Extensive games with
perfect information
ECON5200 Advanced microeconomics
Lectures in game theory
Fall 2009, Part 2
05.10.2009
G.B. Asheim, ECON5200-2
1
Incomplete
information
Imperfect information
Complete information
Perfect information
Lecture 1
Strategic
games
Static
Application:
Bargaining
games

Dynamic
Lecture 2
Extensive games
(multi-stage games)
First: Extensive
games with perfect
information and
without simultaneous moves
05.10.2009

Lecture 1
Bayesian
games
Lecture 3
Extensive games (the
general case)
Application:
Repeated
games
Then: Extensive
games with perfect
information and with
simultaneous moves
(multi-stage games)
2
G.B. Asheim, ECON5200-2
Definition (89.1) of an extensive game w/perf. info.
N = {1, 2}
1
(The set of players)
H = {, T, B, (T, L), (T, R), (B, L'), (B, R')} T
(The set of histories)
2
Z {(T, L), (T, R), (B, L'), (B, R')}
(The set of terminal histories)
H\Z  {, T, B}
B
2
L
R
L
R
1
2
1
1
2
1
0
0
P()  1, P(T)  2, P(B)  2 (Player function)
1(T, L)  1, 1(T, R)  1, 1(B, L')  2, 1(B, R')  0 (Payoff
2(T, L)  2, 2(T, R)  1, 2(B, L')  1, 1(B, R')  0 functions)
A()  {a| (, a)  H}  {T, B}
(Action sets)
A(T)  {a| (T, a)  H}  {L, R} A(B)  {a| (B, a)  H}  {L, R}
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3
1

Definition 92.1 A strategy for player i Strategy
 N in an extensive game w/perf. info. 1
is a function that assigns an action in
A(h) to each non-terminal history
B
T
2
2
h  H\Z for which P(h)  i.
L
Strategies for player 1: T, B
1
Strategies for player 2:
(L, L), (L, R), (R, L), (R, R), 2


R
L
R
1
1
2
1
0
0
What is the difference between an action and a
strategy? A strategy is a rule of action.
In a simultaneous-move game actions cannot be conditioned on anything. Strategy = action in such games.
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G.B. Asheim, ECON5200-2
Every strategy profile determines a termi1
nal history (an outcome)
O(T, (L, L'))  (T, L)
O(T, (L, R'))  (T, L)
O(T, (R, L'))  (T, R)
O(T, (R, R'))  (T, R)
O(B, (L, L'))  (B, L)
O(B, (L, R'))  (B, R)
O(B, (R, L'))  (B, L)
O(B, (R, R'))  (B, R)

B
T
2
2
L
R
L
R
1
2
1
1
2
1
0
0
Let Si denote the
strategy set for player i.
Definition 92.1 A Nash equilibrium of an extensive
game w/perf. info. is a strategy profile s such that,



for each i  N, si  S i , i (O( si , si ))  i (O( si , si )).
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G.B. Asheim, ECON5200-2
Let Si denote the strategy set for player i.
Define ui by ui(s)  i(O(s)) for each si.


