In 2010 I started taking online programming classes and in about 2013 I took an online startup course. For that course we got to use Bitcoin as a payment system. I had to register with an exchange and I had to purchase some bitcoins and then I got to just stare at them wondering what do I do next. So this got me really interested in thinking about Bitcoin, its technology and for I had a winter vacation coming up so I decided I was going to plan a whole study around learning about Bitcoin. But immediately I got really excited about other aspects of the technology. For instance at the time we called them DAX but now we call them DAOs. These are entities that allow us to organize via code, so programmable organizations. So I started studying about that, I joined an online community, started discussing about these types of organizations, how they could work. We also at the time talked a lot about what technology we would need to make these things work. At that time Vitalik Buterin put out his Ethereum white paper. I got an early copy so I sent Vitalik an email saying wow this is really exciting, I'd really like to help in any way I can. He put me into a Skype room with some other Ethereum enthusiasts. It was my winter vacation so I said please is there anything I can do to help, I have time I can make a web page. At the time Charles was there, Charles said I need a secretary, you're going to be my secretary. So I started doing administrative work for Charles and then a couple of months later I quit my job. I flew out to Switzerland and helped set up the Ethereum foundation. Very simply a DAO is an organization. You can think about all of human civilization as organizing and reorganizing, moving, organizing, reorganizing. A DAO is a new type of organization that's going to allow us to program on a blockchain an organization. DAOs are very exciting because they're distributed. Right now we have walled gardens on the internet. We have communities that you have to sign up for. You sign up with those communities with your email. You don't actually own your email but it is your identification for the internet. So the exciting thing about blockchain technology is that you will be able to get your own identification. You'll be able to control that identification. You'll be able to use that as your credential for interacting with other online people and organizations. The exciting thing about a DAO is it allows you to organize in a programmable way. The interactions between people and different parts of an organization can all be programmed. The blockchain is immutable so it can't be corrupted by bad processes and it's transparent. Transparency is something that's really lacking in this day and age.


Please remove this state's CP while your local day is enough. The squad unfortunately uncivilized, unwarranted that at this age that in Nigeria police we constitute themselves to become a killing squad. Instead for them to be protecting the lives and property of Nigerians, so they are even threatening the lives of Nigerians. So I think for us to sustain the peace, the relative peace and the future of this state, they should deploy the commissioner of police. We no longer have confidence in him, we can no longer work with him because as a media we are critical as a coders, we need to work with the police. On sighting them under the flyover bridge we left the office to go and see what was happening and on getting there we discovered that they said they wanted to protest. Well, we could have even left the scene but the elder brother of the disease was supposed to come and address us. They were there, some of them were actually unruly while the others were actually doing their thing without disturbing any other person. About ten minutes later, policemen suddenly alighted from Ikoroku and started shooting.


genomics proteomics and metabolomics. What is genome and what is genomics? The complete set of DNA found in each cell is called as genome and when we study the genome the study is called as genomics. While on the other hand the complete set of proteins found in each cell is called as proteome and the study is called as proteomics. When we study the complete set of genome it is called as genomics and when we study the complete set of proteins of a cell it is called as proteome and the study is called as proteomics. Why we study proteomics? Protein concentrations may be different than the gene expression due to post translational modifications. So it is necessary to study the proteome and because the production of the protein can be different because there is a process in the eukaryotes and also in prokaryotes which is called as post translational modification. Due to post translational modification the protein concentrations as well as the activity of the proteins can be different in a cell. So to study the proteins or the whole protein which is called as proteome it is called as proteomics. While the third terminology is called as metabolomics. Metabolomics is the complete set of metabolites which are found in each cell and the study is called as metabolomics. So metabolome they are present in each cell and the study is called as metabolomics using high throughput mass spectrometry is used to analyze the metabolic components of the cell. Metabolomics is useful for determining the effects of the environment and gene transformation on the metabolism of the plants as well as the metabolism of the animals. So if we conclude that what is genomics, proteomics and metabolomics? Genomics is the study of the genome which is present in each cell of the organism. Proteomics is the study of the proteome all the protein which are present in the cell and metabolomics is the study of the metabolites which are present in the cell. To study all these three things means genomics, proteomics and metabolomics bioinformatics is the key to understand plethora of information and modeling the cell. So bioinformatics are the tools of bioinformatics are required or needed to study genomics, proteomics and metabolomics.


