a presentation of TFNN. The Tom O'Brien Show is produced every business day. Tom takes your phone calls toll-free at 1-877-927-6648. Internationally at<PHONE_NUMBER>. Hey, Robin, how you doing, man? Yeah, good, thank you for taking my call. I wanted to let you know that I've been a subscriber for a couple of years, just different members of your team, and I really enjoy it. But really the reason I'm calling is to express my sincerest gratitude for you, providing that information yesterday on the Small Business Scrantz. I'm a small business owner, primary breadwinner for my family, and if I can get that money, it's gonna really lead a lot to my family, so. That's awesome. Thank you for taking the time to do that. No, listen, man, we appreciate you growling and prowling with us. Now, Tom O'Brien. Welcome, folks. This is Tom O'Brien of TFNN. We're here on five days a week. We go seven hours a day. We go 24 hours a day on the internet at TFNN.com. Always remember, folks, whatever you think about, you bring about whatever. You focus on growth. Hope everyone's having a great day, safe day. Let's make it a great night, folks. Accept others the way they are. You cannot change other people. To try to change them to fit what you want is like trying to change a dog into a cat, or a cat into a horse. They are what they are, and you are what you are. Mockin' wise. Let's take a look at it out here. We have the Dow Industrial's down 199. Nasdaq is off 11. S&P's down 18. Gold contract trading down $27.10 at 17.08 an ounce. We have silver down 84 cents, $18.35 an ounce. Light sweet crude, flat, $96.69 a barrel, notes and bonds, a 10-year note. Trading down 15 ticks at a price point of 118.14. The 30-year of 28 ticks at 139.15, and king dollar. King dollar's up 656 ticks trading out at 108.614. The euro is flat, one US dollar to one euro. You get the pound, a 118 to one US dollar, and you get the yen at 138, almost 139 to one US dollar. Our phone number's<PHONE_NUMBER>. Give us a call, folks. One note's goin' on in your world, in the world of the S&P's. Let's take a look at them. What do you got? Well, this is gonna get so intriguing, folks, and what I mean specifically by this is that we had another rejection of lower price. Now, this could be, wait till you hear this, man. This is one of my thought processes, because this is a trip. So the way this market is trading right here, this is normally how you get into a longer-term low. And what I mean specifically is this, is that when you get, when you come down fast and furious, right? And then, you know, which we did, when we got into this, the spy got into 362, okay? Bottom line, you rally for three or four days, you come back down, it doesn't get into the low.  So, now here's the deviant pot. Is that looking at this market and looking where we're at, July 14th, right? It's like, okay, you know, when you look at the market overall, meaning each year by year, you're coming into the weakest pot of the year, right? And I'm saying, oh, okay. So, the question's gonna be, how far does this rally go? And we'll see how it hits.  Because if we take a look at the queues, what you're gonna see, the queues were down the most today, came back, roared back. You know, I guess you're talking about Apple. I'm gonna get into Apple, because Apple's an ABC up. Apple's taking out the B point, it's an ABC up, man. So, we go to the queues, we take a look at the queues. The queues had a low of 279 or 286. We go to the end queues, and then we take a look at those future contracts. What you're gonna see is, bottom line, you're down hard and fast. We're down at 511, and we're 260 points off of that level. Gold, we're gonna take a look at the gold market. What do you have with the gold? And you know, needless to say this morning, I mean, everything was smoked in a monster way. We take a look at gold. What do you have with gold? Gold got down to a price point of 16.95. You've done, this can't be, let me just look at this again. GC, GC, okay, so, there we go. 238,000 contracts, okay, so let's take a look at this. So, we get down to a price point of 16.95. You're at 17.07, and this had light of volume yesterday. Yesterday we had a big volume. We had 301,000 contracts yesterday. Today you get 238, and the bottom line is it's above 17.0450, and that's all it has to do to get to a higher price. Now, let's go to the good old US dollar. We take a look at the dollar, and what you're gonna see, the dollar's running in the markets, man. That's the bottom line. You're gonna take a look at this dollar, and you can see that when you're basically taking a look at it, that the correlation, and next segment, I'm gonna try to see if I can compare these. So you can see, just tick for tick. I mean, it's tick for tick. That's how this thing's going inside the US dollar. Bottom line is that we made it up to that 109.294. Just backed off a little, and you can see just a back off of a little. So watch this, folks, okay? A back off of just a little on the S&P. What it did is bring the S&P from 3723, okay, to 3788. You're talking about 50 points as it backs off. So, you know, we'll see on a longer term basis, okay? Is this dollar gonna basically back down a bit and give us some relief in the marketplace? Well, it's telling me that it is, and the reason I'm saying that is the aspect that we don't have volume on the currencies, folks, okay? We do have volume in the indices, and the indices don't have any more sellers. So, when you don't have any more sellers, it takes hardly any buying to get to much higher price. That's how it basically shakes out. The wild card in my own head at this particular point is the aspect, okay, are we talking about like a real long-term bottom, meaning two or three months? And we might, you know, because what I've found is that the more that you keep trying to jam bottoms and you cannot get them, and you cannot get any more selling, that's when you get yourself bounce that's pretty extraordinary. And we know by looking at the market, it's all about this good old US dollar. You know, so we'll see what there's gonna be any kind of intervention, and the intervention, I suspect, would be coming in the yen, because you can take a look at the yen. I mean, the yen, you talk about weakness, okay? It's just so weak, it's extraordinary, man. I mean, look at this, and so pitch this, if you had, you know, yen. I mean, it went from four months ago, it was at 114.  That is, they are losing money hand over fist as to what they can buy. Our phone number is<PHONE_NUMBER>. We have the Dow Industries down 150. NASDAQ's off three, S&P's off 12. We'll come right back. Time of glooming inflation, we are purchasing powers eroded, there's no better place to protect your harder and money than in gold. VISTA Gold's flagship asset is the Monk Todd Gold Project in the Northern Territory of Australia. This is Australia's largest undeveloped gold project. We are talking a world-class gold project in a tier one mining district. This is a large-scale, low-cost project with significant existing infrastructure in a politically safe and friendly mining jurisdiction. VISTA Gold just completed the Monk Todd Feasibility Study, which resulted in a 7 million-ounce gold reserve in a 16-year mine life. All of this combined with the approvals of all major operational, as well as environmental permits. This distinguishes Monk Todd as an attractive, devious party, ready development stage gold project. VISTA Gold trades on the New York Stock Exchange under the symbol VGZ. Are you looking for a way to consistently add winning trades to your portfolio? Tom O'Brien is here to help. Tom O'Brien has been successfully trading markets for over 30 years. A frequent contributor to TD Ameritrade Network and CNBC, Tom O'Brien founded TFNN over 20 years ago to help educate investors just like you. Tom's Daily Market Newsletter, Market Insights, is published every morning when the market's open to give you the competitive informational edge you need to succeed. These newsletters are packed full of Tom's advanced technical analysis and are geared to deliver comprehensive strategies for a successful portfolio. Get Tom O'Brien's newsletter, Market Insights, today, and try all of our products and newsletters 30 days risk-free with our money-back guarantee at TFNN.com. TFNN, educating investors. Everything in the universe is governed by the Fibonacci sequence. This mathematical principle is responsible for everything from the most aesthetically pleasing artwork to patterns in the stock market. To stay on top of stock patterns you can take advantage of, sign up for the Fibonacci 24-7 newsletter at TFNN.com. When you subscribe, you'll get a weekly report from veteran day trader Larry Pezzavento on stocks you need to pay attention to, and you can trust Larry's analysis. After all, he's got 45 years experience as a day trader. Larry will also provide daily charts, videos, and data on the key markets that he's tracking. Expect notifications from Larry on market movement you need to act on at any time. First-time subscribers also get a 30-day money-back guarantee. If you're not satisfied, let us know and you'll get a full refund within 30 days of signing up. Subscribe to the Fibonacci 24-7 newsletter today. TFNN.com, educating investors. Toll free at 1-877-927-6648 internationally at<PHONE_NUMBER>. Welcome back, folks, down. Dow industry is right now down 182.  