Lecture 10 Subgame-perfect Equilibrium 14.12 Game Theory Muhamet - - PDF document

Lecture 10 Subgame-perfect Equilibrium

14.12 Game Theory Muhamet Yildiz

1

Road Map

• 1. Subgame-perfect Equilibrium
• 1. Motivation
• 2. What is a subgame?
• 3. Definition
• 4. Example
• 2. Applications
• 1. BankRun
• 2. Infinite-horizon Bargaining

2

A game

1

l~

(2,6) T

B

L R

L

R

(0,1) (3,2) (-1,3 ) (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

,-,-,-

~

(1,-5)

D d (4,4) (5,2) (3,3)

6

And its subgames

1 a

2

a

1 a

~-

~

(1,-5) (1 ,-5)

d

d (3,3) (5,2) (3,3)

7

A game

1

l~

(2,6) T

B

L R

L

R

(0,1) (3,2) (-1,3 ) (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

B

L R L R

(0,1) (3,2) (-1,3) (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 B

L R L R

(0,1) (3,2) (-1,3)

(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 W

DW

W

(r,r)

DW

W

DW DW

W

(D,D) (D,2R-D) (2R-D,D) (R,R) 15

Infinite-horizon Bargaining

T = {l,2, ... , n-l,n, ... }

If t is odd, 1ft is even

Player 1 offers some

• Player 2 offers some

(xt,Yt), (xt,Yt), Player 2 Accept or

• Player 1 Accept or Rejects

Rejects the offer the offer If the offer is Accepted,

• Tfthe offer is Accepted,

the game ends yielding the game ends yielding t 8 (xt,Yt), payoff (xt,Yt), Otherwise, we proceed

• Otherwise, we proceed to

to date t+

1.

date t+ I. 16

n

00 t = 2n - 2k-l

1- 8 2k+! 1- 8 2n t

• n - W )

)

1

X -

• t -

1+8 - 1+8 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

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