And welcome to Lady Aida. Hey everybody, and welcome to another Sunday night Desk of Lady Aida kicking off August. It's me, Lady Aida, with me Phil, on camera control and witty commentary in the background. Sometimes. Do we have a little show for you tonight? I could talk about some LEDs. If there are any news or updates we wanna tell people about. No, last week was the AidaBox unboxing on Wednesday. Thanks for joining us. It was epic. Amazing. Watched it on our video channels and a social network near you. This Wednesday, you and I are not doing show and tell. We are doing Ask an Engineer. That's right. And we have a full week of shows. And just so everybody knows, we're dealing with the same thing everyone else is as a company, as a city, as a state, as a nation. Now there's talk of doing required vaccinations for companies and more. And so this is changing by the day. And so far, most of our team is in a really good spot, but friends and family in all of our circles may or may not be vaccinated quite yet. So we're all just dealing with this. Yeah. Every day we'll publish anything that's helpful for other people who run companies or just so you know. And that's what's going on. It's an occupying thing. And it's all, of course, in the middle of a chip shortage. So we'll be talking about that tonight on the show. Okay. So last we checked in, we were talking about this RGB LED breakout that I finished designing and I ordered the PCVs. You can go to the overhead and I can show it off. I think I showed this up, I showed this up last week or week before. So it's got the IS-31, FL-37, 41, which can control 13 by nine RGB LEDs. Some of the LEDs are hand soldered and I didn't do a great job, so they don't come out. But you can see the basic idea here. They're mostly work. A couple of these here are a little sad, but most of them showed up. And it's got a STEMI QT connectors for chaining together. It's an iSquad C chip, which is really nice. It kind of does everything. You just need a few resistors. Honestly, I don't even know if you need those resistors. I think it's mostly for like safety and stuff. But one of the things that's neat is that this is one of the smallest and easiest ways to connect a lot of RGB LEDs over iSquad C. Usually if you want to connect this many RGB LEDs, you need to have like, you know, shift registers and column drivers and it can get very big. What's nice is that this only needs like a single chip and a few passives. It's much, much easier for controlling the large number of LEDs, which means that one of the projects that I'd wanted to do for a really long time, I was like, oh, we can finally do this. One is I snagged, these were chips that were not in stock anymore, but they were in stock before I snagged them because I can finally do a product I didn't mean to do, which is make LED, like Neopixel glasses. I always wanted to do Neopixel glasses. And the problem is that once you start adding up, you know, any number of Neopixels are about 10 cents a piece.   Also, it was really hard to do anything that beginners or just folks who don't want to spend days trying to figure out how to program these things. And so I feel like we're finally in a good spot now because it'll show up as a USB drive, you'll be able to drag and drop images or just change the text file. And I think that's where we want to be, where before it's like, all right, like just get ready to live in an ID forever. Yeah, so I can go to JLCPCB. Okay, so let's go to the computer and I can show some of the PCBs I've been ordering. So I ordered up the, let's see if it'll render. So this is the matrix breakout. What was that sound? It's like, hi, you should do things. Is that the website made that sound? Yeah, the website, boop boop, boop boop, yeah. So this is what the breakouts look like. They're penalized. So they're meant, you can put them side to side if you want, but I put mounting holes in the bottom. Even though I was like, oh, should I make them? So you can tile them top to bottom. And I was like, you know what, no, like it's too mechanically delicate. So I'd rather have mounting holes. So this is the 13 by nine RGB LEDs, which is, let's see, 13 by, oops. 117 LEDs. And again, what's nice about using raw RGB LEDs is that they're only like two or three cents a piece. So all of, you know, there's always some yield and some loss, but the overall cost is not going to be that much. So you can sell these at a reasonable price. And so after this was done, and I'm like, okay, cool. I feel good about this design. Also I want to like test out the RGB LED placement, you know, making sure that these are, the grid is okay. Like until you actually manufacture these, I'm not going to know, do I need the grid to be larger? I think these are a three millimeter grid and they're two millimeter LEDs, but you know, I might need to expand that a little bit. So this is a pretty tight packing, but hopefully these will be manufacturable. And then so what I started to do is I asked Phil B, like, okay, can you design what these glasses would look like? And this is kind of what he came up with. So we like the idea of having round neopixel like circles. That's kind of like what we wanted, but also maybe the ability to scroll some simple text. And for text, you need to have five character height. Like that's the minimum. You can't really go below that. And above that you can do two, but again, we wanted him one driver chip, which can do 139 LEDs. And so this was kind of an interesting, you know, you can see through it cause it's got these slots, but it isn't just like a plain slotted glasses design cause I wanted to just do something really, not like boring, but I wanted to do something a little more interesting. So we've got 24 LED rings. You know, there's 24 RGB LEDs in a ring. And then, you know, the way it's lined up is, you know, it's kind of grid happy. You know what I mean? If you scroll the text through the ring, it's spaced in a way, like it's spaced in a way that for these five rows, the ring is part of the text scrolling area, but you can also have this separate ring. So it's kind of like, how do you put a circle onto a planar XY grid? Well, this is how we're doing it. And then, you know, this is a vector diagram, but the best way to get it in is to actually raster it. And then I imported it as a raster into Eaglecast. You can see it's rastered all the way here. And then I draw over it. So every place that there is an LED, you can see a sort of like these squares. I dragged the RGB LED over it to sort of lay out to like kind of trace over the design. So that's where I'm at.  