 So, good morning, good afternoon and good evening, depending where you are. Welcome to another online platform seminar. Today we have Lester Chen presenting his very interesting research on equilibrium selection in platform competition. But before I let Chen start, let me just quickly remind about how this new format works. So we have an hour, 40 to 45 minutes is for Lester's talk and then the rest for Q&A. During the talk Lester is going to make a couple of stops for clarifying questions. Those questions you can write them in chat and I will ask them in bulk. For the Q&A at the end, you can just unmute yourself and ask a question. So with that Lester, the screen is yours. You can now start sharing the slides and take it out. Yeah, sure. So let me first share the screen. So let me see just a moment. So can you see my cursor moving? Yes. Oh, great. So thank you Hannah for the introduction. And I also want to thank the organizers for giving me this opportunity to present my paper. And once again, Lester Chen and I'm a job market candidate from Boston University. And today I'm going to present one of my paper, which is about two side markets. So one of the most difficult challenges in two side markets as emphasized by Kyle's and Julian's seminal paper is the chicken and egg problem. In short, typically there are positive cross-size network effects in two side markets, which leads to the multiple equilibria issue. So for the typical market model, state one, and all agents simultaneously make their own and suppose for simplicity that from the same side are identical. Then there can be two equilibria in stage two. All agents join the platform, or no one joins the platform. If they're competing platforms in equilibrium, all agents will coordinate on one of the platforms if the network effects are strong enough. But which platform will they coordinate on? In other words, how should we deal with multiple equilibria? So in the literature, researchers impose various selection criteria to single else and equilibrium. So a popular selection criterion is to select the Pareto dominance equilibrium. However, this criterion often fails under platform competition because coordinating on one of the platform need not Pareto dominates the other. And there are other selection criteria in the literature and I'm not going to go through them one by one. But each selection criterion only works for some cases, but not the others. Moreover, different equilibrium selection criteria often leads to different predictions and implications. And therefore, there's a methodological challenge in selecting a suitable equilibrium. So in response to this challenge and this presentation, I'm going to propose using another approach called the potential game approach to resolve the multiple equilibria issue in two side markets. And as we will see, this approach has so less micro foundations in the game theory literature, wisely supported by experimental results, and it can select a unique equilibrium for many two side market models. So the concept of potential games was formalized by Mondra and Sheppley. And in short, a game is a potential game if it is strategically equivalent to an identical interest game. And first, let us take a look on this example in which the two players simultaneously decides to join platform A or platform B. So coordinating on either platform is an equilibrium. But player one prefers to coordinate on A, while player two prefers to coordinate on B. So clearly, Pareto dominance is not applicable. And in fact, none of the population criteria in the literature can unambiguously select a unique equilibrium for this model. But now let me show you a selection criteria that works. So first of all, observe that this game is strategically equivalent to the following identical interest game in which the two players shares the same utility function. So I just need one number for each cell. So to see this, suppose player two joins A. For player one, joining is better than joining B by three units in the original game and by nine units in the identical interest game. Similarly, if player two joins B, for player one, joining B is better than joining A by one unit in the original game and by three units in the identical interest game. See that the change in player one's payoff from switching actions is proportional to the corresponding change in the identical interest game. And the same logic applies to player two. And therefore, these two games are strategically equivalent. So any game that is strategically equivalent to an identical interest game is called a potential game. And the corresponding common utility function is called the potential function. Moreover, for any identical interest games, generically, there's a unique Pareto dominance equilibrium, which corresponds to the maximizer of the potential function. And for this example, it is both players joining A. As proposed by Mondra and Sheppley, the selection criteria based on potential games is to first transform a game into an identical interest game. And then select the corresponding Pareto dominance equilibrium. And it is called potential maximization. So let me give you an intuition why potential maximization makes sense. Suppose the two players indeed played this identical interest game. Then we probably would expect that they are more likely to coordinate on this Pareto dominance equilibrium. And if we also expect that two strategically equivalent games should have similar strategic behavior, then we should expect that they also coordinate on A in the original game. And in fact, for two player-to-action games, potential maximization always selects the risk dominance equilibrium. So in the game theory literature, many equilibrium selection criteria coincides with potential maximization if a game is a potential game. So as mentioned, risk dominance coincides with potential maximization for two-by-two games. And for more general games, the unique equilibrium under global game