Game Theory
OLLI
Fall, 2015
Day One
Game Theory:
• Not just about games
• About strategies used in daily life
• Concerns cooperation rather than confrontation
• Collaboration rather than competition
• Goes back thousands of years – Talmud and Sun Tzu’s writings
• Modern theory credited to John von Neuman and Oskar
Morgenstern
• John Nash generalized the results and provided a basis for the
modern field
• Game theory builds mathematical models and draws conclusions
from those models.
Rock-Paper-Scissors
Paper covers rock – so paper wins.
Rock breaks scissors – so rock wins.
Scissors cuts paper – so scissors wins.
Rock-Paper
Paper covers rock – so paper wins.
No winner
Ro
C
Paper
Sc
i
o
c
ck
ss
or
Paper covers rock - C wins
s
k
Rock breaks scis sors - R wins
R
Paper covers rock - R wins
ck
o
R
Paper
R
C
Paper
Sc
i
S
c
i
ss
o
No winner
rs
Scissors cuts paper - C wins
s
s
o
r
Rock breaks scis sors - C wins
ck
o
R
s
Paper
C
Sc
i
ss
o
Scissors cuts paper - R wins
rs
No winner
No winner
Ro
C
Paper
Sc
i
o
c
ck
ss
or
Paper covers rock - C wins
s
k
Rock breaks scis sors - R wins
R
Paper covers rock - R wins
ck
o
R
Paper
R
C
Paper
Sc
i
S
c
i
ss
o
No winner
rs
Scissors cuts paper - C wins
s
s
o
r
Rock breaks scis sors - C wins
ck
o
R
s
Paper
C
Sc
i
ss
o
Scissors cuts paper - R wins
rs
No winner
No winner
Ro
C
Paper
Sc
i
o
c
ck
ss
or
Paper covers rock - C wins
s
k
Rock breaks scis sors - R wins
R
Paper covers rock - R wins
ck
o
R
Paper
R
C
Paper
Sc
i
S
c
i
ss
o
No winner
rs
Scissors cuts paper - C wins
s
s
o
r
Rock breaks scis sors - C wins
ck
o
R
s
Paper
C
Sc
i
ss
o
Scissors cuts paper - R wins
rs
No winner
Rock, paper, scissors with scores in terms of (Row, Column):
Column
R
P
S
R
(0, 0)
(-1, 1)
(1, -1)
P
(1, -1)
(0, 0)
(-1, 1)
S
(-1, 1)
(1, -1)
(0, 0)
Row
Rock, paper, scissors (scores in terms of Row):
Column
Row
Political Decision Time
Two candidates for political office must decide to
be for, against, or neutral on a certain
referendum.
Political Decision Time
Pollsters have determined that if candidate R
comes out for the referendum, then he will gain
8000 votes if candidate C also comes out for the
referendum, will lose 1000 votes if candidate C
comes out against, and will gain 1000 votes if
candidate C comes out neutral.
Political Decision Time
If candidate R comes out against, then he will,
respectively, lose 7000 votes, gain 4000 votes,
lose 2000 votes if candidate C comes out,
respectively, for, against, neutral on the
referendum. If candidate R is neutral, then he will
gain 3000 votes if C is for or against and will gain
2000 votes if C is neutral.
Political Decision Time
Pollsters have determined that if candidate R comes out for the
referendum, then he will gain 8000 votes if candidate C also
comes out for the referendum, will lose 1000 votes if candidate
C comes out against, and will gain 1000 votes if candidate C
comes out neutral. If candidate R comes out against, then he
will, respectively, lose 7000 votes, gain 4000 votes, lose 2000
votes if candidate C comes out, respectively, for, against,
neutral on the referendum. If candidate R is neutral, then he will
gain 3000 votes if C is for or against and will gain 2000 votes if C
is neutral.
R gains 8000
Fo
C
Agains t
Ne
o
r
ut
ra
l
R gains 1000
r
F
R gains 3000
Fo
R
Agains t
C
r
Agains t
Ne
N
e
u
C gains 1000
R gains 3000
ut
ra
l
R gains 2000
t
r
a
C gains 7000
l
Fo
r
Agains t
C
Ne
R gains 4000
ut
ra
l
C gains 2000
Vote gain or loss in terms of R:
C
For
Agains t
Neutral
+ 8000
- 1000
+ 1000
Agains t
+ 3000
+ 3000
+ 2000
Neutral
- 7000
+ 4000
- 2000
For
R
Vote gain or loss in terms of (Row, Column):
C
For
Agains t
Neutral
(+ 8000, - 8000)
(- 1000, + 1000)
(+ 1000, - 1000)
Agains t
(+ 3000, - 3000)
(+ 3000, - 3000)
(+ 2000, - 2000)
Neutral
(- 7000, + 7000)
(+ 4000, - 4000)
(- 2000, + 2000)
For
R
Vote gain or loss in terms of R:
C
R
For
Agains t
Neutral
+ 8000
- 1000
+ 1000
Agains t
+ 3000
+ 3000
+ 2000
Neutral
- 7000
+ 4000
- 2000
For
The best strategy is for R to be against and
C to be neutral.
Two television stations, WMBD and WHOI, each have a
game show and a situation comedy to schedule for their
1:00 and 2:00 afternoon time slots. If both schedule
game shows at 1:00, then WMBD will take $5,000 in
advertising revenue away from WHOI. If both schedule
the game show at 2:00, then WHOI will take $3,000 in
advertising revenue from WMBD. If they choose
different hours for their game show, then WMBD will
take $2,000 from WHOI by scheduling the show at 2:00
and will take $1,000 from WHOI if scheduled at 1:00.
What is the best strategy for the television stations?
Two television stations, WMBD and WHOI, each have a
game show and a situation comedy to schedule for their
1:00 and 2:00 afternoon time slots. If both schedule
game shows at 1:00, then WMBD will take $5,000 in
advertising revenue away from WHOI.
If both schedule the game show at 2:00, then WHOI will
take $3,000 in advertising revenue from WMBD. If they
choose different hours for their game show, then
WMBD will take $2,000 from WHOI by scheduling the
show at 2:00 and will take $1,000 from WHOI if
scheduled at 1:00. What is the best strategy for the
television stations?
Scheduling game shows; revenue in terms of WMBD:
The best strategy is for WMBD to schedule the
game show at 1:00 and WHOI to schedule their
game show at 2:00.
Two merchants are planning on building competing stores to serve a
region containing three small cities. The fraction of the total
population that live in each city is shown in the figure below. (Sam’s
and Costco??)
City 1
30%
City 3
25%
City 2
45%
Two merchants are planning on building competing stores to serve a
region containing three small cities. The fraction of the total
population that live in each city is shown in the figure below. If both
merchants locate in the same city, Sam’s will get 65% of the total
business in all three cities.
City 1
30%
City 3
25%
City 2
45%
Two merchants are planning on building competing stores to serve a
region containing three small cities. The fraction of the total
population that live in each city is shown in the figure below. If both
merchants locate in the same city, Sam’s will get 65% of the total
business. If the merchants locate in different cities, each will get 80%
of the business in the city it is in, and Sam’s will get 60% of the
business from the city not containing Costco.
City 1
30%
City 3
25%
City 2
45%
Percentage of business in terms of Sam’s.
Costco
City 1
City 1
City 2
City 3
65%
Sam’s
City 2
City 3
65%
65%
Percentage of business in terms of Sam’s.
Costco
City 1
Sam’s
City 2
City 3
City 1
City 2
65%
80% of 1
20% of 2
60% of 3
City 3
65%
65%
Percentage of business in terms of Sam’s.
