1 1 Lesson overview BA 592 Lesson I.10 Sequential and Simultaneous Move Theory Chapter 6 Combining...

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Lesson overviewLesson overview

BA 592 Lesson I.10 Sequential and Simultaneous Move Theory

Chapter 6 Combining Simultaneous and Sequential MovesChapter 6 Combining Simultaneous and Sequential MovesLesson I.10 Sequential and Simultaneous Move TheoryLesson I.10 Sequential and Simultaneous Move Theory

Each Example Game Introduces some Game TheoryEach 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 ApplicationsLesson I.10 Sequential and Simultaneous Move Applications

2 2BA 592 Lesson I.10 Sequential and Simultaneous Move Theory

Playing a series of games over time Playing a series of games over time requires a strategy for the requires a strategy for the entire series, which defines a strategy for each component entire series, which defines a strategy for each component subgame. For example, the San Diego Chargers have a strategy subgame. For example, the San Diego Chargers have a strategy for playing a season, which defines a strategy for how they play for playing a season, which defines a strategy for how they play each individual game. The series strategy may involve not each individual game. The series strategy may involve not maximizing you chance of winning each subgame if doing so maximizing you chance of winning each subgame if doing so might risk injury to Philip Rivers or another key player.might risk injury to Philip Rivers or another key player.

Example 1: SubgamesExample 1: Subgames


Citigroup and General Electric Citigroup and General Electric must simultaneously choose must simultaneously choose whether to invest $10 billion to buy a fiber-optic network. If 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 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 other does not, then the investor has to make a pricing decision for its telecom services. It can choose either a high price, for its telecom services. It can choose either a high price, generating 60 million customers and a profit per unit of $400, or generating 60 million customers and a profit per unit of $400, or a low price, generating 80 million customers and a profit per unit a low price, generating 80 million customers and a profit per unit of $200. If both firms invest, then their pricing choices become of $200. If both firms invest, then their pricing choices become a second simultaneous-move game, with each choosing high or a second simultaneous-move game, with each choosing high or low price. If both choose the high price, they split the market, 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 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 both choose the low price, they split the market, each with 40 million customers and a profit per unit of $200. If one chooses 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 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.market, with 80 million customers and a profit per unit of $200.



Compute the subgames of this Compute the subgames of this Investment-and-Price Competition Investment-and-Price Competition GameGame, then describe the connections between the subgames. , then describe the connections between the subgames. Should Should Citigroup invest?Citigroup invest? Should Should General Electric invest?General Electric invest?


5 5

Don't InvestDon't 0,0 0,Invest ,0

Citigroup

General Electric


First stage: Investment First stage: Investment GameGame


High LowHigh 2,2 -10,6Low 6,-10 -2,-2

Citigroup

General Electric

Second stage: Pricing GameSecond stage: Pricing Game

14

H ig h

6

L ow

C itig ro up

14

H ig h

6

L ow

G e n e ra l E le c tric

Second stage: General Second stage: General Electric’s Pricing DecisionElectric’s Pricing Decision

Second stage: Citigroup’s Second stage: Citigroup’s Pricing DecisionPricing Decision


Solve the two-stage game by backward induction. The first step Solve the two-stage game by backward induction. The first step is to solve each of the second-stage games.is to solve each of the second-stage games.


7 7

Don't InvestDon't 0,0 0,Invest ,0

Citigroup

General Electric




High LowHigh 2,2 -10,6Low 6,-10 -2,-2

Citigroup

General Electric

Second stage: Solved by Second stage: Solved by dominancedominance

14

H ig h

C it ig ro up

14

H ig h


Second stage: Solved by Second stage: Solved by backward inductionbackward induction



The next step is to input the equilibrium payoffs of each of the The next step is to input the equilibrium payoffs of each of the second-stage games into the first-stage game.second-stage games into the first-stage game.


9 9

Don't InvestDon't 0,0 0,14Invest 14,0 -2,-2

Citigroup

General Electric




High LowHigh 2,2 -10,6Low 6,-10 -2,-2

Citigroup

General Electric

Second stage: Solved by Second stage: Solved by dominancedominance

14

H ig h

C it ig ro up

14

H ig h




10 10

Don't InvestDon't 0,0 0,14Invest 14,0 -2,-2

Citigroup

General Electric


Agreements on one Nash equilibrium are complicated since each player Agreements on one Nash equilibrium are complicated since each player prefers a different equilibrium, so any agreement could be rejected as unfair. prefers a different equilibrium, so any agreement could be rejected as unfair.

