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.
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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
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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.
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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.
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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
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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.
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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
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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.
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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.
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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.
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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?
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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.

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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?
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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.
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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.
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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.
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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.
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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.

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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.

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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.

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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.
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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.
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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)
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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.
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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.
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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.
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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).
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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.
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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.
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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?
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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.