Costco
City 1
City 2
City 3
City 1
65%
80% of 1
20% of 2
60% of 3
80% of 1
60% of 2
20% of 3
City 2
20% of 1
80% of 2
60% of 3
65%
60% of 1
80% of 2
20% of 3
City 3
20% of 1
60% of 2
80% of 3
60% of 1
20% of 2
80% of 3
65%
Sam’s
Adding in percentage of local population in the three cities:
Costco
City 1
City 2
City 3
City 1
30%
65%
80% of 1
20% of 2
60% of 3
80% of 1
60% of 2
20% of 3
City 2
45%
20% of 1
80% of 2
60% of 3
65%
60% of 1
80% of 2
20% of 3
City 3
25%
20% of 1
60% of 2
80% of 3
60% of 1
20% of 2
80% of 3
65%
Sam’s
Doing the math:
Costco
City 1
City 2
City 3
80% of 1
20% of 2
60% of 3
80% of 1
60% of 2
20% of 3
20% of 1
80% of 2
60% of 3
65%
60% of 1
80% of 2
20% of 3
20% of 1
60% of 2
80% of 3
60% of 1
20% of 2
80% of 3
65%
City 1
30%
65%
City 2
45%
City 3
25%
Sam’s
S in City 1 and C in City 2:
80%(30%)
20%(45%)
+ 60%(25%)
48%
Doing the math:
Costco
City 1
City 1
30%
65%
City 2
City 3
S in City 1 and C in
City 2:
48%
Sam’s
City 2
45%
City 3
25%
65%
65%
80%(30%)
20%(45%)
+ 60%(25%)
48%
Percentage of area business in terms of Sam’s:
Costco
City 1
City 2
City 3
City 1
30%
65%
48%
56%
City 2
45%
57%
65%
59%
City 3
25%
53%
47%
65%
Sam’s
Percentage of area business in terms of Sam’s – above or below 50%:
Costco
City 1
City 2
City 3
City 1
30%
+ 15%
- 2%
+ 6%
City 2
45%
+ 7%
+ 15%
+ 9%
City 3
25%
+ 3%
- 3%
+ 15%
Sam’s
Percentage of area business in terms of Sam’s – above or below 50%:
Costco
Sam’s
City 1
City 2
City 3
City 1
30%
+ 15%
- 2%
+ 6%
City 2
45%
+ 7%
+ 15%
+ 9%
City 3
25%
+ 3%
- 3%
+ 15%
Doing the “math” to determine the chance that each merchant will choose a particular city:
City 1 S chooses 27% C chooses 39%
City 2 S chooses 64% C chooses 9%
City 3 S chooses 9% C chooses 52%
Day Two
Following up on three comments/questions:
Thomas Dewey – his political life: MOST interesting;
someone needs to do an OLLI class just on him!
Kroger: who did the research? Why the decision? Marketing
plan? Evidently they don’t want us to know – or I just
couldn’t find it.
Political decision time: really strictly determined? Wouldn’t it
be better to choose each vote 1/3 of the time? (More to
come.)
Game Theory is the logical analysis of situations of conflict and
cooperation.
The elements of a game:
1) Players: how many (there are at least two); how they are chosen
2) Possible actions: each player has a number of possible strategies.
3) Information available to players
4) Payoffs/consequences/outcomes
5) Players’ preferences over payoffs
Obstacles found in games:
1) Any real-world game is very complex.
2) Game theory deals with “rational” moves.
3) Game theory cannot give unique prescriptions for players or play just allows for analysis and strategies.
Strictly determined: game has saddle
point; optimal strategies for both
players
Not strictly determined: game has no
saddle point; mixed strategies for both
players
Political Decision Time
If candidate R comes out against, then he will, respectively,
lose 7000 votes, gain 4000 votes, lose 2000 votes if
candidate C comes out, respectively, for, against, neutral
on the referendum. If candidate R is neutral, then he will
gain 3000 votes if C is for or against and will gain 2000
votes if C is neutral.
Vote gain or loss in terms of R:
C
R
For
Agains t
Neutral
+ 8000
- 1000
+ 1000
Agains t
+ 3000
+ 3000
+ 2000
Neutral
- 7000
+ 4000
- 2000
For
The best strategy is for R to be against and
C to be neutral. (Strictly determined.)
Was suggested that each choice should be
given 1/3 probability.
The math:
1
8000  1000  1000  3000  3000  2000  7000  4000  2000
9
1
 11, 000   1, 222.22
9
This is the GAIN to Row (and the loss to Column).
How about if Row chooses each with 1/3
probability and Column always chooses
against?
The math:
1
 1000  3000  4000 
3
1
  6, 000   2, 000
3
One more time:
Row chooses each with 1/3 probability and
Column chooses against 2/3 and the others 1/6
each.
The math:
1 2
  1000  3000  4000 
3 3
1 1
   8000  3000  7000  1000  2000  2000 
3 6
2
1
1
7
1
  6, 000    5000   1333  277  1611
9
18
3
9
9
Vote gain or loss in terms of R:
C
For
Agains t
Neutral
+ 8000
- 1000
+ 1000
Agains t
+ 3000
+ 3000
+ 2000
Neutral
- 7000
+ 4000
- 2000
For
R
Both choose each option 1/3 of time: +$1222.22 (Row advantage)
Row chooses each option 1/3 of time, Column always is against: +$2000
(Row advantage)
Row chooses each option 1/3 of time, Column is against 2/3 of time: +$1611 1/9
(Row advantage)
Strictly determined: game has saddle
point; optimal strategies for both
players
Not strictly determined: game has no
saddle point; mixed strategies for both
players
Remember: “strictly determined” is the
game theory designation that this is
the best strategy for both involved. It
isn’t necessarily what they actually do.
You are the owner of several greeting-card
stores must decide in December about the
type of displays to emphasize for Mother’s
Day in May. You have three possible
choices:
• emphasize chocolates
• emphasize collectible gifts
• emphasize gifts that can be engraved
Your success is dependent of the state of the
economy in May (this decision is made in
December).
• If the economy is strong, you will do well
with the collectible gifts.
• In a weak economy, the chocolates will do
very well.
• In a mixed economy, the gifts that can be
engraved will do well.
You prepare a matrix for the possibilities, with the
numbers representing your profits in thousands of
dollars.
Weak Econom y
Mixed
Strong Economy
Chocolates
85
30
75
Collectibles
45
45
110
Engraved
60
95
85
What is your strategy?
If you’re an optimist…
Weak Econom y
Mixed
Strong Economy
Chocolates
85
30
75
Collectibles
45
45
110
Engraved
60
95
85
If you’re a pessimist…
Weak Econom y
Mixed
Strong Economy
Chocolates
85
30
75
Collectibles
45
45
110
Engraved
60
95
85
You hear on the news that leading experts believe
there’s a 50% chance of a weak economy in May, a
20% chance of a mixed economy, and a 30% chance of
a strong economy.
Weak Econom y
Mixed
Strong Economy
Chocolates
85
30
75
Collectibles
45
45
110
Engraved
60
95
85
Weak Econom y 50%
Mixed 20%
Strong Economy 30%
Chocolates
85
30
75
Collectibles
45
45
110
Engraved
60
95
85
Chocolates: 85(.50)  30(.20)  75(.30)  71
Collectibles: 45(.50)  45(.20)  110(.30)  64.5
Engraved:
60(.50)  95(.20)  85(.30)  74.5
A classic game from the annals of military
strategy is the Battle of the Bismarck Sea. In early
1943, US military intelligence determined that a
Japanese convoy would be moving from New
Britain to New Guinea. It was up to General
MacArthur's command to intercept the convoy
and do damage.
The Japanese commander had the choice of
sending his convoy either north or south of New
Britain; either route required three days to reach
its destination. Weather predictions were that:
► the northern route would be rainy with poor
visibility
► the southern route would be clear.
MacArthur's general in charge of the mission
could either concentrate most of his search
aircraft on the northern route or on the southern
route. It was up to this commander to find the
convoy and expose it to bombing by as many
aircraft as possible. The Japanese commander, of
course, wanted to limit the exposure to bombing.
The decisions of the two commanders were
completely independent of one another.
The following indicates the reasoning of the two commanders (what
happens during the three-day period:
North: rainy, poor visibility
Japanese convoy goes North
North: rainy,
poor visibility
South: clear
South: clear
Japanese convoy goes South
Weather interferes with the
search, but two days of
US Aircraft goes North bombing is possible because
of greater concentration of
aircraft.
The convoy is in clear
weather,
but most of search was to the
North, so only have time for
two days of bombing.
Poor visibility and loss of time
because of misdirected
US Aircraft goes South aircraft limits bombing time
to one day.
Convoy in clear weather and
search aircraft concentrated
in area provides three days
of bombing.
Japanese convoy goes North
US Aircraft goes North
US Aircraft goes South
Japanese convoy goes South
Weather interferes with the
search, but two days of
bombing is possible because
of greater concentration of
aircraft.
The convoy is in clear weather,
but most of search was to the
North, so only have time for two
days of bombing.
Poor visibility and loss of time
because of misdirected
aircraft limits bombing time
to one day.
Convoy in clear weather and
search aircraft concentrated
in area provides three days
of bombing.
The history books tell us that both commanders
chose the northern route.
A company is being sued (product liability).
