We are alive and welcome back. This is Senate Finance and we've got one last thing today. That's an issue that's been brought to me by Blue Cross, the Chamber of Commerce, and the Health Care Advocate. I've explained to the committee as best I can what's going on, and I've explained to them that because we need to find a vehicle, and time really is of the essence of this, that Sandra Kitchell has said she intends to put, I guess we have language from Jen, that she will put that on the budget adjustment bill. So what my intention today is to let you tell the committee what's up. We can look at Jen's language, and then if we vote to approve, we will approve it with the request that I, I guess, or Jen, that I will communicate that it is the committee's wish that this be added to the budget adjustment so that she has our stamp of approval for having looked at this. I think that's the best solution I can come up with. So I don't know who's going to start. Sarah T. Chow from Blue Shield. I'll start, and then everyone has a piece to talk to you about. So I feel like we, as a group, come to this committee every year with an urgent issue related to something that has happened in Congress, or something that hasn't happened in Congress, and we very much appreciate your extreme responsiveness to these issues and understanding the situation. So this is from last year. Last year we came in at the end of the session and asked you to unmerge the individuals and the small businesses in the exchange marketplace so that the individuals could take advantage of the expanded advanced premium tax credits that were authorized at the federal level and so that also small businesses could benefit from lower rates because of unmerging that risk pool. So we had hoped that by this time this year we would be able to come and say, hey, that's so great, the Congress extended those tax credits. The problem is they were in build back better and their fate is a little bit uncertain. We have heard that there is broad consensus federally for passing those or extending those expanded advanced premium tax credits, but how and when they'll pass is extremely uncertain. So what we're asking, and I think I'll let others explain what the proposal actually is, but we're asking you to allow Congress a little bit of time to do what they should do for Vermonters and the reason that we need this so quickly is that per usual we have to prepare and file our rate filings for 2023 in a couple months and so we need direction on how to approach that. So the quick proposal is to if the federal tax credits are extended, then you would keep the unmerged marketplace in 2023. So that's the sort of quick overview and I will let Mike and Betsy talk more details. Okay. Betsy, you want to go next? Sure. I will just say that the fact that the three of us are here and pretty much united is a good thing. Maybe first. Oh, it's not a first, it's not. I think we were here last year, sort of united. Yes, same thing. So I think that that's good and we have had some conversations offline a little bit to make sure that we're all in unison here, but also I think Jen Carby has actual language that she can walk you through. So yes, we are supportive of extending this date. I think we had this conversation quite a bit last year. We were very hopeful last year at this time. We were hopeful that the feds would do the good and right thing. And I guess maybe this is an attempt to say we're still hopeful, although seeing what's happening at the federal level, I'm going to remain a little bit hopeful, but this would basically give us time until the fall, I think September 1st, or we can talk about that specific date. But in essence, giving the feds a little bit more time, because if that doesn't happen, the trigger that is in the statute now will be triggered. And that will mean that we will have the markets be merged again. And so when I was here last year, I think you all made me pinky swear that I wouldn't Senator Hardy remembers this. I actually think the word pinky swear was used. And that I wear in your group. It's fine. It's Friday afternoon. But that women, if this trigger came to pass, that although we think we might have seen 13 to 17 million dollars of savings for businesses last year and the individuals in the individual market were not harmed by that, that if we had to merge these markets again, we would feel that coming back this way. And so that's a little bit about what we're trying to avoid is feeling that pressure again. Nobody wants that, even though we did pinky swear that one year of savings was better than zero years of savings. And I'll still say that, but feeling that come back and raised rates for the small group market is something that we'd like to avoid. That's the reason for extending the date. So I would say that I can certainly talk to you about what businesses are going through right now as far as struggles and other increases in weight with legislative proposals that are happening right now. But we'll just stick to this for the moment. I do want to be certain in this discussion. I know we're trying to do this quick and get this in the BAA. But what we're talking about when we do this is that there won't be those expanded premium tax credits for all the expanded areas, both in the amount of money as well as in the categories of people that have gotten that. So if this comes to pass and the feds don't act in the way that we hope that they act, if the trigger is done and the markets are merged, not only will small businesses feel it, but having those individuals go back to where they were a year ago, they will feel that as well. But that is what we're not talking about additional protections that would increase the amount on small groups. So I think that's pretty much all I need to get out for now. Be glad to answer any questions. But maybe, Mike, you want to add something to this conversation. Yeah, thank you, Senator Finance. It is becoming more regular for the likes of the Chamber in Blue Cross and the Advocates Office to be on the same page. So and the Health Care Advocates Office is supportive of this approach. I agree with Betsy. It's a really bad scenario on all counts if Congress doesn't act. And I also agree, I think we're all in agreement that if Congress does act that it really makes sense for us to continue the organization of the marketplace as we did this current year with rating the individual and small groups separately.  And I have a little concern. I do have a concern about one piece of language. I think we're all on the same page. If Congress acts and extends the ARPA subsidies, we should divide the market. I think that there is language in here that says or is made substantially similar opportunities available that I have some concern about. I am looking for a tolerance to make sure that Diva has some ability to act in the event that it's not identical, but also wanting to be really clear in the legislature's instructions that we're talking about a small tolerance. So I think it makes sense for Jen to walk through. And if I could get any engagement from the senators on that one point, I would. But we're looking for wording. We don't want to be tied to Congress being Congress, having them drop something $10, $5. But if they drop it $5,000, that's a different story. So we want to, and the question is how do you word wiggle room? OK. Exactly. Jen, that's your field. I'm company of Jen. I tried. I'm the one who put in substantially similar. So you can look at it and see if you have better word choice. Do you? I don't remember what this committee was. Can you share your opinion? Yes, that's what I was just going to ask. You know what? I'm sorry, Jen. But can I just I just want to emphasize something. Betsy a moment ago said we were hopeful a year ago. We had good reason to be hopeful a year ago. Your committee's action saved small businesses, I've been saying, better than $15 million. And that coupled with Congress's action for last year brought in something in the range of $60 million in increased premium tax credits on the individual side. It was it's your committee's action was really successful last year. I think I just wanted to make sure you heard that. Thank you. Hopefully we can be successful this coming year too. Add ourselves on the on the chest. And hopefully they'll help us do that again this year. OK, so Jen, let's see what we've got. OK, can you see that language? Yep. Great. So OK, no, no, no, I can read it without putting everybody off. So that's good. OK, I can make it a little bigger if you need it. No, no, it's the pictures of folks down the side. But we're good on this one. Great. So Jen Carvey, Office of Legislative Council, this is some language, as the chair said, that could be put into Budget Adjustment Act or something else if you so chose. And I modeled it on the language that we used last year. The language that went in last year in Act 25 was just for Plan Year 22. This is just for Plan Year 23. And it gives a purpose that's a little bit different from last year's to reflect where we are in the unknowing of what's coming. But the purpose