We have used the MSNE characterization theorem to find an algorithm to find MSNE and in this module we are going to prove that fundamental theorem which actually gave rise to that algorithm. So, let us reiterate what was this MSNE characterization theorem. It says that a mixed strategy profile sigma i star sigma minus i star is a mixed strategy in Nash equilibrium if and only if these two conditions hold for every player. The utility of that player when it plays the pure strategy Si and the other players pick the mixed strategy sigma minus i star is going to be the same for all these strategies in the support of sigma i star. And the second condition was that this utility is going to be at least as much as the utility any other strategy which is not in the support. So, let us try to prove that. In the proof we will need two results which I will state as observations. These are very simple observations about convex sets and its properties. So, the first thing is that we are going to find the maxima over all the possible mixed strategies of player i. So, what we are saying is that find the maxima over all mixed strategies. So, I can write this expression is this utility expression as follows. So, this is the utility at Si, so strategy pure strategy Si and the other players are picking the mixed strategy sigma minus i and what I am doing is I am looking at sigma i Si and taking it sum over all Si in capital Si. This is what this utility means. Now we are trying to maximize it. Now think about this problem we are trying. So, sigma i Si we know that those are probability distribution. So, this utility for Si there are let us say some k number of strategies in the strategy set of player i. So, all of this individual Si living in that set will give some utility for this player and now we have this k numbers and we are mixing them taking a convex combination of those numbers and we are trying to find out which convex combination will give you the maximum. Do you think of any convex combination which gives you the maximum? It is the maximum value among those k numbers where you can put all the mass and that will give you the maximum convex combination and that is exactly what is written on the right hand side. If you just pick that strategy which gives you the maximum, so this max is nothing but the maximum over all those k numbers that is going to be the maximum value here. No matter in whatever way you want to find this maxima if you try to find out the probability distribution that maximizes this expected utility of this player that is equal to the case where you are putting all the mass on the maximum value. So, therefore this maximum value should be same. So, this is the first observation. This is a very simple observation yet quite important and interesting. The second thing is that let us look at the quantity. So, again the left hand side is the same, but we have replaced only the strategy of the other players as sigma minus i star. So, what is sigma i star? So, sigma i star and sigma minus i star is the mixed strategy Nash equilibrium. So, what is from the definition of it? What do we know? We know that if we use this, this is going to be the maximum over all the other strategies or over all the other mixed strategies. Now, we have already seen this equality to hold from the first observation. Now, what we are going to argue is that this is equal to the maximum where you are looking at the strategies not on the whole set of Si, but just the support of sigma i star. And why is that true? So, suppose this is not true. So, suppose this maxima is not coming from this set from the support, but it is living somewhere outside. So, let us say there exists some Si prime which is not in the support and that has the maximum value. Then what we can say is that we can put the entire probability mass on that strategy profile, on that strategy of that player i. And that because that is going to be the maximum that is certainly is going to give you a higher utility than the expected utility of sigma i star. Because when you are looking at this sigma i star, you do not have the maxima there. All the numbers that you have on the support, all the utilities that you have for the strategies on the support, even if you take the convex combination of that, then also whatever you are going to get at max is the maximum value there. But you already have some other strategies i prime where you have a strictly larger utility value. So, if I put, so this is a different mix strategy, of course, it is a pure strategy. But yes, of course, that is a degenerate mix strategy where I am putting all the probability mass to that Si prime and that is going to give you strictly larger utility, expected utility than the expected utility that you can get on the strategy profile sigma i star sigma minus i star. So, that contradicts the fact that sigma i star, sigma minus i star is a mixed strategy Nash equilibrium. So, therefore, that cannot happen. So, whatever we have assumed is wrong. So, therefore, max must leave within the support of sigma i star. Okay. So, with these two observations, we are now in a position to prove this result. So, there are two parts of this proof because it is a necessary and sufficient condition. First, we will prove the forward direction. That means we will start with assuming that sigma i star sigma minus i star is mixed strategy Nash equilibrium and show these two conditions hold. And in the reverse direction, we will assume these two conditions to hold and show that sigma i star sigma minus i star is a mixed strategy Nash equilibrium. So, the first direction is a forward direction. So, here we are given this is a mixed strategy Nash equilibrium. And we have just used, we are just going to use the definition of mixed strategy Nash equilibrium. So, this is by the definition we are maximizing with respect to sigma i, all the sigma i's. Now, we make use of the first observation that it is going to be the max over that strategy space. So, maximizing the convex combination is equivalent to maximizing with respect to the strategy, the pure strategy. And now we also know from that second observation that this is going to be the maximum over the support of sigma i star. So, let us call this the first observation, first subresult. Now, also we know by definition of the expected utility, you can write this utility at the mixed strategy profile in this form, the expected utility form. Now, because this the support, the definition of this