Chapter 2
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Chapter 2
Dynamic Games with Complete Information
Categories of Social and/or Strategic Situations
Complete Information
Static
Incomplete Information
Players simultaneously make decisions. with clear knowledge
Players simultaneously make decisions;
about other players’ payoff, rationality, etc.
without clear knowledge about other players’ payoff, rationality, etc.
(Chapter 1)
Dynamic
(Chapter 3)
Players can sequentially make decisions.
Players can sequentially make decisions.
with clear knowledge about other players’ payoff, rationality,
without clear knowledge about other players’ payoff, rationality, etc.
etc.
(Chapter 4)
(Chapter 2)
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
This game is dynamic one
Fight
Opera
Fight
Lady-First Game
1
2
Payoff to Pat
0
0
0
0
2
1
Payoff to Chris
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’
R’
L’
R’
0
0
2
1
1
2
3
1
Payoff to player 1
2
Payoff to player 2
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
L
(L’,L’)
(L’,R’)
3,1
2,1
(R’,L’)
(R’,R’)
3,1
1,2
1,2
0,0
2,1
0,0
Player 1
R
You could find Nash equilibria of this game.
The Prisoner’s Dilemma game
Prisoner 2
Cooperation
Cooperation
Deviation
－ 1 －－ 1
－ 9 － 0
0 －－ 9
－ 6 －－ 6
Prisoner 1
Deviation
Note that this game is static; players decide simultaneously.
How can we draw the extensive-form representation of this game ?
Prisoner 1
An information set for
C
D
Prisoner 2
Prisoner 2
C
prisoner 2
D
C
-1
-9
0
-1
0
-9
The Extensive-form of Prisoners’ Dilemma
D
-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’’
Figure 2.4.4.
3
L’’
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 2
Figure 2.4.1. (reused)
0
0
2
1
1
2
3
1
Payoff to player 1
the normal-form representation from its extensive-form representation above.
Player 2
L
(L’,L’)
(L’,R’)
3,1
2,1
(R’,L’)
(R’,R’)
3,1
1,2
1,2
0,0
2,1
0,0
Player 1
R
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,
includes all the decision and terminal nodes following n in the tree (but no nodes that do not
follow n), and
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.
(b)
(c)
Definition (Selten 1965) A Nash equilibrium is a subgame-perfect if the players’ strategies
constitute a Nash equilibrium in every subgame.
Backwards Induction and
Subgame Perfect Outcome
1
R
L
2
R’
2
0
L’
1
1
1
L’’
R’’
0
2
3
0
