Literature
Game Theory
Fun and Games
Wiebe van der Hoek
A Course in Game Theory
Computer Science
University of Liverpool
United Kingdom
Ken Binmore
Martin Osborne en Ariel Rubinstein
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What is it all about?
What is it all about?
game = interaction
traffic
supermarket
employee, employer, board, union
student / teacher
judge and lawyers
George en Osama
marriage and career
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Game Theory
strategic interaction is difficult
because reasoning is circular
If J and M play a game, J’s strategy will typically
depend on his prediction of M’s strategy, which, on its
turn depends on M’s expectation of J’s....
3
does it make sense
terminology of chess and bridge
logic and systematics of interaction
analysis takes you from irrational issues
Surprise and Paradox
2
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4
Strategic voting
!
to vote for a candidate you fancy least? YES!
!!
for a general, to toss a coin?
YES
in poker, place a maximal bid with the ES!!
Y
worst cards?
to throw some goods away before starting
!!
to negotiate about them
YES
to sell your house to the second best
!!
YES
bidder?
5
Boris, Horace and Maurice determine
who can be a member of the Dead Poet
Society
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proposal: allow Alice
amendment: allow Bob, rather than Alice
first vote over amendment, then over
proposal
6
1
Strategic voting
Horace
Borice
Strategic voting
Maurice
first between A, B
Bob
Alice
Nobody
then between A, N
strategic voting H:
Alice
Nobody
Bob
Nobody
Alice
Bob
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7
History
The Theory of Games and Economic
Behaviour (1944)
are rational:
aware of alternatives
form expectations
have preferences
optimize after
deliberation
set A of actions;
set C consequences;
g: A → C
consequence function
preference relation ≥
on A
or: utility function
u: C → R
9
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10
Game Theory
theory of decision
makers
if he accepts: (you,person) get (1.000-x,x)
else (0,0)
are rational
reason strategically
players anticipate on
knowledge and
expectations about
behaviour of other
decision makers
only money counts, and that is known
both are rational: prefer y+1 over y
what will you offer?
!!
£!!!!!
ONE
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Game Theory
Borice
Alice
Nobody
Bob
8
theory of decision
makers
suppose you’re offered £ 1.000
you make a deal with the first person
you encounter: (1.000-x,x) x = 1, 2 ...
M anticipates: vote
for A
Abstracts from `emotions’
first vote for Bob!
result … B, N
Horace
Nobody
Alice
Bob
Nash, Aumann, Shapley, Selten, Harsanyi
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winner Alice
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ideas from economics and mathematics
initially very optimistic, then draw-back
revival since 1970’s
winner Alice
Game Theory
Von Neumann and Morgenstern
Maurice
Bob
Alice
Nobody
11
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12
2
Representation strategic
games
Strategic Games
Definition G = 〈 N‚(Ai)‚(≥i)〉
finite set N (players)
set Ai (actions) for every player i
preference relation ≥i for every player i
ui is utility function: A → R with
13
Representation strategic
games
one-shot
simultaneous
independent
utilities are known
not the choice of
others
L
R
T w1‚w2 x1‚x2
B y1‚y2
z1‚z2
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15
Game Theory
B: Bach
S: Strawinsky
B
S
B
2‚1
0,0
S
0,0
1,2
Battle of the Sexes
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16
Profiles: example
A1, A2 ,... , An are the action sets
(a1, a2 ,... , an) ∈ A1x A2 x ... x An is a
profile
notation: (x), or a*
x-i ∈ A1x A2 x ... x Ai-1 x Ai+1 x ... x An
(x-i,xi) = (x)
focus on i, given the profile of the
others
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z1‚z2
14
N = {1,2}
A1 = {B,S}
A2 = {B,S}
u1, u2 see figure
Profiles
B y1‚y2
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R
Example: BoS
Interpretation
L
T w1‚w2 x1‚x2
a ≥i b ⇔ ui(a) ≥ ui(b)
also called payoff-function
(although not the same)
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N = {1,2}
A1 = {T,B}
A2 = {L,R}
u1(T) =w1, etc
A1, A2 ,... , A7 are bids (∈ R)
(a1, a2 ,... , a7) is a concrete bid
notation: (x)i =(25,22,20,12,0,27,22)=a*
x-6 ∈(25,22,20,12,0,22)
(x-6,x6) = ((25,22,20,12,0,22),27)
17
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(x-6,x’6) = ((25,22,20,12,0,22),26) would have
been better for player 6, given the profiles of the
others
18
3
Nash equilibrium
Nash equilibrium (definitie)
John Nash
equilibrium (“solution”)
every player is rational
ever player plays optimally
no use to devert individually
not an algorithmic approach
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19
Nash equilibrium (alternative)
Given G = 〈 N‚(Ai)‚(≥i)〉
a* ∈ A = A1x A2 x ... x An is Nash
equilibrium iff
∀i∈N ∀ai∈Ai (a*i-1,a*i) ≥i (a*i-1,ai)
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define for every a-i ∈A-i the best
response for i, Bi(a-i)
Bi(a-i)={ai∈Ai | ∀a’i ∈Ai (a-i,ai) ≥i (a-i,a’i)}
N = {1,2}
A1 = {B,S}
A2 = {B,S}
u1, u2 see figure
B
S
B
2‚1
0,0
B: Bach
S: Strawinsky
S
0,0
1,2
a* is N.eq iff ∀i∈N a*i∈ Bi(a*-i)
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21
Example: coordination game
Mozart or Mahler?