1
Definition 92.1 The strategic
T
form of an extensive game
2
w/perf. info. is the strategic
L
R
game G  N, (Si), (ui).
1
Lemma A strategy profile
2
s is a Nash equilibrium
L L
of an extensive game 
1
, 2
T
if and only if it is a
2
,1
B
Nash equilibrium of
the strategic form of .
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1
1
LR 
B
2
L
R
2
1
0
0
R L
1, 2
1, 1
0, 0
2, 1
RR
1, 1
0, 0
6
2
Definition 97.1 The subgame of
Subgame
1
an extensive game w/perf. info.
that follows the history h  H/Z
B
T
is the extensive game where
2
2
 H|h is the set of histories h
for which (h, h)  H,
 Z|h is the set of histories h
for which (h, h)  Z,
 P|h is defined by P|h(h) 
P(h, h) for all h  H|h\Z|h,
 i|h(h)  i|h (h, h)
for all h  Z|h.
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L
R
L
R
1
2
1
1
2
1
0
0
N = {1, 2}
H|T = {, L, R} Z|T {L, R}
H|T\Z|T  {} P()  2
1|T(L)  1, 1|T (R)  1
2|T (L)  2, 2|T (R)  1
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G.B. Asheim, ECON5200-2
Subgame-perfect equilibrium
Let Si|h denote the strategy set
for player i in the game (h).
Definition 97.2 The subgame
perfect equilibrium of an
extensive game w/perf.
info. is a strategy profile s
such that for each player i 
N and each h  H\Z for
which P(h) = i, si  Si |h ,
1
B
T
2
2
L
R
L
R
1
2
1
1
2
1
0
0
(B,(L,L)) is the unique
subgame-perfect equil.
(B,L) is the unique
subgame-perfect equil.
i (Oh ( si |h , si |h ))  i (Oh ( si |h , si )).
outcome
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Lemma A strategy profile s is a subgame perfect equilibrium of an extensive game w/perf. info. if and only
if s|h is a Nash equilibrium of (h) for all h  H\Z.
If the set of histories is finite, then the extens. game is finite.
Proposition 99.2 Every finite extensive game w/perf.
info. has a subgame perfect equilibrium.
Proof is based on backward induction.
Lemma 98.2 (The one deviation property) Let  be a
finite horizon game w/perf. info. A strategy profile is a
subgame perfect equilibrium for  if and only if each
player i  N and each h  H\Z for which P(h) = i,
player i cannot gain by changing his action only at h.
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3
Definition of an extensive game w/simultan. moves
(multi-stage games)
N = {1, 2} (The set of players)
H = {, T, C, (C,(B,B)), (C,(B,S)),
(C,(S,B)), (C,(S,S))}
1
T
C
B
S
2
3
,
1
0
,
0
B
(The set of histories) 2
S 0, 0 1, 3
Z {T, (C,(B,B)), (C,(B,S)), (C,(S,B)),
(C,(S,S))}(The set of terminal histories)
H\Z  {, C}
P()  1, P(C)  {1, 2} (Set-valued player function)
1(T)  2, 1(C,(B,B))  3, 1(C,(B,S))  0, 1(C,(S,B))  0 , 1(C,(S,B))  1
2(T)  2, 2(C,(B,B))  1, 2(C,(B,S))  0, 2(C,(S,B))  0 , 2(C,(S,B))  3
A()  {a| (, a)  H}  {T, C}
(Action sets)
A(C)  {a| (C, a)  H}  A1(C)  A2(C)  {B, S}  {B, S}
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Does actual behavior conform to
subgame-perfection?

The ultimatum game

The take-it-or-leave game

Game of trust (Game of punishment)
The paradox of backward induction:

Why should a player conform to backward induction
at decision nodes where he/she knows that an earlier
player has deviated from backward induction?
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Results


Backward induction identifies a unique strategy profile
in a finite perfect information game without
simultaneous moves and with no payoff ties.
Such a strategy profile is the unique subg.-perf. equil.
Observation

Backward induction generalizes rationalizability to
perfect information games w/out simultaneous moves.
Questions


How to define rationalizability for more general classes
of extensive games.
How to define equilibrium for extensive games so that
equilibrium implies backw. ind. in perf. info. games.
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4
Relationship between equilibrium concepts
Proper
equilibrium
Myerson (1978)
Nash
equilibrium
Extensive form
perfect equil.
Strategic form
perfect equil.
Quasi-perfect
equilibrium
Selten (1975)
van Damme (1984)
Weak sequential
equilibrium
Sequential
equilibrium
Reny (1992)
Kreps & W. (1982)
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Selten (1975)
Subg.-perf. equil.
= Selten (1965) in
multi-stage games
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G.B. Asheim, ECON5200-2
Relationship between rationalizability concepts
1
11
2
2
2120
212
4
1
Rationalizability
22
3
0
111
110
Bernh. (1984)
Pearce (1984)
0
1 003-1
-1
3
3
Schuhm. (1999)
Permissibility
Quasi-perfect
rationalizability
0 (1994)
Börgers
0 (1992)
Brandenb.
Dek. & Fud. (1990)
Asheim & P (2005)
Weak sequential
rationalizability
Sequential
rationalizability
Ben-Porath (1997)
Dekel et al. (1999)
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1
T
Extensive form rationalizability
2
2
Pearce (E.trica, 1984) and Battigalli & Sinichalchi (JET, 2002)
S
B
T 2, 2 2, 2
CB 3, 1 0, 0
CS 0, 0 1, 3
Iterated elimination
of weakly dominated
strategies
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G.B. Asheim, ECON5200-2
Forward induction
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Proper
rationalizability
C
B
S
B 3, 1 0, 0
S 0, 0 1, 3
S
B
T 2, 2 2, 2
CB 3, 1 0, 0
CS 0, 0 1, 3
Iterated elimination of choice
sets under full admissibility
Asheim & Dufwenberg (EJ, 2003)
G.B. Asheim, ECON5200-2
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5
Incomplete
information
Imperfect information
Complete information
Perfect information
Lecture 1
Strategic
games
Static
Dynamic

