chapter05_12ed.pptx.pptx

Walter Nicholson
1
Amherst College
Christopher Snyder
Dartmouth College
PowerPoint Slide Presentation | Philip Heap, James Madison University
©2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Game
Theory
2
CHAPTER
5

Chapter Preview
Ch. 5 • 3
• What happens when decisions are interdependent or involve
strategic interaction?
• Game theory is a way of modeling these types of decisions.
• Cooperative game theory
• Non-cooperative game theory

Basic Concepts of Game Theory
Ch. 5 • 4
• Four elements describe a game.
1. Players
The decision makers in the game.
•
• 2, 3, . . . N players.
2. Strategies: a player’s choice in a game.
• In simple games, they are the same as actions.
• They may be contingent on what another player has done.
• They can involve randomization.

Basic Concepts of Game Theory
Ch. 5 • 5
3. Payoffs
• The utility of both the money earned in the game plus any
other things the player cares about.
• The player’s goal is to obtain the largest possible payoff.
4. Information
• What players know.
• Common knowledge.
• Information in sequential vs. simultaneous games.
• Incomplete information.

Equilibrium
Ch. 5 • 6
• Best response
– A strategy that produces the highest payoff among all possible
strategies for a player given what the other player is doing.
• Nash equilibrium
– A set of strategies, one for each player, that are best responses
against one another.
• If we both play our best response strategy, neither of us has an
incentive to deviate – an equilibrium.
• All games have a Nash equilibrium in either mixed or pure
strategies.

The Prisoners’ Dilemma
Ch. 5 • 7
• The story.
– Two criminals are arrested for a crime.
– They both know that they can only be convicted for a lesser
crime, for which they get 2 years in jail.
– DA puts them in separate rooms and offers each the same deal.
– If you confess and your partner stays quiet you will only get 1
year in jail and they will get 10 years.
– If you both confess you will each get 3 years.
• What would you do?

Illustrating Games: Normal Form
Ch. 5 • 8
-3, -3 -1, -10
-10, -1 -2, -2
Player B
Confess Silent
Confess
Player A
Silent

Illustrating Games: Extensive Form
Ch. 5 • 9

How to Solve the Game
Ch. 5 • 10
• Two ways to find the Nash equilibrium.
– Inspection
• Pick an outcome and see if one or both of the players has
an incentive to deviate. If not, it’s an equilibrium.
– Underline method
• For each player underline the payoff value the player would
get if he played his best response to each of the other
player’s actions. The outcome(s) where there are two
underlines is an equilibrium.

Solving the Game: By Inspection
-3, -3 -1, -10
-10, -1 -2, -2
Player B
Confess Silent
Confess
Player A
Silent
Is {Silent, Silent} an equilibrium?
No. If B (A) chooses Silent,
A (B) should choose
Confess.
Ch. 5 • 11

-3, -3 -1, -10
-2, -2
-10, -1
Player B
Confess Silent
Confess
Player A
Silent
Is {Silent, Confess} an equilibrium?
No. If B chooses Confess,
A should choose Confess.
Ch. 5 • 12

-3, -3 -1, -10
-10, -1 -2, -2
Player B
Confess Silent
Confess
Player A
Silent
Is {Confess, Confess} an equilibrium?
Yes. Neither player can do
better by changing their
strategy.
Ch. 5 • 13

Solving the Game: Underline Method
-3, -3 -1, -10
-10, -1 -2, -2
Player B
Confess Silent
Now do it for Player B.
Confess
Player A
Silent
For each of Player B’s
strategies, underline Player
A’s best response payoff.
Ch. 5 • 14

Other Aspects of the Prisoners’ Dilemma
Game
Ch. 5 • 15
• Confess is a dominant strategy.
• It is the best response to any of the other player’s strategies.
• What is the relationship between a dominant strategy equilibrium
and a Nash equilibrium?
– Every dominant strategy equilibrium is a Nash equilibrium, but
not every Nash equilibrium is a dominant strategy equilibrium.

Other Aspects of the Prisoners’ Dilemma
Game
Ch. 5 • 16
• If the players were allowed to communicate what would seem to
be the best outcome?
• Both staying silent would be better: 2 yrs. vs. 3 yrs.
• But this is a non-cooperative game so binding agreements are not
allowed.
• Other examples of the Prisoners’ Dilemma.

Mixed Strategies
Ch. 5 • 17
• Pure strategy – a single action played with certainty
• Mixed strategy – randomly selecting from several possible actions
• What does it mean to randomly select actions?

