Lesson overview
Chapter 6 Combining Simultaneous and Sequential Moves
Lesson I.10 Sequential and Simultaneous Move Theory
Each Example Game Introduces some Game Theory
• Example 1: Subgames
• Example 2: No Order Advantage
• Example 3: First Mover Advantage
• Example 4: Second Mover Advantage
• Example 5: Mutual Benefit
• Example 6: Off-Equilibrium Paths
Lesson I.10 Sequential and Simultaneous Move Applications
BA 592 Lesson I.10 Sequential and Simultaneous Move Theory
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Example 1: Subgames
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.
BA 592 Lesson I.10 Sequential and Simultaneous Move Theory
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Example 1: Subgames
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.
BA 592 Lesson I.10 Sequential and Simultaneous Move Theory
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Example 1: Subgames
Compute the subgames of this Investment-and-Price Competition
Game, then describe the connections between the subgames.
Should Citigroup invest? Should General Electric invest?
BA 592 Lesson I.10 Sequential and Simultaneous Move Theory
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Example 1: Subgames
First stage: Investment
Game
General Electric
Citigroup
Don't
Invest
Don't
0,0
,0
Second stage: General
Electric’s Pricing Decision
Invest
0,
General Electric
Second stage: Citigroup’s
Pricing Decision
Citigroup
High
Low
14
6
Second stage: Pricing Game
General Electric
High
Low
Citigroup
14
6
High
Low
High
2,2
6,-10
BA 592 Lesson I.10 Sequential and Simultaneous Move Theory
Low
-10,6
-2,-2
5
Example 1: Subgames
Solve the two-stage game by backward induction. The first step
is to solve each of the second-stage games.
BA 592 Lesson I.10 Sequential and Simultaneous Move Theory
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Example 1: Subgames
First stage: Investment
Game
General Electric
Citigroup
Don't
0,0
,0
Don't
Invest
Second stage: Solved by
backward induction
Invest
0,
General Electric
High
14
Second stage: Solved by
backward induction
Citigroup
Second stage: Solved by
dominance
General Electric
High
Citigroup
14
High
Low
High
2,2
6,-10
BA 592 Lesson I.10 Sequential and Simultaneous Move Theory
Low
-10,6
-2,-2
7
Example 1: Subgames
The next step is to input the equilibrium payoffs of each of the
second-stage games into the first-stage game.
BA 592 Lesson I.10 Sequential and Simultaneous Move Theory
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Example 1: Subgames
First stage: Investment
Game
General Electric
Citigroup
Don't
0,0
14,0
Don't
Invest
Second stage: Solved by
backward induction
Invest
0,14
-2,-2
General Electric
High
14
Second stage: Solved by
backward induction
Citigroup
Second stage: Solved by
dominance
General Electric
High
Citigroup
14
High
Low
High
2,2
6,-10
BA 592 Lesson I.10 Sequential and Simultaneous Move Theory
Low
-10,6
-2,-2
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Example 1: Subgames
General Electric
Citigroup
Don't
Invest
Don't
0,0
14,0
Invest
0,14
-2,-2
The final step is to solve the firststage game. It has two Nash
equilibria, and is like the Battle-ofthe-Sexes game.
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 preferredequilibrium strategy, or strategically eliminating some alternative strategies.
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Example 2: No Order Advantage
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.
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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:
Prisoner 1
Prisoner 2
Don't C.
Prisoner 1
Confess
Don't C.
-1,-1
0,-15
Confess
-15,0
-5,-5
Don't C.
Confess
Prisoner 2
Prisoner 2
Don't C.
Confess
Don't C.
Confess
-1,-1
-15,0
0,-15
-5,-5
BA 592 Lesson I.10 Sequential and Simultaneous Move Theory
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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:
Husband
Wife
Football
Football
3,2
Husband
Opera
0,0
Opera
0,0
2,3
Football
Opera
Wife
Wife
Football
Opera
Football
Opera
3,2
0,0
0,0
2,3
BA 592 Lesson I.10 Sequential and Simultaneous Move Theory
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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.
Goalie
Kicker
Left
Right
Left
.1,.9
.7,.3
Right
.8,.2
.3,.7
Goalie
Kicker
Left
Right
Left
Right
Kicker
Kicker
Goalie
Goalie
Left
Right
Left
Right
Left
Right
Left
Right
.9,.1
.3,.7
.2,.8
.7,.3
.1,.9
.8,.2
.7,.3
.3,.7
BA 592 Lesson I.10 Sequential and Simultaneous Move Theory
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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.
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Example 5: Mutual Benefit
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. Congress
Federal Reserve
Low Rate High Rate
B. Balance
3,4
1,3
Congress
B. Deficit
4,1
2,2
Balance
Deficit
Fed.
Fed.
Low
High
Low
High
3,4
1,3
4,1
2,2
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Example 6: Off-Equilibrium Paths
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.
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Example 6: Off-Equilibrium Paths
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
Congress
Balance
Deficit
Fed.
Fed.
Low
High
Low
High
3,4
1,3
4,1
2,2
Federal Reserve
LifB,LifD LifB,HifD HifB,LifD HifB,HifD
B. Balance
3,4
3,4
1,3
1,3
Congress
B. Deficit
4,1
2,2
4,1
2,2
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Example 6: Off-Equilibrium Paths
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).
Congress
Balance
Deficit
Fed.
Fed.
Low
High
Low
High
3,4
1,3
4,1
2,2
Federal Reserve
LifB,LifD LifB,HifD HifB,LifD HifB,HifD
B. Balance
3,4
3,4
1,3
1,3
Congress
B. Deficit
4,1
2,2
4,1
2,2
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Example 6: Off-Equilibrium Paths
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.
Congress
Balance
Deficit
Fed.
Fed.
Low
High
Low
High
3,4
1,3
4,1
2,2
Federal Reserve
LifB,LifD LifB,HifD HifB,LifD HifB,HifD
B. Balance
3,4
3,4
1,3
1,3
Congress
B. Deficit
4,1
2,2
4,1
2,2
BA 592 Lesson I.10 Sequential and Simultaneous Move Theory
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BA 592
Game Theory
End of Lesson I.10
BA 592 Lesson I.10 Sequential and Simultaneous Move Theory
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