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
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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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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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
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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