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game theory, minimax, strategies, dominance

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- 1. BBA3274 / DBS1084 QUANTITATIVE METHODS for BUSINESS Game Theory Game Theory by Stephen Ong Visiting Fellow, Birmingham City University Business School, UK Visiting Professor, Shenzhen
- 2. Today’s Overview
- 3. Learning Objectives After this lecture, students will be able to: 1. 2. 3. 4. Understand the importance and use of game theory in decision making. Understand the principles of zerosum, two person games. Analyse pure strategy games and use dominance to reduce the size of the game. Solve mixed strategy games when there is no saddle point.
- 4. Outline 4.1 4.2 4.3 4.4 4.5 4.6 Introduction Language of Games The Minimax Criterion Pure Strategy Games Mixed Strategy Games Dominance
- 5. Game Theory Models 5 A set of mathematical tools for analyzing situations in which players make various strategic moves and have different outcomes or payoffs associated with those moves.
- 6. Dominant Strategies and the Prisoner’s Dilemma 6 This payoff matrix shows the various prison terms for Bonnie and Clyde that would result from the combination of strategies chosen when questioned about a crime spree.
- 7. Prisoner’s Dilemma – Dominant Strategy 7 A dominant strategy is one that results in the best outcome or highest payoff to a given player no matter what action or choice the other player makes.
- 8. Nash Equilibrium 8 Nash equilibrium is a set of strategies from which all players are choosing their best strategy, given the actions of the other players.
- 9. Game Model Classification Number of players Sum of all payoffs Number of strategies employed Two person (X, Y) Zero sum, where sum of losses by one player = sum of gains by other player
- 10. Example : Duopoly of 2 Stores 10 There are only 2 lighting fixture stores, X and Y. Owner of store X has 2 advertising strategies – radio spots and newspaper ads. Owner of store Y prepares to respond with radio spots and newspaper ads. The 2x2 payoff matrix shows the effect on market shares when both stores advertise.
- 11. Example : Duopoly Game Outcomes Store X Strategy Store Y Strategy Outcome (% Change in Market Share) X1 (Use Radio) Y1 (Use Radio) X wins 3 And Y loses 3 X1 (Use Radio) Y2 (Use Newspaper) X wins 5 And Y loses 5 X2 (Use Newspaper) Y1 (Use Radio) X wins 1 And Y loses 1 X2 (Use Newspaper) Y2 (Use Newspaper) X loses 2 And Y wins 2 1- 11
- 12. Minimax Criterion Used to find the strategy that minimises the maximum loss, or maximizes the minimum payoff (maximin approach). Locate the minimum payoff for each strategy. Select the strategy with the maximum number. The upper value of the game equal to the minimum of the maximum values in the columns. The lower value of the game is equal to the maximum of the minimum values in the rows. 3-12
- 13. Minimax Solution An equilibrium or saddle point condition exists if the upper value of the game is equal to the lower value of the game. This is called the value of the game. Saddle Point STRATEGIES X1 Y1 3 Y2 5 X2 1 -2 MAXIMUM 3 5 Minimum of maximums MINIMUM 3 -2 Maximum of minimums 3-13
- 14. Pure Strategy Game When a saddle point is present, the strategy each player should follow will always be the same regardless of the other player’s strategy. Saddle Point STRATEGIES X1 Y1 10 Y2 6 X2 -12 2 MAXIMUM 10 6 Minimum of maximums MINIMUM 6 -12 Maximum of minimums 3-14
- 15. Mixed Strategy Game When there is no saddle point, players will play each strategy for a certain percentage of the time (P, Q). To solve a mixed strategy game, use the expected gain or loss approach. STRATEGIES Y1 (P) Y2 (1 - P) Expected Gain X1 (Q) 4 2 4P + 2(1-P) X2 (1 - Q) 1 10 1P + 10(1-P) Expected gain 4Q + 1(1-Q) 2Q + 10(1-Q) 3-15
- 16. Mixed Strategy Game The goal of this approach is for a player to play each strategy a particular percentage of the time so that the expected value of the game does not depend upon what the opponent does. This will only occur if the expected value of each strategy is the same. For player Y, 4P + 2 (1 – P) = 1P + 10(1 – P) P = 8/11 For player X, 4Q + 1(1 – Q) = 2Q + 10(1 – Q) Q = 9/11 3-16
- 17. Dominance 17 The principle of dominance can be used to reduce the size of the games by eliminating strategies that would never be played. A strategy can be eliminated if all its game’s outcomes are the same or worse than the corresponding game outcomes of another strategy.
- 18. Example : Dominance Y1 Y2 Y3 Y4 X1 -5 4 6 -3 X2 -2 6 2 -20 1- 18
- 19. Tutorial Lab Practical : Spreadsheet 1 - 19
- 20. Further Reading Render, B., Stair Jr.,R.M. & Hanna, M.E. (2013) Quantitative Analysis for Management, Pearson, 11th Edition Waters, Donald (2007) Quantitative Methods for Business, Prentice Hall, 4 th Edition. Anderson D, Sweeney D, & Williams T. (2006) Quantitative Methods For Business Thompson Higher Education, 10th Ed.
- 21. QUESTIONS?

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