So let's go over to Apple. You get a small ABC structure on the way up, and that says it's NASDAQ, you know, bottom line is that you get, well, apples and everything. So here you go. So what you have is this. Your B point on this is the 64 million shares, 147, 55. What we had done is that on Monday you took it out with 77 million, okay? Bottom line, that's a complex ABC.  So your A point on this is 13372, let's call it 134. So you got 13.8 B, which is gonna get you a generated price projection of 155. And right now you're at 148.  And what I mean specifically is the deviance of it when the timeframe that we're in, the way that we've just tested, yeah, I'm basically bullish that you're gonna get quite a swing here. There's gonna be a lot of opportunity in this period, it's very unusual folks when what we just did the last couple days. It's hard, it's very hard to basically get into lows and have a contraction of volume and a rejection. And when you do it twice in a row, it's basically, you know, inside time of the trade, it's basically, we're in the speculative business, but it's basically proven there's no more sellers men. And when there's no more sellers, it is so easy for the market to rise, it's incredible, okay? Because once the market really generates and looks that, okay, there's no more sellers, and you know, bottom line, I'm trying to figure out myself and send myself, oh, this is gonna be intriguing, man, because we know that J.P. Morgan came out with numbers. Numbers weren't great, but guess what, J.P. Morgan there, that's rejecting that lower price also, J.P. Morgan gap lowered, it was at 106, it's at 107.92, you know? It has volume, so that could get tested, no doubt about that. You're into, you know, the March down draft. You know, the J.P. Morgan had a gap at 105.10. And I suspect that's gonna get filled. You know, we got so close to today, 106.06, I suspect that baby's gonna get filled, but guess what, with J.P. Morgan, that just may be the, where it wants to go, that's what it comes down to. Some of the higher volume equities that we have out here today, you get a till raise, let's get some movement, that's up 42 cents, they're hoping, I don't know, the market's hoping that the federal, well yeah, the Democrats folks are gonna put a bill up, I believe later today, and what that bill's gonna be about is decriminalizing pot on the federal level. The bottom line is not gonna pass because what happens is there's two Democrats, senators that are against it, and all the Republican senators that are against it. So, you know, there's no doubt those pot stocks, I wouldn't be biting on them because the bottom line is that it's quite clear that they need 60 votes. And the amazing part about it is that you have a pot being like the same as heroin or cocaine, which is insane. But you can see how the insanity of our government, this particular point, evidently, they could kill us. Oh, they're so far in a fricking whack, it's unbelievable. But anyway, you know, if you take a look at, let's go take a look at Tillray, because this thing's not going through, man. That's the bottom line. If you take a look at Tillray, no, not that one. Tillray right now is up 42 cents, so it's up over 10 and a half percent. Yeah, it's a bounce, I wouldn't be touching it. Because it's certainly, if it's up on that news, that news was broke earlier today, three or four hours ago. So, let's go take a look at Amazon. Amazon, of course, finished its prime day. Amazon right now is trading out flat, 110 bucks. If in fact we're going to have this type of bounce, the bottom line is that you want to start hunting some of these stocks, man, because I can see Amazon. See that high volume swing point at 128, 99? That's game, man. That's game, it failed, that's where we got down. So it failed there, but guess what? You're in a consolidation, that's where it can go. And what we will get is that, if we finish, we can finish right here, and that's about as bullish as you can get. If we basically, if this spy can turn, I mean the e-mini contract can turn positive, then you really got to watch out because what ends up happening is that the amount, of folks that may be on the wrong side of this could be pretty amazing. Now what's gonna happen here, and the, this is pretty cool, okay? Now this is gonna be so cool watching this, okay? So I have the e-minis up right now, and the e-minis, the number 3797, that's the, we're right at it, right? Now the question's gonna be, let me see this, 10, okay, we're five minutes, four minutes into this bar, but if this bar got taken out with more than 29,000 contracts, you get an ABC structure up.  So we're gonna know in the next, you know, five, 10 minutes, 10, 20 minutes, if in fact it can, oh, look at the NQs are out in ABC up. Okay, hold up, I got these number, man. Oh, this is cool, this is, this is really cool. Okay, so let me pull this up. Okay, so, that's the bar that took it out. Let's take that out, let's take that out. Let's say I'm, okay, one second, yeah, let's take that out. So if I do 621, let me do the top of this first. 752 minus 621 is 131 plus 732, 863, okay. So we're already in an ABC structure up inside the NQs to 11,863, and you're very well, we may have a bigger one when we're approaching the last highs that were up here, which is the 832, and this is where intraday folks, when you get either down drafts or up drafts like this and you're at highs or lows, it's really cool, ripping to the 10 minute, that's what I do anyway, to see if you're gonna get ABC structures intraday because they'll give you a much better gauge as to okay, man, this is gonna make it, this is gonna make topside, this is gonna make the certain number, let's pull this way, the probability gets much higher that that is where you're gonna make it. And of course, in this particular case, it's about the most deviant thing you can do. And the market's job is to take the most amount of money, away from the most amount of people and the least amount of time and it's the most efficient mechanism that I've found in order to do that. Except maybe crypto is probably the real one to just wipe everyone out. Stay right there folks, come right back. If you wanna take advantage of this sector, now is the time to subscribe to my Gold Report. The Gold Report is a comprehensive look at the metal sector as well as the markets that move gold which is the currency and bond markets. New subscribers get a 30 day money back guarantee so you have nothing to lose. Every Monday morning I publish the Gold Report with coverage of gold, silver, bonds, the XAU, HUI, GDX, as well as more than 30 different mining equities. To see for yourself the types of profitable trades that are recommended within the Gold Report, sign up now by visiting TFNN.com. Don't miss out on the next great gold trade. Sign up today. Sharpening your skills as an investor is like getting better at playing a musical instrument. You have to practice, sure, but you also need excellent instruction from experts. At TFNN, you'll get advice and guidance from the authority in technical market analysis and it's not just dry tedious text either. TFNN airs live financial content streamed live on TFNN.com and TFNN's YouTube channel with Tiger TV. Live every market day from 8.30 a.m. to 4.00 p.m. Eastern. For free, each host is an experienced trader