selection, perfect foresight dynamics, and log linear dynamics is the potential maximizer. And even without relying on other selection criteria, the potential maximizer itself is also robust to incomplete information. Moreover, potential maximization is wisely supported by experimental results. Therefore, this selection criterion is justified by many micro foundations and experimental results. Although potential maximization is applicable only to potential games, many two-side market models are indeed potential games. So for example, for the four most cited papers in two-side markets, all of their main models are weighted potential games. And therefore, we can apply potential maximization to resolve the multiple equilibrium issue. And in what else, I will apply potential maximization to a special case of Armstrong's model in which agents from the same sites are identical. And I'm going to derive some novel insight into two-side markets. So maybe I can pause here to see if there are any questions. There are no questions in the chat, but if anyone... Is it possible to ask a question? Yes, go ahead. My question is this. You said that there are many experimental results that support the choice of equilibrium. Can you say something about them? Sure. So basically, these experiments are lab experiments, and then they usually will play some kind of coordination game. So I guess all this research is comparing Pareto dominance and risk dominance in two-by-two games. So those are the experiments. But actually, for more general games, like ours of two-by-two is not ethical. And one of the extensions is potential maximization. And then people find that, like, times will co-ordinate on the potential maximization. I ask because in my experience, typically in coordination games, when you take them to the lab, you don't get clear results. And none of the equilibrium selection criteria that we use in theory actually has any predictive power in the lab. That's my experience. I see. I see. I think that's actually true. I mean, in mixed results. So sometimes you can find that it supports potential maximization. In other contexts, maybe other equilibrium. But at least we have some research that supports potential maximization. And maybe there are also other literature that supports that in other environments. All right. Maybe I can use the model. So in the model, the multi-platform size of agents, side-one agents and end-to-side-two agents. So the payoff of a side-one agent from joining the platform depends also on joining the platform. Measures the per-interact benefit with each participant set by the platform. The payoff of a side-two agent takes a similar form. And if an agent does not have the platform, his payoff is equal to his pro-platform. C and C2 are the marginal cost of an agent. And the timing is standard. The platform prices stage one, and all agents simultaneously make their joining decisions. Interested in the sub-game perfect equilibria of this game. So now analyze this model. And first, that's by the platform are positive or too high. I think we lost for a moment and lost the slide. Welcome back. When do I get lost? I think we're fine on the setup. You run triple D and maybe we lost a couple of seconds, but I think we're good. Maybe I will repeat this slide. So basically, when the prices set by the platforms are both positive and not too high, there will be two equilibria in stage two. So all agents join the platform or not one platform. And under Pareto dominance, all agents will join the developer equilibria. And therefore, under Pareto dominance, the optimal pricing strategy for the platform is to set the highest possible prices of size. So all agents will join the platform in state focus. And therefore, the platform's equilibrium profit is given by this. And actually, for this, we verify that Pareto domination proven that insulating tariffs and for quality with the monoclip platform. So in other words, these are the typical selection criteria. So now let us enter the same model. So first of all, we show that every second in stage two is a weighted potential game with the potential function given by this. And the proof is simply to verify that the function find here is indeed potential function. So to see this, for size one agent, suppose there are n1 size one participants and n2 size two participants. I'm sorry to interrupt you, but there is a clarifying question about a previous slide and I think it may be worthwhile. Previous, even earlier. Yeah, platform optimal pricing P1 is equal to V1 and two and why is it V1 and two and not V1 and one. Because so you can see that's the payoff here. So basically the payoff of a size one agent depends on the number of size two agents. And if everyone, every size two agent joins, then it is V1 and two clear. Michael does that does answer your question. You're muted. I guess into that's not the total number of agents is it that are joining platform one. So small letter and two is those who joined the platform and capital letter and two is those with the agent. And in the first equilibrium, which is all agents joined the platform will small letter and two becomes capital letter and two. What about capital letter and one. I mean, how what is the total number of agents available. Total number of site one agents is capital and one total number of site two agents is. I see, I see those are the different sites. Okay, thanks. Now I'm not confused. No problem. Yeah, welcome. Okay, so let us continue the proof of this lemma by showing that every sub game. Indeed, a way to potential game and the proof again is to verify that this function fine. Indeed, the potential function. So for a size one agent. Part of this and pay off. And join the platform or not for simply you want. And if these sites are one agent joins the platform, there will be M one plus one site one participants. And if not, there will only be M one site one participants. Therefore, the corresponding difference in fine