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Hal Varian Intermediate Microeconomics Chapter Twenty-Eight

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Title: Hal Varian Intermediate Microeconomics Chapter Twenty-Eight


1
Hal VarianIntermediate MicroeconomicsChapter
Twenty-Eight
  • Game Theory

2
Game Theory
  • Game theory models strategic behavior by agents
    who understand that their actions affect the
    actions of other agents.

3
Some Applications of Game Theory
  • The study of oligopolies (industries containing
    only a few firms)
  • The study of cartels e.g. OPEC
  • The study of externalities e.g. using a common
    resource such as a fishery.
  • The study of military strategies.

4
What is a Game?
  • A game consists of
  • a set of players
  • a set of strategies for each player
  • the payoffs to each player for every possible
    list of strategy choices by the players.

5
Two-Player Games
  • A game with just two players is a two-player
    game.
  • We will study only games in which there are two
    players, each of whom can choose between only two
    strategies.

6
An Example of a Two-Player Game
  • The players are called A and B.
  • Player A has two strategies, called Up and
    Down.
  • Player B has two strategies, called Left and
    Right.
  • The table showing the payoffs to both players for
    each of the four possible strategy combinations
    is the games payoff matrix.

7
An Example of a Two-Player Game
Player B
This is thegames payoff matrix.
Player A
Player As payoff is shown first.Player Bs
payoff is shown second.
8
An Example of a Two-Player Game
Player B
L
R
This is thegames payoff matrix.
(1,8)
(3,9)
U
Player A
(0,0)
(2,1)
D
E.g. if A plays Up and B plays Right then As
payoff is 1 and Bs payoff is 8.
9
An Example of a Two-Player Game
Player B
L
R
This is thegames payoff matrix.
(3,9)
(1,8)
U
Player A
(2,1)
(0,0)
D
And if A plays Down and B plays Right then As
payoff is 2 and Bs payoff is 1.
10
An Example of a Two-Player Game
Player B
Player A
A play of the game is a pair such as (U,R) where
the 1st element is the strategy chosen by Player
A and the 2nd is the strategy chosen by Player B.
11
An Example of a Two-Player Game
Player B
Player A
What plays are we likely to see for this game?
12
An Example of a Two-Player Game
Player B
L
R
Is (U,R) alikely play?
(1,8)
(3,9)
U
Player A
(0,0)
(2,1)
D
13
An Example of a Two-Player Game
Player B
L
R
Is (U,R) alikely play?
(1,8)
(3,9)
U
Player A
(0,0)
(2,1)
D
If B plays Right then As best reply is
Downsince this improves As payoff from 1 to
2.So (U,R) is not a likely play.
14
An Example of a Two-Player Game
Player B
L
R
Is (D,R) alikely play?
(3,9)
(1,8)
U
Player A
(2,1)
(0,0)
D
15
An Example of a Two-Player Game
Player B
L
R
Is (D,R) alikely play?
(3,9)
(1,8)
U
Player A
(2,1)
(0,0)
D
If B plays Right then As best reply is Down.
16
An Example of a Two-Player Game
Player B
L
R
Is (D,R) alikely play?
(3,9)
(1,8)
U
Player A
(2,1)
(0,0)
D
If B plays Right then As best reply is Down. If
A plays Down then Bs best reply is Right. So
(D,R) is a likely play.
17
An Example of a Two-Player Game
Player B
L
R
Is (D,L) alikely play?
(3,9)
(1,8)
U
Player A
(0,0)
(2,1)
D
18
An Example of a Two-Player Game
Player B
L
R
Is (D,L) alikely play?
(3,9)
(1,8)
U
Player A
(0,0)
(2,1)
D
If A plays Down then Bs best reply is Right,so
(D,L) is not a likely play.
19
An Example of a Two-Player Game
Player B
L
R
Is (U,L) alikely play?
(3,9)
(1,8)
U
Player A
(0,0)
(2,1)
D
20
An Example of a Two-Player Game
Player B
L
R
Is (U,L) alikely play?
(3,9)
(1,8)
U
Player A
(0,0)
(2,1)
D
If A plays Up then Bs best reply is Left.
21
An Example of a Two-Player Game
Player B
L
R
Is (U,L) alikely play?
(3,9)
(1,8)
U
Player A
(0,0)
(2,1)
D
If A plays Up then Bs best reply is Left. If B
plays Left then As best reply is Up. So (U,L) is
a likely play.
22
Nash Equilibrium
  • A play of the game where each strategy is a best
    reply to the other is a Nash equilibrium.
  • Our example has two Nash equilibria (U,L) and
    (D,R).

