Chapter Six game theory in Operations research .pdf
- 2. Life is full of conflict and competition. Numerical examples involving in conflict include games, military, political, advertising and marketing 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 Game theory is a mathematical theory that deals with the general features of competitive situations in a formal, abstract way. It places particular emphasis on the decision-making processes. 2
- 3. Research on game theory continues to deal with complicated types of competitive situations. However, we shall be dealing only with the simplest case, called two-person, zero sum games. As the name implies, these games involve only two players. 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. 3
- 4. In general, a two-person game is characterized by The strategies of player 1. The strategies of player 2. The pay-off table. 4
- 5. Thus the game is represented by the payoff matrix to player A as a11 a12 ……… a1n a21 a22 …........ A2n am1 am2 ………. amn B1 B2 ……… Bn A1 A2 . . Am 5
- 6. Here A1,A2,…..,Am are the strategies of player A B1,B2,…...,Bn are the strategies of player B aij is the payoff to player A (by B) when the player A plays strategy Ai and B plays Bj (aij is –ve means B got |aij| from A) 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) 6
- 7. 7 Situations are treated as games. ◦ The rules of the game state who can do what, and when they can do it ◦ A player's strategy is a plan for actions in each possible situation in the game ◦ A player's payoff is the amount that the player wins or loses in a particular situation in a game. ◦ Equilibrium (saddle point) is the payoff value that represents both minimax and maximin value of the game. ◦ A player has a dominant strategy if his best strategy doesn’t depend on what other players do
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- 10. Determine the saddle-point solution, the associated pure optimal strategies, and the value of the game for the following game. The payoffs are for player A. 8 6 2 8 8 9 4 5 7 5 3 5 B1 B2 B3 B4 A1 A2 A3 Max Row min Col 8 9 4 8 2 4 3 min max max min Example1 10
- 11. Con’t 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. Similarly, if A plays strategy 2, then whatever B plays, will get at least 4. and if A plays strategy 3, then he will get at least 3 whatever B plays. Thus to maximize his minimum returns, he should play strategy 2. 11
- 12. Con’t Now if B plays strategy 1, then whatever A plays, he will lose a maximum of 8. Similarly for strategies 2,3,4. (These are the maximum of the respective columns). Thus to minimize this maximum loss, B should play strategy 3 and 4 = max (row minima) = min (column maxima) is called the value of the game. 4 is called the saddle-point. 12
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- 14. Specify the range for the value of the game in the following case assuming that the payoff is for player A. 3 6 1 5 2 3 4 2 -5 A1 A2 A3 B1 B2 B3 Col max 5 6 Row min 1 -5 3 2 Example3: Finding the range of values when there is no unique saddle point 14
- 15. Thus max( row min) <= min (column max) We say that the game has no saddle point. Thus the value of the game lies between 2 and 3. Here both players must use random mixes of their respective strategies so that A will maximize his minimum expected return and B will minimize his maximum expected loss 15
- 16. Definition: A strategy S dominates a strategy T if every outcome in S is at least as good as the corresponding outcome in T, and at least one outcome in S is strictly better than the corresponding outcome in T. Dominance Principle: A rational player would never play a dominated strategy. 16