This document discusses game theory concepts including payoff matrices, expected value, strategies, and equilibriums. It provides examples of games of chance with dice and network programming decisions between TV stations. Key points covered include finding optimal strategies that maximize minimum payoff or minimize maximum loss, and determining whether games have a pure strategy equilibrium or require mixed strategies based on payoff matrix properties.

1. A very little Game Theory
Math 20
Linear Algebra and Multivariable
Calculus
October 13, 2004

2. 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?
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3. The Payoff Matrix
Lists one player’s
(call him/her R)
possible outcomes
versus another
player’s (call him/her
C) outcomes
Each aij represents the
payoff from C to R if
outcomes i for R and j
for C occur (a zero-sum
game).
C’s outcomes
1 2 3 4 5 6 R’s outcomes
1 0 -1 -2 -3 -4 -5
2 1 0 -1 -2 -3 -4
3 2 1 0 -1 -2 -3
4 3 2 1 0 -1 -2
5 4 3 2 1 0 -1
6 5 4 3 2 1 0

4. Expected Value
Let the probabilities of R’s outcomes and C’s
outcomes be given by probability vectors
p = [p1 p2 L pn ]
q =
⎡
q1
q2
M
qn
⎢⎢⎢⎢
⎣
⎤
⎥⎥⎥⎥
⎦

5. Expected Value
The probability of R having outcome i
and C having outcome j is therefore
piqj.
The expected value of R’s payoff is
nå
E(p,q) = pi aijqj
i,j=1
= pAq

6. Expected Value of this Game
1
6
1
6
1
6
1
6
1
6
1
6
⎡⎣ ⎢
⎤⎦ ⎥
⋅
⎡
0 −1 −2 −3 −4 −5
1 0 −1 −2 −3 −4
2 1 0 −1 −2 −3
3 2 1 0 −1 2
4 3 2 1 0 −1
5 4 3 2 1 0
⎢⎢⎢⎢⎢⎢⎢
⎣
⎤
⎥⎥⎥⎥⎥⎥⎥
⋅
⎦
⎡
1
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
61
61
61
61
61
⎣
6
⎤
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
= 1
6
⋅ [1 1 1 1 1 1] ⋅
⎡
−15
−9
−3
3
9
15
⎢⎢⎢⎢⎢⎢⎢
⎣
⎤
⎥⎥⎥⎥⎥⎥⎥
⋅ 1
6
⎦
= 0
A “fair game” if the dice are fair.

7. Expected value
with an unfair die
Suppose
Then
p =
1
10
1
10
1
5
1
5
1
5
1
5
⎡⎣ ⎢
⎤⎦ ⎥
E(p,q) =
1
10
1
10
1
5
1
5
1
5
1
5
⎡⎣ ⎢
⎤⎦ ⎥
⋅
⎡
⎢⎢⎢⎢⎢⎢⎢ ⎤
0 −1 −2 −3 −4 −5
1 0 −1 −2 −3 −4
2 1 0 −1 −2 −3
3 2 1 0 −1 2
4 3 2 1 0 −1
5 4 3 2 1 0
⎣
⎥⎥⎥⎥⎥⎥⎥
⋅
⎦
⎡
1
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
61
61
61
61
61
⎣
6
⎤
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
=
1
10
⋅ [1 1 2 2 2 2] ⋅
⎡
−15
−9
−3
3
9
15
⎢⎢⎢⎢⎢⎢⎢
⎣
⎤
⎥⎥⎥⎥⎥⎥⎥
⋅ 1
6
⎦
=
1
60
((−15) + (−9) + 2(−3) + 2⋅ 3+ 2⋅ 9 + 2 ⋅15) =
24
60
=
2
5

8. Strategies
What if we could
choose a die to
be as biased as
we wanted?
In other words,
what if we could
choose a strategy
p for this game?
Clearly, we’d
want to get a 6
all the time!
C’s outcomes
1 2 3 4 5 6 R’s outcomes
1 0 -1 -2 -3 -4 -5
2 1 0 -1 -2 -3 -4
3 2 1 0 -1 -2 -3
4 3 2 1 0 -1 -2
5 4 3 2 1 0 -1
6 5 4 3 2 1 0

9. Flu Vaccination
Suppose there are two
flu strains, and we have
two flu vaccines to
combat them.
We don’t know
distribution of strains
Neither pure strategy is
the clear favorite
Is there a combination of
vaccines that maximizes
immunity?
Strain
1 2
Vaccine
1 0.85 0.70
2 0.60 0.90

10. 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
In other words, R can guarantee a lower
bound on his/her payoff and C can guarantee
an upper bound on how much he/she loses
This value could be negative in which case C
has the advantage

