1.
Lesson 34 (KH, Section 11.4)
Introduction to Game Theory
Math 20
December 12, 2007
Announcements
Pset 12 due December 17 (last day of class)
next OH today 1–3 (SC 323)
2.
Outline
Games and payoﬀs
Matching dice
Vaccination
The theorem of the day
Strictly determined games
Example: Network programming
Characteristics of an Equlibrium
Two-by-two strictly-determined games
Two-by-two non-strictly-determined games
Calculation
Example: Vaccination
Other
3.
A Game of Chance
You and I each have a
six-sided die
We roll and the loser
pays the winner the
diﬀerence in the numbers
shown
If we play this a number
of times, who’s going to
win?
4.
The Payoﬀ Matrix
Lists each player’s
outcomes versus C ’s outcomes
the other’s 1 2 3 4 5 6
1 0 -1 -2 -3 -4 -5
Each aij represents
R’s outcomes
2 1 0 -1 -2 -3 -4
the payoﬀ from C
3 2 1 0 -1 -2 -3
to R if outcomes i
4 3 2 1 0 -1 -2
for R and j for C
5 4 3 2 1 0 -1
occur (a zero-sum
6 5 4 3 2 1 0
game).
5.
Expected Value
Let the probabilities of R’s outcomes and C ’s outcomes be
given by probability vectors
q1
q2
p = p1 p2 · · · pn q=.
..
qn
6.
Expected Value
Let the probabilities of R’s outcomes and C ’s outcomes be
given by probability vectors
q1
q2
p = p1 p2 · · · pn q=.
..
qn
The probability of R having outcome i and C having outcome
j is therefore pi qj .
7.
Expected Value
Let the probabilities of R’s outcomes and C ’s outcomes be
given by probability vectors
q1
q2
p = p1 p2 · · · pn q=.
..
qn
The probability of R having outcome i and C having outcome
j is therefore pi qj .
The expected value of R’s payoﬀ is
n
E (p, q) = pi aij qj = pAq
i,j=1
8.
Expected Value
Let the probabilities of R’s outcomes and C ’s outcomes be
given by probability vectors
q1
q2
p = p1 p2 · · · pn q=.
..
qn
The probability of R having outcome i and C having outcome
j is therefore pi qj .
The expected value of R’s payoﬀ is
n
E (p, q) = pi aij qj = pAq
i,j=1
A “fair game” if the dice are fair.
11.
Strategies
What if we could
choose a die to be
as biased as we C ’s outcomes
wanted? 1 2 3 4 5 6
1 0 -1 -2 -3 -4 -5
In other words,
R’s outcomes
2 1 0 -1 -2 -3 -4
what if we could
3 2 1 0 -1 -2 -3
choose a strategy
4 3 2 1 0 -1 -2
p for this game?
5 4 3 2 1 0 -1
Clearly, we’d want 6 5 4 3 2 1 0
to get a 6 all the
time!
12.
Flu Vaccination
Suppose there are two ﬂu
strains, and we have two
ﬂu vaccines to combat
them.
We don’t know
distribution of strains Strain
Neither pure strategy is 1 2
Vacc
the clear favorite 1 0.85 0.70
Is there a combination of 2 0.60 0.90
vaccines (a mixed
strategy) that
maximizes total
immunity of the
population?
13.
Outline
Games and payoﬀs
Matching dice
Vaccination
The theorem of the day
Strictly determined games
Example: Network programming
Characteristics of an Equlibrium
Two-by-two strictly-determined games
Two-by-two non-strictly-determined games
Calculation
Example: Vaccination
Other
14.
Theorem (Fundamental Theorem of Zero-Sum Games)
There exist optimal strategies p∗ for R and q∗ for C such that for
all strategies p and q:
E (p∗ , q) ≥ E (p∗ , q∗ ) ≥ E (p, q∗ )
15.
Theorem (Fundamental Theorem of Zero-Sum Games)
There exist optimal strategies p∗ for R and q∗ for C such that for
all strategies p and q:
E (p∗ , q) ≥ E (p∗ , q∗ ) ≥ E (p, q∗ )
E (p∗ , q∗ ) is called the value v of the game.
16.
Reﬂect on the inequality
E (p∗ , q) ≥ E (p∗ , q∗ ) ≥ E (p, q∗ )
In other words,
E (p∗ , q) ≥ E (p∗ , q∗ ): R can guarantee a lower bound on
his/her payoﬀ
E (p∗ , q∗ ) ≥ E (p, q∗ ): C can guarantee an upper bound on
how much he/she loses
This value could be negative in which case C has the
advantage
17.
Fundamental problem of zero-sum games
Find the p∗ and q∗ !
The general case we’ll look at next time (hard-ish)
There are some games in which we can ﬁnd optimal strategies
now:
Strictly-determined games
2 × 2 non-strictly-determined games
18.
Outline
Games and payoﬀs
Matching dice
Vaccination
The theorem of the day
Strictly determined games
Example: Network programming
Characteristics of an Equlibrium
Two-by-two strictly-determined games
Two-by-two non-strictly-determined games
Calculation
Example: Vaccination
Other
19.