is given at this. And with some simple occasions will arrive this fall should clearly the change in these sites one agents pay off from switching actions. It's proportional to the corresponding change in. And the same logic applies to a site to agent. And therefore, every sub game in stage two is indeed a way to potential game with the potential function given by fine show after identifying the function. The next step is to identify the potential maximizer. And when there are multiple equilibria in stage two, the potentials of the two equilibria are given by this. So the first one corresponds to all agents and second one corresponds to no one joining the platform. And potential maximization is to select equilibrium with a higher potential. And the following lemma. So basically, under potential maximization, the platform has to leave enough surplus to agents by setting sufficiently low prices in stage one, such that all agents will join. Otherwise, none of them will join the platform. And therefore, potential maximization imposes an additional concern to the platform's profits maximization problem in each one. And generically, and with a loss of generality, assume that the per interaction benefits V one is less than V two. And by solving the above problem, we find that the platform's optimal pricing strategy is to set zero price on site one and the highest possible price on site two. Therefore, the platforms believe by this. So maybe I can pause here for Steve, any questions. There are no more questions in the chat. If anyone else has a question, you can just unmute yourself. Just to clarify, this is also the payoff and the pessimistic belief. Yes, so basically this, if we apply pessimistic beliefs, gets the same outcome here. But let me delay the discussion. To buy the results, I will come back to the pessimistic beliefs. It's the same prediction. All right. So let me first discuss my results and then we'll go back to that point. So this table summarizes and compares the results with the benchmark results. So first of all, under both criteria, the platform charges site the same maximum price and fully extracts their shares. By contrast, and potential maximization, the platform provides access for sites, leaves them lots of shirtless. And therefore, the equilibrium profits is also significantly lower. In this case, I call size one, the site and site two the money side key key implications in this model. The first key implication is that the platform always subsists one side and monetizes the other. So this is actually a very common pricing strategy in two side markets. For example, women enjoy free admissions on ladies nights while men pay an admission fee. Shoppers pay nothing to shopping malls while retailers pay the rent. And consumers are essentially charged to use credit cards while merchants pays for the service. And in the literature, this pricing strategy is caused to divide and conquer strategy. The key in this side is only as V1 or V2 is larger. So as we see size in the total to on each side. So this implies that the platform need not monetize the sites with more agents. So for example, shopping malls have more shoppers than retailers, but only the retailers are charged. And similarly, the money subsidy size is independent of the marginal cost C1 and C2 for serving the agents. So for example, for open SS journals, marginal cost of an additional reader is zero and revealing a paper is costly. But these journals only charge all of this. And the third key implication is about the optimal design of the platform. So oftentimes, and V2 are not extortionist, but rather the platform's endogenous choice. So for example, shopping malls are often decided to maximize the shoppers travel distances by creating popular source far from each other. This benefits the retailers, but harms the shoppers. And therefore, the following discussion is comparative statics by varying V1 and V2. So under political dominance, the optimal design of the platform is to favor both sides. That is to increase both V1 and V2. But this is not true under potential maximization. So as we can see, the platform's equilibrium profits is independent of V1, as long as V1 is less than V2. Therefore, the optimal design of the platform, moreover, under political dominance, the social surplus is just equals to the platform's profit, which means that the optimal design of the platform also maximizes social surplus. By contrast, under potential maximization, the optimal design of the platform is likely to be socially suboptimal, because the platform has no incentive to increase S1's agent surplus. So as we can see, different selection criteria can lead to totally different predictions and implications. So this is the methodological challenge in two-sided markets. Nevertheless, potential maximization gives more realistic predictions in this model platform, for the gene agent model. In particular, the platform always divides and conquers. The optimal design of the platform only depends on per-inch benefits. I mean, the money subsidy size only depends on the per-interaction benefits, and the optimal design of the platform is to favor the money size only. And in fact, these predictions are capturing many distinctive features of two-sided markets, which means that these distinctive features need not rely on platform competition for genius agents. But in fact, it relies on the suitable equilibrium selection criteria. So now I will pause here and I will first answer Bruno's question. So it is true that if we applied pessimistic beliefs on this model, we will get exactly the same results here. But I don't want to overemphasize this point, because this is really a non-generic result for this highly stylized model. So later on when I generalize the model to platform competition, then clearly they are not going to coincide. And also, if we have heterogeneous model, it's not going to coincide. And I can actually tell you the rationale behind why in general