Matching Pennies: Normal Form
Ch. 5 • 18
1, -1 -1, 1
-1, 1 1, -1
Player B
Heads Tails
Heads
Player A
Tails

Matching Pennies: Extensive Form
Ch. 5 • 19

Matching Pennies: Nash Equilibrium
1, -1 -1, 1
-1, 1 1, -1
Player B
Heads Tails
Heads
Player A
Tails
Using the underline method
Ch. 5 • 20
No Nash equilibrium in pure
strategies

Ch. 5 • 21
• Look for a mixed strategy. How would you play the game?
• The mixed strategy equilibrium is for each player to play heads
and tails 50% of the time.
• If true each outcome occurs ¼ of the time.
• A player’s expected payoff is:
– ¼ x (1) + ¼ x (-1) + ¼ x (-1) + ¼ x (1) = 0

Ch. 5 • 22
• Suppose A plays Heads and Tails with equal probabilities. Can B
do better by choosing a different probability mix?
• No. B will always get an expected payoff of 0.
• The same holds for player A.
• Since there is no incentive to deviate when each player chooses a
50:50 mix, we have an equilibrium.

Interpretation of Mixed Strategies
Ch. 5 • 23
• In Matching Pennies what is the point of mixing? What are you
trying to do?
• Mixed strategies in sports: goal kicks in soccer, serves in tennis.
Why not always kick, serve, defend to the right?
• Randomization prevents the other player from being able to
exploit you.

Battle of the Sexes
2, 1 0, 0
0, 0 1, 2
Player B (Husband)
Ballet Boxing
Ballet
Player A
(Wife)
Boxing
Find the Nash equilibrium(ia)
using the underline method.
Ch. 5 • 24

Battle of the Sexes: Mixed Strategy
Ch. 5 • 25
• Want to derive the best response function. A function that shows
one player’s profit maximizing choice for anything the other
player chooses.
• Intuition from the wife’s perspective.
– She wants to end up at the same place as her husband but
doesn’t know where he is going.
– If she believes that he will always go to the ballet (boxing) she
should always go to the ballet (boxing).
– What other beliefs may she have?

Ch. 5 • 26
• Let h be the probability the husband goes Boxing.
• What is the wife’s expected payoff from going to the Ballet:
(h)(2) + (1-h)(0) = 2h
• What is the wife’s expected payoff from going to Boxing:
(h)(0) + (1-h)(1) = 1-h
2, 1 0, 0
0, 0 1, 2
Husband
Ballet Boxing
Wife Ballet
Boxing

Ch. 5 • 27
• The wife will choose to go to the Ballet when expected payoff of
Ballet > expected payoff of Boxing.
2h > 1- h or h > 1/3
• If she believes her husband will go to the Ballet more than 1/3 of
the time she should go to the Ballet.
• If she believes her husband will go to the Ballet less than 1/3 of
the time she should go Boxing.
• If she believes her husband will go to the Ballet 1/3 of the time
she is indifferent between either event.

Ch. 5 • 28
• Do the same analysis for the husband.
• Expected payoff of Ballet = (w)(1) + (1-w)(0) = w
• Expected payoff of Boxing = (w)(0) + (1-w)(2) = 2 -2w
• The expected payoffs from each event are equal when w = 2/3.
• How does the husband behave depending on his beliefs?

w
h
1
2/3
1
1/3
tegy Nash
m (both play
Pure-strategy Nash equilibrium
(both play Boxing)
Pure-stra
Husband’s
best-response
function
equilibriu
Ballet)
Wife’s
best-response
function
Mixed-strategyNash
equilibrium
Battle of the Sexes: Best
Response Diagram
Ch. 5 • 29

Problems with Multiple Equilibria
Ch. 5 • 30
• How do you decide which is the “right” equilibrium?
– Select the one which gives the largest total payoff.
– Select the symmetric equilibrium.
– In BoS either of the pure equilibria would satisfy the first
criteria; the mixed equilibrium would satisfy the second.
– Select the focal point – logical outcome on which to
coordinate, based on information outside of the game.
– What would happen at this university?
– A year after graduating you come back for Alumni Weekend.
You are supposed to meet your friends for a night of
festivities, but can’t remember where or when. What’s the
focal point?

Sequential Games
Ch. 5 • 31
• How does the Battle of Sexes game change if we play it
sequentially? Assume the wife chooses first.
• The wife has two strategies: Ballet or Boxing
• The husband has four contingent strategies:
1. Ballet | Ballet, Ballet | Boxing
2. Ballet | Ballet, Boxing | Boxing
3. Boxing | Ballet, Ballet | Boxing
4. Boxing | Ballet, Boxing | Boxing
• Understand that a strategy is simply a description of what the
player can do: not what they will do.

Subgame Perfect Equilibrium
Ch. 5 • 33
• A proper subgame is part of the game tree starting with a single
node and including everything branching out below it.
• A subgame perfect equilibrium (SPE) is a set of strategies that
form a Nash equilibrium for every proper subgame.
• To find the SPE we want to use backwards induction.