same preferences
two equilibria:
(Mozart,Mozart) and
(Mahler,Mahler)
N.eq right concept?
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Game Theory
20
Example: BoS (N.eq)
`no player i can improve in a*, if the other
players still play a*-i’
two equilibria:
(bach,bach) and
(strawinsky, strawinsky)
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22
Pareto Efficiency
Mo
Ma
Mo
2,2
0,0
Ma
0,0
1,1
23
(Mozart,Mozart) and
(Mahler,Mahler)
N.eq right concept?
(2,2) is (strongly)
Pareto efficient:
¬∃x¬∃y (x,y) > (2,2)
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Mo
Ma
Mo
2,2
0,0
Ma
0,0
1,1
24
4
Pareto Optimality (definition)
Example: prisoner’s dilemma
Given G = 〈 N‚(Ai)‚(≥i)〉
a* = (a1, a2 ,... , an) ∈ A1x A2 x ... x An
is Pareto optimal iff
∀i∈N ∀b* ∈ A1x A2 x ... x An (a*) ≥i (b*)
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25
Example: prisoner’s dilemma
C: cooperate with
the other, keep
silent
D: justify against the
other
Altough cooperate
would be better,
every player has a
preference for
defeat
C
D
C
0,0
-2,3
D
3,-2
-1,-1
Head and Tail
if different, 1 pays a
Pound to 2 if the
same, 2 pays a
Pound to 1
no equilibrium!
game is strictly
competatief
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Game Theory
C: cooperate with
the other, keep
silent
D: justify against the
other
C
D
C
0,0
-2,3
D
3,-2
-1,-1
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26
Example: hawk-dove
preference:
27
Example: Matching Pennies
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hawkish if other is
dovish
dovish if oher is
hawkish
N.eq: (Dove,Hawk)
and (Hawk,Dove)
D
H
D
3,3
1,4
H
4,1
0,0
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28
SCSG
H
T
H
1,-1
-1,1
T
-1,1
1,-1
Strictly Compatitive Strategic Game
if G = 〈{1,2}‚(Ai)‚(≥i)〉,
and ∀a,b ∈A: a ≥1 b ⇔ b ≥2 a
also called zero-sum game:
29
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with u1 and u2 we have
u1 + u2 = 0
30
5
maxminimizer
maxminimizer
let G = 〈{1,2}‚(Ai)‚(≥i)〉 an SCSG
action x* ∈ A1 is maxminimizer for 1:
∀x∈A1 min u1(x*,y) ≥1 min u1(x,y)
y∈A2
y∈A2
y∈A2
action y* ∈ A2 is maxminimizer for 2:
∀y∈A2 min u2(x,y*) ≥2 min u2(x,y)
x∈A1
x∈A1
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x* is a maxminimizer for 1;
y* is a maxminimizer for 2
maxxminyu1(x,y) =
minymaxxu1(x,y) =u1(x*,y*)
solves for 1
maxxminy u1(x,y) =
max{
min{u1(x,y)|y∈A2}
|x ∈A1} =
x1: miny u(x1,y) = 1
...