of the section is to allow for separate individual and small group health insurance markets for Plan Year 2023 in the event that Congress extends increased opportunities for federal advanced premium tax credits to include Plan Year 2023. And that extension is enacted by September 1, 2022. So that would be the date by which we would need to know. We still use the same definitions we had last year that base off definitions in the exchange insurance that the Individuals Small Group Plan statute in 33 VSA section 1811. Subsection C says notwithstanding any provision of that statute, to the contrary, if the Department of Vermont Health Access, Diva, determines on or before September 1, 2022 that Congress has extended the increased opportunities for federal premium assistance originally made available through the American Rescue Plan Act to eligible households, purchasing qualified health benefit plans in the individual market to include Plan Year 2023. And then here's the language Mike was talking about or has made substantially similar opportunities available. Then for Plan Year 2023, a registered carrier shell offers separate health benefit plans to individuals and families in the individual market and to small employers in the small group market. Apply community rating in accordance with the applicable provision to determine the premiums for the carrier's Plan Year 2023 individual market plans separately from the premiums for its small group market plans and filed premium rates with the Green Mountain Care Board separately for the carrier's individual market and small group market plans. That's very similar to last year's language but recognizing that we need a trigger in this case Diva determining that Congress has extended those increased opportunities for federal premium assistance originally made available through ARPA or something substantially similar and that would trigger the separate plans. Okay. Committee questions on the base bill without then we'll get to the wording. Madam Chair, I don't know if you can see people. So I can't, I cannot, I can see four of you but you're not one of them. Okay. Well, I was hoping somebody could remind me of the timeline because I'm sort of thinking back to last year I believe there was some timeline that insurers had to meet in May maybe in order to set your rates and then Diva had to do something so Sarah or someone could do something. Sarah is waving so thank you. Sarah from Blue Cross. So yes, you're absolutely right. The urgency is the sort of long timeline. So we have to submit our rate filings in May somewhere around the 10th and then they get reviewed by the Green Mountain Care Board in their actuaries. The rate hearing is in July. And then the reason we have the trigger of September 1st is because after that date so the rate hearings in July the Green Mountain Care Board approves the rates in August. And then really September is when we start inputting and operationalizing everything with Diva. And that's if it happens after September 1st that's where it gets much more difficult for Diva to get the system set up to get the materials out to customers to do all the interchange of information with the federal government. So that's the urgency now and then why we have that September 1st date in. Okay, so help me understand if we did this which would carry forward basically the work that we did last year and you would set your rates and Green Mountain Care Board or you would request your rates and Green Mountain Care Board would approve your rates. What would happen if Congress doesn't act? Would you then, would it be too late for you to change your rates? So we're gonna submit our rates in a way that the Green Mountain Care Board can analyze all of the factors and then we'll roll it up either together in a merged market or separate in an unmerged market depending on what happens. And so it's a little bit like what we were doing last year but in the reverse. So it's a little more complicated and we have to submit information so that the Green Mountain Care Board can analyze it and it's math to see if the two risk pools would be merged together or unmerged and they'll review the components separately. I don't know if that's... Yeah, no, that does help. So there's a way to do it that you could go either direction and it would work, it's just math. I like that. Yes. It's hard math, but it's still math. And I think you're probably, I think Blue Cross Boot Shield, you're the biggest player but are the other insurance people on board with this too? They're gonna submit their request in the same way so the math will work out. Okay. You should hear from MVP but I believe they do support this. Okay. Great, thank you, that's helpful. Okay. Senator Pearson, did you have a question? Well, now having looked at the language I'm curious about Mr. Fisher's concern. I don't know what better to do but I do agree it's, you know, the tighter we can do it the better but so I didn't know if it's time to turn to that or at some point, I hope we will. Yeah, no, if no one has any questions about what we're doing and why we're doing it I think then we can get into wordsmithing and that substantially similar seems to be the word. I read it in context and I wasn't sure how we could tighten it up unless you wanted to start putting dollar amounts on it at which point we start to risk losing the good for lack of the perfect, you know, trying to get the perfect. So do we trust Diva to decide what is substantially similar, I guess? Michael, you've raised it. Yeah, I think Senator, you started to go down the same track I had, you know, I would say, you know, look, we don't know what Congress is gonna do and if it's a couple of dollars off, you know, no one's gonna, we're all in agreement. But when it starts to be significantly more and I don't know where that line is, you know, there is a breaking point and we're not gonna be around to be a part of that or at least the legislature won't be around to be a part of that decision. So, you know, I suppose brainstorming here, I mean, you could add tighter language using a term like de minimis or something like that to say, hey, we're talking about a tight range. That would be one approach I can think of or another approach could be to require Diva to be in touch with, you know, the likes of me and the carriers and the Green Mountain Care Board in making this decision. Well, that's what I was thinking is doing something like after consultation because, you know, I'm only, I'm getting a $50 premium this year, next year I'm gonna get 45, okay, but I'm gonna get 35, well, is 35 better than paying more? And that may depend on the beholder of that decision. So, I mean, the only one I could think of would be after consultation with. Yeah, and I'm not sure they probably need to consult everybody in the world, right? Well, like we'll all be watching this, but... I was thinking all of you, but... Yeah, and I just, you know, I think this is a real, we don't know, right? Like if they just approve what we've already got everything's good, but I think there's a long way between what we had and what we have now, which is good. I think like Michael said, you know, there was a lot of people who benefited, individuals, but also businesses, but in the premium tax credits there's a lot. So even if they came in, I don't know what halfway looks like, it's going to still be significantly more than what they had before this, but that's going to be certainly more than de minimis. And that's when we start... That's what I'm... I do think that if it's substantially similar, de minimis, something like that, I think we all know what we're talking about, but if it is changed by a lot, even if it still feels better than what you had a year ago, I think there's going to have to be a conversation. Yeah, Pierre. So we're going to have to trust Diva at some point because you're right, if it's 50% less, but it's still 50% more than we had, or that we will have, yeah, than we had. That's when we want to be one man. It's more than de minimis by most standards, but it's probably better... Yeah, better than they get if we unmerged the market and went back to... But again, markets are markets, and we won't know till we get there. Senator Hardy.  And you're all around when we're not hoping I would feel comfortable with that. If you're all willing to do it, just it seems to me that you're a representative group that you're wanting to do the right thing for a whole host of people in organizations. So Jen, could we say substantially similar... Does it say Diva can approve? And say after consultation with? Right. So where I'm thinking, and I can put this back up if that would be easier, where I'm thinking is that it would be if the Department of Remind Health Access determines after consultation with, and I don't know if you want to literally list, you know, organizations or if you want to say interested stakeholders, but, and then, you know, then have the honor before September 1st. I think interested stakeholders that way. If somebody else pops up and says, wait a second, you guys haven't talked to me. One of the other insurers.  