1. 5. Combining simultaneous and sequential moves.
In this section we shall learn
How the tools used to analyze sequential games can be used on simultaneous games and vice versa.
How to understand games that involve both sequential and simultaneous moves.
How a switch from simultaneous to sequential (or vice versa) moves may be used by a player to gain an advantage.
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2. Combining simultaneous and sequential moves.
Many games involve both simultaneous and sequential moves.
The use of game trees (extensive form) and payoff matrices (strategic form) are interchangeable.
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3. Combining simultaneous and sequential moves
Combining simultaneous and sequential moves. From extensive form to strategic form
Recall the senate race game.
The key to converting this from extensive to sequential form is to recognize that Gary has two types of strategy.
Simply play in or out.
Play a contingent strategy. For example; If Arnold plays ads I play out.
Gary
Arnie
Ads
No ads In
Out 1
, 1 3
, 3 2
, 4 4
, 2 ,
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4. Combining simultaneous and sequential moves
Combining simultaneous and sequential moves. From extensive form to strategic form
Strategies.
{In, In} - always enter.
{In, Out} – enter if ads, don’t enter otherwise.
{Out, In} – enter if no ads, don’t enter otherwise.
{Out, Out} – Never enter.
Gary
Arnie
Ads
No ads In
Out 1
, 1 3
, 3 2
, 4 4
, 2 ,
Gary
In, In
In, Out
Out, In
Out, Out
Arnie
Ads
1,1
3,3
No ads
2,4
4,2
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5. Combining simultaneous and sequential moves From strategic form to extensive form
To convert from strategic to extensive form representation requires the use of information sets.
Consider the chicken game
Player #2
Straight
Swerve
Player #1
-2,-2
1,-1
-1,1
0,0
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6. Combining simultaneous and sequential moves
Combining simultaneous and sequential moves. From strategic form to extensive form.
Player #1
Straight
Swerve
Player #2
Payoffs #1
, #2 -2
,-2 1
,-1 -1 ,1 ,0
Information set
Player #2
Straight
Swerve
Player #1
-2,-2
1,-1
-1,1
0,0
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7. Combining simultaneous and sequential moves
Combining simultaneous and sequential moves. Games with both sequential and simultaneous moves.
Product development and pricing game.
Consider a game in which two firms are deciding on their future product lines.
Each must first choose whether to develop and produce one of two similar goods, a fancy expensive to manufacture high-tech model, or a cheaply produced low-tech version.
Their decisions are unknown to each other up until the point the new models are unveiled.
Once the models are unveiled each must then choose a pricing strategy. For simplicity they may choose a high or low priced for each model.
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8. Combining simultaneous and sequential moves
Combining simultaneous and sequential moves. Games with both sequential and simultaneous moves.
Product development and pricing game.
Game tree.
High-Tech
Low-Tech P H L
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9. Combining simultaneous and sequential moves
Combining simultaneous and sequential moves. Games with both sequential and simultaneous moves.
Product development and pricing game.
The players first play a simultaneous product development game.
They then play simultaneously play the pricing game.
But – the two games occur sequentially and thus the solution requires the use of backwards induction.
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10. Combining simultaneous and sequential moves
Combining simultaneous and sequential moves. Games with both sequential and simultaneous moves.
Rules Change.
Switching from a sequential move to a simultaneous move game.
Sometimes this switch can be used as a strategy.
Consider the senate race game as in simultaneous form.
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11. Combining simultaneous and sequential moves
Combining simultaneous and sequential moves. Games with both sequential and simultaneous moves.
Senate race game in simultaneous form.
We see that no ads is a dominant strategy for Arnold. Hence Gary chooses in. In the sequential version Arnold chose ads and Gary stayed out.
Gary In
Out
Arnold
Ads
1,1
3,3
No ads
2,4
4,2
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12. Combining simultaneous and sequential moves
Combining simultaneous and sequential moves. Games with both sequential and simultaneous moves.
Subgame perfection.
Suppose that gateway could commit to playing Foreign. The equilibrium would be {U,U,F}. But the commitment is not credible.
Where the U, U node to be reached Gateway would play U.
At each node the players will maximize.
The continuations of the game from each node must be equilibria.
This implies that the subgame perfect equilibrium is {F,U,U}.
Dell HP
Gateway US
Foreign
Payoffs 1, 5 , 5, 2 3, 4 2, 4, 3
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