Example: Network programming
Suppose we have two
networks, NBC and CBS
Each chooses which
program to show in a
certain time slot
Viewer share varies
depending on these
combinations
How can NBC get the
most viewers?
20.
The payoﬀ matrix and strategies
CBS
es
r
ut
ea
CS r
ivo
in
D
M
rv
s,
I
Ye
Su
60
My Name is Earl 60 20 30 55
NBC
Dateline 50 75 45 60
Law & Order 70 45 35 30
21.
The payoﬀ matrix and strategies
CBS
es
r
ut
ea
CS r
ivo
in
D
M
rv
s,
I
Ye
Su
60
My Name is Earl 60 20 30 55
NBC
Dateline 50 75 45 60
Law & Order 70 45 35 30
What is NBC’s strategy?
22.
The payoﬀ matrix and strategies
CBS
es
r
ut
ea
CS r
ivo
in
D
M
rv
s,
I
Ye
Su
60
My Name is Earl 60 20 30 55
NBC
Dateline 50 75 45 60
Law & Order 70 45 35 30
What is NBC’s strategy?
NBC wants to maximize NBC’s minimum share
23.
The payoﬀ matrix and strategies
CBS
es
r
ut
ea
CS r
ivo
in
D
M
rv
s,
I
Ye
Su
60
My Name is Earl 60 20 30 55
NBC
Dateline 50 75 45 60
Law & Order 70 45 35 30
What is NBC’s strategy?
NBC wants to maximize NBC’s minimum share
In airing Dateline, NBC’s share is at least 45
24.
The payoﬀ matrix and strategies
CBS
es
r
ut
ea
CS r
ivo
in
D
M
rv
s,
I
Ye
Su
60
My Name is Earl 60 20 30 55
NBC
Dateline 50 75 45 60
Law & Order 70 45 35 30
What is NBC’s strategy?
NBC wants to maximize NBC’s minimum share
In airing Dateline, NBC’s share is at least 45
This is a good strategy for NBC
25.
The payoﬀ matrix and strategies
CBS
es
r
ut
ea
CS r
ivo
in
D
M
rv
s,
I
Ye
Su
60
My Name is Earl 60 20 30 55
NBC
Dateline 50 75 45 60
Law & Order 70 45 35 30
What is CBS’s strategy?
26.
The payoﬀ matrix and strategies
CBS
es
r
ut
ea
CS r
ivo
in
D
M
rv
s,
I
Ye
Su
60
My Name is Earl 60 20 30 55
NBC
Dateline 50 75 45 60
Law & Order 70 45 35 30
What is CBS’s strategy?
CBS wants to minimize NBC’s maximum share
27.
The payoﬀ matrix and strategies
CBS
es
r
ut
ea
CS r
ivo
in
D
M
rv
s,
I
Ye
Su
60
My Name is Earl 60 20 30 55
NBC
Dateline 50 75 45 60
Law & Order 70 45 35 30
What is CBS’s strategy?
CBS wants to minimize NBC’s maximum share
In airing CSI, CBS keeps NBC’s share no bigger than 45
28.
The payoﬀ matrix and strategies
CBS
es
r
ut
ea
CS r
ivo
in
D
M
rv
s,
I
Ye
Su
60
My Name is Earl 60 20 30 55
NBC
Dateline 50 75 45 60
Law & Order 70 45 35 30
What is CBS’s strategy?
CBS wants to minimize NBC’s maximum share
In airing CSI, CBS keeps NBC’s share no bigger than 45
This is a good strategy for CBS
29.
The payoﬀ matrix and strategies
CBS
es
r
ut
ea
CS r
ivo
in
D
M
rv
s,
I
Ye
Su
60
My Name is Earl 60 20 30 55
NBC
Dateline 50 75 45 60
Law & Order 70 45 35 30
Equilibrium
30.
The payoﬀ matrix and strategies
CBS
es
r
ut
ea
CS r
ivo
in
D
M
rv
s,
I
Ye
Su
60
My Name is Earl 60 20 30 55
NBC
Dateline 50 75 45 60
Law & Order 70 45 35 30
Equilibrium
(Dateline,CSI) is an equilibrium pair of strategies
31.
The payoﬀ matrix and strategies
CBS
es
r
ut
ea
CS r
ivo
in
D
M
rv
s,
I
Ye
Su
60
My Name is Earl 60 20 30 55
NBC
Dateline 50 75 45 60
Law & Order 70 45 35 30
Equilibrium
(Dateline,CSI) is an equilibrium pair of strategies
Assuming NBC airs Dateline, CBS’s best choice is to air CSI,
and vice versa
32.
Characteristics of an Equlibrium
Let A be a payoﬀ matrix. A saddle point is an entry ars
which is the minimum entry in its row and the maximum
entry in its column.
A game whose payoﬀ matrix has a saddle point is called
strictly determined
Payoﬀ matrices can have multiple saddle points
33.