. .
Ballet
Ch. 5 • 34
Ballet
Ballet Boxing
Boxing Boxing
2, 1 0, 0 0, 0 1, 2
Husband Husband
What are the three proper subgames?
W
.ife
Battle of the Sexes

Ballet
Ballet Boxing
Boxing Ballet Boxing
2, 1 0, 0 0, 0 1, 2
Husband Husband
Find the husband’s optimal strategies at the last two subgames.
W
.ife
. .
Battle of the Sexes
Ch. 5 • 35

.
. .
Ballet Boxing
A (Wife)
Husband plays Ballet
2, 1
Husband plays Boxing
1, 2
Since we have solved the last two stages we can redraw the game:
Now solve this subgame.
The wife will choose to go to the Ballet since she gets 2 instead of 1
Battle of the Sexes
Ch. 5 • 36

Repeated Games
Ch. 5 • 37
• Is there a way in which we can sustain cooperation in the
Prisoners’ Dilemma?
• Repeated game vs. a one-shot game.
• A stage game is a simple game that is played repeatedly.
• Trigger strategy – a strategy in a repeated game in which one
player stops cooperating in order to punish another player for
cheating.

Repeated Games: Definite Time Horizon
• Suppose we repeat the Prisoners’ Dilemma a finite number of
times: 10.
• What is the subgame perfect equilibrium? Start with the 10th
period.
• Each player will Confess in every period. With a finite number od
periods cooperation is not sustainable.
Ch. 5 • 38

Repeated Games: Indefinite Time Horizon
Ch. 5 • 39
• Now suppose we repeat the Prisoners’ Dilemma an unknown
number of times.
• Under what conditions is cooperation sustainable?
• Consider the following trigger strategy:
– A player will stay silent as long as the other player stays silent.
– If one player confesses, both players will confess from then on.
• g - the probability that the game is repeated for another period.

• Payoff from staying Silent (cooperating) each period:
(-2) x (1 + g + g2 + g3 + . . . )
• Payoff from cheating in period 1: then both players confess:
(-1) + (-3) x (g + g2 + g3 + . . . )
Ch. 5 • 40

Ch. 5 • 41
• Cooperation is an equilibrium if:
(-2) x (1 + g + g2 + g3 + . . . ) > (-1) + -3 x (g + g2 + g3 + . . .)
g + g2 + g3 + . . . > 1
g/(1-g) > 1
g > ½
• So cooperation can be sustained as long as the probability of play
continuing is high enough.
• Grim strategy vs. Tit-for-Tat strategy

Continuous Actions and the Tragedy of
the Commons
Ch. 5 • 42
• Discrete vs. Continuous Actions
• Tragedy of the Commons
• There are two shepherds, A and B, who raise s and s sheep
A B
• Benefit from one sheep = 120 – s – s
A B
A B
• Total benefit from raising s and s sheep:
A
s (120 – sA B
– s )
– Shepherd A:
– Shepherd B: B
s (120 – sA B
– s )

Tragedy of the Commons
Ch. 5 • 43
• Marginal benefit from raising one sheep:
A B
– Shepherd A: 120 – 2s – s
– Shepherd B: 120 – sA
– 2sB
• Set the marginal benefit equal to the marginal cost
A B
– Shepherd A: 120 – 2s – s = 0
– Shepherd B: 120 – sA B
– 2s = 0
• Solve for the best response functions
A B
– Shepherd A: s = 60 – ½ s
– Shepherd B: B
s = 60 – ½ sA

Ch. 5 • 44
• To solve for the Nash equilibrium, substitute one shepherd’s best
response function into the other.
A A
– s = 60 – ½ (60 – ½ s )
A
– s = 30 + ¼ sA
A
– s * = 40
B
– s * = 40
• So each shepherd will raise 40 sheep.

Nash equilibrium
B’s best-response
function
SA
12
0
SB
12
0
6
0
6
0
4
0
4
0
A’s best-response
function
Tragedy of the Commons: Best
Response Diagram
Ch. 5 • 45

Ch. 5 • 46
• Why is this referred to as the “Tragedy” of the Commons?

Summary
Ch. 5 • 47
• The basic elements of a game are players, strategies, payoffs, and
information.
• A Nash equilibrium is a set of strategies, one for each player, such
that all players’ strategies are best responses to each other.
• All games have at least one Nash equilibrium, which may involve
mixed strategies.
• If the Prisoners’ Dilemma is repeated an indefinite number of
times, it is possible to sustain the cooperative outcome.

Summary
Ch. 5 • 48
• Sequential games allow the players to use contingent strategies.

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