support is that beyond this support, all this for all the strategies sigma i star has zero probability mass. So, I can just remove those strategies from this summation. And I can just reduce the summation to be the support of sigma i star. Now, I know that this first subresult and the second subresult are equal because the left hand side is the same for both these things. Now, what is this two things saying? It is saying that the max, so this one, so you are looking at the same set of delta sigma i star, the support and the max is exactly equal to the positive weighted average. So, notice this term, it is a positive weighted average. So, all these terms now here is positive. So, there is no zero number there. When can you take a convex combination of all these numbers which is equal to the max of those numbers? It can happen, think about it, it can happen only when all the numbers are equal. So, there is no other way. If there is a, there are dissimilar numbers, one is slightly larger than the other numbers, then the mix of that, the convex combination of that will certainly be the, certainly be smaller than that maximum number. So, it can never happen unless your, all the numbers are exactly equal. And that is exactly what we are going to conclude from here. When all the values are same, that means all this ui si sigma minus i star are going to be the same for all the strategies which is living in this support of sigma i star. So, that essentially proves the first condition, the condition 1. Now, let us go to the condition 2. So, how can we prove condition 2? So, what condition 2 is saying is that for all the strategies which are in support, the utility at that strategy is going to be large, at least as much as the utility for a strategy which is not in the support. So, assume for contradiction that is not true. So, there exists some strategy which is in the support and some strategy si prime which is not in the support, such that the inequality is flipped. So, then again we can use the very similar argument because now we have a competing probability mass. So, we have a strategy of si prime where the utility is going to be larger, then the probability mass that I was earlier giving. So, consider the strategy. So, sigma i star is giving different probability masses to different strategies. So, there is si and there are other strategies also in the support of sigma i star for player i. So, because this si prime has a larger utility now, so whatever probability mass you are giving on this si, you just shift that probability mass that is sigma i star of si to si prime. And because and then you take the expected utility and the because of the fact that this is strictly larger, you can show that the expected utility that you get in the so called mixed strategy Nash equilibrium is going to be smaller than the expected utility that we are just now constructing. And therefore, that is a contradiction to the case that it is a mixed strategy Nash equilibrium, you already found a competing strategy profile which gives you a larger utility than the mixed strategy Nash equilibrium. So, therefore, it is not possible. So, what we assume for contradiction is not true. So, definitely it must be the case that this inequality should be greater than or equal to. So, that proves the condition number two. Now, we have proved the forward direction that if it is an MSNE, the condition one and condition two are getting satisfied. Now, we will have to prove the other direction that is if we are assuming that these two conditions are true, condition one and condition two are true, then it must be a mixed strategy Nash equilibrium. So, how can we prove that? So, just reduce the notation a little bit because we know that condition one is saying that for all the strategies in the delta sigma i star, the support of sigma i star, it is going to be same. So, we can just represent that by a number which only depends on sigma minus i star. So, let us call that Mi of sigma minus i star. And also what we know is that that is going to be at least as large as any other strategy outside that and also inside. Inside it is equal, outside it is at least as much as the utility of those strategies. So, we can without loss of generality say that this it is the maximum over all the strategies in the strategy space of player i. So, this comes from the condition two. So, these two things are essentially rewriting the same conditions.  As before we are writing this as a probabilistically weighted sum of the utilities as at those pure strategies where the pure strategies are living within the support of sigma i star. And this is by the definition of sigma i star for all other cases it is going to be zero. So, we do not really need to care about that. Now, we know that because these numbers are all equal. So, we are looking at only the strategies which is within the support. So, these numbers these infinities are going to be the same which is equal to Mi of sigma minus i star. So, this is a constant it is not it is not dependent on Si anymore. So, we can take that constant term Mi of sigma minus i star outside the summation and the summation will always be equal to 1. So, therefore, it just becomes Mi of sigma minus i star. Now, we use the condition two. So, we are just looking at the maximum value of Ui of Si sigma minus i star. And using the previous observation we know that this when you are trying to maximize something over. So, the maximum value that you get when you are maximizing over this individual strategies or individual numbers that is equivalent to maximizing with respect to a mixed strategy, maximizing with respect to a convex combination of the same things. And that is definitely going to give you this expression here because this is the maximum value of over all the sigma i's. This is going to be larger than if you pick any specific sigma i and this should hold for all sigma i in the mixed strategy space of player i. And that is essentially what we wanted to prove. So, we wanted to prove that if you look at sigma i star sigma minus i star and player i makes a unilateral deviation to sigma i for all such sigma i's, this inequality is going to hold. And we have assumed an arbitrary player i and therefore, this inequality should hold for all i in N. And therefore, we have proved that this particular strategy profile sigma i star sigma minus i star is a mixed strategy Nash equilibrium. So, that proves both the directions of this characterization theorem.


