a very little game theory
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A very little Game Theory
Math 20Linear Algebra and
Multivariable CalculusOctober 13, 2004
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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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 o
utco
mes
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
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
⎡
⎣
⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥
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
E(p,q) = pi aijqji, j=1
n
∑ =pAq
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
€
=0A “fair game” if the dice are fair.
Expected value with an unfair die
SupposeThen
€
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
StrategiesWhat 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 o
utco
mes
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
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 2V
accin
e
1 0.85
0.70
2 0.60
0.90
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 gameIn 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
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 games22 non-strictly-determined games
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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Payoff MatrixCBS shows
60 M
inute
s
Surv
ivor
CS
I
Every
bod
y Lo
ves
Raym
ond
NB
C
Show
s
Friends 60 20 30 55
Dateline 50 75 45 60
Law & Order
70 45 35 30
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 Sur
v CS
I
ELR
F 60 20 30
55
DL 50 75 45
60
L&O 70 45 35
30
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 Sur
v CS
I
ELR
F 60 20 30
55
DL 50 75 45
60
L&O 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
60
M Sur
v CS
I
ELR
F 60 20 30
55
DL 50 75 45
60
L&O 70 45 35
30
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
Pure Strategies are optimal in Strictly-
Determined GamesIf ars is a saddle
point, then erT is
an optimal strategy for R and es is an optimal strategy for C.
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Proof
E(erT ,q) =er
T Aq= ar1 ar2 L arn[ ] ⋅
q1
q2
M
qn
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
=ar1q1 + ar2q2 +L + arnqn
≥arsq1 + arsq2 +L + arsqn
=ars(q1 +L +qn) =ars =E(erT ,es)
Proof
E(p,es ) =pAes = p1 p2 L pm[ ] ⋅
a1s
a2s
M
ams
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
=p1a1s + p2a2s +L + pmams
≤p1ars + p2ars +L + pmars
=(p1 + p2 +L + pm)ars =ars =E(erT ,es)
Proof
So for all strategies p and q:E(er
T,q) ≥ E(erT,es) ≥ E(p,es)
Therefore we have found the optimal strategies
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
Optimal Strategy for 2x2 non-SD
LetThis is between 0 and 1 if A has no
saddle pointsThen €
p1 = p1∗ a22 − a21
a11 + a22 − a12 − a21
; p2 =1− p1
€
E(p,q) =(a12 − a22)(a22 − a21)
a11 + a22 − a12 − a21
+ a22
=a11a22 − a12a21
a11 + a22 − a12 − a21
Optimal set of strategies
We have
p∗=a22 −a21
a11 + a22 −a12 −a21
a11 −a12
a11 + a22 −a12 −a21
⎡
⎣⎢
⎤
⎦⎥
q∗=
a22 −a12
a11 + a22 −a12 −a21
a11 −a21
a11 + a22 −a12 −a21
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
v=a11a22 −a12a21
a11 + a22 −a12 −a21
Flu Vaccination
Strain
1 2V
accin
e
1 0.85
0.70
2 0.60
0.90
p1∗=
.90 −.60.85 + .90 −.70 −.60
=.30.45
=23
p2∗=
13
q1∗=
.90 −.70.85 + .90 −.70 −.60
=.20.45
=49
q2∗=
59
v=(.85)(.90)−(.70)(.60).85 + .90 −.70 −.60
=.345.45
=.766K
Flu Vaccination
Strain
1 2V
accin
e
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
Other Applications of GT
WarBattle of Bismarck
SeaBusiness
Product Introduction
PricingDating
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