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More game theory Today: Some classic games in game theory.
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Transcript of More game theory Today: Some classic games in game theory.
More game theory
Today: Some classic games in game theory
Last time…
Introduction to game theory 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 Games with inefficient equilibria
Prisoner’s Dilemma Public Goods game
Coordination games Battle of the Sexes Chicken
Zero-sum game Matching pennies
Animal behavior Subordinate pig/Dominant pig
Prisoner’s dilemma Why is this
game called prisoner’s dilemma?
Think about a pair of criminals that have a choice of whether or not to confess to a crime
Yes No
Yes –1, –1 +3, –6
No –6, +3 +1, +1
Player 1
Player 2
Prisoner’s dilemma What is the
NE? Let’s
underline
Yes No
Yes –1, –1 +3, –6
No –6, +3 +1, +1
Player 1
Player 2
Prisoner’s dilemma What is the
NE? Let’s
underline Each player
has a dominant strategy of choosing Yes
However, both players get a better payout if each chooses No
Yes No
Yes –1, –1 +3, –6
No –6, +3 +1, +1
Player 1
Player 2
Prisoner’s dilemma and cartels
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
You can decide whether or not you want to contribute to a new flower garden at a local park 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
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) Choose yes Gain $10, lose $200 Choose no Gain $0, lose $0
Public goods game Which is the better choice?
Choose no (Gain nothing vs. net loss of $190)
NE has everybody choosing no Efficient outcome has everybody
choosing yes Why the difference?
Each person does not account for others’ benefits when making their own decision
Battle of the Sexes
Two people plan a date, and each knows that the date is either at the bar or a play
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)
Bar Play
Bar +3, +1
+0, +0
Play
+0, +0
+1, +3
Player 1
Player 2
Battle of the Sexes:Other things to note
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
Bar Play
Bar +3, +1
+0, +0
Play
+0, +0
+1, +3
Player 1
Player 2
Battle of the Sexes Two NE
(Bar, Bar) (Play, Play)
As in cases before when there are multiple NE, we cannot determine which outcome will actually occur
Bar Play
Bar +3, +1
+0, +0
Play
+0, +0
+1, +3
Player 1
Player 2
Battle of the Sexes Battle of the
Sexes is known as a coordination game Both get a
positive payout if they show up to the same place
Bar Play
Bar +3, +1
+0, +0
Play
+0, +0
+1, +3
Player 1
Player 2
Chicken
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
Next step: Underline as before
Swerve
Straight
Swerve +0, +0
–1, +1
Straight +1, –1 –10, –10
Player 1
Player 2
Chicken
Notice there are 2 NE One player swerves and the other goes straight
This game is sometimes referred to as an “anti-coordination” game
NE results from each player making a different decision
Swerve
Straight
Swerve +0, +0
–1, +1
Straight +1, –1 –10, –10
Player 1
Player 2
Matching pennies
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
Heads Tails
Heads +1, –1 –1, +1
Tails –1, +1 +1, –1
Player 1
Player 2
Matching pennies
Underlining shows no NE
A characteristic of zero-sum games Whenever I win,
the other player must lose
Heads Tails
Heads +1, –1 –1, +1
Tails –1, +1 +1, –1
Player 1
Player 2
Subordinate pig/Dominant pig
Two pigs are placed in a cage Left end of cage: Lever to release
food 12 units of food released when lever
is pressed Right end of cage: Food is
dispensed here
Subordinate pig/Dominant pig 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
Who do you think will get more food in equilibrium? Who thinks ?
Who thinks ?
Subordinate pig/Dominant pig
Next: Underline test
The numbers on the previous slide translate to the payoff matrix seenYes No
Yes 3, 7 –1, 12
No 10, 1 0, 0
sub
ord
inate
pig
dominant pig
Subordinate pig/Dominant pig
Exactly 1 NE The dominant
pig presses lever
In Nash equilibrium, the dominant pig always gets the lower payout
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
Yes No
Yes 3, 7 –1, 12
No 10, 1 0, 0
sub
ord
inate
pig
dominant pig
Do people always play as Nash equilibrium predicts?
No 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
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