So it's a little slow going. I did like half yesterday and I'm doing the other half today, but you know, so far so good. And then this is the driver chip here.  And then you could wire it up to any microcontroller. So you can have like a Wi-Fi glasses or Bluetooth glasses or just like RP 2040. So are those two bottom slats empty? Yeah, these are empty. They're just decorative because you, this is how many, you know, you only have a certain number of LEDs. I know I can't put more LEDs on. Like this thing can only drive 139 LEDs, which is a lot, right? Only. Well, we actually, it's 138 because we did test divide by two to be symmetric. So it's 138 RGB LEDs, which is like, you know, it's a lot of RGB LEDs. Again, if you were doing this with dot stars or neopixels, it would like, that'd be really expensive. But I think it's just a good, yeah, I don't think like just cause you can have it like fully filled with LEDs. I think, you know, we could have had a different grid, but I think that this is a good, I don't think there's a lot of benefit to having more. Anyways, this is like kind of what I'm slowly. And you're still rocking out on Eaglecat, right? I'm still a little rounding. You know, you can see here, I have to route all the column lines. The row lines are easy, right? Cause they go across within the column lines. You have to hit each column. You have to snake around. And by the time the second side, I got a little, you can see it's a little more elegant looking than the left side, which is a little bit rougher. Cause I wasn't, I wasn't as sure of how the technique I was going to use by the time I went to the right side. I'm like, okay, I know the technique to use to, to get all these lines around. So they're, they're kind of nicely packed. I might we do the left side eventually, but you know, if it's way out of it, it's way out of it, right? Who cares? Plus the LED glasses that I'm working on. All right, any questions for we move on? No. Okay, cool. So the other thing that we covered was the, what I thought was really interesting is that I had booked some parts, some LSM9 DS1s. I don't know if this is going to be. What do you want to go to the blog? Well, I wanted the article that we wrote here. Yeah, there it is. So it's interesting because this is, you know, like many companies, it's like, okay, if I can't get parts, there's some signs I have that use parts. You can't get the parts, but maybe you can back order the parts, right? You put it in order, you put in a purchase order, and you say, look, schedule it out. I won't need these maybe for a year or two years, but I'm willing to know what I need so that I can kind of like have a place in line. And this is really common. We actually do this for parts all the time. Like we order, you know, raspberry pies, like a year in advance basically, we tell them here's about, here's the prop, you know, of course things can change, but here's approximately how many we're going to buy. It's, you know, not, it's like we can change the scheduling, but we're basically saying we're going to purchase these eventually. And we did this with this LSM9 DS1, which we use in a breakout. And this is the first time I've ever had this happen in like 15, you know, 20 years of buying electronics where I purchased something and after I purchased it, I'm told the price has changed dramatically and I can either take it at the new price or cancel my order. Like I've not, it was different. I've had, you know, oh, I'm about to buy something and the price changes or like I sign up to be notified I want to get notified the price changes. I've never had an existing purchase order changed like not underneath me when I was notified, but it was changed as the purchase order had been placed and was pending. And I think I would place this purchase order like, you know, months ago too. It's the nuisance. It doesn't like, oh, we ordered a day ago and like, you know, the price change underneath. So, you know, it's interesting about this chip is this used to be $3 or $4 a piece, you know, maybe $4 retail, $3 if you negotiated and now it's 20 bucks. And this is from an official distribution, right? Again, that's different. I have, I've definitely seen gray market prices be ridiculous. I've been quoted, you know, nutty prices. This is, we booked in order at a certain price from the official distributor. And then they're just like, you know how you booked it at a certain price? We're going to go ahead and make it 20 times. This is very, very, very unusual. And so the reason I'm making a big deal out of this is like, why am I making a big deal? It's not a punishment. I'm not trying to like call anybody out. What I'm trying to tell is for people who are do, who do like crowdfunding campaigns or pre-orders or budgeting, you can no longer depend on the budgeted price to be the price that you actually get the product at. That's never not been true. This is the first time that I've seen that it hasn't been true, that you order something and you don't get it at that price. Usually when you order parts, the price that you buy it at, even if it ships later. And the price it ships at. Yeah, this is new and unusual. Yes. And it, you know, if we're experiencing this, that means everyone else is too. And we're going to start carrying on it. And this is probably not the only part. This is just the only part I know of because it's the only part. I mean, like, it happened with a couple of the parts, but this one was like extremely, extreme price change, right? Basically eight times more expensive than originally. And so I think if you're, even if you're back ordering parts, you can no longer guarantee that when that back order is filled, it will be filled at the price that you pursue. And we only use a breakout, that chip for a breakout, but. But if you use this in a design and it was like you're just paying, you know, you bought everything else waiting for the design, just.  