23
An Example of a Two-Player Game
Player B
L
R
(3,9)
(1,8)
U
Player A
(2,1)
(0,0)
D
(U,L) and (D,R) are both Nash equilibria forthe
game.
24
An Example of a Two-Player Game
Player B
L
R
(3,9)
(1,8)
U
Player A
(2,1)
(0,0)
D
(U,L) and (D,R) are both Nash equilibria forthe
game. But which will we see? Noticethat (U,L)
is preferred to (D,R) by bothplayers. Must we
then see (U,L) only?
25
The Prisoners Dilemma
  • To see if Pareto-preferred outcomes must be what
    we see in the play of a game, consider a famous
    second example of a two-player game called the
    Prisoners Dilemma.

26
The Prisoners Dilemma
Clyde
S
C
(-5,-5)
(-30,-1)
S
Bonnie
(-1,-30)
(-10,-10)
C
What plays are we likely to see for this game?
27
The Prisoners Dilemma
Clyde
S
C
(-5,-5)
(-30,-1)
S
Bonnie
(-1,-30)
(-10,-10)
C
If Bonnie plays Silence then Clydes bestreply
is Confess.
28
The Prisoners Dilemma
Clyde
S
C
(-5,-5)
(-30,-1)
S
Bonnie
(-1,-30)
(-10,-10)
C
If Bonnie plays Silence then Clydes bestreply
is Confess. If Bonnie plays Confess then
Clydesbest reply is Confess.
29
The Prisoners Dilemma
Clyde
S
C
(-5,-5)
(-30,-1)
S
Bonnie
(-1,-30)
(-10,-10)
C
So no matter what Bonnie plays, Clydesbest
reply is always Confess. Confess is a dominant
strategy for Clyde.
30
The Prisoners Dilemma
Clyde
S
C
(-5,-5)
(-30,-1)
S
Bonnie
(-1,-30)
(-10,-10)
C
Similarly, no matter what Clyde plays,Bonnies
best reply is always Confess. Confess is a
dominant strategy forBonnie also.
31
The Prisoners Dilemma
Clyde
S
C
(-5,-5)
(-30,-1)
S
Bonnie
(-1,-30)
(-10,-10)
C
So the only Nash equilibrium for thisgame is
(C,C), even though (S,S) givesboth Bonnie and
Clyde better payoffs. The only Nash equilibrium
is inefficient.
32
Who Plays When?
  • In both examples the players chose their
    strategies simultaneously.
  • Such games are simultaneous play games.

33
Who Plays When?
  • But there are games in which one player plays
    before another player.
  • Such games are sequential play games.
  • The player who plays first is the leader. The
    player who plays second is the follower.

34
A Sequential Game Example
  • Sometimes a game has more than one Nash
    equilibrium and it is hard to say which is more
    likely to occur.
  • When such a game is sequential it is sometimes
    possible to argue that one of the Nash equilibria
    is more likely to occur than the other.