I guess I'm just wondering, isn't, wouldn't it be, I'm fine with this direction, but couldn't we also just sort of say, if Diva finds it to be advantageous to both, like we're sort of quarreling about how good it'll be, but there's a, the difficult math will suggest it's either better or worse, right? So I wonder if we could just, could we say that, that both markets would be better off than if they went back to a merged market. I think that's, we're saying, you know, they all might only be 10% better, but that's better. That's 10% to the good. Whereas now they may be 50% to the good. Yeah, my is Mr. Fisher. I think given the trick, look in all likelihood, Congress is going to act and they're going to do the same thing where they're not going to act and we're going to be in real trouble and all kinds of levels. But when we start to go down the scenario of who might be better off and what might Congress do and might they go up to 500% as opposed to 300, you know, whatever, there's too many scenarios and we can't possibly, there would be winners and losers. So, I'm coming around to believing that the language we have here substantially similar with consultation. And then, you know, I guess I don't know, you know, probably if Congress did half of the ARPA subsidies, I'd probably oppose it and the chamber would probably be in favor of it, but, you know, and then David gets the side and we'll cross that pass it when we get to it, if it comes to be that way. Okay. So, Jen, you can type that in and yes, I have it. And do you want me to put it back up so you can. I like the idea of interested parties and not naming people as well. And I think that's between that and the substantially similar. I think, I think it's as good as we're going to get. I don't think it can get much better. All right, so I've just put language in here. I wanted to be clear that the determination had to be by September 1st, so notwithstanding any provision, if diva after consultation with interested stakeholders determines honor before September 1st that Congress has extended the opportunities or has made substantially similar opportunities available. And then it goes in. I think. That's as close as we're going to get. I think that's good. And I wouldn't suggest any change, but I just do want to say out loud that I don't think diva needs to consult with us if it's identical. But, but I think the language is fine. It could be a quick consultation. That's right. Okay. Yes. Okay. So I'll just share a bottle of champagne and be done. That's right. Okay. Sandra Brock.  Would that it was always this easy. I think it's a great idea to come back to the association's. All right, send us a rock and. So I have a question. I think it's for Jen. Or anybody can answer it, but it seems like we periodically get these requests. And they're time sensitive. And it has, it seems to me that I've always felt that the rate is not as nimble as it can be. And so things come up after things start like eight or 10 months earlier, and there's no flexibility with the board or diva or whatever to, to do things without calling a special session or coming to us. And I'm just wondering if there's any changes we might think of in the state setting law that could in some ways. Allow for a greater nimbleness and leave us out of it. Michael certainly deferred to the, the people who participate in the process, which is the, you know, the carriers and the healthcare advocate. I don't believe it is what's in the statutory process itself that creates the timeline. I think it may be some combination of federal requirements and the various administrative procedures, but I don't know if Sarah or Mike has more. And I think I'm going to declare this beyond the scope of this bill. No, it's definitely beyond the scope of the bill. This isn't even a bill. And it's Friday afternoon. So I was, you got it. You're in Portugal. That's right. I withdraw the question going back to the river. I think that's something we, we definitely should look, you know, Ken and probably should look at, but I don't think this is really the vehicle to do it. No, I wasn't quite enough. Okay. So committee, any other questions of our honored guests and. We're actually. Seeing us together. If not. I think we could use emotion asking. To send me. To communicate. That we have looked at and approve this language and ask that it be added to the budget adjustment account. And I, if Jen sends me a copy of the language, I will ship that over to Senator Kitchell. Does that. So moved. the further discussion. If not, all those in favor say aye. Aye. Aye. Opposed say no. Senator McDonnell, was that a no? You're muted, Senator. Aye.  Okay, I just started hand waving. I guess you were hitting the button. All right. Okay, that's 7-0-0. I shall communicate that to Senator Kitchell and hopefully we'll keep grasping on this last straw and hoping we can pull out a good deal for the rate payers in these two groups. So thank you all very much. Thank you. Thank you. Well, thank you for being there and bringing that to us. I assume all of your national organizations are in touch with Congress about this and doing their best up there. Okay, good luck. If only I could control my national. Yeah, yeah, that's a good point Betsy. Well, share a name. Yeah, it's one of the hard parts of having the Vermont congressional delegation, at least from most of our point of view, is that it's pretty hard to beat on them because you're pretty sure they're going to do the right thing. So okay, so I will get that out. Thank you all. It is Friday. I'm keeping my promise. I got you out by almost 3.30 on Friday, even if you don't have to drive today. Cummings, this is faith. Yes. We had put on the agenda a discussion about pupil weights and cost equity grants, but it may be that you're done. That's right. And we did that once before. Right now, we are trying to get the school boards in and I asked them to have the superintendents and we asked them if they could send us some superintendents who might be winners and might be losers under, you know, the scenarios and give us feedback as to what the schools would like or not like between the grants and the weights. And they said they needed to have information on winners and losers before their schools could really talk about what the impact would be on their schools. And somewhere in there, I got an email that said Brad James was presenting it on Friday, only wasn't presenting it to us. And I'm not really willing to let information go until this committee has seen it. That's standard protocol. So we're going to have Brad James in. I've asked them to just have the superintendents, the talk to the waiting committee to see if they would talk to us or giving them the option of putting off coming in until we've heard from Brad and then we can ship the information to them. I think that was the end result of many emails this morning. But again, this one unlike Ryegate, we have to do something with. I can't just tell you I'm not taking it up again. So I'm still trying to figure out who, what information, you know, as we go through as you figure out information you need. Let me know. It's still, it's pretty hard. It is to wrap your head around. Are we going to do weights or, you know, equity payments until you see the results? At least maybe we don't name the schools we look at. But we can see what's, you know, in a sampling what they're like. So send a Pearson I saw your hand and then I saw send a hearty. And I'm happy to defer to our resident expert. Okay. But I do have some thoughts to share. I was just going to say that the task force report has the scenarios fully full printouts for all the school districts in the state with cost equity payments and weights. The weights in there are exactly the weights that we were discussing that are in that table that I provided the committee earlier this week. So you can look and see for all of your school districts how the weights would. And then the cost equity payments that are in there are very, very close. They are the cost equity payment amounts that the JFO backed into basically. And before we asked Tammy Colby to provide us with her calculation, and they're really, really close. So that those two scenarios should be able to tell you in general how all the school districts are going. What's different from the report versus what's on the table now is ELL. And that is, again, what we're waiting for education to figure out before we start. But you can look at all the districts in the state. And if any of you have questions and want to want to call me, I'm happy to walk it through with you and explain it when we're off. Yeah, we'll walk it through publicly. Yeah, I mean, it's it's all in the report. And I don't know why the superintendent's association, frankly, is blocking at it. Jeff Francis should know it's all in the report. I don't know. You know, they were they saw that report and came and testified in December and said that they were going to work it through with their association. So I'm not really sure why they haven't. And the superintendents themselves know whether they're