L24_Introduction to Game theory - Graphical solutions and solutions by LP techniques

L24_Introduction to Game theory - Graphical solutions and solutions by LP techniques

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    GAME THEORY Life is full of conflict and competition. Numerical examples involving adversaries in conflict include parlor games, military battles, political campaigns, advertising and marketing campaigns by competing business firms and so forth. A basic feature in many of these situations is that the final outcome depends primarily upon the combination of strategies selected by the adversaries.
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    Game theory is a mathematical theory that deals with the general features of competitive situations like these in a formal, abstract way. It places particular emphasis on the decision- making processes of the adversaries. Research on game theory continues to delve into rather complicated types of competitive situations. However, we shall be dealing only with the simplest case, called two-person, zero sum games.
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    As the name implies, these games involve only two players (or adversaries). They are called zero-sum games because one player wins whatever the other one loses, so that the sum of their net winnings is zero. In general, a two-person game is characterized by The strategies of player 1. The strategies of player 2. The pay-off table.
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    Thus the game is represented by the payoff matrix to player A as a 11 a 12 ……… a 1n a 21 a 22 …........ a 2n . . a m1 a m2 ………. a mn B 1 B 2 ……… B n A 1 A 2 . . A m
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    Here A 1 ,A 2 ,…. .,A m are the strategies of player A B 1 ,B 2 ,…. ..,B n are the strategies of player B a ij is the payoff to player A (by B) when the player A plays strategy A i and B plays B j (a ij is –ve means B got |a ij | from A)
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    Simple Example: Consider the game of the odds and evens. This game consists of two players A,B, each player simultaneously showing either of one finger or two fingers. If the number of fingers matches, so that the total number for both players is even, then the player taking evens (say A) wins $1 from B (the player taking odds). Else, if the number does not match, A pays $1 to B. Thus the payoff matrix to player A is the following table:
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    1 -1 -1 1 1 2 B 1 2 A
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    A primary objective of game theory is the development of rational criteria for selecting a strategy. Two key assumptions are made: Both players are rational Both players choose their strategies solely to promote their own welfare (no compassion for the opponent)
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    Optimal solution of two-person zero-sum games Problem 1(a) Problem set 14.4 A page 534 Determine the saddle-point solution, the associated pure strategies, and the value of the game for the following game. The payoffs are for player A.
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    8 6 2 8 8 9 4 5 7 5 3 5 B 1 B 2 B 3 B 4 A 1 A 2 A 3 Max Row min Col 8 9 4 8 2 4 3 min max max min
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    The solution of the game is based on the principle of securing the best of the worst for each player. If the player A plays strategy 1, then whatever strategy B plays, A will get at least 2. Thus to
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This note was uploaded on 02/23/2011 for the course MATH 112 taught by Professor Ritadubey during the Spring '11 term at Amity University.

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L24_Introduction to Game theory - Graphical solutions and solutions by LP techniques

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