max{
min{u1(x,y)|y∈A2}
|x ∈A1} =
Game Theory
y1
y2
y3
y4
y5
2,-2
2,-2
3,-3 1,-1 1,-1
x2
3,-3
5,-5
4,-4 6,-6 4,-4
x3
5,-5
2,-2
4,-4 3,-3 3,-3
x4
6,-6
8,-8
5,-5 7,-7 5,-5
x5
3,-3
5,-5
4,-4 2,-2 3,-3
x6
4,-4
3,-3
6,-6 5,-5 4,-4
34
maxminimizers
y1
y2
y3
y4
y5
x1
2,-2
2,-2
3,-3 1,-1 1,-1
x2
3,-3
5,-5
4,-4 6,-6 4,-4
x3
5,-5
2,-2
4,-4 3,-3 3,-3
x4
6,-6
8,-8
5,-5 7,-7 5,-5
x5
3,-3
5,-5
4,-4 2,-2 3,-3
x6
4,-4
3,-3
6,-6 5,-5 4,-4
solves for 1
maxxminy u1(x,y) =
max{
min{u1(x,y)|y∈A2}
|x ∈A1} =
x1: miny u(x1,y) = 1
x2: miny u(x2,y) = 3
..: .......... .. = ..
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...
x1
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33
maxminimizers
32
solves for 1
maxxminy u1(x,y) =
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solves for 1 maxxminy u1(x,y)
solves for 2 maxyminx u2(x,y)
‘maximises the minimum that i can
guarantee’
x* is a security strategy for 1
maxminimizers
(x*,y*) is N.eq for G, iff:
y∈A2
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31
Equilibria and maxminimizers
action x* ∈ A1 is maxminimizer for 1:
∀x∈A1 min u1(x*,y) ≥1 min u1(x,y)
...
y1
y2
y3
y4
y5
x1
2,-2
2,-2
3,-3 1,-1 1,-1
x2
3,-3
5,-5
4,-4 6,-6 4,-4
x3
5,-5
2,-2
4,-4 3,-3 3,-3
x4
6,-6
8,-8
5,-5 7,-7 5,-5
x5
3,-3
5,-5
4,-4 2,-2 3,-3
x6
4,-4
3,-3
6,-6 5,-5 4,-4
max = 5 for x* = x4
xn: miny u(xn,y) = 3
35
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36
6
maxminimizers
solves for 1
maxxminy u1(x,y) =
max{
min{u1(x,y)|y∈A2}
|x ∈A1} = 5
solves for 2
maxxminy u2(x,y) =
max{
min{u1(x,y)|x∈A1}
|y ∈A2} =
...
maxminimizers
y1
y2
y3
y4
y5
x1
2,-2
2,-2
3,-3 1,-1 1,-1
x2
3,-3
5,-5
4,-4 6,-6 4,-4
x3
5,-5
2,-2
4,-4 3,-3 3,-3
x4
6,-6
8,-8
5,-5 7,-7 5,-5
x5
3,-3
5,-5
4,-4 2,-2 3,-3
x6
4,-4
3,-3
6,-6 5,-5 4,-4
max{
min{u1(x,y)|x∈A1}
|y ∈A2} = -5!
L
y3
y4
y5
2,-2
3,-3 1,-1 1,-1
x2
3,-3
5,-5
4,-4 6,-6 4,-4
x3
5,-5
2,-2
4,-4 3,-3 3,-3
x4
6,-6
8,-8
5,-5 7,-7 5,-5
x5
3,-3
5,-5
4,-4 2,-2 3,-3
x6
4,-4
3,-3
6,-6 5,-5 4,-4
38
R
l
2,2
0,0
r
1,1
1,1
consider game G
(2,2) looks like `the
optimal’ solution
security strategy of
1 is r, gives 1!
Nash equilibria?
L
R
l
2,2
0,0
r
1,1
1,1
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40
bimatrix games
t1
t2
t3
s1
1,1 2,0 3,-1
s2
2,0 4,0 6,0
π2(si,tj) = (i-2)(j-2)
m x n matrix
1 has strategies s1
and s2, 2 has t1, t2
and t3
Nash equilibrium
(σ,τ):
Game Theory
y2
2,-2
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y1
x1
Equilibrium (5,-5)
39
m x n matrix
1 has strategies s1
and s2, 2 has t1, t2
and t3
payoff π1(si,tj) = ij
solves for 2
maxxminy u2(x,y) =
bimatrix games
max{
min{u1(x,y)|y∈A2}
|x ∈A1} = 5
...
security level vs equilibria
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security level vs equilibria
37
consider cooperative
game G
(2,2) looks like `the
optimal’ solution
security strategy of
1 is r, gives 1!