from districts that would benefit or not benefit. They all know. So it's not where they do. It's not a secret for them. And so I'm very confused by why this morning was confusing. And Jeff is usually a very helpful and cooperative witness, at least when I was on Ed, and I've known Jeff since he was on the city council in Montpelier. Yeah, I mean, I know as an time as an association, it's tough for them to come to a consensus because there are so many. I don't think we're asking them for consensus. I'm asking for feedback to get a sense of the impact pro and con that this would have on schools. I did try and read that chart. Um, I would recommend that you print it out. I I in order to get it big enough so I could see it on my iPad. I had to follow over three screens before I found the impact and sometimes running your finger along that screen doesn't work real well. So there's probably a better way to do it than that. But it was difficult to read just because of the the size that my head is printed out. Um, many pages of it. So take a look at that Senator Pearson. I just, um, you know, this has been just a swirl of things that seem really confusing. Um, and I found Professor Colby's testimony helpful and I kind of heard, um, this, it wasn't a suggestion. She was very careful to be, um, sort of, uh, partial, but, but, um, I guess it left me thinking in terms of this categorical aid versus weights that, and she did say that adjusting the weights is something we can do pretty quickly and does address, um, however imperfectly a certain amount of the challenge we have today. And so it left me thinking, why don't we do that and, and settle or, or move forward with, with a sketch there? But also I've, I've come to appreciate Senator Hardy and others who, who sort of are approaching this as, as a more thorough fix in terms of, um, aid and categorical aid and that approach that also has a more, I like that it creates maybe a more simple process down the road or simple relationship. And so I wonder if there's a way for us to say, look, there's an urgent need. Let's, let's, uh, move forward with some version of the weights that's going to phase in anyway, but also allow, you know, what we heard from Professor Colby anyway was you can't just do the categorical aid today. There, it has to, there's, there's pieces I've finished. There's, there would be more to do with it. And, and I wonder if there's a way to kind of keep that discussion going while we remedy or, or attempt to address some of it. And anyway, that's not a revolutionary question to ask for the the superintendents. Um, and we also have the suggestion and maybe we'll let Senator Hardy walk us through that, that we could look at a hybrid that there's a lot of new categories that are being suggested for weights. And some of them might lend themselves better to categorical aid. And we still have the whole question of pre-K hanging out there. Um, to me, and then, and yeah, and I think the only issue I can see is saying, well, we're gonna, we're gonna do weights now because schools know them, schools are gonna have enough disruption with changes in the weights without changing the process, uh, that, that they use. But I don't think and Senator Hardy, correct me or Senator Brock, that if we want to categorical aid that you might have the same winners and losers. So, so, you know, it might make it even worse. So I do want to, I do want to make a distinction here on one thing and that and, and, and it is a little bit confusing because of the language that Professor Colby was using. And I understand why she was using it. So the cost equity payments, we, they are not the same as a straight categorical aid. And she was calling them a categorical aid, and they're not exactly the same. Categorical aid can be designed in a whole bunch of different ways. And the cost equity payments are basically the lightest touch categorical aid, basically, if you want to put it in to those, into those categories where it would ship money out based on a formula. And it would be, there would be not any kind of paperwork. It would be all completely formulaic. And it would be part of the whole sort of education finance formula process. And that was intentional that we didn't want to create an extra layer of, of burden, quote unquote, for the school districts. And those would be for those four categories, the grade level, school size, poverty, and population density. The one category, and I intentionally say category here, where we felt like a true categorical aid made sense was ELL, because of the complications that come with ELL, and how the weights don't really work for that. And it's really a very specific educational program and identifiable need. And so it's tough because there's this sort of the cost equity payments are sort of in between a weight and sort of straight categorical aid. So that's just one thing. But you're right, Senator Cummings, that the, that what if we did weights versus the cost equity payments for those four categories, there are slightly different winners and losers. And we tried hard not to use that term. But now the cat is out of the bag. Somebody's going to get more and somebody's going to get less.  It creates bigger spikes with the categorical aid. Yes. And with the with the ELL categorical aid, it's a whole different story. And it's differently. So I just want to caution you, I mean, and I think that we were really careful in the task force to be careful with our language on a lot of things like we never use the term winners and losers. We tried really hard to avoid that. It was school districts that gain tax capacity or lost tax capacity is how we phrased it. And then the cost equity payments are different than the type of categorical aid. They're formulaic. They don't have the sort of requirements. And you know, one of the things that that Professor Colby was talking a lot about was sort of regulation and how much regulation are we willing to put into place in order to get the outcomes that we are interested in. And and these are sort of unregulated grants, so to speak. And I know it's subtle and it's hard. And we talk so much about language that it's we sort of, I don't know, Senator Brock, you can say we trained ourselves to to to think about it that way. And it's hard to to to go back. Well, we're trying to catch up in a couple weeks, what took you months. And I know is the thing to say that true categorical aid, you probably have to fill out some paperwork to apply. You get it.  Yeah. And the categorical aid, if we did it on morality, how rural you are, you would just have, we would just the department, I assume would look at some state statistic that says in this section of Orange County, there are fewer than six people per mile. And in this section, there's more than 50 people per mile. And we would send you a check based on that. Yeah, so the the population density is based on census data, a number of people per square mile and Brad has that sort of integrated into his formula. And he can walk you through how he does that. Similar with the school size and the small schools have to be in a very rural area, as I've said before. And one of the things that I think makes a difference is that for categorical aid, they literally are for a category, our current categorical aids are for special education, and transportation, and small schools. And the the categorical aid and centerbrock and attest to this as well, that we heard that pretty much everyone hates is the small school categorical aid. And I think in in large part, no pun intended, they hate it because it's not really a category of expense.   The cost for the teacher is, you know, the same. But the cost for pupil is much higher if you've only got three students. Right, exactly. So it's harder to identify in the same with a rural school, there's evidence and Professor Colby found this in her study, that rural schools are more expensive in large part because the you know, cost of goods and cost of labor are more expensive in the Northeast Kingdom than in downtown Burlington. But it's hard to sort of like say, this thing I'm buying this thing that's costing more. And so they're more ambiguous categories, but they do lead to higher, more expensive schools and higher costs. Teach him and Craftsbury, who are known to be good schools, I'm sure would also be rural schools. Yes. And small schools. Yep. And I think the category and when you're talking about a hybrid, Madam Chair, the category that's sort of in the gray area, at least for me is poverty. You know, small schools grade level and rurality are all these sort of like you can get that statistically, right? They're they're they're just that the way that the school is it's a bit it's based on the school's location or the school's type. Poverty is sort of a mix because in some cases it's just more expensive because you're in a poorer community and you have more expenses. But in some cases, it is actual student supports that you're paying for.  