Do they try to settle for $1.2 million, or do
they go to trial where they could possibly lose
as much as $4 million?
What do the strategists for the company
being sued know?
- $1.2 m illion
Settle
Go
to
trial
innocent
finds
Jury
Jury
finds
guilty
$0
- $1.2 m illion
Settle
Go
to
trial
innocent
finds
Jury
$0
- $4 m illion
Jury
finds
guilty
- $2 m illion
- $1 m illion
- $1.2 m illion
Settle
Go
to
trial
innocent
finds
Jury
25%
$0
- $4 m illion
Jury 75%
finds
guilty
- $2 m illion
- $1 m illion
- $1.2 m illion
Settle
Go
to
trial
innocent
finds
Jury
25%
$0
- $4 m illion
Jury 75%
finds
guilty
30%
40%
- $2 m illion
30%
- $1 m illion
- $1.2 m illion
Settle
Go
to
trial
innocent
finds
Jury
25%
$0
- $4 m illion
Jury 75%
finds
guilty
30%
40%
- $2 m illion
30%
- $1 m illion
What are the chances of the $4 million settlement?
75% X 30% = 22.5%
- $1.2 m illion
Settle
Go
to
trial
innocent
finds
Jury
25%
$0
- $4 m illion
Jury 75%
finds
guilty
30%
40%
- $2 m illion
30%
- $1 m illion
What are the chances of the $1 million settlement?
75% X 30% = 22.5%
- $1.2 m illion
Settle
Go
to
trial
innocent
finds
Jury
25%
$0
- $4 m illion
Jury 75%
finds
guilty
30%
40%
- $2 m illion
30%
- $1 m illion
Expected value if jury finds company guilty: - $2.3 million
Multiply by 75%: -$1.725 million
Strategic Choices
Airlines: sometimes people don’t show and the airline can
sell their ticket (get twice the revenue for the
ticket)
: should the airline overbook?
Price of one-way ticket from Peoria to Orlando: $130
Offer to “overbooked” person to take another flight: $100
Price of one-way ticket from Peoria to Orlando: $130
Offer to “overbooked” person to take another flight: $100
If the airline overbooks by one person, what can happen?
(Assume can sell the seat of a no-show.)
0 no-shows: have to pay overbooked person $100
(- $100 for seat)
1 no-show: can sell the seat again for $130
(+$130 for seat)
Price of one-way ticket from Peoria to Orlando: $130
Offer to “overbooked” person to take another flight: $100
If the airline overbooks by two people, what can happen?
(Assume can sell the seats of any no-shows.)
0 no-shows: have to pay overbooked persons $100 each
(- $200 for two seats)
1 no-show: can sell one seat again for $130; pay
overbooked person $100
(+$130 – $100 = +$30 net for two seats)
2 no-shows: can sell two seats again for $130 each
(+$260 net for two seats)
Price of ticket:
$130
Penalty for overbook:
-$100
Number of no-shows
Don't overbook
Overbook by 1
Overbook by 2
0
0
-100
-200
1
0
+130
+30
2
0
+130
+260
Experience has shown that:
The probability that there WON’T
be a no-show is 25%.
The probability that there will be 1
no-show is 40%.
The probability that there will be 2
no-shows is 35%.
Price of ticket:
$130
Penalty for overbook:
-$100
Number of no-shows
0: 25% 1: 40% 2: 35%
Don't overbook
0
0
0
Overbook by 1 -100 +130 +130
Overbook by 2 -200
+30 +260
Overbook by 1, Expect: 25%(-100) + 40%(+130) + 35%(+130) = $72.50
Overbook by 2, Expect: 25%(-200) + 40%(+30) + 35%(+260) = $53.00
The application of game theory to political science is
focused in the overlapping areas of fair division,
political economy, public choice, war bargaining,
positive political theory, and social choice theory. In
each of these areas, researchers have developed
game-theoretic models in which the players are often
voters, states, special interest groups, and politicians.
Unlike those in economics, the payoffs for games in
biology are often interpreted as corresponding to
fitness. In addition, the focus has been less on
equilibria that correspond to a notion of rationality and
more on ones that would be maintained by
evolutionary forces. The best known equilibrium in
biology is known as the evolutionarily stable
strategy (ESS), first introduced in 1973. Although its
initial motivation did not involve any of the mental
requirements of the Nash equilibrium, every ESS is a
Nash equilibrium.
In biology, game theory has been used as a model to understand
many different phenomena. It was first used to explain the
evolution (and stability) of the approximate 1:1 sex ratios:
suggested that the 1:1 sex ratios are a result of evolutionary
forces acting on individuals who could be seen as trying to
maximize their number of grandchildren.
Additionally, biologists have used evolutionary game theory and
the ESS to explain the emergence of animal communication. For
example, the mobbing behavior of many species, in which a large
number of prey animals attack a larger predator, seems to be an
example of spontaneous emergent organization. Biologists have
used the game of chicken to analyze fighting behavior and
territoriality.
Game theory has come to play an increasingly important role in
logic and in computer science. Computer scientists have used
games to model interactive computations.
The emergence of the internet has motivated the development of
algorithms for finding equilibria in games, markets,
computational auctions, peer-to-peer systems, and security and
information markets.
Game theory has been put to several uses in philosophy. In
ethics, authors have attempted to pursue the project of deriving
morality from self-interest. Since games like the prisoner’s dilemma
present an apparent conflict between morality and self-interest,
explaining why cooperation is required by self-interest is an important
component of this project.
Paradox: reasoning leading to a conclusion
that seems logically
wrong/unacceptable/contradictory
Dilemma: situation with difficult choices
between two or more alternatives (often
equally undesirable)
Monty Hall Problem
Let’s Make a Deal
Monty Hall Problem
Suppose you're on a game show, and you're given the
choice of three doors: Behind one door is a car; behind the
others, goats. You pick a door, say No. 1, and the host, who
knows what's behind the doors, opens another door, say
No. 3, which has a goat. He then says to you, "Do you want
to pick door No. 2?" Is it to your advantage to switch your
choice?
You pick door No. 1, and the host, who knows
what's behind the doors, opens another door, No. 3,
which has a goat. Do you stay with No. 1 or do you
switch?
You pick door No. 1, and the host, who knows
what's behind the doors, opens another door, No. 3,
which has a goat. Do you stay with No. 1 or do you
switch?
Day Three
You pick door No. 1, and the host, who knows
what's behind the doors, opens another door, No. 3,
which has a goat. Do you stay with No. 1 or do you
switch?
