Extensive games with
imperfect information
ECON5200 Advanced microeconomics
Lectures in game theory
Fall 2009, Part 3
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G.B. Asheim, ECON5200-3
1
Incomplete
information
Imperfect information
Complete information
Perfect information
Lecture 1
Strategic
games
Static
Lecture 1
Bayesian
games
Lecture 2
Extensive games
(multi-stage games)
Dynamic
Lecture 3
Extensive games (the
general case)
1
1
1
2
2
2
1
c 
1
2
Multi-stage game
Not multi-stage game
("Almost perfect" information)
(Not "almost perfect" information)
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
2
G.B. Asheim, ECON5200-3
Three reasons why a player when making a move
has only partial information about previous actions

Not perfect recall

2
Not multi-stage
Perfect recall implies that
—
a
player
remembers
what
he once
knew.
 the player 1does not remember
what
he once
did
1
2 does
2 not remember what he once knew
 the player
2
2
2
2
Not OK
OK
 only—
partial
information
about
anhe
opponent’s
a player
remembers
what
once did.
previous action
1
1

2
2
Not complete information
1
2
1
1
1
1
1
 asymmetric
information
about
what nature has done
Not OK
OK
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G.B. Asheim, ECON5200-3
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1
Definition 200.1 An extensive game consists of
 A finite set N (the set of players)
 A set H of sequences (the set of histories)
 A set Z , where Z  H (the set of terminal histories)
 A function P from H\Z to N{c} (the player function)
 A probability distribution fc that for each h with P(h)  c
assigns probability to each action following h.
 For each i  N, a partition Ii of {h  H| P(h)  i} s.t.
A(h)  A(h) if h and h are in the same part of the partition (Ii: information partition; Ii Ii: information set).
 For each i  N, a vNM utility function on the set of
prob. distr. over Z, where i(z) is player i’s utility if z  Z
is realized (payoff function).
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
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G.B. Asheim, ECON5200-3
Definition 203.1 A pure strategy of
player i  N in an extensive game is a
function that assigns an action in A(Ii)
to each information set Ii Ii.
Strategy
Let Xi(h) be the sequence of information sets
that i runs into under the history h, and the
actions that he takes at these information sets.

Definition 203.3 An extensive game has perfect
recall if, for each player i  N, we have that
Xi(h)  Xi(h) if the histories h and h are in the
same information set of player i.
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G.B. Asheim, ECON5200-3
Example
1
L
R
2
2, 1
A
B
1
 I2  {{L}}

r

r
0, 0
1, 2
1, 2
0, 0
 I1  {{}, {(L, A), (L, B)}}
 X2(L)  {L}
 X1()  {}
 X1((L, A))  ({}, L, {(L, A), (L, B)})
 X1((L, B))  ({}, L, {(L, A), (L, B)})
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G.B. Asheim, ECON5200-3
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2
Definition 212.1 A mixed strategy of player i in
an extensive game is a probability measure over the
set of i’s pure strategies. A behavioral strategy of
player i is a collection (  i ( I i )) I i I i of independent
probability measures, where i(Ii) is a probability
measure over Ai(Ii).

A behavioral strategy is a book where the number of
pages equals the number of information sets for the
player and each page specifies a mixed action.
A mixed strategy is a probability distribution over such
books where, for each book, the pages specify pure
actions.
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G.B. Asheim, ECON5200-3
Let   (i)iN be a profile of mixed strategies or a
profile of behavioral strategies. The outcome O() of 
is the probability distribution over terminal histories that
are realized if each player plays according to i.
Two strategies for a player are outcome-equivalent if,
for every pure strategy profile for the opponents, they
generate the same outcome.
 Proposition 214.1 For each mixed strategy of a player
in a finite extensive game with perfect recall, there is an
outcome-equivalent behavioral strategy.
 Definition A Nash equil. in mixed strat. of an ext.
game is a strategy profile  of mixed strategies with the
property that, for each player i  N and for every mixed
strategy i of player i, i (O( i ,  i ))  i (O( i ,  i ))
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G.B. Asheim, ECON5200-3
Examples:
Sequential equilibrium
1
L
NL
1
2, 2
M
1
L
2, 2
R
M
R
2
L
3, 1
R
2
L
0, 0
L
0, 2
R
1, 1
L
R
L
R
3, 1
0, 0
0, 2
1, 1
L
R
R
2 2, 2
L 2,must
set a player
1 0,information
0
MFor3,each
choose a best response givenM
his beliefs.
3, 1 0, 0
R 0, 2 1, 1
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R 0, 2 1, 1
G.B. Asheim, ECON5200-3
9
3
More examples:
Sequential equilibrium (cont)
1
L
NL
1
2, 2
L
2, 2
M
R
1
M
R
2
L
2
R
3, 1
L
1, 0
R
0, 0
0, 1
L
R
L
R
3, 1
1, 0
0, 0
0, 1
L
R
L beRconsistent with the strategy profile.
Beliefs must
L 2, 2 2, 2
M 3, 1 1, 0
M 3, 1 1, 0
What
0, 0 0, 1 should be imposed on beliefs at
R requirements
information sets that are reached with
0 0, 1 0?
R 0,probability
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G.B. Asheim, ECON5200-3
1
L
R
2
L
R
L
3
L
R
R
3
L
R
L
R
Structural consistency?
1
L
1
L
M
NL
R
R
2
L
2
R
L
3
L
R
R
3
L
R
1
M
L
R
3
L
R
L
3
L
R
L
R
R
3
L
R
3
L
R
L
R
Common beliefs?
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Definition 222.1 An assessment in an ext. game is a pair
(, ), where  is a profile of behavioral strategies and  is
a funct. (a belief system) that assigns to each info. set a
probability measure on the set of histories in the info. set.
 Definition 224.1 Consider an ext. game w/perfect recall.
An assessment is sequentially rational if we have that, for
each player i and for each info. set Ii Ii, player i's strategy
is a best resp. to the opp.s’ strategies given i's beliefs at Ii.
 Definition 224.2 Cons. a finite ext. game w/perf. recall. An
n
n 
ass. (, ) is consistent if there is a sequence ((  ,  )) n 1
n
of ass.s that converges to (, ), where each  is compl.
mixed and where n is derived from n using Bayes' rule.
 Definition 225.1 An ass. is a sequential equilibrium of a
finite ext. game w/perfect recall if it is sequentially rational
and consistent.
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G.B. Asheim, ECON5200-3

