Dynamic games: Backward
induction and subgame perfection
Lectures in Game Theory
Spring 2011, Part 4
10.12.2010
G.B. Asheim, ECON4240-4
1
Recall the extensive form: It specifies
Players: {1, ... , i, ... , n}
 What actions an acting player can choose among,
what an acting player knows.
 Payoff for each of the players as a function of the
actions that are realized.

Game tree
2 H
1
Decision node
(initial node)
10.12.2010
H
L
1, 2
1, 1
L 1
2 H 2, 1
Decision nodes L
0, 0
G.B. Asheim, ECON4240-4
Some terms:
 Successor
assigned
Payoffs
Predecessor
players 1 and
2
toImmediate
sucessor
at terminal nodes
 Immediate predecessor
2
1
Tree rule 1

Every node is a successor of the initial node, and
the initial node is the only one having this property.
Not OK
2 H
H
OK
L
H
2
1
L
H
1
L
1
U
3
D
1
L
2 H
L
2 H
H
L
H
2
L
H
L
L
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2 H
L
2 H
L
2 H
L
3
G.B. Asheim, ECON4240-4
Tree rule 2

Each node except the initial node has exactly one
immediate predecessor. The initial node has no
predecessor
predecessor.
2 H
Not OK
H
3
L
1
2 H
1
L
Not OK
L
H
1
L
H
L
2 H
L
10.12.2010
G.B. Asheim, ECON4240-4
4
2
Tree rule 3

Multiple branches extending from the same
node have different action labels.
Tree rule 4

Each information set contains decision nodes
for only one of the players.
Not OK
H
1 H
L
H
1
L
2
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OK
H
2 H
L
H
1
L
2 L
L
5
G.B. Asheim, ECON4240-4
Tree rule 5

All nodes in a given information set must have
the same number of immediate successors and
they must have the same set of action labels on
the branches leading to these successors.
Not OK
H
L
H
1
L
10.12.2010
2 H
Not OK
2
1
2 L
G.B. Asheim, ECON4240-4
2
6
3
Perfect recall

A player remembers what he once knew.
Not OK
2
1
2
2
2
2
2
10.12.2010
OK
2
1
2
2
2
2
2
7
G.B. Asheim, ECON4240-4
Perfect recall

A player remembers what he once did.
Not OK
2
1
2
10.12.2010
1
1
1
1
OK
2
1
2
G.B. Asheim, ECON4240-4
1
1
1
1
8
4
Perfect information

Each information set
contains only one
d i i node.
decision
d
H
1
L
Imperfect information

There is at least one
1
contingency in which an
acting player does not
know exactly where he is.
10.12.2010
H
2 H
L
H
2 L
2 H
L
H
L
2 L
9
G.B. Asheim, ECON4240-4
Analyzing dynamic games
A dynamic game can be analyzed both in
the extensive form
and
the normal form.
Stay out
Entrant
IIncumbent
b t
Accept Fight
0, 2
Fight
- 1, - 1
Enter
Inc mbent
Incumbent
Accept
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1, 1
Enter 1, 1 - 1, - 1
Entrant
Stay out 0, 2 0, 2
Are both NEa reasonable?
Is threat of fight credible?
G.B. Asheim, ECON4240-4
10
5
Another example
2 H
1
H
L
1, 2
1 1
L 1,
2 H 2, 1
L
0, 0
HH HL LH LL
1 2 1,
1 2 1,
1 1 1,
1 1
H 1,
L 2, 1 0, 0 2, 1 0, 0
10.12.2010
G.B. Asheim, ECON4240-4
11
Sequential rationality:

An optimal strategy for a player should maximize his or
her expected payoff, conditional on every information
set at which this player has the move. That is, player i’s
strategy should
h ld specify
if an optimal
i l action
i from
f
each
h off
player i’s information sets, even those that player i does
not believe (ex ante) will be reached in the game.
Backward induction:

The process of analyzing a game backwards in time
(from information sets at the end
d of the tree to information sets at the beginning). At each information set,
one strikes from consideration actions that are dominated, given the terminal nodes that can be reached.
10.12.2010
G.B. Asheim, ECON4240-4
12
6
Illustrating backward induction
0, 2
Stayy out
Entrant
Fight
2 H
- 1, - 1
1
H
Enter
Incumbent
Accept
10.12.2010
L
1, 1
1, 2
L 1, 1
2 H 2, 1
L
G.B. Asheim, ECON4240-4
0, 0
13
Results


Backward induction identifies a unique strategy profile
in a finite perfect information game with no payoff ties.
Such a strategy profile is a Nash equilibrium.
Observation

Backward induction generalizes rationalizability to
perfect information games.
Questions


See Asheim & Perea “Sequential
and quasi-perfect rationalizabilityy in extensive games”
g
(http:/
How tto define
H
d fi rationalizti li
/folk.uio.no/gasheim/AP-GEB05.pdf)
ability for more general classes of extensive-form games.
How to define equilibrium for ext.-form games so that
equilibrium implies backw. ind. in perf. info. games.
10.12.2010
G.B. Asheim, ECON4240-4
14
7
Subgame

Definition: A subgame in an extensive-form game
a) starts with a decision node (the initiating dec. node)
b) includes also all successors (decision nodes that can
be reached from the initiating decision node).
c) splits no information set (no included dec. node is in
an information set that contains excluded dec. nodes).
What are the
subgames?
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15
G.B. Asheim, ECON4240-4
Subgame perfect Nash equilibrium

Definition: A strategy profile is called a subgame perfect
Nash equilibrium if it specifies a Nash equilibrium in
every subgame of the original game.
game
Stay out
Entrant
0, 2
Fight
2 H
- 1, - 1
1
H
Enter
Incumbent
Accept
10.12.2010
L
1, 1
G.B. Asheim, ECON4240-4
1, 2
L 1, 1
2 H 2, 1
L
0, 0
16
8
Results


Any finite extensive-form game has a subgame perfect
Nash equilibrium.
In a perfect information game without payoff ties, the
unique SPNE coincides with the strategy profile
indentified by backward induction.
Algorithm



Consider the normal forms of all subgames.
g
Determine the Nash equilibria of each subgame.
Find the Nash equilibria of the whole game that are also
Nash equilibria of each subgame.
10.12.2010
G.B. Asheim, ECON4240-4
17
Does actual behavior conform to
subgame-perfection?

The ultimatum game

The centipede game

Game of trust (Game of punishment)
The paradox of backward induction:

Why should
Wh
h ld a player
l
conform
f
to b
backward
k d iinduction
d i
at decision nodes where he/she knows that an earlier
player has deviated from backward induction?
10.12.2010
G.B. Asheim, ECON4240-4
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9