Pure Strategies are optimal in Strictly-Determined Games
Theorem
Let A be a payoﬀ matrix. If ars is a saddle point, then er is an
optimal strategy for R and es is an optimal strategy for C.
34.
Pure Strategies are optimal in Strictly-Determined Games
Theorem
Let A be a payoﬀ matrix. If ars is a saddle point, then er is an
optimal strategy for R and es is an optimal strategy for C.
Proof.
If q is a strategy for C, then
n n
E (er , q) = er Aq = arj qj ≥ ars qj = ars = E (er , es )
j=1 j=1
If p is a strategy for R, then
m m
E (er , es ) = pAes = pi ais ≤ pi ars = E (er , es )
i=1 i=1
So for any p and q, we have
E (er , q) ≥ E (er , es ) ≥ E (er , es )
35.
Outline
Games and payoﬀs
Matching dice
Vaccination
The theorem of the day
Strictly determined games
Example: Network programming
Characteristics of an Equlibrium
Two-by-two strictly-determined games
Two-by-two non-strictly-determined games
Calculation
Example: Vaccination
Other
36.
Finding equilibria by gravity
If C chose strategy 2,
and R knew it, R would
deﬁnitely choose 2 1 3
This would make C
choose strategy 1
but (2, 1) is an
2 4
equilibrium, a saddle
point.
37.
Finding equilibria by gravity
Here (1, 1) is an equilibrium 2 3
position; starting from there
neither player would want to
deviate from this.
1 4
38.
Finding equilibria by gravity
2 3
What about this one?
4 1
39.
Outline
Games and payoﬀs
Matching dice
Vaccination
The theorem of the day
Strictly determined games
Example: Network programming
Characteristics of an Equlibrium
Two-by-two strictly-determined games
Two-by-two non-strictly-determined games
Calculation
Example: Vaccination
Other
40.
Two-by-two non-strictly-determined games
Calculation
In this case we can compute E (p, q) by hand in terms of p1 and q1 :
E (p, q) = p1 a11 q1 +p1 a12 (1−q1 )+(1−p1 )a21 q1 +(1−p1 )a22 (1−q1 )
41.
Two-by-two non-strictly-determined games
Calculation
In this case we can compute E (p, q) by hand in terms of p1 and q1 :
E (p, q) = p1 a11 q1 +p1 a12 (1−q1 )+(1−p1 )a21 q1 +(1−p1 )a22 (1−q1 )
The critical points are when
∂E
0= = a11 q1 + a12 (1 − q1 ) − a21 q1 − a22 (1 − q1 )
∂p1
∂E
0= = p1 a11 − p1 a12 + (1 − p1 )a21 − (1 − p1 )a22
∂q
42.
Two-by-two non-strictly-determined games
Calculation
In this case we can compute E (p, q) by hand in terms of p1 and q1 :
E (p, q) = p1 a11 q1 +p1 a12 (1−q1 )+(1−p1 )a21 q1 +(1−p1 )a22 (1−q1 )
The critical points are when
∂E
0= = a11 q1 + a12 (1 − q1 ) − a21 q1 − a22 (1 − q1 )
∂p1
∂E
0= = p1 a11 − p1 a12 + (1 − p1 )a21 − (1 − p1 )a22
∂q
So
a22 − a12 a22 − a21
q1 = p1 =
a11 + a22 − a21 − a12 a11 + a22 − a21 − a12
These are in between 0 and 1 if there are no saddle points in the
matrix.
43.
Examples
1 3
If A = , then p1 = 2 ? Doesn’t work because A has a
0
2 4
saddle point.
2 3
If A = , p1 = 3 ? Again, doesn’t work.
2
1 4
2 3
If A = , p1 = −3 = 3/4, while q1 = −4 = 1/2. So R
−4
−2
4 1
should pick 1 half the time and 2 the other half, while C
should pick 1 3/4 of the time and 2 the rest.
44.
Further Calculations
Also
∂2E ∂2E
=0 =0
∂p 2 ∂q 2
So this is a saddle point!
Finally,
a11 a22 − a12 a21
E (p, q) =
a11 + a22 − a21 − a22
45.
Example: Vaccination
We have
0.9 − 0.6 2
p1 = = Strain
0.85 + 0.9 − 0.6 − 0.7 3
0.9 − 0.7 4 1 2
q1 = =
Vacc
0.85 + 0.9 − 0.6 − 0.7 9 1 0.85 0.70
(0.85)(0.9) − (0.6)(0.7) 2 0.60 0.90
v= ≈ 0.767
0.85 + 0.9 − 0.6 − 0.7
We should give 2/3 of the population vaccine 1 and the rest
vacine 2
The worst case scenario is a 4 : 5 distribution of strains
We’ll still cover 76.7% of the population
46.
Outline
Games and payoﬀs
Matching dice
Vaccination
The theorem of the day
Strictly determined games
Example: Network programming
Characteristics of an Equlibrium
Two-by-two strictly-determined games
Two-by-two non-strictly-determined games
Calculation
Example: Vaccination
Other
47.
Other Applications of GT
War
the Battle of the
Bismarck Sea
Business
product introduction
pricing
Dating
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