By special recording, General Mills, makers of Wheaties, breakfast of champions, and Cheerios, the Oats cereal ready to eat, presents, The Lone Ranger! A fiery horse with a speed of light, a cloud of dust, and a hearty high old silver! The Lone Ranger! With his faithful Indian companion, Toto, the daring and resourceful mask rider of the plains led the fight for law and order in the early western United States. Nowhere in the pages of history can one find a greater champion of justice. Return with us now to those thrilling days of yesteryear. From out of the past come the thundering hoofbeats of the great horse Silver! The Lone Ranger rides again! Now here's the Lone Ranger. You'll hear it said that someone was born to the saddle.  But remember, like anyone else, he had to learn to ride. He probably took many spills doing it. He's good because he practiced, rode every time he had a chance. In anything, not just riding, the winners of the fellows who train, champions are made, not born. I'll agree, Lone Ranger, but is there anything besides practice a person can do to help his training? Absolutely! Eat the right foods. I'd like to pass along something the old pioneers knew. Wheat is one of the best all-around foods you can find for staying power and energy. Today's champions agree with the pioneers about wheat, Lone Ranger. Champions choose wheaties. There's a whole kernel of wheat in every wheatie's flake. Champions are made, not born. Get on your way with wheaties. Breakfast of champions! Slugger Andrews had escaped from territorial prison with the help of a friend who waited with horses. The two men rode all night and most of the next day, finally they walked their horses along a trail a few miles from the town of Hilton. I reckon we're safe enough now, Slugger. You sure took an awful chance, but you made it. Yeah, but I had things figured out close, Jack. I planned that break for months. Watching and checking the location of the guards and all that. What's the idea of heading for Hilton? Got any definite plans? Only living relatives lives there. My cousin Maisie. She's a widow and runs the Hilton Cafe. Is she Maisie Andrews or the Big Chance Cafe? Yeah. Maisie just about runs that town. Yeah, I know. I know. But what about your plans? Hey, Jack, you remember when you were my cellmate a year ago? I told you about the masked man and the red skin who were the cause of my being in prison? Yeah. You said it was an ombre called the Lone Ranger. That's right. And at my trial, the red skin testified against me. Well, Maisie was there and got a good look at him. What about it? I had a letter from Maisie saying she'd spotted that red skin in Hilton a few times during the past couple of months. That means that masked man is snooping around the territory too. Well... Maisie's worried for fear they might get a line on the gang that's working for her. You better be careful. They say that masked ombre is mighty smart. Yeah, maybe. But I'm going to try to get him out of the way before he can interfere with Maisie or me. That's why I'm heading for Hilton. Now, come on. Let's make some time.   Maisie Andrews was going over accounts in her office at the back of a cafe around midnight when she heard a knock at the door. Open that. Who's there? Open up, Maisie. This is Slugger. Slugger, you made it. Come on in. Hiya, Maisie. Meet a friend of mine, Jack Kerry. Hi. Slugger, I'm glad to have you and Jack, but I got to thinking the law might come here looking for you. I'm going to shave off my beard and dye my hair. Jack and I'll hold up somewhere outside of town until I can get a line on that masked man in Indian. There's a room upstairs where you can stay tonight. After you shave and dye your hair, we'll sit down and have a long talk. Long talk. The following evening, Tahoe, Indian friend of the Lone Ranger, stopped at their camp in the nearby hills. Oh, Scott. Oh, clever. Maisie, clever. Any news in town, Tahoe? Ah, me here, fella, break out of prison. Oh, who was the man who escaped? Him, Slugger Andrews. Fella, we helped catch two years ago. Slugger Andrews? Ah, them say him killed guard to get out of jail. Now that he's warned for murder, he'll not hesitate to kill others. Not right. His cousin Maisie Andrews runs a cafe in Hilton. Ah, me know that. I think Slugger's too smart to go there. But just in case, it's worth our while to make sure. Eh, Kimasabi. Yes? Me go to town tonight, watch at cafe. Keep out of sight yourself, Tahoe, as much as possible. We might recognize you. Oh, Scott Hopper, eh? Early that evening, Tahoe went to Hilton and entered the cafe. He stood in the shadows at the back and watched the others into place. He listened intently when Maisie came from her office with a stranger and went to the bar. Tahoe, this is a friend of mine. Give him whatever he wants and charge it to the house. Yes, ma'am. Sure, nice of you, ma'am. Don't mention it. When you go back to the cabin, be sure to give my message. Don't worry, I'll tell him what you said. If there's anything he needs, tell him to be... I forgot something, Jack. Come back into my office a minute. Tahoe hadn't noticed the sudden penetrating look Maisie had thrown his way, and a short time after she and Jack went into the office, the Indian left the cafe and mounted at the hitch-rack. He's got a PC, fella. Get him up, Scott. As Tahoe rode from town and headed toward the Lone Rangers camp, Maisie Andrews and Jack hastily mounted and followed.  