they are not going to coincide. So under potential maximization, if we go back to the previous page, you can see that the platform can actually So it is OK that I charge both sides, but it turns out that if I do the maximization problem, it is not optimal to charge both sides. It is better to charge one side and leave the surface on the other side. But if we applied pessimistic beliefs, basically you cannot charge both sides, because whenever you charge both sides, then there is no participation equilibrium. So that force will force you to have to subsidize one side and charge the other side. So the driving force is quite different, so we will express that they coincide in a more random case. And the third point that I want to make is that, as you can see, this table represents predictions under a typical selection criterion versus potential maximization together with pessimistic beliefs. Suppose if we don't know about potential maximization, which is justified by many game theory literature and experimental results, then we can believe that this is more justifiable on the right-hand side results. So I guess this is my comments. Hi Lester, we lost you for a moment as well again. Really? Yes, the internet connection seems to be bad. So we did not hear the last couple of seconds, just so you know. A couple of seconds, I'm talking about this on the left-hand side. It is representing a prediction under a particular dominant relation proof. And on the left-hand side, this is the right prediction, because it is justified by many discussion criteria. But now we have justified which potential maximizing more micro-fundered equilibrium selection criteria that makes right-hand side more plausible. Yeah, I think you can, there are no other questions? Sure. Okay, then I will continue and go back to... Sorry to interrupt, can you just try to turn off your camera and see if the quality is better for everybody, just to hear you and see your slides? I see, let me see. Let me see. So I'll stop the video and start the video again. Just stop your camera. Just click on the... Go ahead. Oh, you mean I just stopped the video? I see. Okay, sure. Thank you. Go ahead. Okay, so in the remaining time, I'm going to demonstrate how potential maximization can resolve the multiple-equilibrium issue under platform competition. So the base nine model is generalized into a duopoly platform model with platforms A and B. So now, A and B set prices in stage one, and all simultaneously make their joining these stages. Now, in this model, each agent has to join one and only one platform. So basically they decide to join either A or B. The pay of an agent from joining platform M depends on the number of site two agents who also joined platform M. And I allow the site one agents to derive different per interaction benefits at different platforms. And the pay off of a site agent takes a similar form. So for simplicity, I assume a way is the margin cost of serving the agents. And following I'm trying them right, I assume that the price of subscription fees sets by the platforms are not negative. So they argue that this is a reasonable assumption on pure subscription models because strictly subsidizing agents is obvious selection problems moral hazard problems. So now, let us analyze this model. So first of all, I define P1-Telta and P2-Telta as the price differences of platforms for site one and site two, respectively. So when the price differences not too large, there will be two equilibria in stage two. All agents join A or all agents join B. So clearly, and in fact, amongst all popular selection criteria, only for quality is applicable to this model. So for quality, as we have discussed, this is also known as optimistic pessimistic beliefs. So at the end of this talk, I will further compare for quality potential maximization. So now let us analyze this model under potential maximization. So similar to the baseline model, we can easily show that the potential function given by this. And simply to verify that the function phi is defined here in this, the potential function going to skip the proof. The proof is very similar to the proof. So after identifying the potential function, the next step is to identify the potential maximizer. We call that there are two equilibria in stage two when the price differences are not too large, and potential maximization is to select the equilibrium with a higher potential. And it is summarized by the following lemma. Under potential maximization, we can view all side one and side two agents as a representative agent who either joined platform A or platform B. So more precisely, we can view V1 times V2 as the value of the platform. These two terms as the prices of joining the platform. And the representative agent will join platform that gives him a higher net value. And therefore, under potential maximization, when platforms A and B set this in one, analogous to the standard patron competition. So generically, and without loss of generality, assume that your A is higher than that of B. So the standard patron competition would imply that B will set zero both sides so that this will slightly undercut B such that it kept the entire market. And therefore, this inequality sign becomes an equilibrium. With some rearrangements, we will obtain this equation. Project constraint, platform A will maximize its profit by optimally on the two sides. And therefore, this problem will do a unique equilibrium outcome with summarized following. So first of all, these two tiers are both generic and without loss. So the first one is about the second one is about the two sides. So then I've already explained it determines the identity of the dominant platform. And the second one states that the average, this condition determines the money subsidy side of the dominant platform. So in this case, side one is the subsidy side and side two is the money side. So similar to the baseline model, the dominant platform always divides and