Chapter 2

Dynamic Games with Complete Information

Categories of Social and/or Strategic Situations

Complete Information

Incomplete Information

Static

Players simultaneously make decisions.

with clear knowledge about other

players’ payoff, rationality, etc.

(Chapter 1)

Players simultaneously make decisions;

without clear knowledge about other players’ payoff,

rationality, etc.

(Chapter 3)

Dynamic

Players can sequentially make decisions.

with clear knowledge about other

players’ payoff, rationality, etc.

(Chapter 2)

Players can sequentially make decisions.

without clear knowledge about other players’ payoff,

rationality, etc.

(Chapter 4)

These situations are represented by following game theoretic ways.

Normal form

Representation of Games

Extensive form (Game Tree)

Extensive-Form Representation of Games

Chris

Fight

Opera

Pat

Pat

Opera

Payoff to Chris 2

1

Payoff to Pat

This game is dynamic one

Fight

Opera

Fight

0

0

1

0

0

2

Lady-First Game

Definition The extensive-form representation of a game

specifies:

(1) The players in the game,

(2a) when each player has the move,

(2b) what each player can do at each of his or her opportunities

to move,

(2c) what each player knows at each of his or her opportunities

to move,

(3) the payoff received by each player for combination of moves

that could be chosen by the players.

1

L

R

2

L’

2

R’

L’

R’

Payoff to player 1

3

1

2

0

Payoff to player 2

1

2

1

0

Figure 2.4.1.

Definition A strategy for a player is a complete plan of action －

it specifies a feasible action for the player in every contingency in

which the player might be called on to act.

・ Player 1 has two strategies, L or R.

・ Player 2 has following four strategies.

Strategy 1: If player 1 plays L, then play L’ ; if player 1 plays R, then play L’;

denoted by (L’,L’).

Strategy 2: If player 1 plays L, then play L’ ; if player 1 plays R, then play

R’; denoted by (L’,R’).

Strategy 3: If player 1 plays L, then play R’; if player 1 plays R, then play L’;

denoted by (L’,R’).

Strategy 4: If player 1 plays L, then play R’; if player 1 plays R, then play R’;

denoted by (R’,R’).

Given these strategies, we can derive the normal-form representation from its

extensive-form representation(Fig. 2.4.1.)

Player 2

Player 1

(L’,L’)

(L’,R’)

(R’,L’)

(R’,R’)

L

3,1

3,1

1,2

1,2

R

2,1

0,0

2,1

0,0

You could find Nash equilibria of this game.

The Prisoner’s Dilemma game

Cooperation

Cooperation

Prisoner 1

Deviation

－1 ，－1

0 ， －9

Prisoner 2

Deviation

－9 ，0

－6 ，－6

Note that this game is static; players decide simultaneously.

How can we draw the extensive-form representation of this game ?

Prisoner 1

C

D

Prisoner 2

Prisoner 2

C

-1

-1

An information set

for prisoner 2

D

C

-9

0

D

0

-9

The Extensive-form of Prisoners’ Dilemma

-6

-6

Definition An information set for a player is a collection of

decision nodes satisfying:

(i) the player has the move at every node in the information set,

and

(ii) when the play of the game reaches a node in the information

set, the player does not know which node in the information set

has (or has not) been reached.

We refer to a game of which any information set is a singleton set as a game

with perfect information.

Consider the following dynamic game of complete but imperfect information.

1

R

L

2

2

L’

3

L’’

R’’

R’

L’

R’

3

L’’

R’’

3

L’’

Figure 2.4.4.

R’’

L’’

R’’

Subgame-Perfect Nash Equilibrium

Consider the pure-strategy Nash equilibria of a game below.

1

R

L

2

2

L’

L’

R’

R’

Payoff to player 1

3

1

2

0

Payoff to player 2

1

2

1

0

Figure 2.4.1. (reused)

the normal-form representation from its extensive-form representation above.

Player 2

Player 1

(L’,L’)

(L’,R’)

(R’,L’)

(R’,R’)

L

3,1

3,1

1,2

1,2

R

2,1

0,0

2,1

0,0

Nash equilibria :

(R, (R’,L’)),

(L, (R’,R’))

Credible ?

Definition A subgame in an extensive-form game

(a) begins at a decision node n that is a singleton information set,

(b) includes all the decision and terminal nodes following n in the

tree (but no nodes that do not follow n), and

(c) does not cut any information sets (i.e., if a decision node n’

follows n in the game tree, then all other nodes in the

information set containing n’ must also follow n, and so must be

included in the subgame.

Definition (Selten 1965) A Nash equilibrium is a subgameperfect if the players’ strategies constitute a Nash equilibrium in

every subgame.