and gives their take on the market while taking calls and questions live from around the world. From the moment the market opens until the closing bell sounds, Tiger TV has eight different shows with expert hosts to help you make the right moves with your money. Watch online at TFNN.com or on TFNN's YouTube channel and become the investor you were born to be, TFNN. Educating investors. TFNN is excited about our new software charting program, The Art of Timing the Trade Chart. In collaboration with Tom O'Brien and using his best-selling book, The Art of Timing the Trade, Your Ultimate Trading Mastery System, David White has programmed an outstanding piece of software that will complement any trader's methodology. Using this first-of-its-kind program, The Art of Timing the Trade Charts allows you to scan thousands of stocks for Fibonacci formation setups, including guardleafs, ABCs, butterflies, and much more. The Art of Timing the Trade Charts is designed to help you when scouring the markets for stocks just beginning to form the trading patterns that many investors spend days, weeks, or even months searching to find. And right now we're offering licenses available at only $79 a month. We are so confident that you're gonna love this new charting software that will even give you a 30-day unconditional money-back guarantee. Don't miss out on this incredible new piece of software. Get your copy of The Art of Timing the Trade Charts today by visiting TFNN.com. This segment is brought to you by Think or Swim. For more information, just click the Think or Swim banner on the front page of TFNN.com. Welcome back, folks, dao. Dao is down 100 to get the NAS, took up 23. S&Ps are down five and a half, and let's get over there and see if we got a little small ABC structure up inside the E-minis because that 10-minute box should be over, yeah, 330. It's just getting over right now. ES, okay, so, okay, here we go. Let's see, we got 28,000. Look at that, oh man, okay. Load them up, load them up, man. Here we go, this is actually blowing my mind, folks. Let's do this number, man. Oh man, you're gonna love markets. Okay, we got an ABC up. That's the bottom line, just took it out. So, 37.97, 37.97 minus 37.46. So, you got 51, 37.85 plus 37.85, oops, 37.85 plus 38.36, okay. So, where's this number up here? Let me look at this yesterday, 38.73. That's 38.7, where's 38.76? There it is right there, so 38.76. After the downdraft yesterday, folks, at 8.30 prior, that spike that come up there at one o'clock in the afternoon, that's the bottom line, that's the ABC structure up. So, you know, this is what the cool thing is, so watch how this works. The cool thing here is that when you have an intraday ABC structure up, you know, it can do it right now, bottom line, but you always wanna, if it doesn't do it right now, you wanna keep that in mind for tomorrow morning, because that's just how they play out, man. And, yeah, take it for what it's worth, that's the real bottom line. Let's go into the NDX100 and take a look at the strength, versus the weakness out here.  Taken away from it, Moderna's down 6%, you got Airbnb up 4%, and match.com is off 3.4%. So, bottom line is that it's so intriguing, the whole thing's so intriguing, man, it's a mind-blower actually. Notes and bonds, let's go over to the note and bond market here for a second, so we take a look at the active contract. We're at 2.9 on the 10-year, and this here, so you got 149,000 contracts going into 2.1, so this is, the way the 10-year shook out here. Yesterday would happen as this, you know, yesterday it really wasn't, yeah, no, it was basically, it was an update because it took that buyer out and it didn't hold price. That being said, you come down, the 10-year rejected the 117-28, it has 1.4 million contracts versus 2.1, so what that is saying is that the bottom line is that the 10-year wants higher price, so this is really gonna get twisted, man, because the bottom line is that when you take a look at the Fed, the Fed fund rate, okay, the next, let's go take a look at the next meeting. The next meeting, I believe is gonna be the, let's see, calendar, July 27th. That's gonna come pretty quick, actually, yeah. July 27th is the next meeting, and the way that, we'll see how this is set up, you can see the deviants in many of these moves out here. It's just the opposite of what your figure is gonna happen in a longer-term basis, particularly because inflation's still flying, so. We gotta take a look at the Microsoft. Microsoft, yesterday that was coming down hard and fast. Bottom line is that we did, yeah, look at this, man, this is nice, this market's setting up, man. So, picture this. Microsoft came down with a vengeance on Monday. 35 million shares, yet what it didn't do, okay, it didn't break your swing.  We came down with 35 million. Well, then yesterday, you went to a low, a low at 29 million. Now, watch this, folks. You can learn so much about time in the trade on this. This is insane, folks, okay. Look at this, 19 million, so you have no more sellers. It went to a low, a low. You only had 19 million, it rejected a lower price. That's saying Microsoft's gonna go right back top side, man. And the top of that consolidation is out there at 277. And bottom line is that if you get some of these NDX stocks going, what happens is that the NDX rules. That's the real bottom line. If the NDX starts moving, that'll basically get up and go to spy. And I suspect what we're gonna see here, this move is really gonna be deviant. And one of the targets, we just reminded me, let's happen to Mark, hope everything's straight out there, that in the downdraft of 2008, there was a move between March and May that took people by shock before then we went from May all the way back down to October. But the bounce was big, that's the reality. The bounce was big. And that's what happens in these markets, particularly after you've basically come down so fast and you're sold, sold, sold, sold, sold again, sold, sold, sold again. So it's gonna be about this dollar, though. That's the real bottom line. If we go over to the Euro, we take a look at the Euro again. The Euro right now is laying at a buck. One buck to one buck, you know? I'm not quite, yeah, hit 99.50 today. And you know, see, this is the, we're at the point, not necessarily in the Euro, but we're at the point in the yen that they very well could intervene inside the marketplace.  I don't gotta put it up on a longer term basis. And when you see this, this is like absolutely amazing to me that what you have here. So look at this, man.  So the yen, since 2011, has lost 50,000% of its buying power versus the US dollar. Now that is so twisted, it's unbelievable. But guess what? That's who it is. That is one number, man. And the yen is approaching the highs that were generated in 1998, which was 147. There's a spike up there at 147, you know? But I just can't picture. And what does happen, folks, is this, with currencies, what happens is that if you're not leaving Japan, it really doesn't matter, okay? Just like if we weren't leaving the United States, it really wouldn't matter to us how strong the dollar is, cause we're in dollars. What does matter is that if you're leaving Japan and coming to the United States, okay, forget it, man. It's way too expensive. Now, if we're going over to Japan, but guess what? If you're going to Japan, compared to in 2011 when I brought that up, at half price, that's where this comes in in a very large way. Let's go take a look at Intrepid Potash, IPI. So the lowest 27, the high is 121, which was established in April. So this thing crashed. For sure, holy cow. Let me bring this back. Look at this. 35. Yeah, hit 35. This will probably hit 35 again. That's how this is shaken out, man. That's a crash. That's a crash. Stay right there, folks. Come right back. Vista Gold owns and operates the largest undeveloped gold project in Australia, the Mount Todd Gold Project. Vista Gold just completed their feasibility study, resulting in a seven million ounce gold reserve. Vista Gold has all major permits approved and has retained CIBC Capital Market Assistance in evaluating alternatives and in completing an accretive transaction. Vista Gold trades on the NYSE American and TSX under the ticker symbol VGC. Vista Gold executing a strategy to create shareholder value. TFNN has been your trusted source of