They made sure they were not seen in the bright moonlight.   If you'd like to hear, Maisie saw the Indian who testified against you. Good. He was in the cafe this evening. Five. Maisie and I followed him the other side of town and found out were he in the Mastumbria camp. Well, the rest ought to be easy. I know, you better be careful. Maisie says if you want help, she'll get you them in. All right, come on, let's go. While Tahoe told the Lone Ranger what he heard and saw at the cafe. So, Maisie told the strangers to give someone a message when he went back to the cabin. Ah, her say, if him need anything, then her stop talking and then go back into office. It's possible that her cousin, Slugger Andrews, is in hiding outside of town. Ah. I think I'll try a plan that may help us find out. What you do? I'll disguise myself as a Mexican, take Silver and leave him at the livery stable. And I'll hire another horse. Why you do that? I'll go to the cafe tonight and apply for a job. That way I'll be able to find out about Slugger. That's plenty risky. Perhaps, but it's worth trying. You continue to camp here in case I need you, Tahoe. I'll work on this guy. Later that night, Slugger and his friend Jack were talking to Maisie in her private office. This is our big chance to get rid of that snooping masked man, Maisie. I don't want anything to go wrong. That's why I came here to talk it over with you. I'm glad you did, Slugger. Did you tell your men what this is all about, Maisie? All I told them was that they're to take orders from you. That you take them out to gun a couple of hombres who are trying to get a line on our gang. You won't have any trouble with those men. Good. I'll tell the men to meet you at the hitch-rack behind the cafe. And go out the back way and find that camp. I'll show them what it is. Don't get too anxious, Slugger. Move in easy, like, from all sides. And take them by surprise. Don't worry, Maisie. This is one job we're not going to slip up on.  But curtain falls on the first act of our Lone Ranger adventure. Before the next exciting scenes, please permit us to pause for just a few moments. Let's listen to a few seconds of a tune from one of those special Pweeties albums everybody's talking about. Ready? How's that for clear, crisp recording? Well, it's just a sample of what you get in these wonderful Pweeties albums. Each has eight tunes recorded on standard weight, 7-inch, 78 RPM records. But the most amazing part of it is this. You can get one of these tunes. Each has eight tunes recorded on standard weight, 7-inch, 78 RPM records. But the most amazing part of it is this. You can get one of these fine Pweeties record albums for only 25 cents and a Pweeties box top. Four big albums, playtime tunes, folk songs, old favorites, and popular tunes. Take your choice or even get them all for your record collection. Look for directions on the back of the special Pweeties record offer box at your groceries today. Now to continue. The Lone Ranger disguised as a Mexican left the camp and headed for town. Instead of taking a regular trail, he took a shortcut to save time and soon pulled to a stop at the livery stable where he left Silver. Later, he rode a Mustang to the cafe. Hold up, hold up. Here you go. Jeez, Samson, hurry. Well, I have come to see the owner of the cafe, Senor. You mean you want to see Maisie Andrews, Sam? Oh, the owner of the Senorita.  If you call her that, Senorita. These Senoras, Andrews. Is she here, Senor? She's in her office. That door right back there. Gracias. Come in. What do you want? The Senor in the cafe tells me, Senora, that you're the owner, no? Yeah, I'm the owner. So speak up. What's on your mind? Got a complaint? See, Senora, it is that the Senor did not say you were beautiful. Beautiful? Say, sit down, mister. Gracias, Senora. By Sunday, you fellas from across the border should know how to treat a lady. Oh, you flatterer, Senora. Why have you come to see me, mister? The name is Carlos. Juan de Matamoros, San Fernando. But perhaps you will honor me by calling me just Carlos. And alive. I'd never remember the rest of it, so Carlos it is. I'm Maisie