So that's the gray area for me. But so it could go either way. And I think, you know, to Senator Pearson's point of saying, we understand the weights were used to the weights, I think it makes sense to start with the weight for poverty and maybe put language in this needs further research and maybe it in the future it's it it would work better to have some kind of grant. And one of the recommendations of the task force that the Education Committee in the House I think is working on is a grant for what was it trauma informed, Randy Senator Brock. Yeah, so they are looking at that and that sort of plays into this too. So yeah, it's it's complicated. But I understand, you know, we're at a place where what we know might be the easiest path forward. Yeah. Senator Pearson. Yeah, I guess that's my point is is there is an it's not really debated that there's a problem to solve that that we've, you know, that's why we're willing to have this difficult conversation. But there is a real risk that we do nothing. And and that's so so that's where I'm starting to get worried and and sort of I don't pretend to understand it, but I can appreciate Senator Hardy saying that, you know, there's better ways to do this. I don't see us doing it in that way in this year. And so I I've started to, you know, I like to simplify things and simple and break it down. And that's where I think we ought to start shifting into what can we actually do now? And, you know, what I'm trying to suggest is a way that doesn't preclude us taking longer steps in the future. If it's if it's more of a shift towards equity payments. But I do think there's a real need. And I think there's an entirely likely that we do nothing. And that's both of those are unacceptable to me. Yeah. And that's where I'm starting something on. I think, though, that if, for example, the simple approach would be to use the car would be to use the the waiting methodology across the board. And that's I think what we heard probably more from the education community than than anything else. Because again, it's something that people are used to. But at the same time to in effect put that in place permanently when there are probably better ways to if nothing else to simplify and make this education funding more understandable. I think it's a really important thing to do because people don't tend to want to support things that they can't understand. And right now we got essentially a Rube Goldberg diagram. If we if we map this out so that people can see what's going on. You know, I we haven't touched these things since in this century. And I'm not sure how long in the last century it was. And nobody seems to be able to find any rhyme. The reason or logic is to how the weights were set in the first place. Probably on Friday afternoon when everybody was tired. And that is the the prime problem we've got. We need to get the way we reimburse schools for the cost, the extra cost of educating certain students up to some kind of a rational system that reflects modern costs. And that's prime. I'm also very sensitive that schools I don't think they can handle any more stress. And they know weights. And if weights, you know, if we can do that, you know, to sit here and say, well, we haven't looked at these things in 50 years, but we're going to we're going to keep working on it. It's probably a genious on our part, because it's painful. Because some schools are going to get more or less or whatever you want to call them. And we need to find a way to phase that in. And we can't we, it looks like English is the second language is going down as as a grant. And we might want to try one other as our trial balloon, something that's new. So the schools will have to do it anyway, like middle school. That's new or how rural you are. Those are new. And so then the bulk of the money is going to be where it is now. And they at least won't have to deal with the system. And then we'll get a trial balloon as to how well and how simple and understandable the other system is. But the important thing to me is we do something. This has been too long. So Senator Hardy. Thank you, Madam Chair. I mean, I think I said this on Wednesday after Tammy Colby testified that I think that that I want to make sure that we get a bill out of this committee and I think the easiest thing would be to stay to do the weights for everything except for ELL. I still not talking about that house, but the other four categories. And then I think to Senator Brock's point, one of the things that we recommended as a task force was to create this education. Can't remember what we called it, but the sort of oversight body. Randy, do you remember tax education tax advisory? One was evaluation. The other one was a task force. Right. Personally, I'm not sure why the Department of Ed can't do that. Well, they would be part of it and tax would be part of it and JFO. So it would be a sort of like a permanent group. It wouldn't be a that would do the recalculations and things like that. The whole is really important in all of this is to over to have oversight. Yeah, that allows us to go back and look at what we did. We think we may be doing the right thing, but unless you measure it and unless you can report back on did it work, did it do what it was intended to do, be clear on what the intent is and also be clear on how we measure whether or not we've accomplished it. I think that's really important here. Yeah, absolutely. And we've got to do that, then we've got to set up that mechanism. Yeah, and I think we can we can build that in and and the just to be clear that the school districts, all of it would be by formula either way. So at their end, they were just be getting the money. It's a question of how, you know, how the formula work to be getting the money. And one of the things I think we'll hear and we heard in the task force from the districts that are losing the tax capacity that the losers is that they need it rolled in over a long enough period that they can plan. And, you know, it seems to me that five years starting in FY 24 and rolling in in over five years is reasonable. But there may be some districts that need a little bit more help along that line. And then Tammy Colby's suggestion about a floor may also be helpful. And that's another conversation. But the other thing, though, that I'm particularly concerned about is to make sure, you know, when we started at least when I started out on this task force, one of my thoughts was what we're doing is we're rebalancing. We are not increasing costs and spending. And that is one of the things that I think we should be very, very careful about because this isn't federal money that we're talking about that's coming to us for free. And if we bake something into the system that increases the cost of education substantially, simply because we're rebalancing as opposed to actually adding anything of value, that would concern me a lot. Yeah. And I think that's part of what I'm trying to and we won't know. But if it is true that there is a potential for schools to lose up to ten million dollars, I don't know how any school is going to absorb that even if we phase it in over five years. I mean, if they're losing it because they don't have as many poor kids as we've been paying them to have, it's part of their base education program. And it's unless they can cut out an awful lot of their base education program, that's going to go on the tax rate. I mean, the theory that if, you know, you give in categorical aid, I think works is a general theory. But we haven't done anything this big to test that out. And I think some of it will have to come back into the Ed fund. I just don't know how much. And some of it may come out, but it'll probably be the smaller schools. So we'll take a look at that. And we'll talk to. And I've got to find my. All right, I've got all kinds of emails that are popping up. And they this is. Superintendents we should hear from. But you haven't have you posted next week's agenda yet? Yes, I did. Then I if I can get out of this week's agenda, I can find it. Only I'm not being successful at doing that. Madam chair, I think I sent you six superintendent names that were sort of a mix of. And I just got some from. Was Jeff and I and. I'm still looking for this. That those will be only one sided. I'll tell you that. Oh, I'm sure. But everybody everybody gets to get their day. This is not updated. So if I read to. Ah, good tax, which is the other one. OK, so we've got to Wednesday. We've got Dan French and Brad James. And the school boards and the coalition for Vermont student equity. Are they pro or con? They they are the coalition of school districts that is represented by Leonine. That they are a whole mix of school districts around the state that would benefit from changing the weights. I had that feeling. So these are schools that would benefit. So what we need is to find some representatives of schools that would not benefit. And we may. Yeah, I emailed you a couple names. Yeah. And we will we will see if we can get them under out here. And put them in and then we've got. Genetic testing again. OK, I've got feminine hygiene products on here several times next week. Hoping we can again decide if this is a bill that we're going to get out of here. I've got everybody I can think of that come in. And by the end of next week, we're going to have to decide if that bill is a go or no go. And we will get some of the. The the superintendents that stand to lose under this system and listen to all of them and listen to how they think their school might be able to respond and what would also be the best way to transition them. Does that work? OK. All right. So and I thought which I told my town meeting the other night, I thought we were leaning towards waiting. It sounds like we are definitely leaning towards waiting. With maybe one or so exceptions, but. And I think we're going to be staying with the rates we have. So that means we can. We can look at the. Docs that are attached to the report, the runs on our own. OK, good.