35
A Sequential Game Example
Player B
L
R
(3,9)
(1,8)
U
Player A
(2,1)
(0,0)
D
(U,L) and (D,R) are both Nash equilibriawhen
this game is played simultaneouslyand we have no
way of deciding whichequilibrium is more likely
to occur.
36
A Sequential Game Example
Player B
L
R
(3,9)
(1,8)
U
Player A
(2,1)
(0,0)
D
Suppose instead that the game is
playedsequentially, with A leading and B
following. We can rewrite the game in its
extensive form.
37
A Sequential Game Example
A
U
D
A plays first.B plays second.
B
B
L
L
R
R
(0,0)
(2,1)
(1,8)
(3,9)
38
A Sequential Game Example
A
U
D
A plays first.B plays second.
B
B
L
L
R
R
(0,0)
(2,1)
(1,8)
(3,9)
(U,L) is a Nash equilibrium.
39
A Sequential Game Example
A
U
D
A plays first.B plays second.
B
B
L
L
R
R
(0,0)
(2,1)
(1,8)
(3,9)
(U,L) is a Nash equilibrium. (D,R) is a Nash
equilibrium.Which is more likely to occur?
40
A Sequential Game Example
A
U
D
A plays first.B plays second.
B
B
L
L
R
R
(0,0)
(2,1)
(1,8)
(3,9)
If A plays U then B plays L A gets 3.
41
A Sequential Game Example
A
U
D
A plays first.B plays second.
B
B
L
L
R
R
(0,0)
(2,1)
(1,8)
(3,9)
If A plays U then B plays L A gets 3. If A plays
D then B plays R A gets 2.
42
A Sequential Game Example
A
U
D
A plays first.B plays second.
B
B
L
L
R
R
(0,0)
(2,1)
(1,8)
(3,9)
If A plays U then B plays L A gets 3. If A plays
D then B plays R A gets 2.So (U,L) is the
likely Nash equilibrium.
43
Pure Strategies
Player B
L
R
(3,9)
(1,8)
U
Player A
(2,1)
(0,0)
D
This is our original example once more.Suppose
again that play is simultaneous.We discovered
that the game has two Nashequilibria (U,L) and
(D,R).
44
Pure Strategies
Player B
L
R
(3,9)
(1,8)
U
Player A
(2,1)
(0,0)
D
Player As has been thought of as choosingto
play either U or D, but no combination ofboth
that is, as playing purely U or D.U and D are
Player As pure strategies.
45
Pure Strategies
Player B
L
R
(3,9)
(1,8)
U
Player A
(2,1)
(0,0)
D
Similarly, L and R are Player Bs purestrategies.
46
Pure Strategies
Player B
L
R
(3,9)
(1,8)
U
Player A
(2,1)
(0,0)
D
Consequently, (U,L) and (D,R) are purestrategy
Nash equilibria. Must every gamehave at least
one pure strategy Nashequilibrium?
47
Pure Strategies
Player B
L
R
(1,2)
(0,4)
U
Player A
(0,5)
(3,2)
D
Here is a new game. Are there any purestrategy
Nash equilibria?
48
Pure Strategies
Player B
L
R
(1,2)
(0,4)
U
Player A
(0,5)
(3,2)
D
Is (U,L) a Nash equilibrium?
49
Pure Strategies
Player B
L
R
(0,4)
(1,2)
U
Player A
(0,5)
(3,2)
D
Is (U,L) a Nash equilibrium? No. Is (U,R) a Nash
equilibrium?
50
Pure Strategies
Player B
L
R
(1,2)
(0,4)
U
Player A
(0,5)
(3,2)
D
Is (U,L) a Nash equilibrium? No. Is (U,R) a Nash
equilibrium? No.Is (D,L) a Nash equilibrium?
51
Pure Strategies
Player B
L
R
(1,2)
(0,4)
U
Player A
(3,2)
(0,5)
D
Is (U,L) a Nash equilibrium? No. Is (U,R) a Nash
equilibrium? No.Is (D,L) a Nash equilibrium?
No.Is (D,R) a Nash equilibrium?
52
Pure Strategies
Player B
L
R
(1,2)
(0,4)
U
Player A
(0,5)
(3,2)
D
Is (U,L) a Nash equilibrium? No. Is (U,R) a Nash
equilibrium? No.Is (D,L) a Nash equilibrium?
No.Is (D,R) a Nash equilibrium? No.
53
Pure Strategies
Player B
L
R
(1,2)
(0,4)
U
Player A
(0,5)
(3,2)
D
So the game has no Nash equilibria in
purestrategies. Even so, the game does have
aNash equilibrium, but in mixed strategies.
54
Mixed Strategies
  • Instead of playing purely Up or Down, Player A
    selects a probability distribution (pU,1-pU),
    meaning that with probability pU Player A will
    play Up and with probability 1-pU will play Down.
  • Player A is mixing over the pure strategies Up
    and Down.
  • The probability distribution (pU,1-pU) is a mixed
    strategy for Player A.

55
Mixed Strategies
  • Similarly, Player B selects a probability
    distribution (pL,1-pL), meaning that with
    probability pL Player B will play Left and with
    probability 1-pL will play Right.
  • Player B is mixing over the pure strategies Left
    and Right.
  • The probability distribution (pL,1-pL) is a mixed
    strategy for Player B.

