# Game theory and its applications

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### Game theory and its applications

• 1. GAME THEORY AND ITS APPLICATION S Yen 2
• 2. CONTENT hat is game theory pplications efinitions ssumptions ame theory Models with examples (Zero sum game, Prisoner’s Dilemma etc…)
• 3. WHAT IS GAME THEORY John von Neumann heory Oskar Morgenstern research of and is associated with the in early 1940s t deals with Bargaining/Decision analysis. concerned with strategic behavior
• 4. GAME THEORY APPLICATIONS n Economics decision making argaining ilitary strategies omputer networking, network security
• 5. GAME THEORY DEFINITION Theory of rational behavior for interactive decision problems. In a game, several agents strive to maximize their (expected) utility index by choosing particular courses of action, and each agent's final utility payoffs depend on the profile of courses of action chosen by all agents. The interactive situation, specified by the set of participants, the possible courses of action of each agent, and the set of all possible utility payoffs, is called a game; the agents 'playing' a game are called the players.
• 6. DEFINITIONS Eg : Two companies are the only manufactures of a particular product, they compete each other for market share
• 7. DEFINITIONS Definition: Maximin strategy – If we determine the least possible payoff for each strategy, and choose the strategy for which this minimum payoff is largest, we have the maximin strategy.
• 8. FURTHER DEFINITIONS Definition: Constant-sum and non constant-sum game – If the payoffs to all players add up to the same constant, regardless which strategies they choose, then we have a constant-sum game. The constant may be zero or any other number, so zero-sum games are a class of constant-sum games. If the payoff does not add up to a constant, but varies depending on which strategies are chosen, then we have a non-constant sum game
• 9. GAME THEORY :ASSUMPTIONS (1) Each decision maker has available to him two or more well-specified choices or sequences of choices. (2) Every possible combination of plays available to the players leads to a well-defined end-state (win, loss, or draw) that terminates the game. (3) A specified payoff for each player is associated with each end-state.
• 10. GAME THEORY :ASSUMPTIONS (4) Each decision maker has perfect knowledge of the game and of his opposition. (5) All decision makers are rational; that is, each player, given two alternatives, will select the one that yields him the greater payoff.
• 11. GAME THEORY MODELS Two-person, zero-sum games. Two-person, constant-sum game. Two-Person Non constant-Sum Games
• 12. TWO PERSON ZERO-SUM AND CONSTANT-SUM GAMES Two-person zero-sum and constant-sum games are played according to the following basic assumption: Each player chooses a strategy that enables him/her to do the best he/she can, given that his/her opponent knows the strategy he/she is following. A two-person zero-sum game has a saddle point if and only if Max (row minimum) = min (column maximum) all all rows columns
• 13. TWO PERSON ZERO-SUM GAME Company B Payoff Matrix to company A Maximin strategy Company A Strategies  a1,b1- increase advertising Mini max strategy  a2,b2-provide discounts  a3,b3-Extend Warranty Example from text book
• 14. ZERO-SUM GAME • Game theory assumes that the decision maker and the opponent are rational, and that they subscribe to the maximin criterion as the decision rule for selecting their strategy • This is often reasonable if when the other player is an opponent out to maximize his/her own gains, e.g. competitor for the same customers. • Consider: Player 1 with three strategies S1, S2, and S3 and Player 2 with four strategies OP1, OP2, OP3, and OP4
• 15. ZERO-SUM GAME Player 2 OP1 OP2 OP3 OP4 Row Minima S1 12 3 9 8 3 Player 1 S2 5 4 6 5 4 maximin S3 3 0 6 7 0 Column 12 4 9 8 maxima minimax • Using the maximin criterion, player 1 records the row minima and selects the maximum of these (S2) • Player 1’s gain is player 2’s loss. Player 2 records the column maxima and select the minimum of these (OP2) example
• 16. ZERO-SUM GAME Player 2 OP1 OP2 OP3 OP4 Row Minima S1 12 3 9 8 3 Player 1 S2 5 4 6 5 4 maximin S3 3 0 6 7 0 Column 12 4 9 8 maxima minimax • The value 4 achieved by both players is called the value of the game • The intersection of S2 and OP2 is called a saddle point. A game with a saddle point is also called a game with an equilibrium solution. • At the saddle point, neither player can improve their payoff by switching strategies example
• 17. TWO PERSON NON CONSTANT SUM GAME • Most game-theoretic models of business situations are not constant-sum games, because it is unusual for business competitors to be in total conflict • As in two-person zero-sum game, a choice of strategy by each player is an equilibrium point if neither player can benefit from a unilateral change in strategy
• 18. PRISONER’S DILEMMA wo suspects arrested for a crime risoners decide whether to confess or not to confess f both confess, both sentenced to 3 months of jail f both do not confess, then both will be sentenced to 1 month of jail f one confesses and the other does not, then the confessor gets freed
• 19. ormal Form representation – Payoff Matrix Prisoner 2 Confess Not Confess Prisoner 1 Confess -3,-3 0,-9 Not Confess -9,0 -1,-1
• 20. ach player’s predicted strategy is the best response to the predicted strategies of other players o incentive to deviate unilaterally Prisoner 2 Confess Not Confess Prisoner 1 player can benefit from a unilateral change in strategy either Confess -3,-3 0,-9 Not Confess -9,0 -1,-1 trategically stable or self-enforcing
• 21. NASH EQUILIBRIUM Prisoner 2 Confess Not Confess Prisoner 1 Confess -3,-3 0,-9 Not Confess -9,0 -1,-1
• 22. MIXED STRATEGIES probability distribution over the pure strategies of the game ock-paper-scissors game – Each player simultaneously forms his or her hand into the shape of either a rock, a piece of paper, or a pair of scissors – Rule: rock beats (breaks) scissors, scissors beats (cuts) paper, and paper beats (covers) rock o pure strategy Nash equilibrium ne mixed strategy Nash equilibrium – each player plays rock, paper and
• 23. FURTHER STUDIES Will Game Theory give us the optimum or best solution/ decision? When there is n number of players and n number of strategies need to go for Leaner programming.
• 24. Multi agent Systems: Algorithmic, Game-Theoretic, and Logical Foundations by Yoav Shoham, Kevin Leyton-Brown Publisher: Cambridge University Press 2008 ISBN/ASIN: 0521899435 ISBN-13: 9780521899437 Number of pages: 532 Description: Multiagent systems consist of multiple autonomous entities having different information and/or diverging interests. This comprehensive introduction to the field offers a computer science perspective, but also draws on ideas from game theory, economics, operations research, logic, philosophy and linguistics. It will serve as a reference for researchers in each of these fields, and be used as a text for advanced undergraduate and graduate courses.
• 25. Game Theory for Applied Economists Robert Gibbons This book introduces one of the most powerful tools of modern economics to a wide audience:
• 26. Strategy: An Introduction to Game Theory, 2nd Edition Joel Watson Book Description October 16, 2007 | ISBN-10: 0393929345 Publication Date: | ISBN-13: 978-0393929348 | Edition: 2 Strategy, Second Edition, is a thorough revision and update of one of the most successful Game Theory texts available. Known for its accurate and simple-yet-thorough presentation, Joel Watson has refined his text to make it even more student friendly. Highlights of the revision include the addition of Guided (or Solved) Exercises and a significant expansion of the material for political economists and political scientists.