Nash equilibria?
solves for 1
maxxminy u1(x,y) =
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41
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t1
t2
t3
s1
1,1 2,0 3,-1
s2
2,0 4,0 6,0
∀s,t π1(σ,τ) ≥ π1(s,τ)
∀s,t π2(σ,τ) ≥ π2(σ,t)
42
7
domination
strategy sd of 1
dominates si
strongly:
∀t π1(sd,t) > π1(si,t)
and weakly if:
Iterated elimination
t1
t2
t3
s1
1,1 2,0 3,-1
s2
2,0 4,0 6,0
∀t π1(sd,t) ≥ π1(si,t)
∃t π1(sd,t) > π1(si,t)
t1 dominates t2
weakly
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43
Order of elimination
C
C D
AE 2,0 1,1
D
C
s2 of 1 strongly
dominates s1
No further (weak)
domination: all is
left are Nash
Equilibria
this is not generally
so
t1
s1
t2
t3
1,1 2,0 3,-1
2,0 4,0 6,0
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44
elimination: conclusions
D
AE 2,0
1,1
AE 2,0
1,1
AF 0,2
1,1
AF 0,2
1,1
BE 3,3
3,3
BE 3,3
3,3
BF 3,3
3,3
BF 3,3
3,3
lost equilibrium!
strict strategies: no problem
with weakly dominated strategies:
some equilibria can get lost
order of elimination is important
AF 0,2 1,1
BE 3,3 3,3
BF 3,3 3,3
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45
Example: BoS
N = {1,2}
A1 = {B,S}
A2 = {B,S}
u1, u2 see figure
B: Bach
S: Strawinsky
Battle of the Sexes
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46
Ex: coordination game
B
S
B
2‚1
0,0
S
0,0
1,2
Mozart of Mahler?
Same preference
No dominant strategy
Mo
Ma
Mo
2,2
0,0
Ma
0,0
1,1
still two Nash equilibria
no dominant strategies
still (two) Nash equilibira
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Game Theory
47
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48
8
Ex: prisoner’s dilemma
C: cooperate, and
be silent
D: justify against the
other
D dominates C
D dominates C
gives Nash
equilibrium (-1,-1)
Ex: Matching Pennies
C
D
C
D
0,0
-2,3
3,-2
-1,-1
49
don’t always bid in the same way with
poker
being inpredictable can be an
advantage
sometimes a strategy is not dominated
by another pure strategy, but by a
mixed one
maximin for 1: 2 via s3
(s3,t2) is not a
s1
saddlepoint
s2
security level of 1 is
indeed not 2, but 22/3 s3
How?
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Game Theory
1,-1
-1,1
T
-1,1
1,-1
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51
security level and strategy
H
50
security level and strategy
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T
No pure Nash equilibrium
mixed strategies
H
no dominant strategy
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Head and Tail
if different, 1 pays a
Pound to 2; if the
same, 2 pays a
Pound to 1
s1
s2
s3
t1
t2
t3
0,1 1,2 7,3
4,6 2,0 3,2
9,0 0,3
0,4
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52
mixed strategies
t1
t2
t3
1,0 6,4 0,9
2,1 0,2 3,0
3,7 2,3
maximin for 1:
2, via s2
note: (s2,t2) is a
saddlepoint
then 2 is also
security level of
player 1
4,0
53
remove strictly
dominated strategy
s2
2 has no pure
dominating strategy,
but, .....
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s1
s2
s3
t1
t2
t3
1,0 6,4 0,9
2,1 0,2 3,0
3,7 2,3
4,0
54
9
mixed strategies
2 has no pure dominating
strategy, but, .....
q = (1/2,0,1/2) dominates t2
strongly!
π2(s1,q) =
(1/2)—0
mixed strategies
s1
s2
s3
+ (1/2)—9 =4.5 > 4
t1
t2
t3
1,0 6,4 0,9
2,1 0,2 3,0
3,7 2,3
4,0
π2(s3,q) =
(1/2)—7
2 has no pure dominating
strategy, but, .....
q = (1/2,0,1/2) dominates t2
strongly!