Car
Location
Door 1
1
3
Player
picks
Door 1
1
3
Door 2
1
3
Door 3
Car
Location
Door 1
1
3
Player
picks
Door 1
1
3
Door 2
1
3
Door 3
Host
shows
Car
Location
Door 1
1
3
1
2
1
2
1
3
Player
picks
Door 1
Host
shows
Door 2
1
1
3
1
Door 3
Door 2
Car
Location
Door 1
1
3
Player
picks
Door 1
1
3
Door 2
Host
shows
1
2
Door 2
1
2
Door 3
1
1
3
1
Door 3
Car
Location
Door 1
1
3
Player
picks
Door 1
1
3
Door 2
Host
shows
1
2
Door 2
1
2
Door 3
1
1
3
1
Door 3
Door 3
Car
Location
Door 1
1
3
Player
picks
Door 1
1
3
Door 2
Host
shows
1
2
Door 2
1
2
Door 3
1
Door 3
1
3
1
Door 3
Door 2
Car
Location
Door 1
1
3
Player
picks
Door 1
1
3
Door 2
Host
shows
Stay
1
2
1
Door 2 6
1
2
Door 3
1
1
6
Door 3
1
3
Door 2
1
3
1
3
1
Door 3
Switch
Car
Location
Door 1
1
3
Player
picks
Door 1
1
3
Door 2
Host
shows
Stay
1
2
1
Door 2 6
1
2
Door 3
1
1
6
Door 3
1
3
Door 2
1
3
1
3
1
Door 3
Car
Switch
Car
Location
Door 1
1
3
Player
picks
Door 1
1
3
Door 2
Host
shows
1
2
1
Door 2 6
1
2
Door 3
1
1
6
Door 3
1
3
Door 2
1
3
1
3
1
Door 3
Stay
Switch
Car
Goat
Car
Location
Door 1
1
3
Player
picks
Door 1
1
3
Door 2
Host
shows
Switch
Goat
1
2
1
Door 2 6
Car
1
2
Door 3
1
6
Car
1
Door 3
1
3
Door 2
1
3
1
3
1
Door 3
Stay
Car
Location
Door 1
1
3
Player
picks
Door 1
1
3
Door 2
Host
shows
Switch
1
2
1
Door 2 6
Car
Goat
1
2
Door 3
1
6
Car
Goat
1
Door 3
1
3
Door 2
1
3
1
3
1
Door 3
Stay
Car
Location
Door 1
1
3
Player
picks
Door 1
1
3
Door 2
Host
shows
Switch
1
2
1
Door 2 6
Car
Goat
1
2
Door 3
1
6
Car
Goat
1
Door 3
1
3
Door 2
1
3
1
3
1
Door 3
Stay
Goat
Car
Location
Door 1
1
3
Player
picks
Door 1
1
3
Door 2
Host
shows
Switch
1
2
1
Door 2 6
Car
Goat
1
2
Door 3
1
6
Car
Goat
Goat
Car
1
Door 3
1
3
Door 2
1
3
1
3
1
Door 3
Stay
Car
Location
Door 1
1
3
Player
picks
Door 1
1
3
Door 2
Host
shows
Stay
Switch
1
2
1
Door 2 6
Car
Goat
1
2
Door 3
1
6
Car
Goat
Door 3
1
3
Goat
Car
Door 2
1
3
Goat
1
1
3
1
Door 3
Car
Location
Door 1
1
3
Player
picks
Door 1
1
3
Door 2
Host
shows
Stay
Switch
1
2
1
Door 2 6
Car
Goat
1
2
Door 3
1
6
Car
Goat
Door 3
1
3
Goat
Car
Door 2
1
3
Goat
Car
1
1
3
1
Door 3
Car
Location
Door 1
1
3
Player
picks
Door 1
1
3
Door 2
Host
shows
Stay
Switch
1
2
1
Door 2 6
Car
Goat
1
2
Door 3
1
6
Car
Goat
Door 3
1
3
Goat
Car
Door 2
1
3
Goat
Car
1
1
3
1
Door 3
A similar paradox to the Monty Hall Problem
is Bertrand’s Box Paradox.
Bertrand's box paradox is a classic paradox of elementary
probability theory. It was first posed by Joseph Bertrand in 1889.
There are three boxes:
a box containing two gold coins,
a box containing two silver coins,
a box containing one gold coin and one silver coin.
GG
SS
GS
GG
SS
GS
One box is chosen at random, and a coin is withdrawn (without
peeking at the other coin).
The coin is gold.
What are the chances that the other coin is gold, too?
After choosing a box at random and withdrawing
one coin at random, if that happens to be a gold
coin, it may seem that the probability that the
remaining coin is gold is 1⁄2; in fact, the probability
is actually 2⁄3.
GG
SS
GS
One explanation:
The coin chosen is gold. That means it came from the left or right
box.
With a gold coin taken from the left or right box, there are 3 coins
left in the two boxes: 2 gold and 1 silver. The probability is 2/3
that the remaining coin in the box is gold.
GG
SS
GS
Have drawn one gold coin. Now, the probability that the second
coin drawn (from any box) will be gold:
Probability (draw gold if GG)
Probability (draw gold if GG)  Probability (draw gold if SS)  Probability (draw gold if GS)
1
1
2



3
3
1 0  1
2
2
Social Dilemmas
Social dilemmas:
• Prisoner’s Dilemma
• Tragedy of the Commons
• The Free Rider
• Chicken
• The Volunteer’s Dilemma
• The Battle of the Sexes
• Stag Hunt
Understanding of social dilemmas came when John Nash discovered that all of them arise from the same basic
logical trap. This is called the Nash Equilibrium.
It is a position in which both sides have selected a strategy and neither side can then independently change its
strategy without ending up in a less desirable position. It’s a point of balance in a social situation, from which
neither side can escape without loss.
The secret to resolving such situations is for the parties find some way of agreeing to coordinate their actions
and for all parties to stick to the agreement.
.
Prisoners’ Dilemma:
Two thieves, Bill and Fred, have been caught by the police, but the
prosecutor has only enough evidence to sentence them both to two years,
on a charge of carrying concealed weapons, rather than the maximum
penalty of ten years for burglary. As long as they both plead not-guilty,
they will both get two years.
Both plead not-guilty: both get two years
The prosecutor approaches Bill and points out that if Fred pleads guilty,
but Bill does not, then Fred will receive a reduced sentence of four years
for pleading guilty, but Bill will get the maximum sentence of ten years. So
Bill’s best bet, if he believes that Fred will plead guilty, is to also plead
guilty and receive four years, rather than ten.
Both plead guilty: both get four years
The prosecutor also adds that he will give Bill the deal that if he
pleads guilty and Fred doesn’t , then he can go free for turning
state’s evidence. Bill reasons that he can do better for himself if
he pleads guilty, no matter what Fred does. The trouble is that
the prosecutor has made the same deal with Fred, who has come
to the same conclusion. So they both plead guilty, and both get
four years. If they’d both kept quiet, then they would both have
received two years.
Fred
Guilty
Not Guilty
Guilty
Bill
Not Guilty
Both 4
years
Bill 0
Fred 10
Bill 10
Fred 0
Both 2
years
Diner's dilemma is an n-person Prisoner’s dilemma.
Several individuals go out to eat, and prior to ordering,
they agree to split the check equally between all of them.
Each individual must now choose whether to order the
expensive or inexpensive dish. It is presupposed that the
expensive dish is better than the cheaper.
Each individual reasons that the expense s/he adds to
their bill by ordering the more expensive item is very
small (if they’re the only one), and thus their personal
improved dining experience is worth the money.
However, having all reasoned thusly, they all end up
paying for the cost of the more expensive meal, which by
assumption, is worse for everyone than having ordered
and paid for the cheaper meal.
Tragedy of the Commons
Explained by parable of the herders:
A group of herders, each grazing his own animals on
common land, with one herder thinking about adding an
extra animal to his herd. An extra animal will yield a tidy
profit, and the overall grazing capacity of the land will
be only slightly diminished, so it seems perfectly
logical… The tragedy comes when all the other herders
think the same way. They all add extra animals, the land
becomes overgrazed, and soon there is no pasture left.
Tragedy of the Commons
Also explained by teaspoons:
Teaspoons are disappearing from the common coffee
area. The teaspoon users make decisions that their own
utility (benefit to themselves) is improved by removing a
teaspoon from the common coffee area for personal
use, while everyone else’s utility is reduced by only a
fraction per head, since there are so many teaspoons
available. As more and more teaspoon users come to
the same conclusion, the teaspoons in the common area
are soon depleted.
The Tragedy of the Commons
Another example
After the 2004 tsunami in Sri Lanka, funds had been
donated to help people who lived in the affected
areas to move out or to rebuild their houses. Some
people from outside these areas actually moved into
them so they could claim a share of the benefits.
Doing so, they each took a small slice designated for
those who lived there originally.
The Tragedy of the Commons
Another example
Vegetable farmers are restricted in how much water they are allowed to use as a
result of a drought in the area. If they cooperate with the restriction, they will
get a lower yield per acre. (For purposes of illustration, let this yield be 5 tons per
acre rather than the usual 10 tons per acre.) If a few cheat by using water freely,
they could still get 10 tons per acre. If most of them cheat, the reservoirs would
run low, and their yields could drop to 2 tons per acre. More severe restrictions
could also come into force, and individuals who cooperated with the new
restrictions might only then get 1 ton per acre. The outcome depends on how
most of the farmers see themselves: as members of a cooperative group, or as
competing individuals.
The Tragedy of the Commons
Other examples:
DVD piracy, overfishing, pollution and global warming,
benefit cheating, etc.
The Free Rider
• Examples: leaving a mess for others to clean up in a shared
accommodation, the choice between remaining seated or standing
to get up to get a better view, refusing to join a labor union but still
accepting the benefits won by negotiations of members, credit card
fraud, disarmament, etc.
• A new steeple on a church would cost $100,000; everyone is asked
to contribute $100. If a person decides not to contribute and just
reap the benefit of a new steeple, they could assign their benefit
worth $200.
Chicken
For example, in the movie Rebel Without a Cause, the
two characters race their cars toward an abyss. The
first one to jump out is the loser (and the chicken).
Ironically, in this movie, the loser lost even more
when his leather jacket got hung up on the car door
and he went over the cliff with the car.
Chicken
Another example could be the Cuban Missile Crisis –
which could have ended in disaster – when
Khrushchev refused to stop bringing (remove) Soviet
missiles to Cuba and President Kennedy refused to lift
the naval blockade.
