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Today, game theory applies to a wide range of behavioral relations, and is now an umbrella term for the science of logical decision making in humans, animals, and computers. Modern game theory began with the idea regarding the existence of mixed- strategy equilibria in two- person zero- sum games and its proof by John von Neumann. Von Neumann's original proof used Brouwer fixed- point theorem on continuous mappings into compact convex sets, which became a standard method in game theory and mathematical economics. His paper was followed by the 1. Theory of Games and Economic Behavior, co- written with Oskar Morgenstern, which considered cooperative games of several players. The second edition of this book provided an axiomatic theory of expected utility, which allowed mathematical statisticians and economists to treat decision- making under uncertainty. This theory was developed extensively in the 1. Game theory was later explicitly applied to biology in the 1. Game theory has been widely recognized as an important tool in many fields. With the Nobel Memorial Prize in Economic Sciences going to game theorist Jean Tirole in 2. Nobel Prize. John Maynard Smith was awarded the Crafoord Prize for his application of game theory to biology. Representation of games. To be fully defined, a game must specify the following elements: the players of the game, the information and actions available to each player at each decision point, and the payoffs for each outcome. These equilibrium strategies determine an equilibrium to the game. Games here are played on trees (as pictured here). An Introduction to Applicable Game Theory Robert Gibbons ame theory is rampant in economics. Having long ago invaded industrial organization. Nor is economics alone: accounting, finance.
Here each vertex (or node) represents a point of choice for a player. The player is specified by a number listed by the vertex. The lines out of the vertex represent a possible action for that player. The payoffs are specified at the bottom of the tree. The extensive form can be viewed as a multi- player generalization of a decision tree. The game pictured consists of two players. The way this particular game is structured (i. Player 1 . Next in the sequence, Player 2, who has now seen Player 1's move, chooses to play either A or R. Once Player 2 has made his/ her choice, the game is considered finished and each player gets their respective payoff. Courses » Economics » Economic Applications of Game Theory » Lecture Notes Lecture Notes Course Home Syllabus Calendar Readings Lecture Notes. Decision Theory: Lecture notes (PDF) 3: Representation of Games: Lecture. Suppose that Player 1 chooses U and then Player 2 chooses A: Player 1 then gets a payoff of . To represent it, either a dotted line connects different vertices to represent them as being part of the same information set (i. More generally it can be represented by any function that associates a payoff for each player with every possible combination of actions. In the accompanying example there are two players; one chooses the row and the other chooses the column. Each player has two strategies, which are specified by the number of rows and the number of columns. The payoffs are provided in the interior. The first number is the payoff received by the row player (Player 1 in our example); the second is the payoff for the column player (Player 2 in our example). Suppose that Player 1 plays Up and that Player 2 plays Left. Then Player 1 gets a payoff of 4, and Player 2 gets 3. When a game is presented in normal form, it is presumed that each player acts simultaneously or, at least, without knowing the actions of the other. If players have some information about the choices of other players, the game is usually presented in extensive form. Every extensive- form game has an equivalent normal- form game, however the transformation to normal form may result in an exponential blowup in the size of the representation, making it computationally impractical. Characteristic function form. The idea is that the unity that is 'empty', so to speak, does not receive a reward at all. The origin of this form is to be found in John von Neumann and Oskar Morgenstern's book; when looking at these instances, they guessed that when a union C. The balanced payoff of C is a basic function. Although there are differing examples that help determine coalitional amounts from normal games, not all appear that in their function form can be derived from such. Formally, a characteristic function is seen as: (N,v), where N represents the group of people and v: 2. N. It was initially developed in economics to understand a large collection of economic behaviors, including behaviors of firms, markets, and consumers. The first use of game- theoretic analysis was by Antoine Augustin Cournot in 1. Cournot duopoly. The use of game theory in the social sciences has expanded, and game theory has been applied to political, sociological, and psychological behaviors as well. Although pre- twentieth century naturalists such as Charles Darwin made game- theoretic kinds of statements, the use of game- theoretic analysis in biology began with Ronald Fisher's studies of animal behavior during the 1. This work predates the name . The developments in economics were later applied to biology largely by John Maynard Smith in his book Evolution and the Theory of Games. Game- theoretic arguments of this type can be found as far back as Plato. This particular view of game theory has been criticized. It is argued that the assumptions made by game theorists are often violated when applied to real world situations. Game theorists usually assume players act rationally, but in practice, human behavior often deviates from this model. Game theorists respond by comparing their assumptions to those used in physics. Thus while their assumptions do not always hold, they can treat game theory as a reasonable scientific ideal akin to the models used by physicists. However, empirical work has shown that in some classic games, such as the centipede game, guess 2/3 of the average game, and the dictator game, people regularly do not play Nash equilibria. There is an ongoing debate regarding the importance of these experiments and whether the analysis of the experiments