π2(s1,q) =
+ (1/2)—0 =3.5 > 3
after iterated
elimination
what is security level
of 1?
suppose 1 plays
mixed strategy (1r,0,r)
s
s1
s2
s3
t1
t2
t3
1,0 6,4 0,9
2,1 0,2 3,0
(1/2)—7
3,7 2,3
4,0
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57
mixed strategies
m(r) = min{E1,E2, E3}
3
Game Theory
t1
t2
t3
1,0 6,4 0,9
2,1 0,2 3,0
3,7 2,3
4,0
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5
4
58
Head and Tail
if different, 1 pays a
Pound to 2; if the
same, 2 pays a
Pound to 1
H
T
H
1,-1
-1,1
T
-1,1
1,-1
no dominant strategy
1
No Nash equilibrium
0
1/2
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s
s1
s2
s3
Ex: Matching Pennies
2
max for r = 5/6
payoff is E1(r) = 22/3
56
E1(r)= 1(1-r) + 3r = 1 + 2r
E2(r)= 6(1-r) + 2r = 6 – 4r
E3(r)= 0(1-r) + 4r = 4r
6
+ (1/2)—0 =3.5 > 3
suppose 1 playes mixed
strategy (1-r,0,r)
let Ek(r) be payoff of 1 if 2
plays tk:
E1(r)= 1(1-r) + 3r = 1 + 2r
E2(r)= 6(1-r) + 2r = 6 – 4r
E3(r)= 0(1-r) + 4r = 4r
4,0
mixed strategies
3,7 2,3
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55
mixed strategies
+ (1/2)—9 =4.5 > 4
t3
t1
t2
1,0 6,4 0,9
2,1 0,2 3,0
π2(s3,q) =
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(1/2)—0
s1
s2
s3
r→
3/4
5/6
1
59
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60
10
Ex: Matching Pennies
Suppose r plays (0.5,0.5), and
c plays (0.5,0.5)
Ex: Matching Pennies
H
H
T
1,-1
-1,1
T
-1,1
1,-1
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61
Ex: Matching Pennies
Suppose r plays (0.5,0.5), and
c plays (0.5,0.5)
T
0.5 ⋅ 0.5 ⋅ 1 + 0.5 ⋅ 0.5 ⋅ -1 +
1,-1
-1,1
0.5 ⋅ 0.5 ⋅ 1 + 0.5 ⋅ 0.5 ⋅ -1 = 0
T
-1,1
1,-1
π1 would then be:
H
H
Suppose r plays (0.5,0.5), and
c plays (0.5,0.5)
π1 would then be:
0.5 ⋅ 0.5 ⋅ 1 + 0.5 ⋅ 0.5 ⋅ -1 +
would r play (q,1-q) with q > 0.5 then c plays T; π1 =
(q ⋅ 0 ⋅ 1) + (q ⋅ 1 ⋅ -1) +
((1-q) ⋅ 0 ⋅ -1) + ((1-q) ⋅ 1 ⋅ 1) = 1-(2 ⋅ q) < 0
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T
H
1,-1
-1,1
T
-1,1
1,-1
0.5 ⋅ 0.5 ⋅ 1 + 0.5 ⋅ 0.5 ⋅ -1 = 0
this is Nash:
would r play (q,1-q) then π1((q,1-q),(0.5,0.5)) =
(q ⋅ 0.5 ⋅ 1) + (q ⋅ 0.5 ⋅ -1) +
((1-q) ⋅ 0.5 ⋅ -1) + ((1-q) ⋅ 0.5 ⋅ 1) = 0 is not better
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62
Game Theory:
Part II: Extensive Games
Wiebe van der Hoek
Computer Science
University of Liverpool
United Kingdom
no other Nash:
H
63
Rules of the Game
Rules of the Game
root
When?
I
When?
What?