analysis for bonds, metals, stocks, commodities, and options for years. And we are happy to announce that we are bringing that same caliber of analysis for the Forex Market. Teddy Keckstat has 30 plus years of experience in Forex trading, commodity risk management, Forex hedging, volatility, and so much more. Teddy releases his weekly Tiger Forex report every Monday morning with elite coverage of all major currency pairs, including the DXY, Eurodollar, Pounddollar, Aussie Dollar, Dollar Yen, Dollar Swizz Frank, and so much more. Teddy will recommend specific trades when the market presents them and provide updates throughout the week when warranted. For the month of July, inaugural members to the Tiger Forex report will receive 25% off the monthly subscription for as long as they're subscribed. Just use promo code Teddy25 to lock in the added savings. This offer is good only for the month of July, so do not miss your opportunity to save on the Tiger Forex report. TFNN, educating investors. Biotech is booming, but for how long? Whether you think the Biotech bull has room to run or has run its course, trade LABU or LABD. Directions daily S&P Biotech three times bull and bear ETFs. Visit directioninvestments.com slash biotech today. An investor should consider the investment objectives, risks, charges, and expenses of the direction shares carefully before investing. The prospectus and summary prospectus contain this and other information about direction shares. To obtain a prospectus or summary prospectus, please contact direction shares at<PHONE_NUMBER>. The prospectus or summary prospectus should be read carefully before investing. An investment in the funds is subject to risk including the possible loss of principal. The funds are designed to be utilized only by sophisticated investors such as traders and active investors. Distributor, four-side fund services, LLC. TFNN has launched the Tiger's Den, hosted at Discord. TFNN has been educating traders for more than 20 years with live programming hosted by a variety of professional traders during market hours, the Tiger's Den. Available to all tigers and tigeresses for just $1 for the year. There's no catch or added costs when you join our community of traders. Sign up today and become a part of this educational community of traders. Just visit the front page of TFNN.com. This program is brought to you by Vista Gold. Traded on the NYSE American and TSX under the symbol VGZ. I'm O'Brien. Welcome back, folks, down.  And let's go over and we take a look at the NFLX, a good old Netflix. So Netflix made the deal with Microsoft and ordered to sell its new advertising platform and bottom line. You can see Netflix is not doing the thing. Netflix is trading out at 173 and this, no doubt, just has like, this is heavy, man. So you're down from 700. It's amazing what some of these equities have done. Yeah, this thing can go lower, man. I mean, this thing can lay out at 129 pretty quick. It's amazing that it actually stopped right here. The 195 was, you know, some basically support and it's laying out where it traded for about five months in the 2017. 2017, you know, you're talking about some big numbers there. Let's go take a look at the SMHs. For the SMHs, folks, when the NDX, inside the NDX, the SMHs rule the NDX, the NDX rules the market. Bottom line, if you take a look at these SMHs right now, what you're gonna see is that you're going into a swing and you're coming into the swing and you're coming in with volume, man.  So now you didn't need much volume for this B point, but that happens. That's what happens flat out. And, you know, bottom line is you took it out today. Real question, we'll see what it's gonna close above it, which is 209.90. You need the volume of 2.7 million to 3.9. Now it's a small one, but guess what? You start adding these things up and that's how points are gonna start getting added up. So the B point on this is, let's call it 2.10. Your A point is, I'll call it 1.90, because it's 1.89. So you get 10 bucks. That's 209. Oh, hold it. Oh, right, 209. How was that? Yeah, yeah, I see. Interesting, let me do that again. 209, oh no, it's 20, 20, there we go. Yeah, it's 20 points. Okay, so it's 2.20. Yeah, we're 2.10, 2.20. And the next swing point up there is 2.18. So that is basically saying that, you know, the SMHs are gonna start moving to higher price. And you know, bottom line, as I just said, what runs what? Let me get the, was it XLE, wanted to look at it? XLE, okay, so let's go to the XLE for a couple of other tigers. The XLE, okay, what we have out here is that this is the energy sector.  And in this one here, folks, if you can figure out where exon and chevron's going, that's really, that's, well, that is 44% of the XLE. We take a look at this XLE right now. So you rejected, let me put this on a weekly for a second. 68, okay, right there, 6706. Okay, it's not bad. So what you've done is this. Last week you came down with 148 million. This week, last file, we've done 118. What you're doing though is you're coming into 293 million. And the high of that is 6706. So what this is going to be, this will be a rejection also of the highs of that bar. The problem, I mean, you can trade this, but I suspect this is only gonna be good from 67 to 72. Because this bar that came down is good, it's brutal.  It doesn't mean you won't go through it. But what happens is that when you get, when you have large bars, folks, most of the time what happens is you'll drive into it, you'll back off a bit. If it's going to basically drive into it again, you'll know as it backs off, the volume will contract. It'll drive into it again. And a bar like that's gonna take three or four different times to drive into before it gets to higher price. But let's go over to the oil contract, because that's gonna be predicated on the oil contract. So the oil contract right now is at 9635. And this rejected price at 90 bucks today. It's hard to believe that it was moving six and eight dollars a day, but it is. So let me get this. We did 346,000 contracts. You did have light of oil. Yeah, well, oil can bounce. Yeah, I don't see a big bounce here. I don't. You know, 97 is game, but guess what? You're already at 9634. Because it's the same deal. The way that we had come down on price on the 5th of July was brutal. You know, we went from 111 down into this 97. So that's, well, actually let's go look at Exxon and Chevron also. So we do have with Exxon, Exxon out here. This is down off the high of 104, 105. Yeah, back into, I see this is back into a breakout area, okay. Yeah, I wouldn't touch any of these, man. This is volume off the highs. So when I just put this up on a weekly, I'm gonna do a Chevron too, but this is volume off the high, man. When you get volume off the high, it's not good. CVX, let's take a look at this. This one's down from 182. Now, this doesn't have as much volume off the high. It's not that bad actually. It's not like Exxon. And yeah, see this though with Exxon, Chevron is under this 138 area. It's gotta close inside the 138. It's coming into that breakout. See the bottom of the breakout is 138, you're at 136. So that sets up that, yeah, you can get down to like 122. It looks to me, the XLE, Chevron, Exxon, they looks to me like they had their run, man. That's how this thing is shaken out, man. That's how it looks. We go take a look. Let's take a look at that Unity software again for ya. This thing was down yesterday. You're down 40 cents right now. Yeah, don't touch this thing, man. Now, this is, I can see what the Tigers were talking about is that it didn't break the swing number one. The swing is at 29.30. You've only hit 31.73. And that can be a good setup. But I don't like when you have two gaps already. Most of the time when you have two gaps, particularly you have an equity that's losing money handed over fist. You get two gaps, folks. Normally you're gonna get three gaps. And the last one's normally an exhausting gap. What an exhausting gap means, let's say the first gap is 10 bucks, second gap could be 10, 12. The exhausting gap is gonna be double that amount. Dow industrial's right now trading down 118. And as that gets flat, S&P's down 11. Stay right there, folks. Come right back. The technology around us is changing every day. With so much happening, it can seem impossible to keep up with all the information. David White's investment