Yeah, the Nano Connect I think uses the LSM9DS1. So, yeah, it's here. It's happening. So let's, let's move into the great search. Cause we'll. Okay, problem solution. Here we go. The great search brought to you by Digikey and Adafruit where Lady D uses all our powers of engineering and searching for parts to help you find that part that you need, especially when the rug gets yanked out underneath you and your bump cost goes up 1000%. Yeah, not, well, not 1000, but you know, a bunch. So we just, we were just chatting about the LSM9DS1. I thought this was interesting. We wrote about it. The price, the official price of this chip, not only is it, you can't buy this chip, but the official price went from about $4 to about $20. And one of our back orders, you know, not from Digikey, but from another distributor that we had placed months ago, basically got canceled cause it was like, well, we're not gonna sell it to you at the original price. You have to replace it with this new price. And so I thought if there's people who are using this chip, look, I like the LSM9DS1. It's, it's one, I've used it. It's wonderful. I use the LSM9DSO. I love all the ST jars, accelerometers, IMUs. But if you have a board and you are speccing this part at about $3 to $4, and now it's coming back to you at $20, that can really, really mess your bill of materials, right? If you're selling, you know, a gaming thing and it had accelerometer in it, you know, motion sensor, and suddenly the bill of materials costs just went up $15, that could easily double the cost of your product. And so I thought, you know, I think eventually the price will come back down because I don't think this is competitive at this price. However, until that happens, let's see, you know, some alternatives for nine DOF sensors. I wanted to show people, because I had actually had somebody ask me, like, what would you use instead? And I was like, that's a good idea for the great search. However, I'm not gonna make that name public. So that's why there's no quote. So I'm behind the scenes. Okay, so one of the nice things about the LSM9DS1 is, it's a nine axis. It's accelerometer, gyromagnetometer. It also says temperature, which technically would mean 10 DOF, but like temperature is really easy to get. So, you know, we're looking at a nine DOF sensor, which means you get three degree magnetometer, three degree accelerometer, three degree gyro. When you're speccing nine DOF sensors, and you're almost always speccing nine DOF sensors because you wanna do an orientation IMU type thing, there's basically like two kinds of systems, chips that you wanna use. First off, there's not like one, it's not one chip inside. It's actually gonna be like two or three chips that are kind of tied together, all on I squared C or with SPI. So in the LSM9DS1, I don't remember exactly the two parts, but it's basically, I think it's like an LSM303 plus something else. It's basically either an accelerometer plus gyro, and then a separate magnetometer, I think maybe say an LSM2MDL or 3MDL, tied with an LSM330 or something. It's two chips inside, and they're just packaged into one little chip, which is quite nice, it's very nicely and compact. These chips, they have things like filters and 5-files.  And there are also chips out there on the market, like the B&O055, B&O085, and those chips are nine-doll sensors, but they also have a little fusion chip inside of them that does that calculation. So why would you care, like which one you use? Well, first off, there's price. So if you're getting a B&O055, well, now it's gonna be about the same price as the LSM90S1, but in general, if you have a chip that's doing the fusion algorithm for you, you're gonna pay a little bit more because there's this second chip that takes the sensor data and fuses it together to pipe out quaternion orientation data. There's a little microcontroller in there? Yeah, there's a little Lugia Cortex M0. I think the B&O055 and 085, there's actually two different companies, although the partner was very similar. One's Hillcrest, Siva, and one is something else I don't remember, and those have a SAMD21 actually inside. Also, so there's like actually like now three chips inside tied together. It's like our TI calculator that runs Python at a SAMD. They're at a SAMD. So you can get, look, here's the thing, you can't get those chips right now anyways, but yeah, so they're gonna be more expensive. The B&Os are also like totally unavailable. In fact, I think if you search on Digikey for the B&O, I think they're like, we don't even have a time in which these might be available. Yeah, like you can't even accept that. I've never seen that, there's not even an estimate. Wow. Yeah, which is really cool. Well, due to temporary constraints applied, Digikey is unable to accept back orders at the time.  