Who?
l
II
R
w
l
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Game Theory
65
wvdh
l
L
R w
L
I
R
M
II
l
terminal
How much?
r
II
L
r
w
w
w
66
11
Example: tictactoe
Strategies
II
I
o
o x
I
o
I
II
o
o
x
I
o
II
I
I
x
x x
I
xx o
x
x
x
o
o
o
I
x
o
o
x
o
o ox
x x o
x
w
d
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x o o
o ox
x x o
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67
Strategies
A pure strategie for player
p specifies for every
decision note of p what
he will do there
Strategies for I:
lr
rl
68
Strategies
a
I
l
II
r
c
L
w
L
R
M
II e
R w
L
l
b
I
w
A pure strategie for player
p specifies for every
decision note of p what
he will do there
Strategies for II:
a
II
r
c
L
w
wvdh
L
R w
L
l
I
b
R
w
w
r
w
wvdh
69
Strategy profile
70
extensive to strategic
1
a
Example:
II
c
l
L
M
II e
R w
L
l
l
d
II
R
w
A
r
L
I
C
I
l
[lr,RMR]
M
II e
l
LLL, LLR, LML, LMR, LRL, LRR
RLL, RLR, RML, RMR, RRL, RRR
d
II
R
l
w
I
l
w
r
l
rr
d
II
R
l
ll
If all players choose
such a strategy, the
outcome of the
game is determined
x o
o
o ox
x
o
A pure strategie for
player p specifies for
every decision note
of p what he will do
there
I
II
o ox
o
o
II
o
o ox
x x o
o
o ox
o o
o
xx
o
I
x
o
II
x
II
II
o
x
o
b
r
B
AE
2
R
D
C
w
1
w
E
C
d
AF
a c
b c
B
d d
AE
c
F
w
D
AF
BE
BF
D
a c
b c
d d
d d
b
a
reduced strategic form
wvdh
Game Theory
71
wvdh
72
12
Equilibria: example
Strategy profile
Nash equilibria?
via strategic form:
L
1
A
B
2
L
R
L
0,0 2,1
A
B
R
1,2
R
LR
ML
MR
RL
RR
LL
LR
ML
MR
RL
RR
ll
w
w
w
w
w
w
l
l
l
l
l
l
lr
w
w
w
w
w
w
l
l
l
l
l
l
rl
l
w
w
w
w
w
l
w
w
w
w
w
rr
w
w
w
w
w
w
w
w
w
w
w
w
0,0
1,2 1,2
2,1
wvdh
Backward Induction
II
G
II
w
II
l
w
I
w
w
w
w
w
w
L
l
I
l
b
r
w
w
w
w
w
w
w
w
w
75
I can win
wvdh
76
extensive games: definitions
extensive games: G = 〈 N‚H‚P,(≥i)〉
LLL
LLR
LML
LMR
LRL
LRR
RLL
RLR
RML
RMR
RRL
RRR
ll
w
w
w
w
w
w
l
l
l
l
l
l
lr
w
w
w
w
w
w
l
l
l
l
l
l
rl
l
w
w
w
w
w
l
w
w
w
w
w
rr
w
w
w
w
w
w
w
w
w
w
w
w
N: set of players
H histories: ∅, (ak)k=1..K (may be infinite)
again: rr is winning strategy,
since that row only contains a w
Game Theory
II
l
l
II
l
wvdh
wvdh
R w
I
I
l
II e
II
I
w
R
M
I
II
w
G
L
74
w
G
G
R
l
d
II
wvdh
G
w
w
l
I
L
extensive form of G
II
II
w
w
r
c
I
l
II
I
w
l
II
l
I
backward induction
G
I
w
II
a strategy profile: [rr,RLL]
73
I
a
l
w
strategic form of G
G
G
LL
77
wvdh
closed under prefixes
terminals Z: no successor or infinite
P: H\Z → N player who is to move
≥i: preference relation on Z
78
13
extensive games: definitions
N: set of players
H histories: ∅, (ak)k=1..K (may be infinite)
Subgame perfect solutions
closed under prefixes
terminals Z: no successor or infinite
h∈H, a action ⇒ (h,a) ∈H
H is finite ⇒ G is finite
H only contains finite h ⇒ G has finite
horizon
79
Subgames
all continuations of h
wvdh
80
subgame perfect N.-eq
history h
subgame Γ(h)
1
A
B
C
let Γ = 〈N‚H‚P,(≥i)〉 extensive
2
D
s* is N.-eq if ∀i∀si O(s-i*,si*) ≥i O(s-i*,si)
s* is subgame perfect N.-eq if
d
1
c
F
E
b
a
wvdh
81
Equilibria: Example
s*|h is N.-eq for all Γ(h)
wvdh
1
A
B
2
L
0,0
R
1,2
2,1
Game Theory
∀i∀h∈H\Z (P(h)=i ⇒
Oh(s-i*|h,si *|h) ≥i|h O(s-i*|h,si))
for all strategies si for i in Γ(h)
82
equilibria (ctd)
Nash equilibria?