56
Mixed Strategies
Player B
L
R
(1,2)
(0,4)
U
Player A
(0,5)
(3,2)
D
This game has no pure strategy Nash equilibria
but it does have a Nash equilibrium in mixed
strategies. How is itcomputed?
57
Mixed Strategies
Player B
L,pL
R,1-pL
(1,2)
(0,4)
U,pU
Player A
(0,5)
(3,2)
D,1-pU
58
Mixed Strategies
Player B
L,pL
R,1-pL
(1,2)
(0,4)
U,pU
Player A
(0,5)
(3,2)
D,1-pU
If B plays Left her expected payoff is
59
Mixed Strategies
Player B
L,pL
R,1-pL
(1,2)
(0,4)
U,pU
Player A
(0,5)
(3,2)
D,1-pU
If B plays Left her expected payoff isIf B
plays Right her expected payoff is
60
Mixed Strategies
Player B
L,pL
R,1-pL
(1,2)
(0,4)
U,pU
Player A
(0,5)
(3,2)
D,1-pU
If
then
B would play only Left. But there are no Nash
equilibria in which B plays only Left.
61
Mixed Strategies
Player B
L,pL
R,1-pL
(1,2)
(0,4)
U,pU
Player A
(0,5)
(3,2)
D,1-pU
If
then
B would play only Right. But there are no Nash
equilibria in which B plays only Right.
62
Mixed Strategies
Player B
L,pL
R,1-pL
(1,2)
(0,4)
U,pU
Player A
(0,5)
(3,2)
D,1-pU
So for there to exist a Nash equilibrium, Bmust
be indifferent between playing Left orRight i.e.
63
Mixed Strategies
Player B
L,pL
R,1-pL
(1,2)
(0,4)
U,pU
Player A
(0,5)
(3,2)
D,1-pU
So for there to exist a Nash equilibrium, Bmust
be indifferent between playing Left orRight i.e.
64
Mixed Strategies
Player B
L,pL
R,1-pL
(1,2)
(0,4)
U,
Player A
(0,5)
(3,2)
D,
So for there to exist a Nash equilibrium, Bmust
be indifferent between playing Left orRight i.e.
65
Mixed Strategies
Player B
L,pL
R,1-pL
(1,2)
(0,4)
U,
Player A
(0,5)
(3,2)
D,
66
Mixed Strategies
Player B
L,pL
R,1-pL
(1,2)
(0,4)
U,
Player A
(0,5)
(3,2)
D,
If A plays Up his expected payoff is
67
Mixed Strategies
Player B
L,pL
R,1-pL
(1,2)
(0,4)
U,
Player A
(0,5)
(3,2)
D,
If A plays Up his expected payoff isIf A plays
Down his expected payoff is
68
Mixed Strategies
Player B
L,pL
R,1-pL
(1,2)
(0,4)
U,
Player A
(0,5)
(3,2)
D,
If
then A would play only Up.
But there are no Nash equilibria in which Aplays
only Up.
69
Mixed Strategies
Player B
L,pL
R,1-pL
(1,2)
(0,4)
U,
Player A
(0,5)
(3,2)
D,
If
then A would play only
Down. But there are no Nash equilibria in which
A plays only Down.
70
Mixed Strategies
Player B
L,pL
R,1-pL
(1,2)
(0,4)
U,
Player A
(0,5)
(3,2)
D,
So for there to exist a Nash equilibrium, Amust
be indifferent between playing Up orDown i.e.
71
Mixed Strategies
Player B
L,pL
R,1-pL
(1,2)
(0,4)
U,
Player A
(0,5)
(3,2)
D,
So for there to exist a Nash equilibrium, Amust
be indifferent between playing Up orDown i.e.
72
Mixed Strategies
Player B
L,
R,
(1,2)
(0,4)
U,
Player A
(0,5)
(3,2)
D,
So for there to exist a Nash equilibrium, Amust
be indifferent between playing Up orDown i.e.
73
Mixed Strategies
Player B
L,
R,
(1,2)
(0,4)
U,
Player A
(0,5)
(3,2)
D,
So the games only Nash equilibrium has Aplaying
the mixed strategy (3/5, 2/5) and hasB playing
the mixed strategy (3/4, 1/4).
74
Mixed Strategies
Player B
L,
R,
(1,2)
(0,4)
U,
9/20
Player A
(0,5)
(3,2)
D,
The payoffs will be (1,2) with probability
75
Mixed Strategies
Player B
L,
R,
(0,4)
(1,2)
U,
9/20
3/20
Player A
(0,5)
(3,2)
D,
The payoffs will be (0,4) with probability
76
Mixed Strategies
Player B
L,
R,
(0,4)
(1,2)
U,
9/20
3/20
Player A
(0,5)
(3,2)
D,
6/20
The payoffs will be (0,5) with probability
77
Mixed Strategies
Player B
L,
R,
(0,4)
(1,2)
U,
9/20
3/20
Player A
(0,5)
(3,2)
D,
6/20
2/20
The payoffs will be (3,2) with probability
78
Mixed Strategies
Player B
L,
R,
(0,4)
(1,2)
U,
9/20
3/20
Player A
(0,5)
(3,2)
D,
6/20
2/20
79
Mixed Strategies
Player B
L,
R,
(0,4)
(1,2)
U,
9/20
3/20
Player A
(0,5)
(3,2)
D,
6/20
2/20
As expected Nash equilibrium payoff is
80
Mixed Strategies
Player B
L,
R,
(0,4)
(1,2)
U,
9/20
3/20
Player A
(0,5)
(3,2)
D,
6/20
2/20
As expected Nash equilibrium payoff is
Bs expected Nash equilibrium payoff is
81
How Many Nash Equilibria?
  • A game with a finite number of players, each with
    a finite number of pure strategies, has at least
    one Nash equilibrium.
  • So if the game has no pure strategy Nash
    equilibrium then it must have at least one mixed
    strategy Nash equilibrium.
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