If agreements are impossible, finding a focal point is complicated because If agreements are impossible, finding a focal point is complicated because there is no jointly-preferred equilibrium to focus beliefs. there is no jointly-preferred equilibrium to focus beliefs. ReputationReputation becomes becomes important: if players have a mutual history of one player dominating or important: if players have a mutual history of one player dominating or playing tough, players could focus their expectations on the equilibrium that playing tough, players could focus their expectations on the equilibrium that most benefits that player.most benefits that player.

Another solution is a player Another solution is a player strategically committingstrategically committing to his preferred- to his preferred-equilibrium strategy, or strategically eliminating some alternative strategies.equilibrium strategy, or strategically eliminating some alternative strategies.


The final step is to solve the first-The final step is to solve the first-stage game. It has two Nash stage game. It has two Nash equilibria, and is like the Battle-of-equilibria, and is like the Battle-of-the-Sexes game. the-Sexes game.


Changing simultaneous moves into sequential moves Changing simultaneous moves into sequential moves may benefit may benefit neither player, may benefit the first mover, may benefit the neither player, may benefit the first mover, may benefit the second mover, or it may second mover, or it may benefit both playersbenefit both players..

Example 2: No Order AdvantageExample 2: No Order Advantage


Example 2: No Order AdvantageExample 2: No Order Advantage

Don't C. ConfessDon't C. -1,-1 -15,0Confess 0,-15 -5,-5

Prisoner 2

Prisoner 1

Confess is a dominate strategy for each prisoner in the Prisoners’ Confess is a dominate strategy for each prisoner in the Prisoners’ dilemma. Changing simultaneous moves into sequential moves dilemma. Changing simultaneous moves into sequential moves benefits neither prisoner:benefits neither prisoner:

-1 ,-1

D o n 't C .

-1 5 ,0

C o n fe ss

P rison e r 2

D o n 't C .

0 ,-15

D o n 't C .

-5 ,-5

C o n fe ss

P rison e r 2

C o n fe ss

P rison e r 1


Example 3: First Mover AdvantageExample 3: First Mover Advantage

Football OperaFootball 3,2 0,0

Opera 0,0 2,3

Wife

Husband

When a simultaneous move game has multiple Nash equilibria, When a simultaneous move game has multiple Nash equilibria, changing simultaneous moves into sequential moves benefits the changing simultaneous moves into sequential moves benefits the first mover since he can select the equilibrium, as in the Battle of first mover since he can select the equilibrium, as in the Battle of the Sexes:the Sexes:

3 ,2

F o o tb a ll

0 ,0

O p e ra

W ife

F o o tb a ll

0 ,0

F o o tb a ll

2 ,3

O p e ra

W ife

O p e ra

H u sb a nd


Example 4: Second Mover AdvantageExample 4: Second Mover Advantage

Left RightLeft .1,.9 .8,.2

Right .7,.3 .3,.7

Goalie

Kicker

Changing simultaneous moves into sequential moves may benefit Changing simultaneous moves into sequential moves may benefit the second mover, as in the Penalty Kick Game: If the Kicker 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 goes second, he gets .7; but if the Goalie goes second, the Kicker gets only .3.gets only .3.

.1 ,.9

L e ft

.8 ,.2

R ig h t

G o a lie

L e ft

.7 ,.3

L e ft

.3 ,.7

R ig h t

G o a lie

R ig h t

K icke r

.9 ,.1

L e ft

.3 ,.7

R ig h t

K icke r

L e ft

.2 ,.8

L e ft

.7 ,.3

R ig h t

K icke r

R ig h t

G o a lie


Example 5: Mutual BenefitExample 5: Mutual Benefit

Changing simultaneous moves into sequential moves may benefit Changing simultaneous moves into sequential moves may benefit players, as in the players, as in the Budget Balance GameBudget Balance Game: Congress’s fiscal policy : Congress’s fiscal policy can either balance the budget or run a budget deficit, and the can either balance the budget or run a budget deficit, and the Federal Reserve’s monetary policy can either set interest rates Federal Reserve’s monetary policy can either set interest rates low or high. Congress is under pressure to run a deficit, which low or high. Congress is under pressure to run a deficit, which causes inflation. The Federal Reserve want to set low interest causes inflation. The Federal Reserve want to set low interest rates unless there is inflation, when it wants to set high interest rates unless there is inflation, when it wants to set high interest rates. rates.


