We define zero-sum games and show that they can be modeled with matrices. We find optimal strategies for two types of such games: (1) strictly determined games which have a saddle point, and (2) 2x2 non-strictly determined games, for which a calculus computation finds the optimal strategy

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
C ’s outcomes
outcomes versus
1 2345 6
the other’s
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 3210 -1
occur (a zero-sum
6 5 4321 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.

9. Expected value of this game
pAq
0 −1 −2 −3 −4 −5
  
1/6
−1 −2 −3 −4  1/6
1 0
  
0 −1 −2 −3  1/6
2 1
1/6 1/6 1/6 1/6 1/6 1/6 
=  
0 −1 −2  1/6
3 2 1
  
0 −1  1/6
4 3 2 1
54 3 2 1 0 1/6
−15/6
 
 −9/6 
 
 −3/6 
1/6 1/6 1/6 1/6 1/6 1/6 
= 
 3/6 
 
 9/6 
15/6
=0

10. Expected value with an unfair die
1/10 1/10 1/5 1/5 1/5 1/5
Suppose p = . Then
pAq
0 −1 −2 −3 −4 −5
 
1/6
1 0 −1 −2 −3 −4  1/6
 
0 −1 −2 −3
2 1  1/6
= 1/10 1/10 1/5 1/5 1/5 1/5  
0 −1 −2
3 2 1  1/6
 
0 −1
4 3 2 1  1/6
54 3 2 1 0 1/6
−15
 
 −9 
 
 −3  24 2
1 1
· 1 1 2 2 2 2
= =
 60 = 5

10 6
3
9
15

11. Strategies
What if we could
choose a die to be
C ’s outcomes
as biased as we
1 2345 6
wanted?
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 3210 -1
Clearly, we’d want 6 5 4321 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
1 2
Neither pure strategy is
Vacc
1 0.85 0.70
the clear favorite
2 0.60 0.90
Is there a combination of
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
arj qj ≥
E (er , q) = er Aq = ars qj = ars = E (er , es )
j=1 j=1
If p is a strategy for R, then
m m
pi ais ≤
E (er , es ) = pAes = 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  
1 3
definitely choose 2
 
This would make C  
 
choose strategy 1  
 
but (2, 1) is an
 
2 4
equilibrium, a saddle
point.

37. Finding equilibria by gravity
 
2 3
Here (1, 1) is an equilibrium
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 = p
and q1 = q:
E (p, q) = pa11 q + pa12 (1 − q) + (1 − p)a21 q + (1 − p)a22 (1 − q)

41. Two-by-two non-strictly-determined games
Calculation
In this case we can compute E (p, q) by hand in terms of p1 = p
and q1 = q:
E (p, q) = pa11 q + pa12 (1 − q) + (1 − p)a21 q + (1 − p)a22 (1 − q)
The critical points are when
∂E
= a11 q + a12 (1 − q) − a21 q − a22 (1 − q)
0=
∂p
∂E
= pa11 − pa12 + (1 − p)a21 − (1 − p)a22
0=
∂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 = p
and q1 = q:
E (p, q) = pa11 q + pa12 (1 − q) + (1 − p)a21 q + (1 − p)a22 (1 − q)
The critical points are when
∂E
= a11 q + a12 (1 − q) − a21 q − a22 (1 − q)
0=
∂p
∂E
= pa11 − pa12 + (1 − p)a21 − (1 − p)a22
0=
∂q
So
a22 − a12 a22 − a21
p= q=
a11 + a22 − a21 − a22 a11 + a22 − a21 − a12
These are in between 0 and 1 if there are no saddle points in the
matrix.

43. Examples
13
, then p = 2 ? Doesn’t work because A has a
If A = 0
24
saddle point.
23 3
If A = , p = 2 ? Again, doesn’t work.
14
2 3
, p = −3 = 3/4, while q = −4 = 1/2. So R
−2
If A = −4
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
1 2
0.9 − 0.7 4
q1 = =
Vacc
1 0.85 0.70
0.85 + 0.9 − 0.6 − 0.7 9
2 0.60 0.90
(0.85)(0.9) − (0.6)(0.7)
≈ 0.767
v=
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