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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.

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5 combining simultaneous and sequential moves
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.

Games People Play.


Combining simultaneous and sequential moves
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.

Games People Play.


Combining simultaneous and sequential moves from extensive form to strategic form

Gary

Arnie

,

In

1

, 1

Gary

Ads

3

, 3

Out

Arnie

In

2

, 4

Gary

No ads

4

, 2

Out

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; IfArnold plays ads I play out.

Games People Play.


Combining simultaneous and sequential moves from extensive form to strategic form1

Gary

Arnie

,

In

1

, 1

Gary

Ads

3

, 3

Out

Arnie

In

2

, 4

Gary

No ads

4

, 2

Out

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.

Games People Play.


Combining simultaneous and sequential moves from strategic form to extensive form
Combining simultaneous and sequential movesFrom strategic form to extensive form

  • To convert from strategic to extensive form representation requires the use of information sets.

  • Consider the chicken game

Games People Play.


Combining simultaneous and sequential moves from strategic form to extensive form1

Payoffs

#1

, #2

Player #2

-2

,-2

Straight

Straight

Swerve

1

,-1

Player #1

-1

,1

Straight

Swerve

0

,0

Player #2

Swerve

Information set

Combining simultaneous and sequential moves.From strategic form to extensive form.

Games People Play.


Combining simultaneous and sequential moves games with both sequential and simultaneous 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.

Games People Play.


Combining simultaneous and sequential moves games with both sequential and simultaneous moves1

P

H

P

H

P

L

High-Tech

P

H

P

L

P

L

P

H

P

H

High-Tech

P

L

Low-Tech

P

H

P

L

P

L

P

H

P

H

P

L

High-Tech

P

H

Low-Tech

P

P

L

L

P

H

P

H

Low-Tech

P

L

P

H

P

L

P

L

Combining simultaneous and sequential moves.Games with both sequential and simultaneous moves.

Product development and pricing game.

Game tree.

Games People Play.


Combining simultaneous and sequential moves games with both sequential and simultaneous moves2
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.

Games People Play.


Combining simultaneous and sequential moves games with both sequential and simultaneous moves3
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.

Games People Play.


Combining simultaneous and sequential moves games with both sequential and simultaneous moves4
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 forArnold. Hence Gary chooses in. In the sequential version Arnold chose ads and Gary stayed out.

Games People Play.


Combining simultaneous and sequential moves games with both sequential and simultaneous moves5

Payoffs

1,

5

,

5

US

Gateway

US

HP

5,

5

,

2

Foreign

5,

2

,

5

US

Gateway

US

Foreign

3,

4

,

4

Dell

Foreign

US

Gateway

2,

5

,

5

US

Foreign

HP

Foreign

4,

3

,

4

US

4,

4

,

3

Gateway

Foreign

4,

4

,

4

Foreign

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}.

Games People Play.


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