Lecture 2
Extensive games
(multi-stage games)
First application:
Bargaining games

Second application:
Repeated games
Extensive games with
perfect information and
with simultaneous moves
(multi-stage games)
Extensive games with
perfect information
and without simultaneous moves
05.10.2009
Lecture 1
Bayesian
games
Lecture 3
Extensive games (the
general case)
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G.B. Asheim, ECON5200-2
Bargaining games of alternating offers

Version: Split-the-pie w/linear payoff function
N = {1, 2}
Players:
Histories: I
 or (x0, R, x1, R, … , xt, R)
(x0, R, x1, R, … , xt)
(x0, R, x1, R, … , xt, A)
(x0, R, x1, R, … )
II
III
IV
Terminal histories: Histories of type III and IV
Player fn.: P(h) = 1 for hist. of type I and II if h =  or t is odd.
P(h) = 2 for hist. of type I and II if t is even.
Payoff fn.: 1(h) = 1txt if h  Z and h is finite, where 0  1t  1.
2(h) = 2t (1xt) if h  Z and h is finite, where 0  2t  1.
1(h) = 2(h) = 0 if h  Z and h is infinite.
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2
1
x
0
2

Player 1 is in the
same situation in
periods 0 and 2.
A x0 , 1  x0
R
2
xt is 1’s
share in
period t.
 is a com-
x1
1
R
A
x1 ,  (1  x1 )
 2 x2  2 
1
1
Rubinstein’s
model
mon dis 2 x 2 ,  2 (1  x 2 )
bargaining
count factor
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6

Proposition 122.1 The bargaining game of
alternating offers (split-the-pie w/linear payoff
functions) has a unique subgame perfect
equilibrium. This equilibrium is characterized by:
1
1 2
x
1 demands x t 
1  1 2
x
2 A
1-x
1 (1   2 )
R
t
1 accepts if and only if x 
2
1  1 2
x
1  1
1 (1   2 )
t
1 A
 1
2 offers x 
1  1 2
1  1 2
R
1
 (1  1 )
2 accepts if and only if 1  x t  2
1  1 2
0
0
0
1
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1x1
2(1-x1 )
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G.B. Asheim, ECON5200-2
Properties of the subgame-perfect equilibrium

Efficiency. Agrement is reached at once.

Stationary strategies. Each player proposes the same
each time he offers, and the uses the same criterion
to decide whether to accept an offer. BUT: Note
that it is not imposed a priori that strategies must be
stationary; this is a result of the analysis.

First-mover advantage. Player 1 (who moves first)
receives the biggest share:
1 2
1
1
If 1   2   , 1 (O ( s )) 


1  1 2 1   2
Increased impatience reduces a player’s share of the pie.

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G.B. Asheim, ECON5200-2
Infinitely repeated games
Prisoners’
dilemma
D
D 1, 1
C
3, 0
C 0, 3 2, 2
u2
3
2
1
Can repetition discipline
1
the players to cooperate?
Deviating yields a short run gain.
Deviating yields a loss of reputation
that undermines future cooperation.
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2
u1
3
Yes, if gain
now  PV of
future loss
21
7

●
Definition 137.1 Let G  N, (Ai), (ui) be a strategic
game; let A  i  N Ai where, for each i  N, Ai is
compact and ui is continuous. A -discounted
infinitely repeated game of G is an extens. game
w/perfect information and simultan. moves, where
H  t0 At ( A0  {} contains only the empty history)
 Z  A (the set of terminal histories)
 P(h)  N for each nonterminal history h  H
● ui (( a t ))  (1   ) t1 t 1ui (a t ) ; (at)  A and 0    1.
Feasible payoff profile for G: v is such that there exists
(a)aA with a  0 and aAa  1 s.t. v  aA a u(a) .
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
22
G.B. Asheim, ECON5200-2
(Average) payoff in a repeated game
(1   )t 1  t 1vt


Discounting
p: Probability that the game ends.
r: Rate of time preference.