Andrews, Mrs. Andrews. See, I have ear, you are a Senora. The fair ones are always taken, no? You said that as if it made a difference. But I... hold on, just why did you come in here, Carlos? Senora, I come to this town. I say Carlos, this is such nice town. I come into the cafe and say Carlos, this is a very fine place. This is where I shall work, no? So I tell myself, see, then I came to see you, Senora. You mean to say you made up your mind you were... By thunder, Carlos, you have a way with you. So you want to work here, huh? See, Senora. What can you do? You're good at handling cards? See, Senora, but I am also excellent at handling a gun, I'm told. Now you talk like a gun slinger. How do I know the law isn't chasing you? Many times, Senora, others will say that Carlos is perhaps the outlaw. I do not deny, I do not agree. But right now the law does not know I am here. I know as much now as I did before. I think I can use you, though, Carlos. Of course, if you're afraid of getting into trouble... Trouble? She is something I go hunting for, Senora. He keeps one on his toes, no? You'll do. Come see me in the morning. I'll let you meet some of the men who work for me. See, Senora, that's it. Slugger and the men followed the regular trail until they neared the grove in which the Lone Ranger's camp was located. Tato had rolled into his blanket on the ground and went to sleep while he waited for the Lone Ranger. Suddenly he was awakened by a whinny from his horse scout. Good morning. Tato walked the scout's side. Then carefully scanned the surrounding shadows for some sign of an intruder. He slowly drew his gun to be ready for trouble. Then a voice startled him. Freeze it in! Who's that? Drop that gun. The camp is around. Tato whirled in the direction from which the voice had come. He saw first one figure, then another move into the clearing until several men ringed the camp and were closing in on him. Realizing he had no chance against so many, Tato spoke. Me drop gun. The engine's here alone. That's right. Where's the masked man? Come on, speak up, engine. I'm not here. Where is he? Me not know. We'll soon loosen your tongue and get the truth.  No, we'll take him to my hideout, Captain. Jack, you go tell Major what happened.  Right. We'll keep this red skin covered and gun him if he makes a move to get away. All right, let's get to our horses and ride. Get along, Indians. At the café, the Lone Ranger disguised as a Mexican, south of the table talking to Maisie. Wait a minute, Carlos. Here comes one of my ma'am. Hey, Maisie, I came to... Oh, maybe we better talk in private. No, go ahead. Carlos is all right. He's gonna work for me. Oh, well, listen. Slugger sent me to tell you we got the engine, but the masked army wasn't at the camp. Too bad you didn't have sense enough to find out if the masked man was there before you moved in. When the Lone Ranger gets back there, he'll know something is wrong now. And I have ear of the masked man and his Indian friend, but not be good if you have armed the engine. The masked man will be... Ah, we didn't hurt the red skin yet. Oh, where is it? They took him to hideout. They figure on forcing him to tell where the masked man is. I've learned the engines do not talk if they do not want to, senor. Especially that one. We would get him to talk one way or another. Slugger want you to come to the cabin, man. I never reckon I might as well. If I don't, Slugger might get riled enough to plug the Indian for not talking. I'm sorry for the interruption, Carlos. What the... Am I not one of your workers now? Why should I not come along with you and meet the others tonight? Oh, perhaps I can help you find the masked man you seek. Come on if you want to. Jack, get my horse ready. All right, ma'am. I'll go get ready to ride, Carlos. Eh? Then I'll meet you up front in ten minutes. The Lone Ranger waited until Jack and Maisie left. Then he nonchalantly left the cafe. Once outside, he hurried into the shadows and quickly reached the back door of the sheriff's office. He stood a moment, scribbling a note on a small piece of paper, but she wrapped around a silver bullet. Then he pounded on the door. What's the matter? Who's there? I have bring a message for the sheriff. It is most important, senor.  