We're just gonna pretend this never happened. Yeah, this is basically, they don't even have the price. Again, they don't even know. They don't wanna say, they never want anyone to say, hey, look, why did the price change? I don't know, I don't know why they did it. I think that they really do not know when it's going to be a time. I wouldn't keep the price on it and I wouldn't put any time or date right now because everyone's familiar. I think they're like, we don't even want to accept the order to carry the risk of what's gonna happen. Who knows, maybe this, it could be many years and they don't wanna hold that order. It's a ghost town. So those chips are, they're very specialty and those go faster. There's also sometimes you can get external chips that are like fusion chips and they'll do the work for you, but yeah, so the B&Os are kind of famous for like they do all the work for you and running this algorithm. You know, again, you can write, you can code the algorithm and we have a tutorial here on how to run AHS on a microcontroller, but you do have to have a microcontroller kind of constantly pulling data and you can't, you know, you can't miss any data, right? You have to be very consistent because you'll start to skew a little bit. You'll start to get y'all from missing data. So you have to be like, you know, every 50 hertz or whatever on the dot, you have to get the data. So if you can't get these, you know, again, you can always do it yourself by combining accelerometer, magnetometer and gyroscope. Of those, you want a pretty good magnetometer, but honestly, there's a lot of good magnetometers as long as it's got, you know, a good, you know, it's got range and it isn't affected too much by external magnetic fields. Magnetometer, I'm not too worried about. Magnetometer is what we're pretty good at. Accelerometer is also, we really figured out men's accelerometers, like everyone's got accelerometer now. We're pretty good at it. You know, you can use it to do tilt and motion. The part that's really going to dictate whether you get good sensor data out is gonna be the gyroscope. The gyroscope is what has drift. You don't have drift in anything else. The gyroscope is the only thing that like, it'll have offset that will add accumulate error on your algorithm. So we do have an article check out comparing gyroscope data sheets and you can look at some of the gyroscopes. NXP has really great gyroscopes as well. I think they've just continued them now. I mean, like it's one of those things where it's like, here's a great sensor, no longer available. Here's another great sensor, no longer available. But you can piece together, if you can't get one of these, you can piece together an accelerometer, gyroscope.  They do not have to be from the same company. They can be separate companies or whatever, distributors, and then you convert them to SI units and then plug them into the AHRS algorithm to get orientation data. That said, let's look, you know, there's really only one option, but let's look at what we can get. Now, nothing's in stock, but I can show you what I would recommend. There's only one option. Is that technically an option or is that, how do you describe when there's one? It's not a choice. Maybe it is an option because the option is either bite or don't. Yeah, so if you search for, for example, in stock IMU chips, which I think is anything with at least one accelerometer and gyroscope, there's really slim pickings and the prices are quite, quite high. I think just the demand is really high right now, but a lot of these are modules, and they get very expensive very quickly because they are, you know, you get into like these industrial IMUs. Like a car, I mean, like, you know, bill of material and cost of cars. I mean, these are used in robot, the problem is that they're used in like Wii's and like Switch game controls, and they're used in like industrial robotics and automotive, right? These are sensors that are, they're so popular that every, like everyone's competing because everyone wants this thing. In the beginning of the pandemic, no one was saying, hey, when this is, a year later we're gonna have a chip shortage. Like that wasn't one of the things, you know, it was like quick stock up on toilet paper. Yeah. So let's not look at stuff in stock because, you know, we're just gonna be disappointed. However, let's look at surface mount components and let's look at active. So at least, you know, when they do, you know, it will likely come back into stock. And then if you're one of the, if you want something that's a more close size comparison to the LSM9DS1, you want nine axis, so let's pick, I don't know why it says gyroscope nine axis, but I'll click it anyways. But we definitely want gyro and magnetometer and accelerometer. So let's see what's available. And today you can see the B&L 085. Oh, these you could at least maybe back order. Again, these are the ones that have, the B&L 080 and 085, which are almost identical. There's one is a little bit, the one fixes a little bit of an SPI bug. Those are available. Then there's the LSM9DS1 that we referred to. And then after that, there's a lot of modules.  It's like five sensors, you know, put together. It's like someone on their magnetometer gyroscope put together on a board. But if you want it all in one, I would actually recommend going with the ICM 2948. This sensor also can do its own fusion algorithm inside. I believe it's under NDA, you might have a site NDA, or you might have to download a binary blob and then compile it in. But I'm not 100% sure, but I know that InvenSense, which is now a TDK, they made very good nine-doff sensors. They were used in Nintendo and Apple products and before Nintendo and Apple both, I think changed over to ST, which means InvenSense got half their business taken away. But it doesn't mean that they're not good sensors. They're very, very good sensors. They're just, you know, they didn't win in a gigantic fight between elephants, right? Which is how it goes with your small company. Good news though is the price isn't bad, right? Like once you get to 100 pieces or 500 pieces, the price is down to $4, 450. So it's much, much closer to where the LSM90S1 was. You can actually get it at a price that's comparable. If you design the LSM90S1 into something and you're like, I just need something that's about the same size, same functionality, I squared C can give me nine-doff IMU data out. Whether or not you use this motion plus fusion thing, this would be a good alternative. It's not pin compatible and definitely not firmware compatible. But it's I-squared T and you give it some power. I think it might be 1.8 volt, but you know, maybe you could put a level shifter on there and I would recommend this chip. I use it, we have a breakout for it. And it's quite easy to use and it's pretty reliable. You can use our libraries to get your design kick started as well. And while I don't know 100% that this is true, it does say that there's gonna be at least a thousand in stock. Oops, in September, if you type in a quantity, says estimate September. So you have a shot of actually getting these. So that's what we're at as engineers, just praying. Hoping.  