More game theory
Today: Some classic games in
game theory
Last time…
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Introduction to game theory
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Games have players, strategies, and
payoffs
Based on a payoff matrix with
simultaneous decisions, we can find Nash
equilibria (NE)
In sequential games, some NE can be ruled
out if people are rational
Today, some classic game
theory games
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Games with inefficient equilibria
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Coordination games
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Battle of the Sexes
Chicken
Zero-sum game
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Prisoner’s Dilemma
Public Goods game
Matching pennies
Animal behavior
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Subordinate pig/Dominant pig
Prisoner’s dilemma
Player 2
Yes
Player
1
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No
Why is this
game called
prisoner’s
dilemma?
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Yes
–1, –1
+3, –6
No
–6, +3
+1, +1
Think about a
pair of
criminals that
have a choice
of whether or
not to confess
to a crime
Prisoner’s dilemma
Player 2
Yes
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No
What is the
NE?
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Player
1
Yes
–1, –1
+3, –6
No
–6, +3
+1, +1
Let’s
underline
Prisoner’s dilemma
Player 2
Yes
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No
What is the
NE?
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Player
1
Yes
–1, –1
+3, –6
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No
–6, +3
+1, +1
Let’s underline
Each player
has a
dominant
strategy of
choosing Yes
However, both
players get a
better payout
if each
chooses No
Prisoner’s dilemma and cartels
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Cartels are usually unstable since each
firm has a dominant strategy to charge
a lower price and sell more
See Table 11.4 (p. 327) for an example
Public goods game
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You can decide whether or not you
want to contribute to a new flower
garden at a local park
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If you decide Yes, you will lose $200, but
every other person in the city you live in
will gain $10 in benefits from the park
If you decide No, you will cause no change
to the outcome of you or other people
Public goods game
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What is each person’s best response,
given the decision of others?
We need to look at each person’s
marginal gain and loss (if any)
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Choose yes  Gain $10, lose $200
Choose no  Gain $0, lose $0
Public goods game
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Which is the better choice?
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Choose no (Gain nothing vs. net loss of $190)
NE has everybody choosing no
Efficient outcome has everybody choosing yes
Why the difference?
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Each person does not account for others’ benefits
when making their own decision
Battle of the Sexes
Player 2
Bar
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Play
Bar +3, +1 +0, +0
Player 1
Play +0, +0 +1, +3
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Two people plan a date,
and each knows that
the date is either at the
bar or a play
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Neither person knows
where the other is
going until each person
shows up
If both people show up
at the same place, they
enjoy each other’s
company (+1 for each)
Battle of the Sexes:
Other things to note
Player 2
Bar
Player 1
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Play
Bar +3, +1 +0, +0
Play +0, +0 +1, +3
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Player 1 gets additional
enjoyment from the bar
if Player 2 is there too,
since Player 1 likes the
bar more
Player 2 enjoys the play
more than Player 1 if
both show up there
As before, we underline
the best strategy, given
the strategy of the
other player
Battle of the Sexes
Player 2
Bar
Player 1
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Play
Bar +3, +1 +0, +0
Play +0, +0 +1, +3
Two NE
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(Bar, Bar)
(Play, Play)
As in cases before
when there are
multiple NE, we
cannot determine
which outcome will
actually occur
Battle of the Sexes
Player 2
Bar
Player 1
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Play
Bar +3, +1 +0, +0
Play +0, +0 +1, +3
Battle of the Sexes
is known as a
coordination game
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Both get a positive
payout if they show
up to the same place
Chicken
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Two cars drive toward each other
If neither car swerves, both drivers
sustain damage to themselves and their
cars
If only one person swerves, this person
is known forever more as “Chicken”
Chicken
Player 2
Player 1
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Swerve
Straight
Swerve
+0, +0
–1, +1
Straight
+1, –1
–10, –10
Next step: Underline as before
Chicken
Player 2
Player 1
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Swerve
Straight
Swerve
+0, +0
–1, +1
Straight
+1, –1
–10, –10
Notice there are 2 NE
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One player swerves and the other goes straight
This game is sometimes referred to as an “anticoordination” game
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NE results from each player making a different decision
Matching pennies
Player 2
Heads
Player
1
Tails
Heads +1, –1 –1, +1
Tails
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–1, +1 +1, –1
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Two players each
choose Heads or
Tails
If both choices
match, Player 1 wins
If both choices
differ, Player 2 wins
This is an example
of a zero-sum game,
since the sum of
each box is zero
Matching pennies
Player 2
Heads
Player
1
Tails
Heads +1, –1 –1, +1
Tails
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–1, +1 +1, –1
Underlining shows
no NE
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A characteristic of
zero-sum games
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Whenever I win, the
other player must
lose
Subordinate pig/Dominant pig
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Two pigs are placed in a cage
Left end of cage: Lever to release food
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12 units of food released when lever is
pressed
Right end of cage: Food is dispensed
here
Subordinate pig/Dominant pig
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If both press lever at the same time, the subordinate
pig can run faster and eat 4 units of food before the
dominant pig “hogs” the rest
If only the dominant pig presses the lever, the
subordinate pig eats 10 of the 12 units of food
If only the subordinate pig presses the lever, the
dominant pig eats all 12 units
Pressing the lever exerts a unit of food
Subordinate pig/Dominant pig
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Who do you think will get more food in
equilibrium?
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Who thinks
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Who thinks
?
?
Subordinate pig/Dominant pig
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subordinate pig
dominant pig
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Yes
No
Yes
3, 7
–1, 12
No
10, 1
0, 0
Next:
Underline test
The numbers on the
previous slide translate to
the payoff matrix seen
Subordinate pig/Dominant pig
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dominant pig
subordinate pig
Yes
No
In Nash equilibrium, the
dominant pig always gets the
lower payout
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Yes
3, 7
–1, 12
No
10, 1
0, 0
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Exactly 1 NE
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The dominant
pig presses lever
Why?
The subordinate pig has a
dominant strategy: No
The dominant pig, knowing that
the subordinate pig will not press
the lever, will want to press the
lever
Do people always play as
Nash equilibrium predicts?
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No
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Many papers have shown that people often
are not selfish, and donate into public
goods
Norms are often established to make sure
that people are encouraged to act in the
best interest of society
Summary
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Today, we looked at some well-known
games
Some games have NE; others do not
However, people do not always behave
as NE would predict