newsletter, the Technology Insider, is designed to give you all the information you need to understand the technology that shapes today's markets and tomorrow's future. David White has made his living staying on the cutting edge of technology. His weekly newsletter will give you specific recommendations for value tech stocks, as well as entry prices, target prices, and stops to set for each trade. Dave delivers his weekly newsletters every Friday with updates throughout the week. You can get the Technology Insider at tfnn.com for only $37.50. Sign up for Dave's newsletter, the Technology Insider, and get an inside look at everything the technology sector has to offer. Try it risk-free today with our 30-day money-back guarantee. Tfnn, educating investors. You might think that if you want to be successful at trading in the stock market, you're going to need a crystal ball. After all, it's impossible to predict the future, right? Like any endeavor in life, before you decide it's impossible, get some advice from the experts. You might find that it's not so impossible for daily market overviews that give you direction on the key indices, selective stocks, and commodities. Subscribe to the opening call newsletter at tfnn.com. The opening call newsletter is written by Basil Chapman, creator of the trading methodology known as the Chapman Wave. The Chapman Wave up-down sequence gives you an edge in identifying price turns, finding the peaks and valleys in stock prices.  First-time subscribers also get a 30-day money-back guarantee. If you're not satisfied, let us know and you'll get a full refund within 30 days of signing up. tfnn.com, educating investors. Everything in the universe is governed by the Fibonacci sequence. This mathematical principle is responsible for everything from the most aesthetically pleasing artwork to patterns in the stock market. To stay on top of stock patterns you can take advantage of, sign up for the Fibonacci 24-7 newsletter at tfnn.com. When you subscribe, you'll get a weekly report from Veteran Day Trader Larry Pezzavento on stocks you need to pay attention to, and you can trust Larry's analysis. After all, he's got 45 years' experience as a day trader. Larry will also provide daily charts, videos, and data on the key markets that he's tracking. Expect notifications from Larry on market movement you need to act on at any time. First-time subscribers also get a 30-day money-back guarantee. If you're not satisfied, let us know and you'll get a full refund within 30 days of signing up. Subscribe to the Fibonacci 24-7 newsletter today. tfnn.com, educating investors. Don't forget, you can listen to tfnn live on your mobile device 24 hours per day. Go to tfnn.com and hit Watch Tiger TV. Welcome back, folks, to Dow. Dow Industries right now, down to 128, and Aztec is a flat S&Ps are down 10. Is it going to do it? Is it going to make it? Don't forget tomorrow, folks, okay, tomorrow morning. Bottom line, if you do trade these S&Ps, the market in general, the SPY, the future, it has a small ABC structure up. So that, you know, what happens is the way I trade these things is that if you do have a flat market tomorrow, you already, at the open, you already know that you have a small ABC structure up.  We take a break and then we start trading again. So we'll see where the real thing shakes out. Let's go to the Dachshund, Germany for a second. Take a look at this, because Europe's a mess. And the bottom line, yeah, see, Europe, that didn't break the low either. So Europe is at the bottom line. So Europe, is that the March low? No, the 22nd low. The February 22nd, February of this year low. That's where the Dachshund is. Let's take a look at UKX, let me see, the FTSEE. The FTSEE is at 7,039. Yeah, see, this is going to get interesting, man. So the European equities that are digging into the February of this year's lows. So it's going to be pretty cool, because what ends up happening, of course, is that they have four to five hours ahead of us. You can take a look at where they are in the morning and that's going to tell you quite a bit. You know, that's how this baby shakes out, man. So I think you, you know, there's a lot of opportunity out here, man. That's a real bottom line. Don't forget, we got our man, Teddy Kegstad. He's got a new Tiger Forex letter and a currency letter. And folks, the currencies are running everything, man. You want to learn currencies, you know. You don't have to jump into everything at once, but you want to learn currencies. I'm telling you, man, because what ends up happening at certain points, bonds run everything and currencies run, and it goes back and forth. Right now, the currency market is running the whole ball of wax. Always remember, folks, the bear can claw your heart out, the bull can run you over, and thank God, there's always another trade. Health, happiness, and prosperity. Have a great night, have a safe night. Come back and visit Tommy tomorrow morning. Kicks us off nine in the morning. Great show, folks. Real, look at him, folks.


Welcome everyone. Let us recall what we were talking about in the last class. We had started talking about what is called the Oman's model of incomplete information. What is this comprise of? The Oman's model comprises of three different elements. The first element is the state of nature or the states of the world. This is what you can say is the true configuration of the world that we are modeling comprises of a set of players 1 to n. And for each player we have a partition of the states of the world. So, this fancy fi was a partition of capital y for player i. And the understanding was that a player can only know the state of the world up to the granularity that is described in the partition. So, for instance here in this case this omega here was the true state of the world. And the player who had the black partition for him the he can only know omega up to the up to the detail that is available in this element of the partition that he know what it what that means is he only knows that one of these yellow states is the true state of the world but does not know which one. Likewise for another player let us say player j he could potentially have a different partition in this case these are blue colored boundaries as are were his partitions and when omega is the true state of the world all he knows is that one out of this green shaded region is the true state of the world but not which one with specific one. This was the limitation that players had about knowing anything about the world. So, this is how they would this is the the the the coarseness to which they know what the actual state of the world is. Then from there we then said let us we took this example of two players where John and Paul we knew that the everyone it was John was colorblind and everyone knew that there were two possibilities for Paul that he could be colorblind or not colorblind but only Paul knew that whether he was colorblind or not. And there were three cars that they were looking at potentially which could have one there were three cars in the race and one of them could have won the race the colors of the cars were purple red and green and the definition of a colorblind person that the was that the colorblind person cannot distinguish between red and green. So, with this definition we decide we said let us let us try to understand what the partitions for the various players are. So, for John for instance it meant that because John did not know if Paul was colorblind or not John could not distinguish between these two states which is colorblind and purple and not colorblind and purple he only could tell if the car was purple or not purple. So, he could not distinguish between this these two states and likewise he could not distinguish between any of these four states which is the four the four states here capture whether Paul was colorblind or not and whether the red or green car won right. So, all four combinations are present here and he could not distinguish between any of these. So, his partition therefore of the states of the world which comprise of these six states were partitioned into two sets for John into this the first set here and the second set here. For Paul the states of the world were partitioned into 1, 2, 3, 4, 5 different subsets because Paul had a different way of knowing about the world. So, Paul could