4
Refinements: Perfect equil. in strategic games
i : mixed strategy of player i,  : profile of mixed strategies



A
B
C
Definition 248.1 Consider a finite strategic game.
0 0, 0 0, 0
  (1, …, n)Ais a0,perfect
equilibrium if there is a
k 
sequence ( ) kB1 of0,completely
strategy profiles
0 1, 1 2, mixed
0

such
that
for
each
i
 N, i is a best
that converges to
C 0, 0 0, 2 2, 2
response to  ki for all k.
Proposition 248.2 A strategy profile in a finite 2-player
strategicIfgame
is a perfect
equilibrium
if and
the players
are cautious,
would
notonly if it is a
mixed strategy
Nash
equilibrium
of
the Nash
equilibria
(A, A)and
andthe
(C,strategy
C)
neither player be
is weakly
dominated.
strategically
unstable?
Proposition 249.1 Every finite strategic game has a
perfect equilibrium.
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G.B. Asheim, ECON5200-3
Refinements: Perfect equil. in strategic games (cont)
L
R
L
R
T 1,1,1 1,0,1
T 1,1,0 0,0,0
B 1,1,1 0,0,1
B 0,1,0 1,0,0

r
”Structural consistency?”
L
R
L
R
T 1,1,1 1,1,0
T 0,1,1 0,0,2
B 1,0,1 0,2,0
B 2,0,0 0,0,0

r
Common beliefs?
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G.B. Asheim, ECON5200-3
Refinements: Perfect equil. in extensive games
T'
1
T
3, 0
L 2
T'
L' 1
0, 2
t
l
1, 0
0, 3
3, 0
T l 3, 0
3, 0
L t 0, 2
1, 0
L l 0, 2
0, 3
The agent strategic form

3
L'
T 3,0,3 3,0,3
T 3,0,3 3,0,3
L 0,2,0 1,0,1
L 0,2,0 0,3,0
l
t
17.09.2009
1 4
 1 
T'
L'
T'
L'
T t 3, 0
G.B. Asheim, ECON5200-3
15
5
Refinements: Perfect equil. in extensive games (cont)
A
B 2
1, 1
R
L
1
L
R 1
0, 2
a
b
2, 0
3, 3
Aa 1, 1
1, 1
Ab 1, 1
1, 1
Ba 0, 2
2, 0
Bb 0, 2
3, 3
The agent strategic form 1  



L
R
L
1 2
R
A 1,1,1 1,1,1
A 1,1,1 1,1,1
B 0,2,0 2,0,2
B 0,2,0 3,3,3
b
a
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G.B. Asheim, ECON5200-3
Refinements: Perfect equil. in extensive games (cont)



Definition 251.1 A perfect equilibrium in a finite
extensive game w/perfect recall is the profile of
behavioral strategies that corresponds to a perfect
equilibrium of the agent strategic form of the game.
Proposition 251.2 For every perfect equilibrium  of a
finite extensive game w/perfect recall there is a belief
system  such that (, ) is a sequential equilibrium.
Corollary 253.2 Every finite extensive game w/perfect
recall has a perfect equilibrium and thus also a
sequential equilibrium.
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G.B. Asheim, ECON5200-3
There are two perfect equilibria in pure
strategies (i.e. the agent strategic
L
form has two perfect equilibria
in pure strategies). The
2
strategic form has only
R
L
one perfect equilibrium.
0, 0
1, 1
1
R
1

0, 0
r
1, 1
1
L
R
2
L
1, 0
17.09.2009
R
1, 1

0, 0
There is one perfect equilibrium (i.e.
the agent strategic form has one
perfect equil.). The strategic
1
form has two perfect equir
libria in pure strategies.
1, 1
G.B. Asheim, ECON5200-3
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6