conquers. Now, the money subsidy size depends on the per interaction benefits across average per interaction benefits across the two platforms, rather than its own per interaction benefits V1A and V2A. And therefore, accomplish first the money subsidy size of a dominant and I'm going to use the following example to illustrate this point. So in this example, A favor side one more than side two, but B favor side two much more than side one. And suppose initially that A is a monopolist. And by the previous analysis, we know that it will monetize side one and subsidize side two. Suppose B now enters the market and under platform competition by the previous proposition we know that A still dominates the market. But now A subsidizes side one and monetizes side two. And therefore, the money subsidy side of a dominant platform is reversed under platform competition. Observe that if A and B are separate monopolist, then B actually makes a higher profit than A because B can extract more surplus from one side. So this implies that the optimal design of a monopoly platform might not work well under platform competition. So under platform competition, if the two platforms are very competitive, then the optimal design of the platforms tends to favor both sides because the platform with a smaller product of V1 and V2 is defeated in equilibrium and has zero market share. But if one of the platforms is inferior, say if both V1B and V2B are close to zero, then these two terms are close to zero. And these whole term is close to V2A, meaning that's the optimal design of A tends to favor only the money side in order to extract more surplus from that side. So let me pause here to see if there are any questions. No questions in the chat. So if anyone has any questions, you can just unmute yourself. Okay. I think at this point, maybe you should continue and then we will move to Q&A. Sure, sure. Yeah. Okay. So lastly, I want to spend some minutes to discuss the quality and potential maximization. So for the quality, most of us know that the quality is to assume that all agents always coordinate on the pre-specified platform whenever there are multiple equilibria. And therefore, the quality trees platform is symmetrically. By contrast, potential maximization trees platforms are symmetric in the sense that the identity of the format for with a specific is has to determine which platforms should be the platform. So potential maximization. So in the previous example, the one with a larger product of one and two. Calibra is dominant focal because all agents coordinate on this one, but this is an equilibrium rather than as on the start. Let's say I just wanted to let you know that we keep losing you. So, so unfortunately, yep, I think we lost Lester all together. I hope he comes back soon for Q&A. And at least what we can say is that I think we got the most of his presentation in. So, I guess we'll wait a little bit longer. And in case he's not coming back in another minute, I think we'll need to call the seminar short. Yeah, so I guess let me thank Lester in absentia for for the, you know, I think it's an interesting concept. I had a couple of questions I wanted to ask him at the end, but I guess I'll ask him privately. I will share a question with you. So something to ponder. He mentioned that some of the platform competition games can be represented as potential games and wonder to what extent this concept and how general this concept is and which platform competition games can be characterized as potential games and which cannot and where we can apply this concept or not. Otherwise, I thought it was, it was interesting and again, thinking Lester in absentia since he's still not back. And with that, I think we are going to end the seminar. I had a question to more general question. We may want to think about and that is, as far as I know, websites, dating websites for heterosexuals usually charge both sides, men and women. And the competition doesn't seem to force a zero price on one side and not positive price on the other. Maybe they're both willing to pay an equal amount. I don't know what the reason might be. So I can actually chime in on that. Having done some research on dating sites specifically as platforms, for legal reason in most countries, for legal reason of gender discrimination, they are not allowed to charge different prices to the two sides. And in fact, the bars nights out, policy has been challenged as gender discrimination and in some jurisdictions they had won. So nights, ladies nights out are banned from in some countries in U.S. But for only dating sites, it's just illegal. Well, I'm also, I do remember a very long time ago when I was dating. They, I was there, there were platforms of specializing women. They weren't online, but they were still platforms and they, and they kept inviting men please, please join. It's free for men. And, but I guess they had women that had a very high willingness to pay. All right, guys, I think we have Lester back. So maybe we should give him the opportunity to continue. Okay, so let me just finish the last part. So I'll continue at this. Please. Lester, we still can't hear you. At least I can. No, we can't. Lester, I actually have a suggestion. Oh, he's gone. Maybe he can wrap up without slides. I think we can hear him well without slides if he comes back. I think the lesson here is that the internet has to be before is very important. Yeah, we did. Yeah, we did testing. And was there any issues yesterday? No. So I guess it happens. So, Andre, I'm thinking about just, just calling it. I agree. I think it's, I mean, honestly, it seems like his internet is very bad. It looks like it's red, whatever, red color, which means it's not working well. Okay. So, yeah, so, so thank you very, very much. Anyone, everyone who participated and thanks to Lester and I still think we got some interesting ideas here and obviously questions. And I'm looking forward to another seminar in two weeks. Thank you.