Merci, c'est un joint-work avec des collègues belgiennes de Monts, Thomas Briez, Marion Hallé et Bruno Quentin, et Gilles Guerrard, qui est de Bruxelles. Je vais aussi travailler sur les jeux, mais sur des jeux différents. En particulier, mes principales applications sont les networking, les perspectives routées. Je ne suis pas quelqu'un d'une théorie de networks, mais on verra que c'est très naturel. On peut peut-être utiliser nos techniques, comme les techniques de simulation, afin de solider ces problèmes naturels dans la théorie de network. Laissez-moi juste parler de ce que sont les problèmes de théorie de game. Vous connaissez tous les jeux de ciseaux de rock. La seule façon de modéliser ce jeu est d'utiliser des matresses où vous avez les joueurs, donc si un joueur joue le papier et l'autre joue les ciseaux, vous allez là-bas et vous savez que l'un des joueurs perd et l'autre le vint. Vous avez une façon de modéliser cela comme un jeu de matrice. Dans la théorie classique, vous avez trois assumptions sur les joueurs. Les joueurs sont clairs, donc ils peuvent raisonner parfaitement. Ils sont rationnels, ils veulent maximiser leur payage, donc ils veulent gagner. Ils sont selfies, ils ne savent pas ce qu'il y a d'autres personnes qui vont gagner ou pas. Ils veulent juste gagner eux-mêmes. Dans l'approche de la théorie de statique, et ce n'est pas ce que Team Ravgarden a parlé de dans le tutoriel, ils really carent des notions de l'équilibrium. C'est une approche de la théorie de statique. Vous voulez juste trouver quelque chose qui est un bon équilibrium ou vous voulez étudier le prix de l'anarchie, ce qui est complètement statique. Vous fixez ce équilibrium tout à l'heure. Les joueurs font ça et vous allez bien. Une autre façon de penser sur la théorie de jeu est une approche dynamique. En particulier, vous ne savez pas ce que les joueurs vont faire et vous n'avez aucune façon de communiquer avec eux-mêmes. Vous devez toujours trouver une bonne stratégie. A priori, la première fois que vous jouez le jeu, vous n'aurez pas immédiatement une bonne stratégie parce que vous devez apprendre un peu d'informations sur ce que les joueurs veulent faire. C'est possible de trouver immédiatement une bonne stratégie, mais si vous jouez plusieurs temps, vous pouvez apprendre des choses. Vous pouvez apprendre un petit peu d'informations et vous then improvez votre stratégie de fois en fois, quand vous continuez de jouer le jeu. La question est, est-ce que cette dynamique va stabiliser ou pas? Est-ce possible que si vous avez suffisamment de joueurs, à un moment, vous trouverez quelque chose de bon, un bon profil de stratégie? Et si oui, est-ce que la stratégie que vous obtenez après la convergence est une bonne stratégie ou pas? C'est la question et c'est une idée utilisée dans plusieurs places, comme dans la théorie de jeu, où la machine est en train de jouer contre elle-même et elle apprend des erreurs d'une manière de penser. C'est vraiment les choses qu'ils utilisent pour apprendre les jeux. Mais aussi, si vous pensez sur les algorithmes d'improvement de stratégie afin d'assurer quelque chose, comme parité-games, c'est aussi une sorte de dynamique que vous repliez encore et encore jusqu'à trouver quelque chose d'une bonne stratégie. Et c'est aussi le principal objectif de la théorie de jeu évolutionnaire où maintenant le jeu fonctionne en temps continu, donc c'est plus avec des équations différenciées, mais c'est encore le même genre de choses. Donc, maintenant il y a un lien. C'est le principal objectif. Il y a un lien entre l'approche statique et l'approche dynamique, qui est que, si vous regardez un équilibrium, comme un Nash équilibrium, pour exemple, alors c'est le même, si vous regardez quelque sorte de point stable avec respect à la dynamique. À un moment, vous verrez quelque sorte de stratégie où vous n'avez pas changé votre mind et c'est quelque sorte d'équilibrium. Donc, ici est l'exemple du jeu Rock, Paper, Scissor, où vous commencez, à un moment, si vous followz ces curves, vous verrez le single Nash équilibrium, qui est en train de jouer une troisième, et probablement une troisième pour toutes les trois possibilités. Ok, donc maintenant, ce que je veux faire dans ce talk c'est d'appeler ce genre d'idées, d'improvements, ou de dynamique dans les jeux, des jeux de matrix, mais des jeux joués sur les graffes. Donc, c'est là où c'est plus et plus comme le premier talk. Et ce que nous voulons faire c'est d'être capable de prouver la termination de ces dynamiques dans ces jeux, en utilisant des théories formales comme réduction en particulier et nous verrons aussi la notion de minors. Et pour vous donner un sentiment sur ce qui pourrait être l'applicabilité de ces techniques, je vais commencer avec l'application, de montrer des liens avec la route interdomaine qui est un problème naturel dans la théorie network. Imaginez que vous êtes en Europe, c'est Polen et France, et vous voulez tourner la paquette vers la Texas, vers les États-Unis. Donc, c'est une simple façon de tourner une paquette, c'est d'utiliser la connexion transatlantique. Mais ensuite vous devez payer la paquette très loin. Et bien sûr, vous pouvez imaginer une stratégie claire qui est que le professeur français pourrait envoyer la paquette vers le professeur polonais et puis le professeur polonais va payer la paquette en passant par les Atlantiques. Vous devez voir que c'est un jeu sur une paquette que j'ai maintenant joué. Il sera clair dans une minute. Mais nous pouvons modéliser les préférances des joueurs avec les routes que la paquette est suivie. Pour exemple, le professeur français préfère envoyer la paquette vers le professeur polonais. Il préfère la paquette v1, v2, v bottom et il y a aussi une autre route qui est très rapide qui est celle qui s'arrête entre la paquette et le professeur polonais. Parce que le professeur polonais aussi veut envoyer la paquette vers le professeur polonais et donc si ils envoyent la paquette vers le professeur polonais c'est une situation très rapide où la paquette n'intervient pas. Et il y a un truc similaire pour les joueurs. Donc maintenant, let's remove the map and we just with a game on a graph and here is the situation. So now this is really a game on a graph it's a very simple graph but you could imagine that such a situation could happen on more difficult graphs. And there are some problematic stuff that I will show you now. So first let me say that in this particular very simple game there is I mean the biggest problem is that there is not a single Nash equilibrium. So even if you don't know what is Nash equilibrium it's very easy to see. I mean these two profiles of strategy so the C1S2 means that the V1 in V1 the player will decide to send the paquette through V2 and V2 is willing to pay the price of going through the Atlantic. And the other situation where somehow the play is going through the vertex V1 so now you have two strategies, two profiles of strategies and let's see that this one is Nash equilibrium, why? Because this player has no incentive to deviate alone because otherwise he will pay the price of going through the Atlantic which is worse for him and this player also has no incentive to deviate because if he is the only one to deviate from C1S2 then the play will become C1C2 which is worse for him. So this is a Nash equilibrium and there is an over one completely symmetric. So this is some kind of the static vision of the game, you study the game and you see, oops, there are two Nash equilibria which one to choose. So now the dynamic approach will be let's for a given number of times and let's see whether or not we stabilize at some point towards one of those Nash equilibria. So let's say at the beginning you start with the two players that are willing to pay the price of going through Atlantic. And now in a asynchronous matter the two players can decide to change their strategy or not. And let's imagine a situation both players would prefer to switch and so it's very natural to think that at the next step somehow at the next iteration of your algorithm, of your routing algorithm now you are in a very very bad situation where the packet does not arrive anymore it loops forever and so at some point at this point two players want to fix the situation and because of the asynchronousity I mean they cannot