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Joseph ThomasMath 101Professor PetersonFebruary 23, 2016John Nash Game Theory and EquilibriumThe name John Nash is a name that many people may recognize when it is said outload. The general public maybe familiar with this name due to a film from featured in 2001 called “A Beautiful Mind”. This is a biographical film about famous Mathematician and Nobel Prize winner John Nash (Howard). While this film observes various aspects of his life, especially his struggles with mental illness; it only briefly touches on his development of Game Theory and Equilibrium. Game theory is the main aspect that lead to John receiving the Nobel Prize Award in Economics in 1994 (). Now this is the everyday knowledge of John Nash, but how many people really understand what exactly his work accomplished. If I had to guess the number is rather low, and for this reason I have chosen to explore Game theory and specifically Equilibrium. The question is, are his theories still relevant today and what areas are it is used for.First it is important that we come to an understanding to what exactly game theory is. Game Theory in simple terms is the study of interactive decision making and, where the outcome for each participant or "player" depends on the actions of all (Sherrerd). In the most basic of terms it can be described as “what is the best reply for another person’s strategy.” Now within Game Theory we have a subject that is known as Equilibriums. Equilibrium by definition is the state at which opposing forces are balanced. John Nash’s goal was to figure out how to solve equilibria in zero sums games by focusing on an opponent’s strategy in order to find solutions to what their payoff is. The reason we focus on the strategy, is that your opponent’s payoffs are assumed to already be known because they are the in complete opposition of too yours. “Mixed Strategy Nash Equilibria” and in this requires us to consider what the opponents’ payoffs are as well as their strategies (Rosenthal). The technique for solving each these equilibriums is a little different. I want to attempt to explain exactly how to find a basic solution of Nash equilibriums, using a game created by a man named Von Neumann. This is a non-zero sum game called “Appendix Calculating a Nash Equilibriums”. This example is from the Book “A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature”This chart above is a nonzero sum game where Alice wins exactly what Bob loses and vice versa. The payoffs in this game are representing the amount Bob is going to Alice. This means Bob’s payoffs will be represented by a negative value of the number indicated on the chart. To calculate this equilibrium, we need to find the strategy that shows best possible payoff when the other player is also seeking the best possible strategy. In this case Alice choosing the bus is going to be represented by probability “p” and walking probability will be “1 - p”. Bob choosing the bus will be represented by probability “q” and walking probability will be “1- q”. Now I am going to attempt to walk you through solving this equilibrium and find the expected payoffs when Alice chooses the bus and walks. First we will start with the bus, and Alice’s payoff from the bus this is going to be multiplied by the probability that Bob plays the Bus which is represented as “3 multiplied by q”. Plus, her payoff when she chooses the bus and Bob chooses to walk will be multiplied by the probability that Bob plays walk which is 6 multiplied by (1-q). Now the second step to setting up this equilibrium is too find the expected payoff if Alice chooses the walk option. The payoff of Alice choosing to walk and Bob playing Bus is represented by multiplying the probability of Bob choosing to walk which is 5 times q; Plus, Alice’s payoff from walk when Bob also plays walk with the probability being 4 multiplied by (1-q). Now as an equation these two payoffs are represented by these two equations. Alice’s expected payoff with the bus is equal to 3q?+ 6(1 –?q) and the expected payoff for her with walking is equal to 5q?+4(1 –?q). Now in order to find the equations for bob we need to remember what I said regarding your opponent’s payoffs. They are in complete opposition to yours. So what we need to is change the positive integers in to negative using the same two equations. So Bob’s expected payoff for the bus is represented as, –3p?+ –5(1 –?p). Then we have his expected payoff for walking which will represented as –6p?+ –4(1 –?p). Equations.Alice’s expected payoff for Bus = 3q + 6(1 – q)Alice’s expected payoff for Walk = 5q +4(1 – q)Bob expected payoff for Bus = –3p?+ –5(1 –?p)Bob expected payoff for Walk = –6p?+ –4(1 –?p)Now we need to use these equations to find the probability of each choice for both the Bus and walking by using skills from elementary algebra. In order for one of these probability to be a Nash equilibrium the probability must be equal. First we will start with Alice and set her two equations equal to each other. Now we need to distribute the 6 and the 4, combine like terms in order to isolate q and this is our result. This solution shows that Alice should choose to play the bus 1/2 of the time and choose walk 1/2 of the time. The payoffs are equal so we have an Equilibrium, and therefore Alice would want not want to change her strategy. Now if we take a look at Bob’s strategy we get an equation such as this.To solve the equation, we use the same strategy that we used for Alice by using elementary algebra skills.p = 1/4 This solution shows that Bob should choose the bus 1/4 of the time and choose walk 3/4 of the time. This is not an Equilibrium because the payoffs are not equal, so Bob would be better off choosing another strategy. I hope that made sense, it is a somewhat complicated concept but I hope you understand.Now that I have showed some calculation of Game Theory and Equilibrium, it is important to answer the question “How does this concept relate to real life?’ What kind of job can you get using these concepts? After doing research I was shown that Game Theory can lead to careers in economics or even computer sciences (Indeed). One of the aspects of computer sciences using game theory is developing games. One area that game theory is linked to that appeals to me as “Fantasy Sports” such as football where you can use Nash Equilibriums to find a strategy that minimizes your opponents possible score, which is essentially the same as the non-zero sums games I discussed earlier (Duronio). There are several other examples of game theory in everyday life as well, such as in real instate negotiations, or saving money to buy a car.I hope after presenting this information, you have a least a basic idea of Game Theory in relation to Nash Equilibriums. Despite the struggles John Nash had with his mental illness he was still able to come with an idea that touches on so many aspects of life. It was this reason he was given the Nobel Prize and is remembered as one of the best mathematicians to this day. If you are interested in learning more about John Nash’s personal life and struggles mentally, I highly recommend you check the Film “A Beautiful Mind”. Hopefully you found this information interesting or informative.Works CitedA Beautiful Mind. Dir. Ron Howard. Perf. Russell Crowe. Universal Studios, 2001. DVD. Duronio, Ben. "7 Easy Ways To Use Game Theory To Make Your Life Better." Business Insider. Business Insider, Inc, 2012. Web. 17 Mar. 2016. "Game Theory Jobs." . Web. 17 Mar. 2016."John F. Nash Jr. Biography." . A&E Networks Television. Web. 16 Mar. 2016. Rosenthal, Edward C. "The Complete Idiot's Guides | How to Articles for Everyone, Everywhere." Idiot's Guides. Web. 16 Mar. 2016. Sherrerd, John. "Game Theory Explained." PBS. PBS. Web. 16 Mar. 2016. ................
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