1 1 Lesson overview BA 592 Lesson I.10 Sequential and Simultaneous Move Theory Chapter 6 Combining Simultaneous and Sequential Moves Lesson I.10 Sequential and Simultaneous Move Theory Each Example Game Introduces some Game Theory Example 1: SubgamesExample 1: Subgames Example 2: No Order AdvantageExample 2: No Order Advantage Example 3: First Mover AdvantageExample 3: First Mover Advantage Example 4: Second Mover AdvantageExample 4: Second Mover Advantage Example 5: Mutual BenefitExample 5: Mutual Benefit Example 6: Off-Equilibrium PathsExample 6: Off-Equilibrium Paths Lesson I.10 Sequential and Simultaneous Move Applications

2 2 BA 592 Lesson I.10 Sequential and Simultaneous Move Theory Playing a series of games over time requires a strategy for the entire series, which defines a strategy for each component subgame. For example, the San Diego Chargers have a strategy for playing a season, which defines a strategy for how they play each individual game. The series strategy may involve not maximizing you chance of winning each subgame if doing so might risk injury to Philip Rivers or another key player. Example 1: Subgames

3 3 BA 592 Lesson I.10 Sequential and Simultaneous Move Theory Citigroup and General Electric must simultaneously choose whether to invest $10 billion to buy a fiber-optic network. If neither invests, that is the end of the game. If one invests and the other does not, then the investor has to make a pricing decision for its telecom services. It can choose either a high price, generating 60 million customers and a profit per unit of $400, or a low price, generating 80 million customers and a profit per unit of $200. If both firms invest, then their pricing choices become a second simultaneous-move game, with each choosing high or low price. If both choose the high price, they split the market, each with 30 million customers and a profit per unit of $400. If both choose the low price, they split the market, each with 40 million customers and a profit per unit of $200. If one chooses the high price and the other the low, the low-price gets the entire market, with 80 million customers and a profit per unit of $200. Example 1: Subgames

4 4 BA 592 Lesson I.10 Sequential and Simultaneous Move Theory Compute the subgames of this Investment-and-Price Competition Game, then describe the connections between the subgames. Should Citigroup invest? Should General Electric invest? Example 1: Subgames

5 5 BA 592 Lesson I.10 Sequential and Simultaneous Move Theory First stage: Investment Game Example 1: Subgames Second stage: Pricing Game Second stage: General Electric’s Pricing Decision Second stage: Citigroup’s Pricing Decision

6 6 BA 592 Lesson I.10 Sequential and Simultaneous Move Theory Solve the two-stage game by backward induction. The first step is to solve each of the second-stage games. Example 1: Subgames

7 7 BA 592 Lesson I.10 Sequential and Simultaneous Move Theory First stage: Investment Game Example 1: Subgames Second stage: Solved by dominance Second stage: Solved by backward induction

8 8 BA 592 Lesson I.10 Sequential and Simultaneous Move Theory The next step is to input the equilibrium payoffs of each of the second-stage games into the first-stage game. Example 1: Subgames

9 9 BA 592 Lesson I.10 Sequential and Simultaneous Move Theory First stage: Investment Game Example 1: Subgames Second stage: Solved by dominance Second stage: Solved by backward induction

10 BA 592 Lesson I.10 Sequential and Simultaneous Move Theory Agreements on one Nash equilibrium are complicated since each player prefers a different equilibrium, so any agreement could be rejected as unfair. If agreements are impossible, finding a focal point is complicated because there is no jointly-preferred equilibrium to focus beliefs. Reputation becomes important: if players have a mutual history of one player dominating or playing tough, players could focus their expectations on the equilibrium that most benefits that player. Another solution is a player strategically committing to his preferred- equilibrium strategy, or strategically eliminating some alternative strategies. Example 1: Subgames The final step is to solve the first- stage game. It has two Nash equilibria, and is like the Battle-of- the-Sexes game.

11 BA 592 Lesson I.10 Sequential and Simultaneous Move Theory Changing simultaneous moves into sequential moves may benefit neither player, may benefit the first mover, may benefit the second mover, or it may benefit both players. Example 2: No Order Advantage

12 BA 592 Lesson I.10 Sequential and Simultaneous Move Theory Example 2: No Order Advantage Confess is a dominate strategy for each prisoner in the Prisoners’ dilemma. Changing simultaneous moves into sequential moves benefits neither prisoner:

13 BA 592 Lesson I.10 Sequential and Simultaneous Move Theory Example 3: First Mover Advantage When a simultaneous move game has multiple Nash equilibria, changing simultaneous moves into sequential moves benefits the first mover since he can select the equilibrium, as in the Battle of the Sexes:

14 BA 592 Lesson I.10 Sequential and Simultaneous Move Theory Example 4: Second Mover Advantage Changing simultaneous moves into sequential moves may benefit the second mover, as in the Penalty Kick Game: If the Kicker goes second, he gets.7; but if the Goalie goes second, the Kicker gets only.3.

15 BA 592 Lesson I.10 Sequential and Simultaneous Move Theory Example 5: Mutual Benefit Changing simultaneous moves into sequential moves may benefit players, as in the Budget Balance Game: Congress’s fiscal policy can either balance the budget or run a budget deficit, and the Federal Reserve’s monetary policy can either set interest rates low or high. Congress is under pressure to run a deficit, which causes inflation. The Federal Reserve want to set low interest rates unless there is inflation, when it wants to set high interest rates.

16 BA 592 Lesson I.10 Sequential and Simultaneous Move Theory The normal form below includes payoffs consistent with the data above. Under simultaneous moves, Budget Deficit is a dominate strategy for Congress, which makes Federal Reserve respond with High Rates, for payoffs 2,2. But if Congress moves first, backward induction has the Federal Reserve setting high interest rates if there is a deficit, so the deficit is rejected. Example 5: Mutual Benefit

17 BA 592 Lesson I.10 Sequential and Simultaneous Move Theory Simultaneous moves are usually analyzed in the normal form, and sequential moves in a game tree. It is possible to represent simultaneous moves in a game tree, and some sequential moves in normal form. The latter has an advantage if you believe that the extra detail in a game tree is not essentially to solving the game. For example, if you believe that Nash equilibria or rationalizeability are the definitive solutions to games. Example 6: Off-Equilibrium Paths

18 BA 592 Lesson I.10 Sequential and Simultaneous Move Theory Write the Budget Balance Game with Congress moving first into normal form. First, identify strategies: Congress can choose Balance or Deficit, and the Federal Reserve can choose these: L if B, L if D (Low interest rates if Balance, Low if Deficit) L if B, H if D H if B, L if D H if B, H if D Example 6: Off-Equilibrium Paths

19 BA 592 Lesson I.10 Sequential and Simultaneous Move Theory There is only one rollback solution: Balance with (L if B, H if D) (Low interest rates if Balance, High if Deficit) There are two Nash Equilibria: the rollback solution of Balance with (L if B, H if D), and the equilibrium Deficit with (H if B, H if D). Example 6: Off-Equilibrium Paths

20 BA 592 Lesson I.10 Sequential and Simultaneous Move Theory The strategies Deficit with (H if B, H if D) is not a rollback equilibrium because H if B is not optimal for the Federal Reserve if the opportunity to play actually arises. But those strategies Deficit with (H if B, H if D) are a Nash equilibrium because, given that Congress chooses Deficit, it does not matter whether the Federal Reserve choose L if B or H if B. Example 6: Off-Equilibrium Paths

21 End of Lesson I.10 BA 592 Game Theory BA 592 Lesson I.10 Sequential and Simultaneous Move Theory