Like, hey, we're out of lithium everybody. We're gonna have to deal with that. We're not there yet, but this is reminding me of all the futures that we're gonna have. Yeah. Oh, cool. Pandemic dystopia, what's the end result? Well, you're out of chips. All right, that's the great tip. All right, check it out. The ICI. We're gonna send us your request as a part you can't get, want us to find the best option. We can... Yeah, we'll always be able to help out if you're... Do the best we can. So send us, you can do it on Twitter and you can do it in other things, send me an email, whatever, all those things on which part you'd like to have a great search segment about. We always have one every week, but we also like to do ones that people send us. Okay, well, thanks everybody. That's it for DiscoVladiator this week. We will, yeah. Yeah, it wasn't meant to be a downer message. It's just like, this is what it is now. It's like... Well, we're half, you know, this part search could be two years long. It's kind of devastating. Yeah, it is teaching us to continue to diversify our product offering and be nimble. So this is how companies are forged. That's what they are. All right, thanks everybody. We'll see everybody next week. Bye.


Welcome back to NPTEL course on game theory, so we continue our discussion about mixed strategies. So let us recall once again there are 2 players we are talking about mixed strategy, the strategy spaces X and Y, mixed strategy of player 1 is probability distribution on X, mixed strategy for player 2 is probability distribution on Y. So this space we denoted by PX, this will denote by Py. Now as I said last time if X is a continuum then describing this PX requires the knowledge of measure theory or probability theory, so we avoid going into those technicalities. So we concentrate on the case where X and Y are finite sets. So let us say X and Y are finite sets. So this is where the game becomes, this game can be expressed by matrices, matrix, so look at it. So X is let us assume 1, 2, M, Y is 1, 2, maybe K then what the payoff function is basically pi from X cross Y to R, so we have the following thing, pi 11, pi 12, pi 1K, pi 21, pi 22, pi M1, pi M2, pi Mk, now think about this as a M by K matrix, so let me denote this by A. So the player 1 is choosing the rows and player 2 this is about the player 1 and player 2 choice are basically the columns, the player 1 chooses one of the row and player 2 chooses one of the column that is exactly the X is basically the set of rows of this matrix and Y is the set of columns of this matrix and the payoff function is simply the entries in this matrix. This entry tells you that if the player 1 chooses the row 1 and player 2 chooses the column 1 then this is the amount that player 1 gets. So these are called matrix games, so they are given by a single matrix. So there are 2 players, row player and column player, so the row player is nothing but our player 1, column player is nothing but player 2, so you have a matrix, player 1 is choosing the rows, player 2 is choosing the columns and the corresponding entry gives you the payoff of player 1 or the row player and the same thing is what paid by the player 2. Now matching pennies is a matrix game because there are only 2 choices, so in this case the matrix will be 1 minus 1 minus 1 1, so if the player 1 chooses row 1, player 2 chooses column 1 then player 1 gets 1, player 2 gets minus 1, similarly this thing and similarly the rock papers is just game is another matrix game. So here let us recall this is rps, rps, this is the diagonal entries will be 0, if when rock and paper means the rock can be hidden by paper, so therefore this is minus 1, this is 1 paper and rock, so this is minus 1 and paper and scissors, this is plus 1 and scissors and rock means this is minus 1, scissors and paper means this is 1, this is the matrix game where the matrix is given by 3 by 3 matrix which is given like this, so this is a rock paper scissors game, so like that there are so many matrix games. So now we concentrate on matrix games, so these are given by a single matrix, we have already seen that in these matrix games for example as an example is matching pennies there is no saddle point equilibrium and that is where we started discussing the mixed strategies. There is a mixed strategies, player 1 is choosing rows, so he can choose any of the rows with certain probabilities, so he chooses any row with certain probabilities. So let us say row 1, let us say he chooses this with probability x1, row 2 with probability x2, so row m with probability xm and we want all this sum to 1 and of course all of them are non-negative, this is a mixed strategies and what is a mixed strategy for column player? Basically he choose column 1 with probability y1, column 2 with probability y2 like this column k he will choose with probability yk and we want summation yj to be 1, of course j runs from 1 to k and all these things are non-negative, so this is a mixed strategy. So we will say the mixed strategies of player 1, let me denote it by delta m and player 2 delta k, so the delta m of course