communicate synchronously because of the asynchronousity it's also possible that both players decide to switch together and come back to the previous situation. So because of the asynchronous nature of the dynamics we could block the packets forever or decide to choose a strategy which is not a Nash equilibrium so you avoid completely the Nash equilibria. So you are in an undesirable cycle of the dynamics somehow. So let me a little bit formalise that because what is the question we want to ask is you start with a game you obtain some kind of a graph of dynamics and I will come back to that a little bit later because we will consider two kinds of dynamics so you obtain a graph of the dynamics where the vertices now are the profile of strategies of the players and there is a link if you can switch as synchronously from one to the other. Another question in particular in the network community one of the question was identify some necessary and sufficient conditions on the game, I mean on the network directly on the network such that this particular graph of the dynamics has no cycles meaning that no matter how you evolve at some point you will reach a stable point and hopefully this stable point will be exactly one kind of equilibrium for instance a Nash equilibrium. So of course from a network point of view the conditions should be algorithmically simple locality stable I mean it should not be too evolved so in particular you cannot construct the graph of the dynamics and ask your question directly on this graph of the dynamics and there are propositions and I will come back at the end of the talk on those propositions to see how it differs from ours. Ok so now let me focus on the fact that you could imagine two dynamics very easily you could imagine two dynamics in those games so here is the dynamics that we have studied so far with the asynchronous choice where both players can switch together somehow and there is also the case where only one single is allowed to switch at each time I don't know which one but at each single step only one player can switch and so let me just formalize this a little bit just to be clear about what we are talking about so first let me stress out that I will only focus on positional strategies during the talk and in the paper there is something more about non positional strategies so positional strategies means again that you don't care about the history you don't know where the packet is coming from you just react when you obtain the packet and you know where it is supposed to go and so the first simplest dynamics is the one step dynamics meaning that only one player is allowed to change during one improvement during one step dynamics and this player moreover is supposed to improve his outcome so if he switches he must switch to something that is better for him this is quite reasonable and so this is exactly what you obtain here so only one single player is changing so for instance from C1, C2 which is the worst situation possible player 2 can decide to switch and this is much better for him and by the way it's also much better for the over player but it doesn't care and so here is the complete graph that you obtain and you see that here you have no cycles so you have but you have two stable points which are exactly the Nash equilibria in the static approach the asynchronous game that I presented before is what we call the concurrent semantics the concurrent dynamics now several players can decide to change their mind not all of them but at least you can have one or several players that change they all change at a single node so you could imagine a game now a more general game where several, I mean one player owns several vertices but still we ask in the dynamics that he makes a change at a single node and now here is the tricky part since you don't know whether or not over players will change you must just improve your outcome with respect to your single change so with respect to the current strategy the current profile of strategy that you know you must provide a change that is preferable that is good for you but maybe when all the synchronous changes are made together maybe this will be worse for everyone so it's exactly what happens here because so if you again start from C1, C2 which is the worst situation possible now both players can switch together and you go there in S1, S2 but because of the previous cycle that we have already seen even if we are in S1, S2 which is not the best possible choice because it's not an hash equilibrium both players could decide to switch because it's preferable for them but if they do it both together then it's the worst situation again that you recover ok ok so now this asks questions I mean just formalizing that with some different dynamics over games on graphs ask some new questions and so the first question is the network questions so if I give you a game what is the possible condition so that the graph of the dynamics has no cycle meaning that at some point the iteration will terminate but you could ask several of our questions like if I give you a single game but two different dynamics what should be the relationship between those two dynamics so that one is terminating if and only if the other one is also terminating ok and vice versa you could fix the dynamics once and for all but now compare two different games so I give you two games G1 and G2 one single dynamics like the concurrent dynamics for instance and now could you compare those games in order to be sure that one is terminating if and only if the other one is also terminating and of course these are questions that are I mean more precise than the first one that we are looking for ok and so in order to give you a feeling of what we have been able to obtain I will first present the tools that we have been using which is simulation and minor in games so simulation you all know so it's going to be very quick and it's on graphs not on games right so it's on the graph of dynamics not on the games that I use simulations so a graph of dynamics G is able to simulate a graph of dynamics G prime somehow if all that you can do with G prime you can mimic it in G so very natural very as usual simulation as usual whatever you can do in G prime if you are able to simulate this profile of strategy in G then you must be able to mimic it in G ok now there is some kind of a folklore result that I can use here is that of course if the graph of dynamics in G1 simulates a graph of dynamics on G2 then of course if a graph of dynamics terminates on the simulator then it is not possible that you have an infinite path in G2 otherwise it would also mean you have an infinite path in G1 meaning that also the dynamics terminates on G2 so this simulation is able to help you proving termination and vice versa it's also proving non termination so if the smallest terminating then the biggest one was also non terminating so it's two tools one for proving termination and the other one for proving non termination ok so it's kind of folklore so the one that is not folklore is the next one which is using the classical notion of minors in graphs but not on games so here we have to be a little bit careful it's a little bit more difficult than just minors in graphs so until we say that a game G prime is a minor of an over game G if G prime can be obtained from G by a succession of operations which feels like minor in graphs so one is the deletion of an edge so you can choose an edge and delete it ok easy if a node becomes isolated so there is no edge that is touching him then you can remove it safely now the problem is how you remove a node if it is not isolated at some point and you can do it in a particular case and let me show it on the example it's easier so if you start with this game I don't tell you what all the players preferences and so on that is just an example so you can remove edges ok so you can decide to remove this edge now in this particular game there is no isolated node but still I am able to remove the node D4 somehow and look at what is happening by the way when I remove D4 I make the shortcuts ok so in the minor I obtain all the shortcuts so if I have an edge from V2 to D4 I have also an edge now from V2 to V5 ok so I put all the shortcuts and now exactly the condition that I will put is to be sure that this shortcut was not already present in the game before because otherwise with respect to the preferences of the players I could be unable to retaliate from those conflicting preferences so it's exactly the fact that there is a single outgoing edge from this node and no shortcuts from all its predecessors kind of technical obligation you have to fulfill in order to be able to delete a node and you can continue I mean then you can continue deleting edges and nodes that are either isolated or fulfill this condition ok so now the theorem is what it's relating somehow this minor operation with the simulation operation and then we just apply the folklore theorem on simulation which is that if you have a game G prime that is a minor of G then everything that G can do with respect to the positional one step dynamics everything