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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 oneplayer game.
There are two such subgames.
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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 (PT & ND),
2.
Don’t Prosecute (I) if Anna trespasses and Do Nothing (N) if she doesn’t. Denote
this as (IT & ND).
It is straightforward to see that there are two Nash equilibria:
1.
(T,
IT & ND)
2.
(D,
PT & ND).
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 unmodeled 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.
 Winter '08
 notsure
 Economics, Game Theory

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