we can easily see that this is a subset of rm, this is a subset of rk. Now let us look at some interesting properties, delta m is convex and compact, of course same way is convex and compact. So why is this convex? We will only prove for this case and the other things similar, so xz let us take 2 in delta m and we also take some lambda in 0 comma 1. So what is x? x is nothing but x1, x2, xm and we know that the summation xi is 1 and of course all of them are non-negative. Similarly z is z1, z2, zm and of course zi they are all non-negative and summation zi is 1. Now what is lambda x plus 1 minus lambda z, this easily you can calculate, so this calculation is straight forward and now we can easily verify that the summation of all these things lambda x1 plus 1 minus lambda z1 plus lambda x2 plus 1 minus lambda z2 plus so and so lambda xm plus 1 minus lambda zm, this one can actually verify this, it becomes 1 and leave this as a simple exercise, we can show it easily. So this says that this delta m is convex. Now also recall this delta m it is a simple exercise again once you know little bit of real numbers the delta m is closed as well as bounded. The boundedness is trivial because the entries of delta m all of them are less than or equals to 1. Any vector in delta m if you take it it is a vector in m dimensional rm and each entry in this vector is a non-negative number less than or equals to 1 and all of them sum to 1. So therefore all the entries are less than or equals to 1 and then that you can use it to show that the boundedness, what exactly the boundedness that is a little bit of a topological thing. So let us not go into those details and closeness again comes from multiple ways one can see it. So for example this delta m is basically all those vectors such that the sum of the components is 1 so one can also see that as a hyperspace. So hyperplane a linear relation satisfying and then that is the solution space of that I will leave it as simple exercises. So the delta m is convex similarly delta k is also convex of course compact. So that is next what is the suppose player 1 chooses a mixed strategy x in delta m and player 2 chooses y in delta k how much player 1 will get this we have not yet defined. So we need to define this one we only know this the payoff function only when they use their choices if they choose rho 1 or rho 2 or anything but now we are we are saying that the they can choose the those rows with certain probabilities. So now let us look at it so we need to define let me call it as a pi x comma y is this is what we need to get an expression for this. So let us do that one what is pi x y pi x y rho 1 is chosen with probability x 1 or let me put as rho i with x i column j is chosen with probability x not x y j therefore I and column j are chosen with probability x i into y j why is this product x i y j because as I as we discussed earlier in this scheme the most important thing is that the when a player is choosing his strategy the other players are completely unaware of it. So they are choosing completely independently simultaneously. So therefore if the rho i is chosen with a probability x i and column j is chosen with a probability y j rho i and column j together are chosen with probability x i into y j the independence is really playing a whole here. Therefore player 1 will get a i j the i j entry of the payoff matrix with probability x i y j therefore player 1 receives a i j with probability x i into y j this is as now sum over all j 1 to k and sum over all i 1 to m. So let us look at understand this one. So he receives a i j with probability x i into y j that is basically the his expected payoff. This is expected utility that player 1 gets under this mixed strategy x, y another mixed strategy pay. So the player 1 has chosen x player 2 has chosen y and then what player 1 will get is that summation i is equals to 1 to m summation j is equals to 1 to k a i j into x i y j. So let us understand once again why is this term because player 1 let us go back to this player 1 is getting a i j when player 1 chooses rho i and column j by player 2. Now the rho i is chosen with a probability x i and column j is chosen with a probability y j therefore the rho i and column j together are chosen with probability x i into y j and hence he will get a i j with probability x i into y j. And now this should be sum over all rows and columns that is why rho i over rho i and over columns j. Now this is known as the expected profit. This is let me denote it by pi x y. So let us try to understand this little more.  So in fact the people can see it as the following thing. This is a matrix A this is m by k then y 1 y 2 y k this is the column vector I am writing it and I will also write here x 1 x 2 x m as a rho vector this is m by 1. So this is 1 by m this is k by 1. So now this is a like a matrix multiplication this we can also write it as the vector x A y or we also can put it as x transpose A y x transpose A y. So remember when I say y we generally write every vector as a column vector. So that is why y and x transpose whenever we are writing we are writing it as a row then I put it as a x transpose these are the various ways people do is. So if we understand this notation very clearly we avoid this transpose but you can write it as x A y or whatever way we like. So this is basically the matrix notation that we have used. This is known as expected payoff that player 1 receives from player 2 when player 1 follows the mixed strategy x and player 2 follows the mixed strategy y.  