Perfect Information and Backward Induction
Georg Nöldeke
Wirtschaftswissenschaftliche Fakultät, Universität Basel
Advanced Microeconomics, HS 11
Lecture 6
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Perfect Information
Definition (Games of (Im)Perfect Information)
An extensive form game Γ is a game of perfect information if
I(x) = {x} holds for all x ∈ D. All other extensive form games are
games of imperfect information.
In words: In a game of perfect information every player is
perfectly informed of all previous actions whenever it is his
turn to move.
In particular, there are no “simultaneous” moves in a game
of perfect information.
To simplify notation,
information sets are not listed when describing a game of
perfect information.
strategies are written as Si : Xi → A, where si (xi ) ∈ A(xi ) is
the action chosen by player i after history xi .
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Backward Induction
The backward induction algorithm for a game of perfect
information Γ:
Call a decision node x penultimate in Γ if for every a ∈ A(x)
the history (x, a) is terminal.
Construct a new game Γ0 by eliminating penultimate nodes
from Γ as follows:
For every penultimate node x in Γ with ι(x) = 0, define
ui (x) = ∑a∈A(x) π(a | x)ui (x, a) and then eliminate all nodes
following x from the game tree, so that x becomes an end
node.
For every penultimate node x in Γ with ι(x) = i > 0, let si (x)
be an action that maximizes player i’s payoff ui (x, a) over
the actions in A(x). Define u j (x) = u j (x, si (x)) for all players
j = 1, . . . , N and then eliminate all nodes following x from the
game tree, so that x becomes an end node.
Repeat the above procedure until the initial node of the
game tree is reached.
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Backward Induction
Definition (Backward Induction Solution)
A strategy profile s for a game Γ of perfect information is a
backward induction solution if it can be obtained from the
backward induction algorithm.
In a finite game of perfect information the backward
induction algorithm must reach the initial node, implying
the following result.
Remark: An extensive form game is finite if its strategic
form is finite.
Theorem (Existence of Backward Induction Solutions)
Every finite game of perfect information has a backward
induction solution.
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Backward Induction
By definition, a backward induction solution is a strategy
profile.
As every strategy profile, the backward induction solution
results in an outcome. This outcome is known as the
backward induction outcome.
Observe: In a diagrammatic representation of the
backward induction algorithm in a game without chance
moves, the backward induction outcome is obtained by
following the arrows forward from the initial node until an
end node is reached. This end node is the backward
induction outcome.
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Backward Induction and Nash Equilibrium
The proof of the following result is a bit tricky.
Theorem (Backward Induction and Nash Equilibrium)
Every backward induction solution of a finite game Γ of perfect
information is a pure strategy Nash equilibrium of Γ.
Observe that the combination of the two preceding results
implies
Theorem
Every finite game of perfect information has a pure strategy
Nash equilibrium.
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Remarks
If a finite game of perfect information is generic, that is it
has the property that for for all players i the payoff of player
i satisfies ui (x) 6= ui (x0 ) whenever x and x0 are two different
terminal histories, then the backward induction solution is
unique.
Otherwise it may happen that there are multiple backward
induction solutions.
While every backward induction solution in a finite game of
perfect information Γ is a pure strategy Nash equilibrium of
Γ, the reverse does not hold: there may be additional pure
strategy Nash equilibria not corresponding to any
backward induction solution.
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Some Problems with Backward Induction
Standard Motivation for Backward Induction: Players are
rational and anticipate that other players will make rational
decisions in the continuation of the game.
Problems with this motivation:
Your opponents may be irrational. Even if you are rational,
you might then be better off not playing according to your
strategy in the backward induction solution.
In most situations that economists model as game of
perfect information, there is actually incomplete information
about other players’ preferences.
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