11. Fundamental Problem of
Zero-Sum games
Find the p* and q*!
In general, this requires linear
programming. Next week!
There are some games in which we can
find optimal strategies now:
Strictly-determined games
2 2 non-strictly-determined games

12. 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?
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13. Payoff Matrix
CBS shows
60 Minutes
Survivor
CSI
Everybody Loves Raymond
NBC Shows
Friends 60 20 30 55
Dateline 50 75 45 60

14. 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
60 M
Surv
CSI
ELR
F 60 20 30 55
DL 50 75 45 60
LO 70 45 35 30

15. 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
60 M
Surv
CSI
ELR
F 60 20 30 55
DL 50 75 45 60
LO 70 45 35 30

16. Equilibrium
(Dateline,CSI) is an
equilibrium pair of
strategies
Assuming NBC airs
Dateline, CBS’s
best choice is to air
CSI, and vice versa
60 M
Surv
CSI
ELR
F 60 20 30 55
DL 50 75 45 60
LO 70 45 35 30

17. 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

18. Pure Strategies are optimal
in Strictly-Determined Games
If ars is a saddle
point, then er
T is an
optimal strategy for
R and es is an
optimal strategy for
C.
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TIFF (Uncompressed) decompressor
are needed to see this picture.

19. Proof
T ,q) = e r T
Aq = [ar1 ar2 L arn ]×
E(er
q1
q2
é
êêêê
úúúú
M
qn
ë
ù
û
= ar1q1
+ar2q2
+L +arnqn
³ arsq1
+arsq2
+L +arsqn
= ars(q1
+L + qn ) = ars = E(er T
,es)

20. Proof
E(p,es ) = pAes = p1 p2
[ L pm ]×
a1s
a2
é
êêêê
úúúú
s
M
ams
ë
ù
û
= p1 a1s + p2
a2
s +L + pmams
£ p1
ars + p2
ars +L + pmars
= ( p1
+ p2
+L + pm )ars = ars = E(er T
,es)

21. Proof
So for all strategies p and q:
E(er
T,q) ≥ E(er
T,es) ≥ E(p,es)
Therefore we have found the optimal
strategies

22. 2x2 non-strictly determined
In this case we can compute E(p,q) by
hand in terms of p1 and q1
E(p,q) = [p1 p2 ] ⋅
⎡
a11 a12
a21 a22
⎣ ⎢
⎤
⋅
⎦ ⎥
⎡
q1
q2
⎣ ⎢
⎤
⎦ ⎥
= p1a11q1 + p1a12q2 + p2a21q1 + p2a22q2
= p1a11q1 + p1a12(1−q1) + (1− p1)a21q1 + (1− p1)a22(1−q1)
= (a11 + a22 − a12 − a21)p1 −(a22 − a21[ )]q1 + (a12 − a22 )p1 + a22

23. Optimal Strategy for 2x2 non-SD
∗ a22 − a21
a11 + a22 − a12 − a21
Let
This is between 0 and 1 if A has no
saddle points
Then
p1 = p1
; p2 =1− p1
E(p,q) =
(a12 − a22 )(a22 − a21)
a11 + a22 − a12 − a21
+ a22
=
a11a22 − a12a21
a11 + a22 − a12 − a21

24. Optimal set of strategies
We have
ù
û ú
2
a22 - a21
1 + a22 - a1
2 - a21
a11 + a22 - a12 - a2
p* =
1
a11 - a1
a1
é
ë ê
úúúú
1
êêêê
q* =
a22 - a12
a11 + a22 - a12 - a2
a11 - a21
1
a11 + a22 - a12 - a2
ù
é
û
ë
1a22 - a12a2
1
a1
a11 + a22 - a12 - a2
v=
1

25. Flu Vaccination
Strain
1 2
Vaccine
1 0.85 0.70
2 0.60 0.90
* =
p1
.90 - .60
.85 + .90 - .70 - .60
=
.30
.45
=
2
3
* =
p2
1
3
* =
q1
.90 - .70
.85 + .90 - .70 - .60
=
.20
.45
=
4
9
* =
q2
5
9
v=
(.85)(.90) - (.70)(.60)
.85 + .90 - .70 - .60
=
.345
.45
= .766K

26. Flu Vaccination
Strain
1 2
Vaccine
1 0.85 0.70
2 0.60 0.90
So we should give
2/3 of the
population vaccine
1 and 1/3 vaccine 2
The worst that
could happen is a
4:5 distribution of
strains
In this case we
cover 76.7% of pop

27. Other Applications of GT
War
Battle of Bismarck
Sea
Business
Product Introduction
Pricing
Dating
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