Outline

2. Outline • 1. Motivation • 2. Nash Equilibrium 3. The Prisoner's Dilemma 4. Dominant Strategy Equilibrium • 5. Limitations of the Nash Equilibrium • 6. Sequential Move Games and the Value of Limiting One's Options.

3. “Two suspects are arrested and charged with a crime. They are hold in separate cells and are explained the consequences of their actions. If neither confesses then both will be convicted of a minor offense and sentenced to one month in jail. If both confess, then both will be sentenced for 6 months. Finally, if one confesses but the other does not, then the confessor will be released immediately, but the other will be sentenced to 9 months” Ford and GM are selling almost identical cars. If both set high price, each gets a profit of $500 million. If both set low price, the profits to each drop to $300 million. If one of them charges a high price, and the other one low, then $100 m. and $700 m. are the corresponding profits. What do the Two Stories have in common?

4. Prisoners’ Dilemma

5. Set of players Strategies available to each player Payoff received by each player for each combination of strategies that can be chosen by the players Strategic (Normal) form Games

6. Strategy s is dominant for player i if for each feasible combination of the other player’s strategies, player i’s payoff from playing any other strategy is less than the payoff from playing s. Example: it is a dominant strategy for each player in the prisoners’ dilemma to confess Dominant Strategy

7. Coordination Game

8. A profile of strategies is a Nash equilibrium if for each player i is the best response to the strategies specified by this profile for the other players Example: there are two Nash equilibria in coordination game: (left,left) and (right,right). Nash Equilibrium

9. What is the likely outcome of this game? Toyota versus Honda Game Matrix: A Capacity Expansion Game

10. Elements of this game: Players: agents participating in the game (Toyota, Honda) Strategies: Actions that each player may take under any possible circumstance (Build, Don't build) Payoffs: The benefit that each player gets from each possible outcome of the game (the profits entered in each cell of the matrix)

11. Information: A full specification of who knows what when (full information) Timing: Who can take what decision when and how often the game is repeated (simultaneous, one-shot) Solution concept of the game: "What is the likely outcome"? (Dominant Strategy Equilibrium, Nash Equilibrium)

12. Nash Equilibrium A Nash Equilibrium occurs when each player chooses a strategy that gives him/her the highest payoff, given the strategy chosen by the other player(s) in the game. ("rational self-interest") Toyota vs. Honda: A Nash equilibrium: Each Firm Builds a New Plant

13. Why? • Given Toyota builds a new plant, Honda's • best response is to build a new plant. • Given Honda builds a new plant, Toyota's • best response is to build a plant. • Why is the Nash Equilibrium plausible? • It IS "self enforcing"… • Even though it DOES NOT necessarily maximise joint profit…

14. Note that this is the same game as the Prisoner’s dilemma, in which Nash Equilibrium is (confess, confess). Pareto Dominant Point: Neither confesses Other examples: Bertrand and Cournot Equilibria

15. Dominant and Dominated Strategies • A dominant strategy is a strategy that is better than any other strategy that a player might choose, no matter whatstrategy the other player follows. • Note: When a player has a dominant strategy, • that strategy will be the player's Nash • Equilibrium strategy. • Example: prisoners’ dilemma only Nash equilibrium is • (confess, confess).

16. Example: Equilibrium in Dominant Strategies May not Exist

17. Marutti does not have a dominant strategy, but a Nash Equilibrium exists: Amby builds, Marutti Doesn't Definition: A player has a dominated strategy when the player has another strategy that gives it a higher payoff no matter what the other player does. Example: "Do not build" in Honda-Toyota game. Example: "Do not build" for Amby only in Amby-Marutti game.

18. Why look for dominant or dominated strategies? A dominant strategy equilibrium is particularly compelling as a "likely" outcome Similarly, because strictly dominated strategies are unlikely to be played, these strategies can be eliminated from consideration in more complex games. This can make solving the game easier.

19. Game Matrix 4: Dominated Strategies • Toyota

20. "Build Large" is dominated for each player By eliminating the dominated strategies, we can reduce the game to matrix #1!

21. Nash Equilibrium need not be unique Example: the game of Chicken Slick

22. Nash Equilibria: (Swerve, Stay) and (Stay, Swerve) Example: Natural Monopoly markets

24. 2.Nash Equilibrium in pure strategies need not exist • Example: Matching Pennies • Player 1 • But Nash equilibrium in mixed strategies exists (for normal form games)!

25. Sequential Move Games Definition: A game tree shows the different strategies that each player can follow in the game and the order in which those strategies get chosen. Example. Consider Prisoner’s dilemma played over time. At every period both players choose either cooperate or “cheat” simultaneously. In the next period, having observed the outcome, they make the choice again. Grim trigger strategy: choose cooperate unless the partner cheats, in which case cheat forever after. Cheat after having cheated at least once.

26. Besanko & Braeutigam/Microeconomics: An Integrated Approach Chapter 14, Figure 14-01 FIGURE 14-1 Payoffs in the Repeated Prisoners’ Dilemma under the Grim Trigger Strategy

27. Besanko & Braeutigam/Microeconomics: An Integrated Approach Chapter 14, Figure 14-03 FIGURE 14-3 Game Tree for Learning-By-Doing Exercise 14.3

28. Subgame perfection • Game trees often are solved by starting at the end of the tree and, for each decision point, finding the optimal decision for the player at that point. • Ensures optimality at each point. • The solution to the revisited game differs from that of the simultaneous game. Why? • First mover can force second mover's hand • Illustrates the value of commitment (i.e. limiting one's own actions) rather than flexibility

29. Besanko & Braeutigam/Microeconomics: An Integrated Approach Chapter 14, Figure 14-02 FIGURE 14-2 Game Tree for the Sequential-Move Capacity Expansion Game Between Toyota and Honda

30. Summary • 1. Game Theory is the branch of economics concerned with the analysis of optimal decision making when all decision makers are presumed to be rational, and each is attempting to anticipate the actions and reactions of the competitors • 2. A Nash Equilibrium in a game occurs when each player chooses a strategy that gives him/her the highest payoff, given the strategies chosen by the other players in the game. • 3. The Nash Equilibrium may be a good predictor when it coincides with the Equilibrium in Dominant Strategies.

31. 4. When there are multiple Nash Equilibria, we must appeal to other concepts to choose the "likely" outcome of the game. • 5. An analysis of sequential move games reveals that moving first in a game can have strategic value if the first mover can gain from making a commitment.