about the paper he was talking about some issues of patent licensing and i asked him some questions and it turned out i was all wrong and then i looked at the paper and there's a lot of matrix algebra which i'm not very good at so it's the whole thing seemed at first kind of like oh jeez what am i going to do with this um and so i'm really pleasantly surprised having spent some time with the paper because in my opinion this is a beautiful simple model that illuminates the theory of i o and quite a fundamental way um and so you know i learned a lot but i plan to continue studying this paper and hope to you know expect to learn to learn more um just to put it in context um you know you start off with basic oligopoly theory corno the corno substitutes versus corno compliments and you know if you do a good job you study these two models and understand the the driving forces behind each one that leads uh when you go from monopoly in the direction of substitutes it leads prices to go down and when you go in the monopoly from from monopoly in the direction of compliments it leads prices to go up and as tim alluded to we've been spending years you know decades a couple of decades extending the sort of oligopoly model to the case where there are network effects and um summarizing sort of what we know about all the network effect competition with substitutes when you increase the number of firms on the one hand that causes market power of firms to go down so that pushes prices down but on the other hand there's a tendency under monopoly to to provide users with a discount to discount prices in order to internalize externalities and as you um add more substitute competitors the incentive to provide this externality discount goes down it kind of gets the discount gets divided roughly speaking by the number of um competitors um and so in particular there's a recent paper by Tan and Joe in the restud that um illustrates this point very very cleanly and one of the things that the Tan and Joe papers are their main result is the possibility of what they call the perverse pattern where if you take the price of equilibrium prices in a um in an oligopoly model with network effects and you increase the number of competing firms you might get prices to go up because the um reduction in the externality discount resulting from additional competition may outweigh the reduction in market power that comes from um additional competition and so what you know what you can see this model is doing is looking at the same question but going in the direction of compliments um at a very high level they're saying when you go from having a monopoly and Tim is using the word I think it would be clearer to use the word like components producers of multiple components or something like that rather than saying two monopolist three monopolist when you go from having centralized control with a single monopoly platform to decentralize control of the components in terms of pricing of course we get the one effect of double marginalization pushing prices up but we also get a stronger effect for prices to go down due to the externalities so as you increase the competitors the strength with which price uh externalities drive the prices down goes up and so it really has this very nice um almost symmetric relationship to the world of substitutes and that's sort of a good way to think about it at a high level um and to sort of convey how fundamental this is but at the same time it's a lot more it's a lot subtler and more interesting than just that I think Tim mentioned that the key for them to be able to move forward from a technical perspective is to linearize the demand um and by doing that they can even in the monopoly case so usually when you have a setup like this under monopoly taking uh maximizing with respect to prices doesn't really get you anywhere it's not tractable but under this linear setup it becomes tractable and they get this very nice decomposition of the network effects into um outbound and inbound externalities so that in and of itself seems to me is kind of as a bit of a contribution it's not really a new solution to a problem but it's a different it's a different view on a solution to the monopoly problem where there's this decomposition between outbound and inbound effects which people have been aware of but it's really the linearization of the problem that makes um that that kind of gets you over the over the hump and seeing what's going on and the key insight you get there um well okay so so then you go you go from that basic setup in monopoly to the compliments world and then you you see that the um the due to this decomposition when you decentralize control from monopoly to multiple compliment producers the force of the the relative force of outbound externalities increases compared to the force of inbound externalities so the importance of outbound externalities grows with the number of producers but the importance of inbound externalities stays the same so you get this nice um increase in the um in the weight of the externalities that's proportional to the number of producers which is exactly the mirror image of what we see in the substitutes in the substitutes world where where you also get something that's proportional to the number of producers but the effect is getting smaller and smaller so I'm very excited to try to understand these different forces better um they have nice examples um comparing the star network where you tend to you know assume positive externalities the hub you have this hub and spokes you can think about that way right so the in the star network the hub tends to get um discounted and the spokes are the the cash cows or they're the they're the ones that the sellers try to exploit um and that that all makes sense that's kind of a reality check whereas in the ring structure what they find is that the prices don't change compared to the no externalities case um a couple of questions minor questions there um is is that result about the prices not changing um purely due to linearity that would be interesting I suspect if you had a non-linear world you'd still get the prices the same the symmetric across the different um devices but that they wouldn't necessarily stay at the one half level so these are types of things that'd be interesting as robustness checks if possible but I understand that you have to be linearizing in order to make it tractable um but I think the sort of final bigger point I would make is can you go for a bit of a deeper result in terms of the um comparative statics on prices as you increase the number of complement producers because you're going for this thing where you where you looked for the possibility of a price drop for one of the devices um and you know a nice thing to notice about this is that we actually do have variable usage here right so in the tan and joe substitutes model in order to make the uh the model tractable they have to assume full market coverage so they can look at prices going up and down but no matter what happens to prices it's always you know a measure the same exact set of consumers who are consuming the goods but here in this complement setting you get for free the fact that as prices change for each device the usage goes up and down so could you come up with a relevant measure of aggregate usage and um you know convince come up with a convincing explanation or convincing justification for why this particular measure of aggregate usage is the right one or a relevant one and then look for conditions under which um as you add um go go for monopoly towards more complement producers the aggregate usage goes up as a result of this strengthened externality effect something like that would really i think make this paper extremely strong and very important you know it would help help um help build the the contribution of this paper so i probably henna i probably used up my time so well twice but uh but i'm sorry if you're if you're wrapping up you know i think it was worth it yes okay well thank you thank you very much um so i really enjoyed it