not distinguish when he was colorblind he could not have distinguished between the red whether the red car winning or the green car winning, but he could have told if the purple car was winning. So, he could have when the when he was colorblind and the purple car won he could he could actually say he could know that the purple car won when he is not colorblind he can of course distinguish between every possible color. So, that is that therefore gives you these three separate sets right. So, this is how we set up the partitions for this particular example. We then from there went to what it meant for a player to know about something first what is that something that we are that that a player needs to know. So, that something is what is what we called an event. So, an event is simply a subset of why when we are talking of knowing things we are talking of knowing about events. So, it event is then therefore a collection of the states of the world and you know an event if you know that any one of them has happened if you if you know for sure that that at least one of them has occurred. So, what does it what does that mean? So, here in this case we took this event A as this event what is this event stand for it was the event that Paul is colorblind and the green car as one or Paul is not colorblind and the green car as one or Paul is colorblind and the purple car as one right it is one of these three possibilities this for us was an event. So, which succinctly you can say that either the green car one or Paul is colorblind and the purple car one this is what this event captures and the and then we said what do these players know about this event in each in a particular state of the world. So, we fixed a particular state of the world which was omega star in this case the omega star is that Paul is colorblind and the red car is one and then we asked what does each player know about this event and our definition of knowing was that we said player I knows event A in state of the world omega star if you look at the partition that of the element of the partition that omega star lies in for player I that element is completely enclosed in the event A. So, you have these various partitions and every omega belongs to one element of this partition right. So, in particular for the yellow for the for when this is your true state of the world omega for the black partition it omega belongs to this yellow this yellow region here and for the blue partition it belongs to this green region here. So, what we know so what since a player knows that actually one of that element has occurred but not which one if that if that element if that entire element is a subset of A then we say that that player knows A ok. So, what he what he knows is therefore that he definitely knows that certainly one of these has occurred and therefore from there he knows that at least one of the elements in A has occurred ok. So, and therefore he knows A ok. So, so Fi of omega remember is the element of the partition that contains omega ok. So, remember this definition as well. So, this is where we ended last time. So, this gives us basically a framework for saying when does player I know an event in a certain state of the world ok. So, you have a certain state of the world omega star you have a player whose who has a certain partition which tells him what he could know and then from and you have a certain event to which we are referring. So, when does player I know that event in the state of the world omega star that is answered by looking at this inclusion. So, he knows player I knows event A in state of the world omega star if Fi of omega star is a subset of A. So, based on what we have discussed so far we can talk of questions like this we can discuss when does does player I know A in state of the world omega star right this kind of thing can we can be discussed. But we said that we would also want to discuss something more complex like for example, does player I know that player J knows something right these are the sort of things we would want to ask like for example, does Paul know that John knows that Paul knows the color of the car and so on. Now, how do we answer something like this for this that is where this framework is so elegant that it comes up with a with in one shot away for you to generate not just one level of knowing or knowledge about a particular event but also about what others know about that particular event and so on an entire hierarchy can be generated from this very easily. So, define what we call this is what is called a knowledge operator. So, let A be an event and let I be a player now we define Ki of A as follows. Ki of A is defined as the states of the world omega in which player I knows A. So, it is all possible states of the world in which it is true that player I knows A. So, this is set of states of the world player I knows A. So, as you change A you will of course, keep getting different sets Ki of A. Ki of A is the set of states of the world in which player I knows A. Now, what kind of a set is Ki of A? So, if A is any event what sort of a set is Ki of A? Yes, exactly. So, if A is an event then Ki of A is also an event it is a subset of the states of the world. So, an event is simply a subset of the states of the world. So, if A is an event then Ki of A is also an event. So, then once it is an event I can again talk of for any other player I can ask whether that player knows this event. So, for example, I can now ask does player J know Ki of A? So, what does this mean? For example, when would player J know Ki of A? He would know Ki of A if f J in state of the world omega star. So, he would know Ki of A if f J of omega star is a subset of Ki of A. So, if f J of omega star is a subset of Ki of A then it means that player J knows Ki of A and but what is Ki of A itself? But what is Ki of A itself? Ki of A itself is the event that player I knows A. So, if p J, player J knows Ki of A in omega star then he effectively knows that player then effectively we can say that player J knows that player I knows event A. So, this is effectively so which would mean that player J knows that player I knows A. But now I can go recurse on this further. I can talk so effectively. So, now I can talk of one more level. For example, I can write KJ of Ki of A. What is this? What is KJ of Ki of A? It is also an event and what event is it? It is the set of events. It is a set of states of the world in which player J knows that player I knows A. The set of one second. States of the world in which player J knows that player I knows A. What was the question? Where are you referring to? Is the element of the partition for player I? That element has to be completely enclosed in A. Yes. So, FJ of omega star is a subset of Ki of A is simply saying that player J knows Ki of A in state of the world omega star. So, what is it that you what was the other thing? Yes. So, you can also write that. So, let me write that actually here. So, KJ of Ki of A is the set of states of the world in which player J knows that player I knows A. So, if player J knows that player I knows A in state of world omega star. That is basically saying that omega star belongs to KJ of Ki of A. That is actually this is equivalent. So, is this clear? So, therefore, now what this knowledge operator is basically doing is now from here you can simply you can create any length of such hierarchies of knowledge. What does player I know about player J knows about player L about player M and so on and so forth. You can create any lengths of these hierarchies and answer and have a framework to discuss all of them because all of this knowledge of something has been reduced to just one which has been reduced to simply an event. So, let us do an example further. So, let us say for example, let us take the simple event. The event is that the purple car one. So, color blind comma purple and not color blind comma purple. So, this is the event that the purple car has one. So, now let us ask ourselves. So, go back to the partitions that we had written. So, what are the states of the world in which Paul knows that the purple car has one? What are the states of the world in which Paul knows that the purple car has one? Paul knows that the purple car has one in this event. Why? Because this is a subset of the event A that I am referring to. He also knows that the purple car has