So now this is compact and convex. So this is also compact and convex and what about this what kind of function this is. So let us check that what is this pi pi x y is nothing but I write I generally prefer writing inner products x A y. So we can easily see we can verify pi is linear in x when y is fixed. Similarly it is linear in y when x is fixed. So in fact the most important what is the linear in x means what here it is a if for example because this is the inner product if you have you might have already seen in some courses that if inner product x plus x prime A y is same as x A y plus x prime A y. So this is the linearity of course if I put alpha here beta here then this will be alpha this will be beta that is the linearity then similarly with respect to y. Now if you observe this is much stronger than the convexity or concaveity. So in fact what we can say here is that pi is convex in y for sure and it is concave in x. This we can I will leave it as an exercise just follow the definition of convexity and concavity and show that pi is convex in y and concave in x. In fact the linearity actually says that the pi is both concave as well as convex. So we can see. Now because it is if you recall the Minmax theorem can be applied. So what does the Minmax theorem say if the strategy sets x and y are compact and convex and the payoff function is concave in the maximizing variable convex in the minimizing variable then there is a saddle point equilibrium. Now all that conditions hold true here and hence from Minmax theorem saddle point equilibrium exists. So the Minmax theorem for this matrix game Minmax theorem for matrix games is basically due to von Neumann I think around 1918 has proved this result around this time and this is basically the first result formal result which actually paid way for the subject game theory. So now once we understand this matrix games and this notion of equilibrium here and we know that this saddle point equilibrium always exists from the Minmax theorem. Now what remains for us now is to prove this Minmax theorem. So there are multiple ways of proving this result. So we will prove some way which does not use much technical details because any of this existence of equilibrium they require a fixed point. We will try to avoid the fixed point theorems here. Of course in the matrix game case this can be proved using the separation theorems in the convex in the convexity we use something called separation theorems. So that requires us to introduce more notation. So instead of that we use basic calculus and optimization. So we will try to prove the Minmax theorem under the convex assumption. So now we will try to see the proof of it and we will continue the in the next session.


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September is observed annually as Suicide Prevention Month. This year, it was commemorated under the FEMA, creating hope through action. During an appearance on NTN's morning updates, Naomi Grandison, family life educator in the Bureau of Health Education, revealed that many St. Lucian families are affected by suicide. Based on 2019 statistics, St. Lucia's suicide rate stood at 7.9 for 100,000, well above many countries, the region. It goes to show that for me, this is my personal opinion, that there is something wrong. There is something that we need to shift. There is something that needs to change. Because the overall statistic worldwide is 100 out of 100, and that will be about 703,000 deaths by suicide. So it shows that one, persons are suffering with mental health challenges, with coping skills, with hopelessness, with services, inadequate services, to deal with those things. With the heaviness of the impact of stigma, which is that you, I don't want to deal with you, cast you on a side for want of a better world, or feeling that you would be judged or not given certain privileges, otherwise, because you declare that you have some sort of mental health problem, even if it doesn't mean that you have lost touch with reality, or even if it doesn't mean that you're psychotic. So herein lies the problem.  It could have been a school or movement thing. But for the most part, there are signs and persons are suffering. St. Lucian artist George Zenkwa, best known as Platinum George, shared his experience managing depression and suicidal ideations throughout his life. He said that after years of constant struggle, he found solace in God. Zenkwa is now using his experience to assist others. Going through the music industry made it worse again in terms of now I'm investing in music, but yet still I'm not getting any returns. Then you would see that I have a family to maintain. I have bills to pay. Then spending a whole bunch of money, and you're not even making half of this. It basically had me on a cloud of darkness. My transformation came through. I got invited to church. Imagine I'm the kind of person that I always go to church. No matter what you invite me, I would go. I'm not that kind of person who's picky in religion because I am trying to get that emptiness filled. Nurse Grandison indicated that mental health care plays a pertinent role in combating depression and decreasing St. Lucian's suicide rate. I want to say that again, the skills building, the love for each other, the learning how to communicate, the learning how to get help, knowing where to go to get help. You understand knowing how to love yourself is all part of what can help to mitigate the actual completing of suicide, even if some people may be a passion, thought other people are haunted with it for a very long time. So I just want to add to that. If we can skill build and encourage our children to build certain skills and let them know the importance of communicating, expressing your emotions similarly to rush in your tea for the morning, clothing yourself, beving. You need to be able to communicate and talk and offload. It is a part of your overall well-being. And I think that can help to avoid. And I think in our society, I just want to say this, and again this is my personal opinion, in our society we're just too judgmental. And being judgmental is extremely toxic.    And we think like that because we don't know better in some cases. We think that it's going to help the person to change, but it only serves to bring a person deeper into that pit.