that G prime is going to be able to do G can mimic it so G simulates G prime with respect to the positional one step dynamics and then you use the folklore theorem and boom you obtain that if it terminates for G, if the positional one step dynamics terminates for G then it also terminates for G prime and the same results with some kind of different proof techniques but the same results also apply for the positional concurrent dynamics ok so it's a relationship between a condition on games minors with respect to simulation techniques that you are able to use in order to prove termination in various dynamics with various dynamics so it's positional concurrent so it's the one with where all players can switch together and P1 is positional one step where a single player is able to change ok, now it could be the end but it's not completely the end because it does not at all match with what network people have been doing because somehow the conditions that I have been using in my game theoretic model somehow is not really realistic so first it's kind of easy to understand that in network theory you would like to be able to impose fairness so in the dynamics it's not safe if one server, one provider wants to switch but never is able to do it so somehow this is possible in my graph of dynamics I can follow an infinite pass where one of the provider is never able to change his own strategy but this is not realistic from a network point of view and so you would be able to instead of speaking about termination speak about fair termination this cannot happen another second condition that is not really realistic I don't really like that but I suppose network guys have good reasons to do that is not to consider any possible switch of strategy for the players but if they switch they switch to their best reply so if they switch in a given node in a given router then they switch to the best possible reply to the current situation so this changes the dynamics and you could imagine that still you have the best reply with single step with only one player changing and the best reply with the concurrent so now you have not only two dynamics you have four because of the best reply variance ok so now the landscape is a little bit more blur maybe but our theorem does not apply so it's not completely trivial because it does not apply to fair termination and it does not apply to the restriction to best reply the notion of minor is not strong enough in order to capture that so what we have been doing in the paper is to find some other notion of minor which is as easy as dealing with the classical notion of minor which is you can only remove edges if they are dominated by some over edges and if you stick to that then the result that you are able to obtain is very good because now you prove not only that result that the dynamics terminates for G if and if it terminates for G prime but also what you are able to so it should be fairly terminated but moreover it's for the best reply dynamics ok so just before concluding what I want to show you is that there is now in the network if I just put the result of the network theory into this setting it is what they are able to obtain they have a sufficient condition for termination which is that they have a necessary condition sorry first to apply concurrent dynamics partially fairly terminated then necessarily you have only one stable point in your dynamics so it's not possible that you have two Nash equilibria in your game they also have a sufficient condition that is that if you remove some kind of patterns so imagine that you do not want to have a special game sub game as a minor then they are able to prove that the best reply dynamics fairly terminates but they do not have a matching condition so this is exactly what they are looking for unfortunately I am not able to solve that but I was we were able to solve this problem for the non best reply dynamics so we are able to find a kind of a stronger pattern that is a necessary and sufficient condition in order to get termination of the positionnal concurrent dynamics and at the end of the day it's really just means that you have to be able to check that your game does not contain this nasty game which is non terminating because you have two Nash equilibria and so it's really just checking whether or not you have this game as a minor or not if you have it you are not terminating if you don't have it you are terminating ok so to summarize it's a first article in order to study the dynamics in graph games and so in order to look for equilibria some kind of equilibria like Nash equilibria we have studied different possible dynamics I've gave you four dynamics and we studied some conditions in order to prove termination or fair termination by using the tools that we are using is some kind of game minor some pattern inside the game and graph simulations as perspective so ok there is the open problem that I told you about in the network theory finding a necessary and sufficient matching condition on some kind of forbidden minor in the games in order to prove fair termination of the best reply concurrent dynamics and from our perspective from the perspective of game theory on games maybe one interesting things could be to model some malicious router that maybe is able to cheat a little bit and this could be modeled with imperfect information so again what happens with dynamics on such games we have modeled a synchronicity with the concurrence in the dynamics but this is not a really realistic model so maybe we could think about over kind of models and again for the fairness we use the same fairness as for LTL for instance but we could also look for probabilistic way of modeling fairness and this could be another possible perspective thank you no no so not for the fairness fairness would be rather than you want to converge with probability 1 so it's rather in the dynamics that you obtain probabilities but not necessarily in the original game but of course you could also consider probabilistic arenas and I don't know exactly what is the question now but definitely something to look at so since you were talking about dynamics in this game changing the strategy does it make sense to subgame perfect equilibrium in this game? subgame perfect equilibria so this is something that we have tried to do to catch exactly what is the kind of equilibria that we recover and in this this is the case at least for the positionnal strategies so if you are asking whether you are subgame perfect equilibrium with respect to just the positionnal strategies I don't know if it really makes sense but then I suppose that the concurrent dynamics is really catching that the subgame perfect equilibria like the one player strategy one step dynamics is exactly catching the Nash equilibria so there is the relationship this is why this is the relationship with concurrents now it's more and more like evolutionary game theory this is the main idea this is where the main idea comes from and then we are a bit afraid of continuous dynamics we were trying to study what happens in a discrete time but definitely now that we have done that maybe it's worth coming back to the real maybe of probabilistic arenas where you add probabilities also in the arena and now you need probabilities in your strategies and maybe a continuous evolution is worth studying


What's up everybody? JC here for jazby's case rates comm. This is our last and only filler guys Just unlock the last eight spots and impeccable football RMB Break with this 2014 prison football Valley pack number 36 and again guys pretty simple do the break first randomized Cuz their names team to strip the hits out then at the end will randomize the 32 names top eight get a T a spot in RMB and the top four get into the Pittsburgh Steelers RMB and the bottom four get into the Kent City Chiefs RMB So let's do the break first. Let's roll One and two three times. Good luck one two three We got Jeff down to Jeff Three here one two three Packers down to the Rams All right, again guys stick to what you got no trading to do this now Eric Ebron Marquis Lee Jarvis Landry Ruggie. We got Kenny stills. You got a Scott's whole zine Vincent Jackson Dwayne bow and Kurt Warner All right now let's switch scenes. What's up Rex? And let's do the giveaway remember top eight get into the impeccable top four get into Pittsburgh next one's get into Kansas City Let's click dice roll and it is a one and a five six times. Good luck one two three four five and Six in final time Boom six times So Nick you're in Jeff John Jeff Allen a Aeron Jeff and Nick so boom I'm gonna paste in all the names Just all of you guys are there But just both just the top eight get in unfortunately So Nick you are in the Pittsburgh Steelers Jeff as well a Aeron and Allen Murdoch And then Jeff and you go into the Kansas City as well as John Jeff and Nick there you go guys So appreciate it coming up next is the rmb's which like I said It's gonna take me a little bit of time just to kind of do a couple rmb randomizes But then we'll be able to do impeccable soon guys. So appreciate it. Jasper's case breaks calm