fully captures all aspects of the relevant situation. Price, have turned to evolutionary game theory in order to resolve these issues. These models presume either no rationality or bounded rationality on the part of players. Despite the name, evolutionary game theory does not necessarily presume natural selection in the biological sense. Evolutionary game theory includes both biological as well as cultural evolution and also models of individual learning (for example, fictitious play dynamics). Prescriptive or normative analysis. Since a strategy, corresponding to a Nash equilibrium of a game constitutes one's best response to the actions of the other players . This normative use of game theory has also come under criticism. Economics and business. A common assumption is that players act rationally. In non- cooperative games, the most famous of these is the Nash equilibrium. A set of strategies is a Nash equilibrium if each represents a best response to the other strategies. If all the players are playing the strategies in a Nash equilibrium, they have no unilateral incentive to deviate, since their strategy is the best they can do given what others are doing. One or more solution concepts are chosen, and the author demonstrates which strategy sets in the presented game are equilibria of the appropriate type. Naturally one might wonder to what use this information should be put. Economists and business professors suggest two primary uses (noted above): descriptive and prescriptive. In each of these areas, researchers have developed game- theoretic models in which the players are often voters, states, special interest groups, and politicians. Early examples of game theory applied to political science are provided by Anthony Downs. In his book An Economic Theory of Democracy. In the Downsian model, political candidates commit to ideologies on a one- dimensional policy space. Downs first shows how the political candidates will converge to the ideology preferred by the median voter if voters are fully informed, but then argues that voters choose to remain rationally ignorant which allows for candidate divergence. Game Theory was applied in 1. Cuban missile crisis during the presidency of John F. Taking the simplest case of a monarchy, for example, the king, being only one person, does not and cannot maintain his authority by personally exercising physical control over all or even any significant number of his subjects. Sovereign control is instead explained by the recognition by each citizen that all other citizens expect each other to view the king (or other established government) as the person whose orders will be followed. Coordinating communication among citizens to replace the sovereign is effectively barred, since conspiracy to replace the sovereign is generally punishable as a crime. Thus, in a process that can be modeled by variants of the prisoner's dilemma, during periods of stability no citizen will find it rational to move to replace the sovereign, even if all the citizens know they would be better off if they were all to act collectively. In contrast, it is difficult to know the intentions of nondemocratic leaders, what effect concessions will have, and if promises will be kept. Thus there will be mistrust and unwillingness to make concessions if at least one of the parties in a dispute is a non- democracy. One example would be Peter John Wood's (2. Wood thought this could be accomplished by making treaties with other nations to reduce green house gas emissions. However, he concluded that this idea could not work because it would create a prisoner's dilemma to the nations. In addition, the focus has been less on equilibria that correspond to a notion of rationality and more on ones that would be maintained by evolutionary forces. The best known equilibrium in biology is known as the evolutionarily stable strategy (ESS), first introduced in (Smith & Price 1. Although its initial motivation did not involve any of the mental requirements of the Nash equilibrium, every ESS is a Nash equilibrium. RYOBI 3. 6V in Lawnmowers. 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To accomplish this we can use the mjw. PDF classes PDFGet. Page. Width function. Brian Vaughn (left) as Benedick and Victoria Adams as Beatrice in the Utah Shakespeare Festival’s 2003 production of Much Ado About Nothing. Much Ado About Nothing is a comedic play by William Shakespeare thought to have been written in 15, as Shakespeare was approaching the middle of his career. This will give us the full width of the page we then subtract 3. If you run the code you should see your PDF pop up with a beautiful header at the top of the page. Another fun thing is to create shapes in your PDF files. This can be used to create bar graphs or to highlight certain areas. Here is some sample code that creates a square. Notice that you can specify the line style and the line width. Also notice that there is no draw square function. Instead there is a draw polygon function. It takes one parameter, but that parameter is an array of points specified in x y coordinates. X being how far from left to right to draw the point and Y being how far from top to bottom. So in our example we are specifying 4 points (the four corners of the square). Point 1 is 3. 00 pixels to the right, 1. Point 2 is 4. 00 pixels to the right, 1. Point 3 is 4. 00 pixels to the right, 2. Point 4 is 3. 