Low Rate High RateB. Balance 3,4 1,3B. Deficit 4,1 2,2

Federal Reserve

Congress

3 ,4

L ow

1 ,3

H ig h

F e d .

B a lan ce

4 ,1

L ow

2 ,2

H ig h

F e d .

D e fic it

C o n g re ss

The normal form below includes payoffs consistent with the data The normal form below includes payoffs consistent with the data above.above.

Under simultaneous moves, Budget Deficit is a dominate strategy Under simultaneous moves, Budget Deficit is a dominate strategy for Congress, which makes Federal Reserve respond with High for Congress, which makes Federal Reserve respond with High Rates, for payoffs 2,2. But if Congress moves first, backward Rates, for payoffs 2,2. But if Congress moves first, backward induction has the Federal Reserve setting high interest rates if induction has the Federal Reserve setting high interest rates if there is a deficit, so the deficit is rejected. there is a deficit, so the deficit is rejected.

Example 5: Mutual BenefitExample 5: Mutual Benefit


Simultaneous moves are usually analyzed in the normal form, Simultaneous moves are usually analyzed in the normal form, and sequential moves in a game tree. It is possible to represent and sequential moves in a game tree. It is possible to represent simultaneous moves in a game tree, and some sequential moves simultaneous moves in a game tree, and some sequential moves in normal form. The latter has an advantage if you believe that 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 the extra detail in a game tree is not essentially to solving the game. For example, if you believe that Nash equilibria or game. For example, if you believe that Nash equilibria or rationalizeability are the definitive solutions to games.rationalizeability are the definitive solutions to games.

Example 6: Off-Equilibrium PathsExample 6: Off-Equilibrium Paths


LifB,LifD LifB,HifD HifB,LifD HifB,HifDB. Balance 3,4 3,4 1,3 1,3B. Deficit 4,1 2,2 4,1 2,2

Federal Reserve

Congress

3 ,4

L ow

1 ,3

H ig h

F e d .

B a lan ce

4 ,1

L ow

2 ,2

H ig h

F e d .

D e fic it

C o n g re ss

Write the Budget Balance Game with Congress moving first into Write the Budget Balance Game with Congress moving first into normal form. First, identify strategies: Congress can choose normal form. First, identify strategies: Congress can choose Balance or Deficit, and the Federal Reserve can choose these: 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, L if D (Low interest rates if Balance, Low if Deficit)L if B, H if DL if B, H if DH if B, L if DH if B, L if DH if B, H if D H if B, H if D




Federal Reserve

Congress

3 ,4

L ow

1 ,3

H ig h

F e d .

B a lan ce

4 ,1

L ow

2 ,2

H ig h

F e d .

D e fic it

C o n g re ss

There is only one rollback solution: Balance with (L if B, H if D) There is only one rollback solution: Balance with (L if B, H if D) (Low interest rates if Balance, High if Deficit)(Low interest rates if Balance, High if Deficit)

There are two Nash Equilibria: the rollback solution of Balance 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 with (L if B, H if D), and the equilibrium Deficit with (H if B, H if D). if D).



3 ,4

L ow

1 ,3

H ig h

F e d .

B a lan ce

4 ,1

L ow

2 ,2

H ig h

F e d .

D e fic it

C o n g re ss

The strategies Deficit with (H if B, H if D) is not a rollback 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 equilibrium because H if B is not optimal for the Federal Reserve if the opportunity to play actually arises. if the opportunity to play actually arises.

But those strategies Deficit with (H if B, H if D) are a Nash But those strategies Deficit with (H if B, H if D) are a Nash equilibrium because, given that Congress chooses Deficit, it does equilibrium because, given that Congress chooses Deficit, it does not matter whether the Federal Reserve choose L if B or H if B. not matter whether the Federal Reserve choose L if B or H if B.



Federal Reserve

Congress

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End of Lesson I.10End of Lesson I.10

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