Lecture notes 6

# Lecture notes 6 - Economics 112 Lecture Notes Lecture 6....

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© David Scoones 2010 Page 1 Economics 112 Lecture Notes Lecture 6. Sequential Games In sequential games players move in sequence, one at a time. This assumes that players are committed to their moves. Adding a sequential structure clarifies the information available to players, and can lead to simpler predictions. A special case of sequential game is an embedded game . I Key terms: extensive form, subgame, proper subgame, refinement, subgame perfect equilibrium, backward induction, embedded game, forward induction, commitment device. 1. The trespass game The very first example we saw this term was a sequential game: the trespass game. In that story Brandon’s Beach Resort controls access providing a simple shortcut to a public beach. Anna’s Adventure Tours sets up dive tours just down the beach from Brandon’s access, and Anna would find it convenient to cut through Brandon’s access. Brandon doesn’t like people cutting across the property and has posted a no trespassing sign to prevent it. If Anna ignores the sign, Brandon can sue for damages due to the trespass, but the case is costly to prove. If she did trespass and Brandon prosecutes, Anna will be convicted and face a small fine. The extensive form is: 2. Subgames A subgame is a part of a sequential game that itself forms a game: i.e. it has a well defined set of players, strategies and payoffs. Subgames can be defined very formally, but the basic idea is reasonably intuitive. Think of snipping off pieces of a game tree, pieces that themselves look like game trees. Each subgame must begin at a well defined decision node. You mustn’t remove actions from any decision nodes, and cannot split information sets. The trespass game has three subgames, indicated by the red boxes below. After Anna moves, Brandon faces what amounts to a one-player game. There are two such subgames.

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© David Scoones 2010 Page 2 The third subgame is the whole game. This is of course not very “sub”, and so the subgames that are not the whole game are sometimes distinguished with the label proper subgame . 3. Nash Equilibria In the trespass game, Anna has two strategies: 1.Trespass (T) and 2. Don’t trespass (D) Brandon has two (contingent) strategies: 1. Prosecute (P) if Anna trespasses and Do Nothing (N) if she doesn’t. Denote this with the and sign (P|T & N|D), 2. Don’t Prosecute (I) if Anna trespasses and Do Nothing (N) if she doesn’t. Denote this as (I|T & N|D). It is straightforward to see that there are two Nash equilibria: 1. (T, I|T & N|D) 2. (D, P|T & N|D). Be sure to convince yourself this is true. 4. Refinements A refinement is an additional restriction placed on the predicted outcomes of a game, beyond the requirement that they must be Nash equilibria. These restrictions reduce the number of predictions, and so “refine” the set of equilibria. We saw refinements of a sort when we considered focal points, where some un-modeled information made one of the game’s Nash equilibria more likely to be chosen. Here we will instead evaluate the
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## This note was uploaded on 01/28/2011 for the course ECON 112 taught by Professor Notsure during the Winter '08 term at University of Victoria.

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Lecture notes 6 - Economics 112 Lecture Notes Lecture 6....

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