1 p
1 r
Why multiply with (1  )?
Suppose that vt  v for all t. Then

(1   )t 1  t 1vt  (1   )(1     2  )v  v
Special case 1: v1  v , but vt  0 for all t  1.

(1   )t 1  t 1vt  (1   )v
Special case 2: v1  0 , but vt  v for all t  1.

(1   )t 1  t 1vt  v
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Strategies as machines

Example: Simple strategy profiles (Abreu, 1988)
Consider the following paths:
(a (0)t )  a(0)1 , a (0) 2 , a(0) 3 , a (0) 4 , 
(a (1)t )  a(1)1 , a(1) 2 , a (1)3 , a(1) 4 , 

(a (n)t )  a(n)1 , a (n) 2 , a(n)3 , a (n) 4 , 
Rules: Start with (a(0)t).
If i  N (and only i) deviates from (a(j)t), then (re)start (a(i)t).
Write:
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 (a (0)t ), (a (1)t ),  , (a (n)t ) 
G.B. Asheim, ECON5200-2
24
8

Definition 153.1 (The one-deviation property)
A strategy profile is a subgame perfect equilibrium
for the -discounted infinitely repeated game of G
if and only if there is no history after which a
player can gain by deviating for a single period.
Examples of infinitely
repeated games
Proof: (Only if ) Trivial.
(If ) If there is a profitable multi-period deviation of infinite
duration, then there is a profitable multi-period deviation of
finite duration. (This is due to the discounting: What happens
in the distant future is of little importance, implying the deviation is profitable even if truncated.) But if there is a profitable
multi-period deviation of finite duration, then there is a
profitable one-period deviation (see proof of Lemma 98.2).

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


G.B. Asheim, ECON5200-2
25
u2
Set of subgame
Lemma 153.3 The set of payoff
perfect equilibrium
profiles that can be realized in subg.
payoffs
perf. equilibria for the -disc. infinitely
repeated game of G is compact.
m(2)
Corollary Consider the -disc.
u1
m(1)
infinitely repeated game of G.
For each player i  N, there is (a) a minimal payoff m(i)
that can be realized as a subg. perf. equilibr. payoff, and
(b) a subg. perf. equilibr. path (p(i)t) s.t. ui (( p (i) t ))  m(i)
Proposition 154.1 Consider the -disc. inf. repeated game
of G. Then (at) is a subg. perf. equilibr. path if and only if
the simple strategy profile  (a(0) t ), ( p(1) t ), , ( p (n)t ) 
is a subgame perfect equilibrium.
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26
Let G  N, (Ai), (ui) be a strategic game.
Player i's minmax payoff in G, vi, is the lowest
payoff that the other players can force upon i:
vi  min max ui ( ai , ai )
a i Ai ai Ai
.
A payoff profile w in G is enforceable if wi  vi for all
i  N. A payoff profile w in G is strictly enforceable if wi  vi for all i  N.
If a  A is an outcome of G for which u(a) is
(strictly) enforceable, then we refer to a as a
(strictly) enforceable outcome of G.
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9
A Folk Theorem

Proposition 151.1 Let a be a strictly enforceable outcome of G. Assume that there is a collection (a(i))iN
of strictly enforceable outcomes of G such that for
each player i  N we have
u2
ui(a)  ui(a(i)) and
u (a (1))
ui(a(j))  ui(a(i))
for all j  i. Then there exists
u (a  )
   1 such that for all     u(b , p )
u (a (2))
there is a subg. perf. equilibrium
u1
of the  -discounted infinitely
u( p , b )
repeated game of G that generates
the path (at) where at  a for all t.
1
2
1
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28
10