Chicken
Two people walking along a sidewalk toward each
other have the choice of stepping to one side or not.
If they both step aside, then the outcome is “neutral”.
If neither steps aside, then it is a “bad” outcome. The
“good” outcome (for one or the other it’s good) is if
one steps aside and the other doesn’t.
Don't Step
Aside
Don't Step
Aside
Step Aside
Step Aside
The hawk-dove version of the game imagines two players
(animals) contesting an indivisible resource who can choose
between two strategies, one more escalated than the other.
They can use threat displays (play Dove), or physically attack
each other (play Hawk). If both players choose the Hawk
strategy, then they fight until one is injured and the other wins.
If only one player chooses Hawk, then this player defeats the
Dove player. If both players play Dove, there is a tie, and each
player receives a payoff lower than the profit of a hawk
defeating a dove.
The Volunteer’s Dilemma
Group situations in which the person making the
first move risks losing out – while the others
gain. But if no one volunteers, then the loss can
be disastrous.
The Volunteer’s Dilemma
Who should jump out of the life boat to keep it from sinking?
Who should take the blame for a group offense so not all are
punished?
Migrating wildebeest herds coming to a river with crocodiles.
A person throws himself on a grenade to save the others.
The Volunteer’s Dilemma
A science magazine conducted an experiment: they
invited readers to send a card requesting either $20
or $100. The offer was for everyone to receive what
they asked for – as long as no more than 20% of the
requests were for $100 – in which case no one would
get anything. (The magazine backed out of their offer
before the results came in, but they would have been
fine: 35% asked for $100.)
The Battle of the Sexes
A couple decide independently whether to go to a
movie or a ball game. Each person likes to do
something together with the other, but one of them
prefers movies and the other ball games. The payoffs
are in terms of satisfaction: being at the preferred
place gives satisfaction, and being with the other
person gives even more satisfaction.
Movie
Ball Game
Ball Game
Ball Game
Movie
Movie
The Stag Hunt
• The “inverted” Prisoner’s Dilemma; with the Prisoner’s Dilemma,
the reward to the individual is always greater for cheating/telling.
With the Stag Hunt, the reward is greater for cooperating/not
cheating.
• A group of villagers are hunting a deer. It’s agreed that, in order to
get the deer, everyone has to stay at their assigned place. The
temptation comes when a hare comes within reach of a villager. If
he goes for the hare (a sure thing), then the others will lose the
chance to get a deer (from the commotion).
Some Other Games
The Three Prisoners
Three prisoners, A, B and C, are in separate cells and sentenced to death. The governor has
selected one of them at random to be pardoned. The warden knows which one is pardoned,
but is not allowed to tell. Prisoner A begs the warden to let him know the identity of one of the
others who is going to be executed. "If B is to be pardoned, give me C's name. If C is to be
pardoned, give me B's name. And if I'm to be pardoned, flip a coin to decide whether to name
B or C."
The warden tells A that B is to be executed. Prisoner A is pleased because he believes that
his probability of surviving has gone up from 1/3 to 1/2, as it is now between him and C.
Prisoner A secretly tells C the news, who is also pleased, because he reasons that A still has
a chance of 1/3 to be the pardoned one, but his chance has gone up to 2/3. What is the
correct answer?
The answer is that prisoner A didn't gain information about his own fate. Prisoner A, prior to
hearing from the warden, estimates his chances of being pardoned as 1/3, the same as both
B and C. As the warden says B will be executed, it's either because C will be pardoned (1/3
chance), or A will be pardoned (1/3 chance) and the B/C coin the warden flipped came up B
(1/2 chance; for a total of a 1/6 chance B was named because A will be pardoned). Hence,
after hearing that B will be executed, the estimate of A's chance of being pardoned is half that
of C. This means his chances of being pardoned, now knowing B isn't, again are 1/3, but C
has a 2/3 chance of being pardoned.
The ultimatum game is a game in economic experiments. The first player (the proposer) receives a sum
of money and proposes how to divide the sum between himself and another player. The second player
(the responder) chooses to either accept or reject this proposal. If the second player accepts, the money
is split according to the proposal. If the second player rejects, neither player receives any money. The
game is typically played only once so that reciprocation is not an issue.
Two people, Alice and Bob, play the game. An experimenter puts 100 one dollar bills on a table in front of
them. Alice may divide the money between herself and Bob however she chooses. Bob then decides
whether to accept her division, in which case each keeps the money as Alice divided it, or to reject the
division, in which case neither receives any money.
For example, Alice divides the money into one stack worth 65 dollars and one worth 35 dollars. She offers
the smaller amount to Bob. If he accepts, he keeps 35 dollars and Alice keeps 65 dollars. If Bob rejects the
division, neither he nor Alice receive anything.
If Bob acts rationally according to Rational choice theory, he should accept any division in which Alice
offers him at least one dollar, since doing so leaves him with more money than he would have had
otherwise. Even a division which gives Alice 100 dollars and Bob zero costs Bob nothing, so he has no
purely rational reason to reject it. If Alice knows that Bob will act rationally, and if she acts rationally
herself, then she should offer Bob one dollar and keep 99 for herself. In practice, divisions which Bob
regards as unfair are generally rejected.
Blotto games (or Colonel Blotto games, or "Divide a Dollar" games) constitute a class of two-person
zero-sum games in which the players are tasked to simultaneously distribute limited resources over
several objects (or battlefields). In the classic version of the game, the player devoting the most
resources to a battlefield wins that battlefield, and the gain (or payoff) is then equal to the total
number of battlefields won.
The Colonel Blotto game was first proposed and solved by Emile Borel in 1921, as an example of a
game in which "the psychology of the players matters". It was studied after the Second World War by
scholars in Operation Research, and became a classic in Game Theory.
The game is named after the fictional Colonel Blotto from Gross and Wagner's 1950 paper. The
Colonel was tasked with finding the optimum distribution of his soldiers over N battlefields knowing
that:
1.on each battlefield the party that has allocated the most soldiers will win, but
2.both parties do not know how many soldiers the opposing party will allocate to each battlefield,
and:
3.both parties seek to maximize the number of battlefields they expect to win.
As an example Blotto game, consider the game in which two players each write down
three positive integers in non-decreasing order and such that they add up to a
pre-specified number S.
Subsequently, the two players show each other their writings, and compare
corresponding numbers. The player who has two numbers higher than the
corresponding ones of the opponent wins the game.
For S = 6 only three choices of numbers are possible: (2, 2, 2), (1, 2, 3) and (1, 1, 4).
It is easy to see that:
Any triplet against itself is a draw
(1, 1, 4) against (1, 2, 3) is a draw
(1, 2, 3) against (2, 2, 2) is a draw
(2, 2, 2) beats (1, 1, 4)
It follows that the optimum strategy is (2, 2, 2) as it does not do worse than breaking
even against any other strategy while beating one other strategy.
There are however several Nash equilibria. If both players choose the strategy
(2, 2, 2) or (1, 2, 3), then none of them can beat the other one by changing strategies,
so every such strategy pair is a Nash equilibrium.
Day Four
Matching Problems
Matching problems can include:
Matching men and women – when there are the same
number of each
Matching volunteers with jobs at a “work day”
Matching professional teams and “draft picks”
Matching medical schools and applicants
A stable matching system is not necessarily a
system under which everyone is satisfied. A
matching system is stable when no unmatched pair
will find it beneficial to deviate from the matching and
form their own match.
For any preference structure, there is at least one
stable matching system.
The method used: Gale-Shapley algorithm
Gale-Shapley Algorithm
First Stage: Every “chooser” turns to the first on their list and issues an
“invitation”. Everyone who receives more than one invitation selects their
favorite of the “choosers” and tells the others that they will never be a choice.
Second Stage: Every “chooser” who has been rejected selects the second on
their list and makes an invitation. Everyone who receives more than one
invitation (including invitations from the previous stage) chooses their favorite
and rejects the others.
Third Stage: Every “chooser” who has been rejected now goes to the next on
their list and is screened through the same process.
The procedure continues until no “chooser” is rejected – everyone has a
match.
Consider the four teams: Atlanta (Hawks), Bulls, Celtics, and Detroit (Pistons)
who are trying to draft: Julius Erving, Kareem Abdul-Jabbar, Larry Bird, and
Michael Jordan. The following charts show the preferences of the teams and
the players.