wvdh
N: set of player
H histories: ∅, (ak)k=1..K (may be infinite)
P: H\Z → N player to play
≥i: preference relation on Z
subgames: Γ(h) = 〈 N‚H|h‚P|h,(≥i|h)〉
wvdh
extensive games: Γ = 〈 N‚H‚P,(≥i)〉
83
so: (A,R) and (B,L)
interpretation (B,L):
given that 2 plays L
after A, 1 better
choose B
intuitive?
what is optimal for
1?
wvdh
1
A
B
2
L
R
1,2
1
0,0
2,1
A
B
2
L
0,0
R
2,1
1,2
84
14
equilibria (ctd)
equilibria (ctd)
1
so: (A,R) and (B,L)
interpretation (B,L):
given that 2 plays L
after A, 1 better
choose B
AR is the only
subgame perfect
equilibrium
A
B
2
L
R
1,2
1
0,0
2,1
A
B
2
R
L
0,0
wvdh
1,2
2,1
A
L
R
wvdh
1
0,0
2,1
chain k and n
competitors
every competitor
can either enter
challenge k (i), or
not (o)
if so, k chooses
between cooperate
(c) and fight (f)
F
0
0
0
0
C
i
F
0,0
o
i
k
5
0 F
C
0
2 1
4
0
0
0
2 2
2
0 2
2
0
Game Theory
F
5
0
1
0
7
0
2
1
C
3
o
i
2,2
o
i
k
o
k
F
C
10
0 F
C
1
7 1
4
0
2
1
2 0
2
2 2
0
0
i
o
4
2
2
0
7
2
0
1
o
C
9
2
2
6 1
2
2
2
i
i
k
k
F
F
C
7
1
1
0
9
2
1 12
2 2
1
1
3
5
1
0
0
o
C
i
i
k
o
k
10
1 F
C
0
7 1
9
1
1
0
7 2 12
2
1 2 1
2
2
0
1
F
o
C
15
10
1
1
1
1
12 1
0
1
1
2
i
k
F
0
0
0
0
3
o
C
i
i
i
k
o
k
5
0 F
C
0
2 1
4
0
0
0
2 2
2
0 2
2
0
F
5
0
1
0
7
0
2
1
C
3
3
o
i
F
3
3
i
k
o
k
F
C
10
0 F
C
1
7 1
4
0
2
1
2 0
2
2 2
0
0
o
k
F
C
3
3
2
o
k
F
C
3
o
i
o
k
F
3
3
2
2
2
k
F
i
k
5,1
C
C
F
i
o
o
o
k
1
k
o
86
n
i
i
o
2
k
i
k
2,1
88
C
F
3
1,2
k
2
3
R
L
0,0
k
C
B
2
1
3
A
wvdh
87
i
F
1,2
shop-chain game
chain k and n
competitors
every competitor
can either enter
challenge k (i), or
not (o)
if so, k chooses
between cooperate
(c) and fight (f)
i
B
2
wvdh
85
shop-chain game
1
so: (A,R) and (B,L)
interpretation (B,L):
given that 2 plays L
after A, 1 better
choose B
AR is the only
subgame perfect
equilibrium
not BL!