00 pixels to the right, 2. Next lets draw an ellipse. An ellipse is simply a circle that can be squeezed either vertically or horizontally. To define it correctly we have to use some mathematical terms. If you remember from geometry class a circle has a radius. The radius is the distance from the center of the circle to the edge of the circle. An ellipse has two radiuses. One is horizontal the other is vertical. So the code for our ellipse is this. Lets draw an ellipse. PDF. PDFSet. Draw. Color = vb. Yellow. PDF. PDFSet. Line. Color = vb. Black. PDF. PDFSet. Line. Style = p. PDF. The new line is the PDFDraw. Ellipse call. Its a very simple. However, you would be wrong. Instead the first to parameters correspond to the upper left corner of the square that bounds the ellipse. The next two parameters specify the horizontal radius and the vertical radius respectively. If this seems confusing just run the program and you will see what I mean. The x & y parameters for our ellipse are the same as the x & y parameters for our first point in the square so you will see how it works. If you run the program you should see this: Lets step back to text manipulation in PDF documents again. One thing you usually see in a professional document is the header like we did above. Another thing is usually page numbers in the footer. We can use the same logic we used for our header to add page numbers. I would like to add the numbers in the footer of the page on the right side, like most documents have. I'm not going to walk through how you can do this step by step, but here is the code for a visual basic subroutine that adds the page number to the bottom right corner of your PDF document. When you are done with the first page, simply call the PDFEnd. Page method. Next call the PDFNew. Page method to start the next page. Than just call the commands to add your text or shapes to the next page. You can do this as many times as you want. Don't forget to call the Add. Page. Number method on each page though. Another useful feature of PDF documents is adding bookmarks. Bookmarks allow you to jump from section to section in a PDF document easily. When the user views a PDF document with bookmarks, they are able to see a table of contents type tab on the left side of the screen. Note: if you want that pane to be visible by default you should add this line of code to the initializing section of your program. For instance lets add four bookmarks to our first page of our document. Call these anywhere in your code before you call the PDFEnd. Page method. The next parameter is the depth of this bookmark. Note: Start your depth at 0. The last parameter is the y position to where the bookmark will move the page. So the first call created a bookmark titled . The next call creates a bookmark titled . It has a depth of 1 (so it will be a child under our first bookmark) and it will scroll the page 3. If you run the program you will see all the bookmarks created like this screen shows. Another necessity to learn when creating PDF documents is how to add images to them. The mjw. PDF class allows you to add any . PDF document. If the image is in a different format you will need to convert it to . PDF file. However, if the image is a jpeg it is very easy to add it to the PDF doc. In the sample source code included with this tutorial you will see a logo. Below is the code to end our first page and to start our second page. On the second page we add our logo to the upper left corner of the page. We call the PDFImage function. The first parameter is the path to the jpeg file. The next two parameters are the x and y coordinates for the logo. The next two parameters specify the width and height of the image. These parameters can be left off and then it will just display the picture in its original size. You can also specify just the height or width and it will scale the other side of the picture to keep it in proportion. The last parameter is also optional, but it allows you to specify a web site to go to if someone clicks on the image. If you run your program now you will see a PDF file that has all the properties of a complete PDF document. Headers, shapes, images, and page numbers. Combining all these techniques you should be able to do just about anything you would want to. Download the Advanced PDF VB Tutorial source code to see the full sample. Why would I buy a PDF control? This is a common question. Although the mjw. PDF class does allow you to do some pretty cool things you still may wish to buy a control for one or more of the reasons below: You need to add other images besides JPGs on the fly. You want to compress or encrypt your pdf file. Send me an email and I can help you out. Otherwise if the mjw. Class works for you feel free to use it. I would still ask you to send me an email just so I know what your using it for. Note: The original source code for this class (before modifications) can be found here. Free Footed Baby Pants Pattern. As promised, today I’m sharing a free footed baby pants pattern! If you hate trying to keep socks on a baby’s feet as much as I do, you’ll love these little pants! They’re super comfy and will keep a baby’s feet nice and toasty. The pattern is sized for newborns, from 6- 1. Hattie is just over 1. In 6 easy steps, you’ll have a pair whipped up in no time. Let’s get started! What you’ll need: knit fabric – if buying by the yard, you’ll need 1/2 yd, which will be enough for several pairs of pants (but you might as well buy some extra for a Seafarer or Skipper Top ; ))1. Please remember this pattern is for PERSONAL USE ONLY.*When cutting pattern pieces out, fold fabric in half and cut each set of 2 front/back pieces at once through both layers. This will result in mirror images of each front/back piece.**Use a serger or a long zig zag stitch to allow the fabric to stretch once sewn. Always use a ball point needle when sewing knits.**All seam allowances are 3/8. Right sides together, pin and stitch each foot (top) piece to each pant front at ankle curve, matching ankle marks and stretching ankle curve to fit foot piece. The convex curve on the pant front ankle will stretch outward to fit the foot top piece. Right sides together, stitch each front piece to each back piece at inseams and outseams. Right sides together, place one pant leg inside the other, matching raw edges of front and back crotch curves. Pin and stitch crotches together with one continuous seam, from top of front to top of back (or vice versa). Turn each pant leg wrong side out. Right sides together, pin each sole piece to bottom of each leg opening, matching heel and toe marks. Stitch together, stretching bottom of leg opening as needed to fit curves of sole. Mark both elastic and top of pants in quarters and pin together, matching quarter marks and placing elastic close to raw edge of top of pants. Stretching elastic to fit top of pants, stitch elastic to pants with zig zag stitch, 1/4. Fold elastic and top of pants toward wrong side of pants. Stitch top folded edge in place by zig zag stitching around folded edge (stretching elastic to fit), close to edge of elastic. And that’s it! A cute, quick pair of baby pants that will guarantee snuggly toes : ). 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