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Game Theory Part I

This slide explain about Game Theory. this slide is divided into five parts. this is the first part.

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Game Theory Part I

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  1. Chapter 12 Strategy and Game Theory (Part I) © 2004 Thomson Learning/South-Western

  2. Basic Concepts • Any situation in which individuals must make strategic choices and in which the final outcome will depend on what each person chooses to do can be viewed as a game. • Game theory models seek to portray complex strategic situations in a highly simplified setting.

  3. Basic Concepts • All games have three basic elements: • Players • Strategies • Payoffs • Players can make binding agreements in cooperative games, but can not in noncooperative games, which are studied in this chapter.

  4. Players • A player is a decision maker and can be anything from individuals to entire nations. • Players have the ability to choose among a set of possible actions. • Games are often characterized by the fixed number of players. • Generally, the specific identity of a play is not important to the game.

  5. Strategies • A strategy is a course of action available to a player. • Strategies may be simple or complex. • In noncooperative games each player is uncertain about what the other will do since players can not reach agreements among themselves.

  6. Payoffs • Payoffs are the final returns to the players at the conclusion of the game. • Payoffs are usually measure in utility although sometimes measure monetarily. • In general, players are able to rank the payoffs from most preferred to least preferred. • Players seek the highest payoff available.

  7. Equilibrium Concepts • In the theory of markets an equilibrium occurred when all parties to the market had no incentive to change his or her behavior. • When strategies are chosen, an equilibrium would also provide no incentives for the players to alter their behavior further. • The most frequently used equilibrium concept is a Nash equilibrium.

  8. Nash Equilibrium • A Nash equilibrium is a pair of strategies (a*,b*) in a two-player game such that a* is an optimal strategy for A against b* and b* is an optimal strategy for B against A*. • Players can not benefit from knowing the equilibrium strategy of their opponents. • Not every game has a Nash equilibrium, and some games may have several.

  9. An Illustrative Advertising Game • Two firms (A and B) must decide how much to spend on advertising • Each firm may adopt either a higher (H) budget or a low (L) budget. • The game is shown in extensive (tree) form in Figure 12.1

  10. An Illustrative Advertising Game • A makes the first move by choosing either H or L at the first decision “node.” • Next, B chooses either H or L, but the large oval surrounding B’s two decision nodes indicates that B does not know what choice A made.

  11. FIGURE 12.1: The Advertising Game in Extensive Form 7,5 L B H 5,4 L A L 6,4 B H H 6,3

  12. An Illustrative Advertising Game • The numbers at the end of each branch, measured in thousand or millions of dollars, are the payoffs. • For example, if A chooses H and B chooses L, profits will be 6 for firm A and 4 for firm B. • The game in normal (tabular) form is shown in Table 12.1 where A’s strategies are the rows and B’s strategies are the columns.

  13. Table 12.1: The Advertising Game in Normal Form

  14. Dominant Strategies and Nash Equilibria • A dominant strategy is optimal regardless of the strategy adopted by an opponent. • As shown in Table 12.1 or Figure 12.1, the dominant strategy for B is L since this yields a larger payoff regardless of A’s choice. • If A chooses H, B’s choice of L yields 5, one better than if the choice of H was made. • If A chooses L, B’s choice of L yields 4 which is also one better than the choice of H.

  15. Dominant Strategies and Nash Equilibria • A will recognize that B has a dominant strategy and choose the strategy which will yield the highest payoff, given B’s choice of L. • A will also choose L since the payoff of 7 is one better than the payoff from choosing H. • The strategy choice will be (A: L, B: L) with payoffs of 7 to A and 5 to B.

  16. Dominant Strategies and Nash Equilibria • Since A knows B will play L, A’s best play is also L. • If B knows A will play L, B’s best play is also L. • Thus, the (A: L, B: L) strategy is a Nash equilibrium: it meets the symmetry required of the Nash criterion. • No other strategy is a Nash equilibrium.

  17. Two Simple Games • Table 12.2 (a) illustrates the children’s finger game, “Rock, Scissors, Paper.” • The zero payoffs along the diagonal show that if players adopt the same strategy, no payments are made. • In other cases, the payoffs indicate a $1 payment from the loser to winner under the usual hierarchy (Rock breaks Scissors, Scissors cut Paper, Paper covers Rock).

  18. TABLE 12.2 (a): Rock, Scissors, Paper--No Nash Equilibria

  19. Two Simple Games • This game has no equilibrium. • Any strategy pair is unstable since it offers at least one of the players an incentive to adopt another strategy. • For example, (A: Scissors, B: Scissors) provides and incentive for either A or B to choose Rock. • Also, (A: Paper, B: Rock) encourages B to choose Scissors.

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