one in this event. Because the element of the partition that this event lies that this state of the world lies in itself is a singleton element and that is a subset of A. Likewise, this state of the world is in a singleton element which is also a subset of A. So, the states of the world in which Paul knows that the purple car has one is both CBP and NCBP. In both of these states Paul knows that the purple car has one. In summary, Paul knows that the purple car has one when the purple car has one. Is that clear? Because the states of the world in which Paul knows that the purple car has one is actually the A itself is the set A itself. What about John? When does John know that the purple car has one? Again, when this is when the true state of the world is this and when the true state of the world is this? Because these are both the element of the partition that they would end up defining is this and that is actually A itself. So, it is a subset of A. So, therefore, in both of these states of the world John knows that the purple car has one. So, once again John knows that the purple car has one when the purple car has one. So, Kp of A is equal to Kj of A is equal to A. Now, what is this? There is something really interesting that has happened here that the states of the world in which these in which both these players in which each of these players know this event A is equal to A itself. So, now what would happen if I ask one more level of the knowledge hierarchy? When would Paul know that John knows that the purple car has one? It would be A itself and when would Paul know that John knows that Paul knows that purple car has one? It would be A itself. So, in other words, if I can actually construct this I can have any such say Paul knows that John knows that Paul knows that John knows etc etc etc of A that would be equal to A. Now, what does this remind you of? This means that the event A which is the event that the purple car has one is common knowledge. So, this event A is in fact common knowledge. So, this is an example of how certain things can end up becoming common knowledge. For an event to end up becoming common knowledge, it would have to be there for that this sort of recursion eventually ends up with an invariant set. I will define what it means to be common knowledge in a moment, but this is let us just to give you an example of how this whole thing works out because we started off with common knowledge as our motivation. So, this is an example of that. So, actually maybe let us just do that itself now. Let us just define also welcome everyone. Let us recall what we were talking about in the last class. We had started talking about what is called the Oman's model of incomplete information. What does this comprise of? The Oman's model comprises of three different elements. The first element is the state of nature or the states of the world. This is what you can say is the true configuration of the world that we are modeling comprises of a set of players 1 to n and for each player we have a partition of the states of the world. So, this fancy fi was a partition of capital Y for player i and the understanding was that a player can only know the state of the world up to the granularity that is described in the partition. So, for instance here in this case this omega here was the true state of the world and the player who had the black partition for him he can only know omega up to the detail that is available in this element of the partition. What that means is he only knows that one of these yellow states is the true state of the world but does not know which one. Likewise, for another player let us say player j he could potentially have a different partition. In this case these are blue colored boundaries as were his partitions and when omega is the true state of the world all he knows is that one out of this green shaded region is the true state of the world but not which one with specific one. This was the limitation that players had about knowing anything about the world. So, this is how they would this is the coarseness to which they know what the actual state of the world is. Then from there we then said let us as we took this example of two players where John and Paul we knew that the everyone it was John was colorblind and everyone knew that there were two possibilities for Paul that he could be colorblind or not colorblind but only Paul knew that whether he was colorblind or not and there were three cars that they were looking at potentially which could have one there were three cars in the race and one of them could have won the race the colors of the cars were purple red and green and the definition of a colorblind person that the was that the color blind person cannot distinguish between red and green all right. So, with this definition we said let us try to understand what the partitions for the various players are. So, for John for instance it meant that because John did not know if Paul was colorblind or not John could not distinguish between these two states which is colorblind and purple and non not colorblind and purple he only could tell if the car was purple or not purple. So, he could not distinguish between this these two states and likewise he could not distinguish between any of these four states which is the four the four states here capture whether Paul was colorblind or not and whether the red or green car won right. So, all four combinations are present here and he could not distinguish between any of these. So, his partition therefore of the the states of the world which comprised of these six states were partitioned into two sets for John into this the first set here and the second set here for Paul the states of the world were partitioned into one, two, three, four, five, five different subsets because Paul could call Paul had a different way of knowing about the world. So, Paul could not when he was colorblind he could not have distinguished between the red whether the red car winning or the green car winning but he could have told if the purple car was winning. So, he could have when the when he was colorblind and the purple car won he could he could actually say he could know that the purple car won. When he is not colorblind he can of course distinguish between every every possible color. So, that is that therefore gives you these three separate sets. So, this is how we set up the partitions for this particular example. We then from there went to what it meant for a player to know about something. First what is that something that we are that that a player needs to know. So, that something is what is what we called an event. So, an event is simply a subset of why when we are talking of knowing things we are talking of knowing about events so it event is then therefore a collection of the states of the world and you know an event if you know that any one of them has has happened. If you if you know for sure that that at least one of them has occurred. So, what does it what does that mean? So, here in this case we took this event A as this event what is this event stand for? It was the event that Paul is colorblind and the green car as one or Paul is not colorblind and the green car as one or Paul is colorblind and the purple car as one. It is one of these three possibilities this for us was an event. So, which succinctly you can say that either the green car one or Paul is colorblind and the purple car one this is what this event captures and the and then we said what do these players know about this event in each in a particular state of the world. So, we fixed a particular state of the world which was omega star in this case the omega star is that Paul is colorblind and the red car is one and then we asked what does each player know about this event and our definition of knowing was that we said player i knows event A in state of the world omega star if if you look at the partition that of the element of the partition that omega star lies in for player i that element is completely enclosed in the event A. So, you have these various partitions and every omega belongs to one element of this partition right. So, in particular for the yellow for the for when when this is your true state of the world omega for the black partition it omega belongs to this yellow this yellow region here and for the blue partition it belongs to this green region