Imperfect Information and Subgame
Perfection
Georg Nöldeke
Wirtschaftswissenschaftliche Fakultät, Universität Basel
Advanced Microeconomics, HS 11
Lecture 7
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Motivation
The idea is to extend the logic of backward induction to
games with imperfect information.
The difficulty in doing so, is that even when all subsequent
actions are taken as given it is not clear how to determine
an “optimal action” for a player at an information set.
Suppose, for instance, two players moves simultaneously.
As we have seen in our analysis of strategic form games,
what is optimal for one player in such a situation may
depend on the action chosen by the other player.
The concept of a subgame perfect equilibrium offers one
solution to this difficulty by applying the logic of backward
induction to a subset of the decision nodes.
A more powerful generalization of backward induction is
provided by the concept of sequential equilibrium.
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Motivation
Reinhard Selten
Winner of the Nobel Prize in
Economics 1994
Photo: Volker Lannert/University of
Bonn
http://www.myscience.de/image/
db/menu_17164.jpg
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Subgames
To prepare for the following definition of a subgame, we
say that a node y follows node x in an extensive form game
if there is a sequence of feasible actions such that starting
with the history x we can reach history y.
In a game of perfect information any decision node x
together with all nodes following x and the associated
payoffs as well as information sets defines a game of
perfect information which is called the subgame defined by
x.
In a game of imperfect information, we can use the same
idea – but should make sure that the subgame defined by
x respects the information structure of the original game.
Advanced Microeconomics, HS 11
Lecture 7
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Subgames
Definition (Subgame)
A decision node x defines a subgame of an extensive form
game Γ if
I(x) = {x} and
for all decision nodes y following x: z ∈ I(y) implies that z
also follows x.
The corresponding subgame Γx has x as initial node,
contains all nodes following x in its game tree, and inherits
its information sets and payoffs from the original game Γ.
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Subgames
Observe:
The initial node x0 of any extensive form game Γ define a
subgame, which is nothing but the original game Γ. All
other subgames are called proper subgames.
Some extensive form games have no proper subgames.
For instance, if all players move simultaneously, then there
are no proper subgames.
In a game of perfect information every decision node
defines a subgame.
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Pure Strategy Subgame Perfect Equilibria
Every pure strategy profile for an extensive form game Γ
defines actions for all information sets appearing in any
subgame Γx of Γ and in this sense induces a strategy
profile for every subgame Γx of Γ.
Definition (Pure Strategy Subgame Perfect Equilibrium)
A pure strategy profile s for an extensive form game Γ is a pure
strategy subgame perfect equilibrium of Γ if it induces a Nash
equilibrium in every subgame of Γ.
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Lecture 7
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Pure Strategy Subgame Perfect Equilibria
The concept of a pure strategy subgame perfect
equilibrium generalizes backward induction:
Theorem
If Γ is a game of perfect information then s is a pure strategy
subgame perfect equilibrium if and only if it is a backward
induction solution.
The concept of a pure strategy subgame perfect
equilibrium refines pure strategy Nash equilibrium:
Theorem
If s is a pure strategy subgame perfect equilibrium of an
extensive form game Γ then s is a pure strategy Nash
equilibrium of Γ.
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How to find Pure Strategy Subgame Perfect Equilibria
To find pure strategy subgame perfect equilibria proceed as
follows:
Identify all those subgames Γx that do not have any proper
subgames.
Determine a pure strategy Nash equilibrium for all those
subgames.
Replace the initial node x of all those subgames with a end
node with the payoffs corresponding to the Nash
equilibrium payoffs in Γx .
As is the backward induction algorithm repeat the above
steps until the initial node of Γ is reached.
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Mixed Strategy Subgame Perfect Equilibria
The algorithm described on the previous slide may fail
because one (or more) of the subgames of Γ has no pure
strategy Nash equilibrium.
For finite extensive form games this problem may be
solved by considering mixed strategy Nash equilibria of the
relevant subgames.
The result will be a mixed strategy subgame perfect
equilibrium.
Warning:
There are some subtleties in defining mixed strategies for
extensive form games – see the section on mixed
strategies, behavioral strategies, and perfect recall in the
textbook.
For the purpose of these lectures, you will not have to worry
about these.
Advanced Microeconomics, HS 11
Lecture 7
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