Backwards Induction and

Subgame Perfect Outcome

1

R

L

2

R’

L’

2

1

0

L’’

1

R’’

1

3

0

0

2

Dynamic Games with Complete Information

Categories of Social and/or Strategic Situations

Complete Information

Incomplete Information

Static

Players simultaneously make decisions.

with clear knowledge about other

players’ payoff, rationality, etc.

(Chapter 1)

Players simultaneously make decisions;

without clear knowledge about other players’ payoff,

rationality, etc.

(Chapter 3)

Dynamic

Players can sequentially make decisions.

with clear knowledge about other

players’ payoff, rationality, etc.

(Chapter 2)

Players can sequentially make decisions.

without clear knowledge about other players’ payoff,

rationality, etc.

(Chapter 4)

These situations are represented by following game theoretic ways.

Normal form

Representation of Games

Extensive form (Game Tree)

Extensive-Form Representation of Games

Chris

Fight

Opera

Pat

Pat

Opera

Payoff to Chris 2

1

Payoff to Pat

This game is dynamic one

Fight

Opera

Fight

0

0

1

0

0

2

Lady-First Game

Definition The extensive-form representation of a game

specifies:

(1) The players in the game,

(2a) when each player has the move,

(2b) what each player can do at each of his or her opportunities

to move,

(2c) what each player knows at each of his or her opportunities

to move,

(3) the payoff received by each player for combination of moves

that could be chosen by the players.

1

L

R

2

L’

2

R’

L’

R’

Payoff to player 1

3

1

2

0

Payoff to player 2

1

2

1

0

Figure 2.4.1.

Definition A strategy for a player is a complete plan of action －

it specifies a feasible action for the player in every contingency in

which the player might be called on to act.

・ Player 1 has two strategies, L or R.

・ Player 2 has following four strategies.

Strategy 1: If player 1 plays L, then play L’ ; if player 1 plays R, then play L’;

denoted by (L’,L’).

Strategy 2: If player 1 plays L, then play L’ ; if player 1 plays R, then play

R’; denoted by (L’,R’).

Strategy 3: If player 1 plays L, then play R’; if player 1 plays R, then play L’;

denoted by (L’,R’).

Strategy 4: If player 1 plays L, then play R’; if player 1 plays R, then play R’;

denoted by (R’,R’).

Given these strategies, we can derive the normal-form representation from its

extensive-form representation(Fig. 2.4.1.)

Player 2

Player 1

(L’,L’)

(L’,R’)

(R’,L’)

(R’,R’)

L

3,1

3,1

1,2

1,2

R

2,1

0,0

2,1

0,0

You could find Nash equilibria of this game.

The Prisoner’s Dilemma game

Cooperation

Cooperation

Prisoner 1

Deviation

－1 ，－1

0 ， －9

Prisoner 2

Deviation

－9 ，0

－6 ，－6

Note that this game is static; players decide simultaneously.

How can we draw the extensive-form representation of this game ?

Prisoner 1

C

D

Prisoner 2

Prisoner 2

C

-1

-1

An information set

for prisoner 2

D

C

-9

0

D

0

-9

The Extensive-form of Prisoners’ Dilemma

-6

-6

Definition An information set for a player is a collection of

decision nodes satisfying:

(i) the player has the move at every node in the information set,

and

(ii) when the play of the game reaches a node in the information

set, the player does not know which node in the information set

has (or has not) been reached.

We refer to a game of which any information set is a singleton set as a game

with perfect information.

Consider the following dynamic game of complete but imperfect information.

1

R

L

2

2

L’

3

L’’

R’’

R’

L’

R’

3

L’’

R’’

3

L’’

Figure 2.4.4.

R’’

L’’

R’’

Subgame-Perfect Nash Equilibrium

Consider the pure-strategy Nash equilibria of a game below.

1

R

L

2

2

L’

L’

R’

R’

Payoff to player 1

3

1

2

0

Payoff to player 2

1

2

1

0

Figure 2.4.1. (reused)

the normal-form representation from its extensive-form representation above.

Player 2

Player 1

(L’,L’)

(L’,R’)

(R’,L’)

(R’,R’)

L

3,1

3,1

1,2

1,2

R

2,1

0,0

2,1

0,0

Nash equilibria :

(R, (R’,L’)),

(L, (R’,R’))

Credible ?

Definition A subgame in an extensive-form game

(a) begins at a decision node n that is a singleton information set,

(b) includes all the decision and terminal nodes following n in the

tree (but no nodes that do not follow n), and

(c) does not cut any information sets (i.e., if a decision node n’

follows n in the game tree, then all other nodes in the

information set containing n’ must also follow n, and so must be

included in the subgame.

Definition (Selten 1965) A Nash equilibrium is a subgameperfect if the players’ strategies constitute a Nash equilibrium in

every subgame.

Backwards Induction and

Subgame Perfect Outcome

1

R

L

2

R’

L’

2

1

0

L’’

1

R’’

1

3

0

0

2

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