The sheriff's busy and is dead. Well, I have reasons for not being seen. You give him this, please, senor. He's very urgent. Oh, all right. I'll give it to him, but... You better come in anyway. How do you, senor? Hey, wait a minute. Hey! You sure scurried out of sight in a hurry. Well, I'll see what the sheriff makes of this note he gave me. When Maisie and Jack came to the front of the cafe, the Lone Ranger was already winging beside the horse he had hired at the livery stable. The two men and Maisie mounted and took the trail leading to the hideout cabin. Later, they entered the cabin where Slugger and Maisie's three gun slicks were waiting. Taco was tied to a chair. I'm glad you got here, Maisie. Who's that with you? This is Carlos. He's working for me. Carlos, meet my cousin, Slugger Andrews. Oh, it's the last time I meet your cousin. Glad to meet you, Carlos. Look, Maisie, we haven't been able to make that Indian talk. He's showing honorary red skin. Even some slapping around don't have any effect. Yeah, I'm almost tempted to... Can you show your temper, Slugger? There must be some way to get him to tell where that mask man is. Well, I have the other way to make Indians talk, Maisie. You have? How? He's with the knife. They fear the knife at the back. Wait, I shall prove my point. See, I hold the knife in my hand. I walk behind the Indian. The others watched expectantly as the Lone Ranger walked slowly behind the chair to which Tato was tied with his hands and back. All of the crooks had holstered their guns. Well, Carlos, go on. We're waiting to see how you make out. If you get him to tell where the Lone Ranger is, I'll pay you a bonus. I did not think I'll fail, Maisie. Hey, what are you doing with that knife? The point of the knife drawn along the wrist, like so, Senor, is most effective. Let him alone, Slugger. Go ahead, Carlos. Maisie and the men were several feet away and in front of Tato, the Indians body hid the knife in the Lone Ranger's hand. He will soon break down and talk of me goes. Watch. As he talked, the masked man drew the keen blade across the rope which bound Tato's wrists. Tato winced as if in pain. Hey, look at that, the Indian's showing he's afraid. Keep it up, Carlos. Get him talking. Tato felt the ropes give. Then the Lone Ranger purposely dropped the knife to the floor. Oh, Carlos, I'm getting so clumsy. As the masked man bent to pick up the knife, he slipped one of his guns into Tato's hand. Slugger suddenly spoke out. I saw that, Carlos. I told you you're double crossing it. Reach all of you. We both have guns. That's right. Hey, that boy, Slugger, I knew it anyway. Now, gun him. Hold it. Now, will you? You not shooting? Oh, no! The Lone Ranger and Tato fired simultaneously as Slugger and one of the crooks started to draw, wounding them both. Hold it! That umbrake tricked me out. Get him. Got that gun, Maisie. Now, don't bite under our plug yet.  I think you would at that. Keep them all covered, man. I got your note, mister, with the silver bullet in it. We followed you out here, and we rode up quiet light till we heard shooting. That Slugger Andrew's all right. He's wanted this time for murder. Now, me hear him talk, Sheriff. Woman, leader, gang. Well, we'll take Maisie along with the rest of them. But get them all to talk, and when the truth comes out, she'll go to jail for a long time. I never thought I'd be fool enough to fall for flattering words from any, Andre. Then he came along, talking and looking like a Mexican and tricked me. How'd I hate him for that? You might have been a fine woman if you hadn't turned crooked, Mrs. Andrews. As Carlos would say, you have a certain amount of beauty and charm. Sheriff, I'll now leave now. We'll see you later after I remove my disguise. Adios, everybody. Adios. Oh, yeah. Hey, did you hear what he said about me? Beauty and charm. Ha! Look, you chunkhead. Maybe he didn't mean it. But it'll be something I can think about while I'm sitting in jail. I was dreaming of a hacienda in old Mexico an hour after I met Carlos. I mean him. Ha! Ha! Ha! Well, you'll have plenty of time to dream, Maisie. Plenty of time. You and Slugger should have learned long ago that you can't outthink or outdo the lone rangers. And Jimmy is eight years old. He is strong and he is bold. He can capture outlaws because he knows. He's got gold power from Cheerios. Yes, he's got gold. Because it's made from toasted oats all ready to eat with milk. And the gold power it gives you. You see, each spoonful of Cheerios and milk is packed with vitamins, proteins, and minerals. The very things your body needs for healthy nerves, good red blood, strong bones, and muscles. Yes, Cheerios is real muscle-building food. Have Cheerios every morning. Then you'll hear people say, He's feeling his Cheerios. The Lone Ranger, a copyrighted feature of the Lone Ranger Incorporated, is created by George W. Crandall, produced by Crandall Campbell Muir Incorporated, directed by Charles D. Livingston, and edited by Fran Stryker. The part of the Lone Ranger is played by Brace Beamer, your announcer, Fred Foy. The Lone Ranger is brought to you by General Mills every Monday, Wednesday, and Friday at this same time. Be sure to listen. This recorded program has come to you from Detroit. This is ABC Radio Network.


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