Atlanta
Bulls
Celtics
Detroit
Julius
Kareem
Larry
Michael
1
Julius
Julius
Kareem
Michael
1
Detroit
Bulls
Detroit
Celtics
2
Kareem
Michael
Julius
Kareem
2
Celtics
Detroit
Atlanta
Bulls
3
Larry
Larry
Larry
Larry
3
Atlanta
Atlanta
Bulls
Atlanta
4
Michael
Kareem
Michael
Julius
4
Bulls
Celtics
Celtics
Detroit
First Stage: Every “chooser” turns to the first on their list and issues an
“invitation”. Everyone who receives more than one invitation selects their
favorite of the “choosers” and tells the others that they will never be a
choice.
Julius
Kareem
Larry
Michael
First Stage: Every “chooser” turns to the first on their list and issues an
“invitation”. Everyone who receives more than one invitation selects their
favorite of the “choosers” and tells the others that they will never be a
choice.
Julius
Kareem
Atlanta
Bulls
Celtics
Larry
Michael
Detroit
First Stage: Every “chooser” turns to the first on their list and issues an
“invitation”. Everyone who receives more than one invitation selects their
favorite of the “choosers” and tells the others that they will never be a
choice.
Julius
Kareem
Atlanta
Bulls
Celtics
Larry
Michael
Detroit
Julius chooses Atlanta over the Bulls.
Julius
Kareem
Atlanta
Bulls
Celtics
Larry
Michael
Detroit
Second Stage: Every “chooser” who has been rejected selects the second on their list
and makes an invitation. Everyone who receives more than one invitation (including
invitations from the previous stage) chooses their favorite and rejects the others.
Julius
Kareem
Atlanta
Bulls
Celtics
Larry
Michael
Detroit
Second Stage: Every “chooser” who has been rejected selects the second on their list
and makes an invitation. Everyone who receives more than one invitation (including
invitations from the previous stage) chooses their favorite and rejects the others.
Julius
Kareem
Larry
Michael
Atlanta
Bulls
Celtics
Detroit
Atlanta
Bulls
Celtics
Detroit
Bulls
Second Stage: Every “chooser” who has been rejected selects the second on their list
and makes an invitation. Everyone who receives more than one invitation (including
invitations from the previous stage) chooses their favorite and rejects the others.
Julius
Kareem
Larry
Michael
Atlanta
Bulls
Celtics
Detroit
Atlanta
Bulls
Celtics
Detroit
Bulls
Michael chooses the Bulls over Detroit.
Julius
Kareem
Larry
Michael
Atlanta
Bulls
Celtics
Detroit
Atlanta
Bulls
Celtics
Detroit
Bulls
Detroit makes its second choice, since Michael chose the Bulls over that team.
Julius
Kareem
Larry
Michael
Atlanta
Bulls
Celtics
Detroit
Atlanta
Bulls
Celtics
Detroit
Bulls
Atlanta
Bulls
Celtics
Detroit
Detroit
Bulls
Kareem chooses Detroit over the Celtics.
Julius
Kareem
Larry
Michael
Atlanta
Bulls
Celtics
Detroit
Atlanta
Bulls
Celtics
Detroit
Bulls
Atlanta
Bulls
Celtics
Detroit
Detroit
Bulls
Now the Celtics make their second choice.
Julius
Kareem
Larry
Michael
Atlanta
Bulls
Celtics
Detroit
Atlanta
Bulls
Celtics
Detroit
Bulls
Atlanta
Bulls
Celtics
Detroit
Detroit
Bulls
Atlanta
Bulls
Celtics
Celtics
Detroit
Detroit
Bulls
Julius chooses the Celtics over Atlanta.
Julius
Kareem
Larry
Michael
Atlanta
Bulls
Celtics
Detroit
Atlanta
Bulls
Celtics
Detroit
Bulls
Atlanta
Bulls
Celtics
Detroit
Detroit
Bulls
Atlanta
Bulls
Celtics
Celtics
Detroit
Detroit
Bulls
So now Atlanta needs to make their second choice.
Julius
Kareem
Larry
Michael
Atlanta
Bulls
Celtics
Detroit
Atlanta
Bulls
Celtics
Detroit
Bulls
Atlanta
Bulls
Celtics
Detroit
Detroit
Bulls
Atlanta
Bulls
Celtics
Celtics
Detroit
Detroit
Bulls
Atlanta
Bulls
Celtics
Celtics
Detroit
Atlanta
Detroit
Bulls
Kareem chooses Detroit over Atlanta – stays with Detroit.
Julius
Kareem
Larry
Michael
Atlanta
Bulls
Celtics
Detroit
Atlanta
Bulls
Celtics
Detroit
Bulls
Atlanta
Bulls
Celtics
Detroit
Detroit
Bulls
Atlanta
Bulls
Celtics
Celtics
Detroit
Detroit
Bulls
Atlanta
Bulls
Celtics
Celtics
Detroit
Atlanta
Detroit
Bulls
So Atlanta makes their third choice, Larry.
Julius
Kareem
Larry
Atlanta
Bulls
Celtics
Detroit
Atlanta
Bulls
Celtics
Detroit
Bulls
Atlanta
Bulls
Celtics
Detroit
Detroit
Bulls
Atlanta
Bulls
Celtics
Celtics
Detroit
Detroit
Bulls
Atlanta
Bulls
Celtics
Celtics
Detroit
Atlanta
Detroit
Bulls
Celtics
Detroit
Atlanta
Michael
Bulls
The final matching:
Julius
Kareem
Celtics Detroit
Larry
Michael
Atlanta Bulls
A stable matching system is not necessarily a system under
which everyone is satisfied.
Julius
Kareem
Larry
Michael
(Celtics choice 2)
(Detroit choice 2)
(Atlanta choice 2)
(Bulls choice 2)
Celtics
Detroit
Atlanta
Bulls
(Julius choice 2)
(Kareem choice 2)
(Larry choice 3)
(Michael choice 2)
Medical School Admissions Problem
In many countries a large number of applicants seek
admission to a small number of medical schools. The
competition for this limited number of places poses
several problems. Because the number of
candidates is greater than the medical school
admissions quotas, many candidates apply to
several medical schools.
Medical School Admissions Problem
The number of applications submitted to the
admissions office of each school often exceeds its
quota. The offices must evaluate all applicants and
decide which ones to accept and which to reject.
They will then offer to some applicants if there are
vacancies (first acceptances don’t commit) and
refuse admission to other (less-qualified) even if
there are vacancies.
Medical School Admissions Problem
One cannot assume that all who are offered admission
won’t accept an offer from what they consider to be better
offers from other schools. The admission quota of a school
who gets rejections from first offers won’t meet their quota.
The medical schools, to fill their quotas, may offer
admission to a number of applicants greater than their
quota. But, if enough students don’t decline, the number of
acceptances will exceed the quota.
Proposed solution to medical school problem:
Create an independent “placement center” where data is
collected and assignments made in the following manner:
1) Each applicant submits a list of schools he is willing to
accept.
2) Each applicant ranks the schools in order of preference.
3) The placement center sends each medical school a list of all
who have applied to that school.
Proposed solution to medical school problem:
4) Each school announces its quota.
5) Each medical school submits to the placement center a list
of the applicants ranked in order of preference and lists those
applicants it will not accept, even if there are vacancies.
Proposed solution to medical school problem:
6) The placement center implements the Gale-Shapley algorithm by:
a. All applicants are referred to the medical school of their first choice
according to the new preference order. If the number of candidates
is greater than the quota, the applicants are selected according to
the school’s preference order until the quota is filled. Any applicants
left over are “rejected”.
b. Rejected applicants are then referred to their next-choice medical
school. They are added to the list from the previous stage. If this
creates a list at the medical school that exceeds the quota, then the
school’s preference order is used and those exceeding the quota
rejected.
c. The process is repeated until the quota at each school is filled. At
this point, all the medical schools admit everyone on their current
list.
Game Theory in the
Movies and on TV
Rebel Without a Cause: the two characters race their cars toward an
abyss. The first one to jump out is the loser (and the chicken).
Ironically, in this movie, the loser lost even more when his leather
jacket got hung up on the car door and he went over the cliff with the
car.
Princess Bride
The story centers on Buttercup, a former farm girl who has
been chosen as the princess bride to Prince Humperdinck of
Florian. Buttercup does not love him, she who still laments
the death of her one true love, Westley, five years ago.
Westley was a hired hand on the farm, his stock answer of
"as you wish" to any request she made of him which she
came to understand was his way of saying that he loved her.
