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)

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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.

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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

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In a finite, perfect-information game, ...

... the set of subgame-perfect equilibria is the set of strategy profiles that are computed via backward induction.

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A subgame-perfect equilibrium?

x 1~ ___ (2,6)

T B

L R L R

(0,1) (3,2) (-1 ,3) (1 ,5)

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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

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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)

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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

t8 (xt,Yt), payoff (xt,Yt), Otherwise, we proceed - Otherwise, we proceed to to date t+ 1. date t+ I.

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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) .

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14.12 Economic Applications of Game TheoryFall 2012

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