here. So, what we know so, what since a player knows that one of that element has occurred but not which one if that if that element if that entire element is a subset of A then we say that that player knows A. So, what he what he knows is therefore that he definitely knows that certainly one of these has occurred and therefore, from there he knows that at least one of the elements in A has occurred. So, and therefore, he knows A. So, Fi of omega remember is the element of the partition that contains omega. So, remember this definition as well. So, this is where we ended last time. So, this gives us basically a framework for saying when does player i know an event in a certain state of the world. So, you have a certain state of the world omega star you have a player who is who has a certain partition which tells him what he could know and then from and you have a certain event to which we are referring. So, when does player i know that event in the state of the world omega star that is that is answered by looking at this inclusion. So, he knows player i knows event A in state of the world omega star if Fi of omega star is a subset of A. So, based on what we have discussed so far we can talk of questions like this we can discuss when does does player i know A in state of the world omega star. This kind of thing can we can be discussed, but we said that we would also want to discuss something more complex like for example, does player i know that player j knows something. These are the sort of things we would want to ask like for example, does Paul know that John knows that Paul knows the color of the car and so on. Now, how do we answer something like this. For this that is where this framework is so elegant that it comes up with in one shot a way for you to generate not just one level of knowing or knowledge about a particular event, but also about what others know about that particular event and so on. An entire hierarchy can be generated from this very easily. So, define what we call this is what is called a knowledge operator. So, let A be an event and let i be a player. Now, we define Ki of A as follows. Ki of A is defined as the states of the world omega in which player i knows A. So, it is all possible states of the world in which it is true that player i knows A. So, this is set of states of the world player i knows A. So, as you change A, you will of course keep getting different sets Ki of A. Ki of A is the set of states of the world in which player i knows A. Now, what kind of a set is Ki of A? So, if A is any event what sort of a set is Ki of A? Yes, exactly. So, if A is an event then Ki of A is also an event it is a subset of the states of the world. So, an event is simply a subset of the states of the world. So, if A is an event then Ki of A is also an event. So, then once it is an event I can again talk of for any other player I can ask whether that player knows this event. So, for example, I can now ask does player J know Ki of A? So, what does this mean? For example, when would player J know Ki of A? He would know Ki of A if fj in state of the world omega star. So, he would know Ki of A if fj of omega star is a subset of Ki of A. So, if fj of omega star is a subset of Ki of A then it means that player J knows Ki of A and but what is Ki of A itself? But what is Ki of A itself? Ki of A itself is the event that player I knows A. So, if pj player J knows Ki of A in omega star then he effectively knows that player then effectively we can say that player J knows that player I knows event A. So, this is effectively so which would mean that player J knows that player I knows A. But now I can go recurse on this further. I can talk so effectively. So, now I can talk of one more level. For example, I can write kj of Ki of A. What is this? What is kj of Ki of A? It is also an event and what event is it? It is the set of events. It is a set of states of the world in which player J knows that player I knows A. The set of one second. States of the world in which player J knows that player I knows A. What was the question? Where are you referring to? Here. Is the element of the partition for player I? That element has to be completely enclosed in A. So, fj of omega star is a subset of Ki of A is simply saying that player J knows Ki of A in state of the world omega star. So, what is it that you what was the other thing? Yes. So, you can also write that. So, let me write that actually here. So, kj of Ki of A is the set of states of the world in which player J knows that player I knows A. So, if player J knows that player I knows A in state of world omega star. That is basically saying that omega star belongs to kj of Ki of A. This is actually this is equivalent. So, is this clear? So, therefore, now what this knowledge operator is basically doing is now from here you can simply you can create any length of such hierarchies of knowledge. What does player I know about player J knows about player L about player M and so on and so forth. You can create any lengths of these hierarchies and answer and have a framework to discuss all of them because all of this knowledge of something has been reduced to just one which has been reduced to simply an event eventually. So, let us do an example further. So, let us say for example, let us take the simple event. The event is that the purple car one. So, color blind comma purple and not color blind comma purple. So, this is the event that the purple car has won. So, now let us ask ourselves. So, go back to the partitions that we had written. So, what are the states of the world in which Paul knows that the purple car has won? What are the states of the world in which Paul knows that the purple car has won? Paul knows that the purple car has won in this event. Why? Because this is a subset of the event A that I am referring to. He also knows that the purple car has won in this event because the element of the partition that this state of the world lies in itself is a singleton element and that is a subset of A. Likewise, this state of the world is in a singleton element which is also a subset of A. So, the states of the world in which Paul knows that the purple car has won is both CBP and NCBP. In both of these states Paul knows that the purple car has won. In summary, Paul knows that the purple car has won when the purple car has won. Is that clear? Because the states of the world in which Paul knows that the purple car has won is actually the A itself, is the set A itself. What about John? When does John know that the purple car has won? Again, when the true state of the world is this and when the true state of the world is this because these are both the element of the partition that they end up defining is this and that is actually A itself. So, it is a subset of A. So, therefore, in both of these states of the world John knows that the purple car has won. So, once again John knows that the purple car has won when the purple car has won. So, Kp of A is equal to Kj of A is equal to A. Now, what is this? There is something really interesting that has happened here that the states of the world in which these, in which both these players, in which each of these players know this event A is equal to A itself. So, now what would happen if I ask one more level of the knowledge hierarchy? When would Paul know that John knows that the purple car has won? It would be A itself and when would Paul know that John knows that Paul knows that purple car has won? It would be A itself. So, in other words, if I can actually construct this, I can have any such thing Paul knows that John knows that Paul knows that John knows etc. of A that would be equal to A. Now, what does this remind you of? This means that the event A which is the event that the purple car has won is common knowledge. So, this event A is in fact common knowledge. So, this is an example of how certain things can end up becoming common knowledge. For an event to end up becoming common knowledge, it would have to be therefore that this sort of recursion eventually ends up with an invariant set. I will define what it means to be common knowledge in a moment, but this is let us just to give you an example of how this whole thing works out because we started off with common knowledge as our motivation. So, this is an example of that. So, actually maybe let us just do that itself now. Let us just define also