But Westley went away to sea, only to be killed by the Dread
Pirate Roberts. One scene humorously highlights both
strategic manipulation of the rules of the game and the
unrealistic assumption of ”common knowledge”.
An item of information in a game is common
knowledge if all of the players know it and all
of the players know that all other players
know it and all other players know that all
other players know that all other players know
it, and so on. This is much more than simply
saying that something is known by all, but
also implies that the fact that it is known is
also known by all, etc.
Common Knowledge
Consider a simple example of two allied armies situated on
opposite hilltops waiting to attack their foe. Neither commander
will attack unless he is sure that the other will attack at exactly
the same time. The first commander sends a messenger to the
other hilltop with the message "I plan to attack in the morning."
The messenger's journey is perilous and he may die on the way
to delivering the message. If he gets to the other hilltop and
informs the other commander - can we be certain that both will
attack in the morning? Note that both commanders now know the
message, but the first cannot be sure that the second got the
message. Thus, common knowledge implies not only that both
know some piece of information, but can also be absolutely
confident that the rest no it, and that the rest know that we know
it, and so on.
In the Princess Bride
Our hero Westley, in the guise of the Dread Pirate
Roberts, confronts his foe-for-the-moment, the Sicilian,
Vizzini. Westley challenges him to a Battle of Wits. Two
glasses are placed on the table, each containing wine
and one purportedly containing poison. The challenge,
simply, is to select the glass that does not lead to
immediate death.
Roberts: “All right: where is the poison? The battle of wits has begun. It ends
when you decide and we both drink, and find out who is right and who is
dead.”
Vizzini: “But it's so simple. All I have to do is divine from what I know of you.
Are you the sort of man who would put the poison into his own goblet, or his
enemy's? Now, a clever man would put the poison into his own goblet,
because he would know that only a great fool would reach for what he was
given. I'm not a great fool, so I can clearly not choose the wine in front of you.
But you must have known I was not a great fool; you would have counted on
it, so I can clearly not choose the wine in front of me.”
Roberts: “You've made your decision then?”
Vizzini: “Not remotely. Because iocane comes from Australia, as everyone
knows. And Australia is entirely peopled with criminals. And criminals are used
to having people not trust them, as you are not trusted by me. So I can clearly
not choose the wine in front of you.”
Roberts: “Truly, you have a dizzying intellect.”
The scene, beyond providing some comic relief on the theme of common
knowledge, also has an important lesson on strategic moves; if the rules of the
game may be changed, then the game can be rigged to one player's advantage:
Vizzini: “Let's drink -- me from my glass, and you from yours.”
[allowing Roberts to drink first, he swallows his wine]
Roberts: “You guessed wrong.”
Vizzini (roaring with laughter): “You only think I guessed wrong -- that's what's so
funny! I switched glasses when your back was turned. You fool. You fell victim to
one of the classic blunders. The most famous is ‘Never get involved in a land war
in Asia.’ But only slightly less well known is this: ‘Never go in against a Sicilian
when death is on the line.’”
[He laughs and roars and cackles and whoops until he falls over dead.]
[Roberts begins to rescue Buttercup, the girl over whom this battle was staged in
the first place]
Buttercup: “To think -- all that time it was your cup that was
poisoned.”
Roberts: “They were both poisoned. I spent the last few years
building up an immunity to iocane powder.”
The movie contains several other scenes with game-theoretic themes,
including many on bluffing.
The Good, the Bad and the Ugly
In the final scene: a game set up so that it cannot be lost, and can we really
trust Clint Eastwood?
The final scene in this Clint Eastwood movie is an example of game
theory. Three men in a triangle -- each with a gun, a rock at the center
of the three. It is up to each man to evaluate his situation. All are
excellent shots. Who do they shoot?
Clint has supposedly put a message on a rock that holds the key to
everything, but do the other two trust Clint to have actually written the
correct answer? As the other two evaluate the situation, they realize
they can't trust Clint to have written the answer on the rock -- therefore
they can't shoot Clint who likely still has the answer. That means the
other two can only shoot each other, but only one will likely hit before
the other.
What they don't know is that Clint has given one an unloaded gun...
Clint can ignore this one. The one Clint has to worry about with the
loaded gun will try to kill the one with the unloaded gun. Neither will fire
at Clint. Clint will fire at the one with the loaded gun. As the camera
passes from one face to the other the audience is meant to figure out
what each would do.
The guy with the loaded gun shoots at the guy with the unloaded gun -Clint shoots the guy with the loaded gun. Game over. As with the
hangings in the movie, he has dangled Duco out as bait while Clint
takes the money.
The game is decided before it starts.
Clint sets up a situation where each evaluates their possible moves,
but in reality, Clint has already won the game. Its a brilliant example of
people making the best decisions based on the information available to
them...and somebody manipulating the information available to them.
Butch Cassidy and the Sundance Kid
The leaders of the Hole-in-the-Wall Gang, Butch and Sundance, are fond of robbing
trains with a particular fancy for a rail line run by Mr. E. H. Harriman. They learn that
Harriman, having had enough, hired a group to pursue the robbers. "He resents the
way you've been picking on him so he's outfitted a special train and hired special
employees," they are told by a friend who also suggests that the gesture is quite
flattering.
After eluding the gang, Butch recognizes how the outcome is Pareto dominated,
though using a different vocabulary:
"A set-up like that costs more than we ever took...That crazy Harriman. That's
bad business. How long do you think I'd stay in operation if every time I pulled
a job, it cost me money? If he'd just pay me what he's spending to make me
stop robbin' him, I'd stop robbin' him."
[Screaming out the door at E. H. Harriman.] "Probably inherited every penny
you got!"
"Those inherited guys - what the hell do they know?"
The Pareto principle (also known as the 80–20 rule, the law of the vital
few, and the principle of factor sparsity) states that, for many events,
roughly 80% of the effects come from 20% of the causes.
Essentially, Pareto first showed that approximately 80% of the land in
Italy was owned by 20% of the population; Pareto developed the
principle by observing that 20% of the peapods in his garden
contained 80% of the peas. And it is a common rule of thumb in
business; e.g., "80% of your sales come from 20% of your clients."
Ransom
Irrationality improves Mel Gibson's bargaining power as he turns the tables on
the kidnappers of his son.
Movie scene: "Instead, I am offering this money as a reward on your head."
Ransom
You only want to pay the ransom if you trust the kidnappers are trustworthy to spare the loved one.
A 2-person Kidnapping is a classic example of a finite sequential game with imperfect information.
Let’s assume the following:
1. You have seen the kidnapper’s face, or the kidnapper has other reasons to believe that murdering you
poses less risk of capture than releasing you.
2. You have the money to pay the kidnapper, and will be able to transfer payment to the kidnapper without
any issue.
3. Your public Image is important (known to both players), and/or you have some skeletons in the closet
you’d rather stay hidden (known generally by both players).
Don Eppes is an FBI agent of the all I need to know I learned on the streets ilk. His
brother, Charlie, is a brilliant, reclusive university professor brother. Charlie helps
solve crimes by relating the crime to seemingly complicated (though generally
nonsensical) mathematical theories. Two episodes of Numb3rs involved game
theoretic themes directly.
One of the episodes is "Assassin" (Season 2, Episode 5)
Charlie and Don are in pursuit of an assassin, attempting to calculate where the killer
would strike.
Charlie: “Hide-and-seek.”
Don: “What are you talking about, like the kids' version?”
Charlie: “A mathematical approach to it, yes. See, the assassin must hide in order to
accomplish his goal, we must seek and find the assassin before he achieves that
goal.”
Megan: “Ah, behavioral game theory, yeah, we studied this at Quantico.”
Charlie: “I doubt you studied it the way that Rubinstein, Tversky and Heller studied
two-person constant-sum hide-and-seek with unique mixed strategy equilibria.”
Megan: “No, not quite that way.”
Don: “Just bear with him.”
Simpsons
Rock-Paper-Scissors is often employed on The
Simpsons to settle disputes. In one game (episode
9F16), Bart thinks "Good ol' rock. Nuthin' beats that!"
which Lisa of course predicts, highlighting the
importance of unpredictability in mixed strategies.
In episode 7F21, Bart and two friends are deciding how to share among them a single copy of
Radioactive Man #1, a comic book of great value to all three:
Martin: How about this, guys? Bart can have it Mondays and Thursdays, Milhouse will get it
Tuesdays and Fridays, and yours truly will take it Wednesdays and Saturdays.
