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Submitted by:ROLL NO: MEB16062, MEB16063,MEB16064SUB: INDUSTRIAL ENGINEERING AND OPERATIONAL RESEARCHDefinitionTerminologySimple example An introduction to GAME THEORYGAME THEORY INTRODUCTIONMEB16063Introduction: Game theory is the study of how people behave in a strategic situation. Where a strategic situation refers to actions which one person should take considering how others might respond to that action. Game theory is a mathematical model used by rational decision makers in a situation where there is strategic interaction among decision makers. So, it is a study of conflict and cooperation between intelligent rational decision makers. Usually decision makers face a conflict of interest situation and each of them aims to get the best benefit out of it. Situations like this occur in various day to day activities. Game theory is usually used in a strategic choice of firm, governments policy determination processes, or buyer-seller relationship etc. The major publication in game theory was the book, Theory of Games and Economic Behaviour by Von Neumann and Oskar Morgen stein, in 1944. Later, in the early 1950s John Nash published, The Nash Equilibrium. Nash equilibrium made game theory a very useful tool. In 1994 John Nash also received the Nobel prize as he has introduced Nash equilibrium. An American Biographical movie was also released on 2001 based on the life of John Nash, played by Russell Crowe.What is game: A game is a situation where the participant’s payoffs depend not only on their decisions, but also on their rival's decisions. Optimal decision of one firm will depend on what others do in the game. Elements to describe a game- 1)Players, 2)Rules (when each player moves, what are the possible moves, what is known to each player before moving), 3)Outcomes of the move, 4)Payoffs of each possible outcome (How much each player receives for any specific outcome).Game theory and Economics: If the market is composed by a small number of firms, each firm must act strategically. So, depending upon the course of actions taken by one firm such as price hike or reduction, increase or decreases in production etc. other firm will decide steps they should take to counter the actions of the other firm. Each firm affects the market price changing the quantity produced. Suppose 2 firms are producing 100 units and if one of the firms decides to increase the production by 10 units. The market supply will increase from 200 to 210 and the price has to drop to reach an equilibrium. Therefor it also affects the profits of the firms. Each firm knows that its profit depends not only on how much it produced but also on how much the other firms produce. TERMINOLOGIES IN GAME THEORYMEB160641) Players: Generally, there are two players in game theory.2) Strategy: There are two types of strategy namely pure strategy and mixed strategy. Pure strategy means a player will choose only one strategy and ignore other strategies. While in mixed strategy a player follows more than one strategy. In both the cases sum of the probability of all strategy is one.3)Maximin principle: Maximizes the minimum guaranteed gain of a player.4)Minimax principle: Minimizes the maximum loss.5)Saddle point: When maximin value is equal to minimax principle then the game is said to have a saddle point.6)Value of the game: If the game has a saddle point then the value of the cell at the saddle point is called the value of the game.7) Two-person zero sum game: In a game with two players, if the gain of one player is equal to the loss of another player, then that game is called two-person zero sum game.8)Two-person constant sum game: In a game with two players if the sum of the payoffs of the players is constant the it is called as two-person constant sum game.Solutions of the game: To predict what will be the solution/outcome of the game we need some tools: 1)Dominated and dominated strategies (I will do whatever I want to do no matter what others do)2)Nash Equilibrium (I will do whatever I want to, provided what you do).Simple example on how can we use game theoryMEB16062We are going to take a very basic and simple problem on Game Theory which is known as Prisoners’ Dilemma. The prisoners’ dilemma is a standard example of a game analysed in game theory that shows why two completely rational individuals might not cooperate, even if it appears that it is in their best interest to do so. It was originally framed by Merrill Flood and Melvin Dresher while working at RAND in 1950. Albert W. Tucker formalized the game with prison sentence rewards and named it “prisoners’ dilemma”, presenting it as follows:Two members of a criminal gang are arrested and imprisoned. Each prisoner is in solitary confinement with no means of communicating with the other. The prosecutors lack sufficient evidence to convict the pair on the principal charge, but they have enough to convict both on a lesser charge. Simultaneously, the prosecutors offer each prisoner a bargain. Each prisoner is given the opportunity either to betray the other by testifying that the other committed the crime, or to cooperate with the other by remaining silent. The possible outcomes are:If A and B each betray the other, each of them serves two years in prisonIf A betrays B but B remains silent, a will be set free and B will serve three years in prison (and vice versa)If A and B both remain silent, both of them will serve only one year in prison (on the lesser charge).It is implied that the prisoners will have no opportunity to reward or punish their partner other than the prison sentences they get and that their decision will not affect their reputation in the future. Because betraying a partner offers a greater reward than cooperating with them, all purely rational self-interested prisoners will betray the other, meaning the only possible outcome for two purely rational prisoners is for them to betray each other. The interesting part of this result is that pursuing individual reward logically leads both of the prisoners to betray when they would get a better individual reward if they both kept silent.It is assumed that both prisoners understand the nature of the game, have no loyalty to each other, and will have no opportunity for retribution or reward outside the game. Regardless of what the other decides, each prisoner gets a higher reward by betraying the other ("defecting"). The reasoning involves an argument by?dilemma: B will either cooperate or defect. If B cooperates, a should defect, because going free is better than serving 1 year. If B defects, a should also defect, because serving 2 years is better than serving 3. So either way, a should defect. Parallel reasoning will show that B should defect.From this outcome we can see that no matter what the other one choses the next player will always choose the same strategy. So here the defecting strategy is the dominant one and the cooperating one is the dominated one. To reach the NASH equilibrium both parties will choose to defect.REFERENCES:WIKIPEDIA ................
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