Lecture 10 Extensive Form Games2010 3

# Lecture 10 Extensive Form Games2010 3 - ECON 4109H LECTURE...

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ECON 4109H LECTURE 10 Extensive Form Games November 21, 2010 ECON 4109H LECTURE 10

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Entry Game An incumbent faces the possibility faces the possibility of entry by a challenger. The challenger may be a ﬁrm considering entry into an industry currently occupied by a monopolist. The challenger may be a politician competing for the leadership of a party. The challenger may enter or not. If he enters, the incumbent may either acquiesce or ﬁght. ECON 4109H LECTURE 10
Extensive Games At the start of an extensive game, and after any sequence of events a player chooses an action. For example, at the start of the entry game, the challenger, may choose to go In or Out . After that, the incumbent may choose Acquiesce Fight . ECON 4109H LECTURE 10

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Terminal Histories How can we represent the possible sequences of moves? In the entry game there are three possible sequences: ( Out ), ( In , Acquiesce ), ( In , Fight ). The above are called terminal histories . From the terminal histories, it is possible to infer what actions are available at each point in time . ... ... but it is not possible to infer who moves when. For example, given the above terminal histories, it is possible to infer, that ﬁrst there is a decision between In and Out . If Out is chosen the game ends. If In is chosen, there is a choice between Fight and Acquiesce . ECON 4109H LECTURE 10
Terminal Histories Contrast the following two sets of histories: 1 ( Out ), ( In , Acquiesce ), ( In , Fight ). 2 ( Out ), ( In ), ( In , Acquiesce ), ( In , Fight ). In (1), it is clear what the possible sequences of moves are. In (2), it is not so clear. If In is chosen in the ﬁrst stage, will there be another move or not? On the one hand there are two sequences ( In , Acquiesce ), and ( In , Fight ), which suggests that a choice between Acquiesce and Fight follows the move In . On the other hand, there is the lone sequence ( In ), which suggests that the game ends after the move In . ECON 4109H LECTURE 10

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Terminal Histories Contrast the following two sets of histories: 1 ( Out ), ( In , Acquiesce ), ( In , Fight ). 2 ( Out ), ( In ), ( In , Acquiesce ), ( In , Fight ). The answer is that ( In ) is a history but not a terminal history . In general if ( C , D ) is a terminal history, then ( C ) cannot be a terminal history. If ( C , D , E ) is a terminal history, then neither ( C ) nor ( C , D ) can be terminal histories. ( C ) and ( C , D ) are subhistories of ( C , D , E ) . (1) could be a list of terminal histories, but (2) could not. ECON 4109H LECTURE 10
Subhistories For any history, (i.e., a ﬁnite sequence of actions), ( a 1 , a 2 , . . . , a k ) , (the empty history , representing the start of the game, before any move has been made) is a subhistory of ( a 1 , a 2 , . . . , a k ) . Also for every m such that 1 6 m 6 k , ( a 1 , a 2 , . . . , a m ) is a subhistory of ( a 1 , a 2 , . . . , a k ) . In particular ( a 1 , a 2 , . . . , a k ) is a subhistory of itself. All subhistories of ( a 1 , a 2 , . . . , a k ) except ( a 1 , a 2 , . . . , a k ) are proper subhistories of ( a 1 , a 2 , . . . , a k ) . ECON 4109H LECTURE 10

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Subhistories of Inﬁnite Histories Consider an inﬁnite history ( a 1 , a 2 , . . . ) is a subhistory of ( a 1 , a 2 , . . . ) For all m , ( a 1 , a 2 , . . . , a m ) is a subhistory of ( a 1 , a 2 , . . . ) ( a 1 , a 2 , . . . ) is a subhistory of itself.
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Lecture 10 Extensive Form Games2010 3 - ECON 4109H LECTURE...

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