Bart: Perfect!
Milhouse: Wait a minute! What about Sundays?
Bart: [suspiciously] Yeah, what about Sundays?
Martin: Well, Sunday possession will be determined by a random number generator. I will take the
digits 1 through 3, Milhouse will have 4 through 6, and Bart will have 7 through 9.
Bart: Perfect!
Milhouse: Wait a minute! What about 0?
Bart: [suspiciously] Yeah, what about 0?
Milhouse: Yeah!
Martin: Well, in the unlikely event of a 0, possession will be determined by Rock Scissors Paper
competition, best 3 out of 5. How's that?
Bart: Oh, okay.
Milhouse: Yeah, all right.
Big Bang Theory
"The Dumpling Paradox" is the seventh episode of the
first season. When the gang orders Chinese, their order of
dumpling appetizer comes with four dumplings which is
non-divisible by the three of them.
"The Lizard-Spock Expansion" is the eighth episode of the second
season. This episode first aired on Monday, November 17, 2008.
Social Justice
Bankruptcy
Analysis of a Bankruptcy Problem from the Talmud
Many times one encounters a bankruptcy situation where there are claims
against a given estate, and the sum of the claims against the estate exceeds its
worth. In such situations, one would like to know what would be a “fair” way of
dividing the estate among the claimants.
What seems fair in one case may seem less so in another.
Analysis of a Bankruptcy Problem from the Talmud
The following is an interesting method of division that has its origin in the Babylonian Talmud. It
involves a man who married three women and promised them in their marriage contract the
sums of 100, 200 and 300 units of money to be given them upon his death. The man died, but
his estate amounted to less than 600 units. The recommendation in the Mishna (one portion of
the Talmud) was the following:
Estate Worth →
100
200
300
100
33 1/3
50
50
200
33 1/3
75
100
300
33 1/3
75
150
Claims ↓
For many years, this arrangement was not understood. Some
thought this division reflected special circumstances whose
description was neglected. No solid explanation was found
until quite recently. Two game theorists (Aumann and
Maschler) examined the rule. They found that one solution
concept, called the nucleolus, gave precisely the numbers in
the table. The problem was, though, that the nucleolus was not
discovered until 1969. Instead, a consistency axiom gives a
hint as to how this was created and shows how more problems
with creditors and various claims can be resolved.
Consider the case where a garment is worth 100 units
of money.
One claims that his share is 50 units.
The other claims that his share is 80 units.
The claimant asking for 50 units of money declares, in
effect, that he has no claim to the second 50 units and,
as far as he is concerned, the other claimant can have
them.
The claimant to 80 units declares that he has no claim to the
remaining 20 units and, as far as he is concerned, the first
claimant can have them.
Value of garment
The two claims
100
80
Says other can
have 20
Uncontested division
Gets 50
50
Says other can
have 50
Gets 20
That leaves 30 remaining (100 – 50 – 20 = 30)
Value of garment
The two claims
100
80
Says other can
have 20
Uncontested division
50
Says other can
have 50
Gets 50
Gets 20
That leaves 30 remaining (100 – 50 – 20 = 30)
Divide the remaining 30 equally: 15 and 15
Add to uncontested.
Value of garment
The two claims
100
80
Says other can
have 20
Uncontested division
Gets 50
50
Says other can
have 50
Gets 20
Divide the remaining 30 equally: 15 and 15
Add to uncontested.
50
20
+ 15
+ 15
65
35
Analysis of a Bankruptcy Problem from the Talmud
The three women were promised 100, 200 and 300 units of money. The recommendation in the
Mishna (one portion of the Talmud) was the following. The first column (100) is an equal
division. The third column (300) is in proportion to the promise. The middle column follows the
nucleolus.
Estate Worth →
100
200
300
100
33 1/3
50
50
200
33 1/3
75
100
300
33 1/3
75
150
Claims ↓
Aumann and Maschler show that there is in fact only one division that is
consistent. The solution can be described by the following seven-step algorithm:
1. Order the creditors from lowest to highest claims.
2. Divide the estate equally among all parties until the lowest creditor receives
one-half of the claim.
3. Divide the estate equally among all parties except the lowest creditor until the
next lowest creditor receives one-half of the claim.
4. Proceed until each creditor has reached one-half of the original claim.
5. Now, work in reverse. Start giving the highest-claim money from the estate
until the loss, the difference between the claim and the award, equals the loss
for the next highest creditor.
6. Then divide the estate equally among the highest creditors until the loss of the
highest creditors equals the loss of the next highest.
7. Continue until all money has been awarded.
Analysis of a Bankruptcy Problem from the Talmud
Estate Worth →
100
200
300
100
33 1/3
50
50
200
33 1/3
75
100
300
33 1/3
75
150
Claims ↓
Value of estate
200
The claims
100
Lowest gets
one-half of
claim
50
200
300
Divide the remaining between the other two: 200 – 50 = 150
50
Can stop. No more creditors
75
75
Social Justice - Voting Paradox
A certain amount of municipal budget is unspent, and the city council must decide how to
invest it. There are three options: investment in education, investment in security, investment
in health. On the council are representatives of three parties:
Left party:
3 members
Center party:
4 members
Right party:
5 members
The parties’ list of preferences (in order from top to bottom):
Center (4)
health
security
education
Left (3)
education
health
security
Right (5)
security
education
health
No issue has a majority if everyone votes for their favorite.
If there were votes in “pairs” - choose two of the three items to vote on:
security vs. education: vote is 9 to 3
(center and right put security over education)
health vs. security:
vote is 7 to 5
(center and left put health over security)
education vs. health:
vote is 8 to 4
(left and right put education over health)
Center (4)
health
security
education
Left (3)
education
health
security
There is a cyclic preference relation:
security
education
Right (5)
security
education
health
health
security
This paradox was first noted in 1785 by mathematician and philosopher Marquis de
Condorcet. A decision rule in the form of a social choice function determines the
preference order of the society with regard to all the alternatives under discussion.
The task is not just to choose the most-preferred from
among the alternatives, but also to determine the
preference order of the society with regard to all the
alternatives under discussion.
There is not always a decisive answer. There is no social
choice function that satisfies all the constraints and goals.
Cooperative Game
Buyer-Seller Exchange Game
Player A has a house he is willing to sell, and Player B is interested in
buying the house. A deal can be made only if both parties “profit” (the
selling price is higher than the lowest the seller will take and the selling
price is lower than the highest the buyer will pay).
Suppose a house is worth $100,000. The seller will not sell it for lower
than this amount but hopes to get a higher payment.
Also, suppose the worth of the house to the buyer is $150,000. He will
not pay more than this but hopes to pay less.
A deal can be made if the house is sold for $120,000. The seller will
“profit” $20,000, and the buyer will “profit” $30,000.
The Traveler’s Dilemma (Nash equilibrium)
An airline loses two suitcases belonging to two different travelers. Both
suitcases happen to be identical and contain identical items. An airline
manager tasked to settle the claims of both travelers explains that the airline is
liable for a maximum of $100 per suitcase (he is unable to find out directly the
price of the items), and in order to determine an honest appraised value of the
suitcases the manager separates both travelers so they can't confer, and asks
them to write down the amount of their value at no less than $2 and no larger
than $100. He also tells them that if both write down the same number, he will
treat that number as the true dollar value of both suitcases and reimburse both
travelers that amount. However, if one writes down a smaller number than the
other, this smaller number will be taken as the true dollar value, and both
travelers will receive that amount along with a bonus/plus: $2 extra will be paid
to the traveler who wrote down the lower value and a $2 deduction will be taken
from the person who wrote down the higher amount. The challenge is: what
strategy should both travelers follow to decide the value they should write
down?
Application of Dilemmas (Nash equilibrium)
Write down the amount of their value at no less than $2 and no larger than
$100.
If both write down the same number, he will treat that number as the true dollar
value of both suitcases and reimburse both travelers that amount.
If one writes down a smaller number than the other, this smaller number will be
taken as the true dollar value, and both travelers will receive that amount along
with a bonus/plus: $2 extra will be paid to the traveler who wrote down the
lower value and a $2 deduction will be taken from the person who wrote down
the higher amount.
One might expect a traveler's optimum choice to be $100; that is, the traveler
values the suitcase at the airline manager's maximum allowed price.
Remarkably, and, to many, counter-intuitively, the traveler's optimum choice is
in fact $2; that is, the traveler values the suitcase at the airline manager's
minimum allowed price.
List of Games