Lecture 10
Subgame-perfect Equilibrium
14.12 Game Theory
Muhamet Yildiz
1
Road Map
1. Subgame-perfect Equilibrium
1.
2.
3.
4.
Motivation
What is a subgame?
Definition
Example
2. Applications
1. BankRun
2. Infinite-horizon Bargaining
2
A game
1
l~
(2,6)
T
L
(0,1)
B
R
(3,2)
L
(-1,3 )
R
(1,5)
3
Backward induction
• Can be applied only in perfect information
games of finite horizon.
How can we extend this notion to infinite
horizon games, or to games with imperfect
information?
4
A subgame
A subgame is part of a game that can be
considered as a game itself.
• It must have a unique starting point;
• It must contain all the nodes that follow the
starting node;
• If a node is in a subgame, the entire
information set that contains the node must
be in the subgame.
5
A game
1
A
2
a
1
a
,-------------,-------,---~
d
D
(4,4)
(1,-5)
(5,2)
(3,3)
6
And its subgames
1
a
~-~
(1,-5)
2
d
a
1 a
(1 ,-5)
d
(3 ,3)
(5 ,2)
(3 ,3)
7
A game
1
l~
(2,6)
T
L
(0,1)
B
R
(3,2)
L
(-1,3 )
R
(1,5)
8
Definitions
A substrategy is the restriction of a strategy to
a subgame.
A subgame-perfect Nash equilibrium is a
Nash equilibrium whose sub strategy profile
is a Nash equilibrium at each subgame.
9
Example
1
l~
(2,6)
T
L
(0,1)
B
R
(3,2)
L
(-1,3)
R
(1,5)
10
A "Backward-Induction-like" method
Take any subgame with no proper subgame
Compute a Nash equilibrium for this subgame
Assign the payoff of the Nash equilibrium to
the starting node of the subgame
Eliminate the subgame
Yes
The moves computed as a part of
any (subgame) Nash equilibrium
11
In a finite, perfect-information
game, ...
... the set of subgame-perfect equilibria is the
set of strategy profiles that are computed via
backward induction.
12
A subgame-perfect equilibrium?
x
1~
_ _ _ (2,6)
T
L
(0,1)
B
R
(3,2)
L
(-1 ,3)
R
(1 ,5)
13
Bank Run
• Alice and Bob each deposit D = $lM in a bank
• Bank invests the money in a project, which pays
2r if liquidated at t= 1, 2R if waited to t=2, where
R > D > r > D/2
• Either player has the option of withdrawing at
either date, getting D if bank has the money
• Ifthey do not withdraw, bank pays R to each
14
Bank Run
A
R > D > r > D!2
DW
W
W
(r,r)
DW
W
(D,D)
DW
W
(D,2R-D) (2R-D,D)
DW
(R,R)
15
Infinite-horizon Bargaining
T = {l,2, ... , n-l,n, ... }
If t is odd,
Player 1 offers some
(xt,Yt),
Player 2 Accept or
Rejects the offer
If the offer is Accepted,
the game ends yielding
8t(x t,Yt),
Otherwise, we proceed
to date t+ 1.
1ft is even
- Player 2 offers some
(xt,Yt),
- Player 1 Accept or Rejects
the offer
- Tfthe offer is Accepted,
the game ends yielding
payoff (xt,Yt),
- Otherwise, we proceed to
date t+ I.
16
n
00
t = 2n - 2k-l
X t -
1- 8 2k +! 1- 8 2n - t
---1+8 - 1+8
n- W )
)
1
1+8
A SPE: At each t,
• proposer offers 8/(1 +8) to the other
•
and keeps 1/(1 +8) for himself;
• responder accepts an offer iff
•
she gets at least 8/(1 +8) .
17
MIT OpenCourseWare
http://ocw.mit.edu
14.12 Economic Applications of Game Theory
Fall 2012
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.