4
2
2
0
7
2
0
1
o
C
9
2
2
6 1
2
2
2
i
i
k
o
k
F
F
C
7
1
1
0
9
2
1 12
2 2
1
1
C
3
3
5
1
0
0
o
C
i
3
i
k
o
k
10
1 F
C
0
7 1
9
1
1
0
7 2 12
2
1 2 1
2
2
0
1
F
o
C
15
10
1
1
1
1
12 1
0
1
1
2
15
1
1
i
i
o
k
k
C
F
2
i
o
F
3
F
o
i
k
C
F
5
0 F
C
0
2 1
4
0
0
0
2 2
2
0 2
2
0
0
0
0
0
o
i
k
o
C
3
3
i
i
k
o
k
F
10
0 F
C
1
7 1
4
0
2
1
2 0
2
2 2
0
0
7
0
2
1
o
i
F
9
2
2
6 1
2
2
2
7
2
0
1
F
C
7
1
1
0
9
2
1 12
2 2
1
1
o
i
k
k
C
4
2
2
0
3
i
i
k
o
k
C
F
10
1 F
C
0
7 1
9
1
1
0
7 2 12
2
1 2 1
2
2
0
1
5
1
0
0
o
C
15
10
1
1
1
1
12 1
0
1
1
2
i
k
F
3
o
i
o
i
k
o
F
5
0 F
C
0
2 1
4
0
0
0
2 2
2
0 2
2
0
C
3
3
k
C
0
0
0
0
i
i
i
k
o
k
F
10
0 F
C
1
7 1
4
0
2
1
2 0
2
2 2
0
0
7
0
2
1
F
3
3
C
5
0
1
0
o
i
F
9
2
2
6 1
2
2
2
7
2
0
1
F
C
7
1
1
0
9
2
1 12
2 2
1
1
i
o
i
k
F
o
i
i
F
5
0 F
C
0
2 1
4
0
0
0
2 2
2
0 2
2
0
C
3
o
i
k
k
C
0
0
0
0
o
i
i
k
o
k
F
10
0 F
C
1
7 1
4
0
2
1
2 0
2
2 2
0
0
7
0
2
1
o
i
F
9
2
2
6 1
2
2
2
7
2
0
1
F
C
7
1
1
0
9
2
1 12
2 2
1
1
o
i
k
k
C
4
2
2
0
3
i
i
k
o
k
C
F
10
1 F
C
0
7 1
9
1
1
0
7 2 12
2
1 2 1
2
2
0
1
5
1
0
0
o
C
15
10
1
1
1
1
12 1
0
1
1
2
i
k
F
0
0
0
0
3
o
C
i
i
i
k
o
k
F
5
0 F
C
0
2 1
4
0
0
0
2 2
2
0 2
2
0
5
0
1
0
7
0
2
1
C
3
3
o
i
F
3
3
i
k
o
k
F
C
10
0 F
C
1
7 1
4
0
2
1
2 0
2
2 2
0
0
o
k
F
C
3
3
2
o
k
F
C
3
o
i
o
k
F
3
3
C
5
0
1
0
i
o
k
F
o
o
2
2
2
k
k
3
C
15
10
1
1
1
1
12 1
0
1
1
2
C
F
2
3
F
10
1 F
C
0
7 1
9
1
1
0
7 2 12
2
1 2 1
2
2
0
1
o
1
C
F
C
3
i
k
o
k
2
3
i
k
C
5
1
0
0
i
o
k
F
3
o
i
k
k
C
4
2
2
0
C
3
o
1
i
i
o
k
F
C
3
3
2
o
k
F
C
3
o
i
o
k
F
3
3
C
5
0
1
0
i
o
k
F
C
3
i
k
i
2
2
2
o
k
k
C
F
2
i
o
4
2
2
0
7
2
0
1
o
C
9
2
2
6 1
2
2
2
i
i
k
o
k
F
F
C
7
1
1
0
9
2
1 12
2 2
1
1
C
3
3
5
1
0
0
o
C
i
3
i
k
o
k
10
1 F
C
0
7 1
9
1
1
0
7 2 12
2
1 2 1
2
2
0
1
F
o
C
15
10
1
1
1
1
12 1
0
1
1
2
1
i
o
shop-chain game
k
C
F
2
2
i
i
o
F
3
0
0
0
0
o
C
i
i
k
5
0 F
C
0
2 1
4
0
0
0
2 2
2
0 2
2
0
Game Theory
F
5
0
1
0
7
0
2
1
C
3
3
i
k
o
o
i
3
k
F
C
4
2
2
0
7
2
0
1
o
C
9
2
2
6 1
2
2
2
i
k
F
F
C
7
1
1
0
9
2
1 12
2 2
1
1
C
3
3
i
k
o
5
1
0
0
o
C
subgame perfect equilibrium:
F
3
i
k
o
10
0 F
C
1
7 1
4
0
2
1
2 0
2
2 2
0
0
o
k
F
C
3
F
o
k
k
i
k
2
i
3
i
k
o
k
10
1 F
C
0
7 1
9
1
1
0
7 2 12
2
1 2 1
2
2
0
1
F
o
C
15
10
1
1
1
1
12 1
0
1
1
2
all shops play i, chain k playc c
not realistic, if many more shops to
fight
solution: shops should be uncertain
about the motives of k
wvdh
96
16
Backward Induction
r
1
2
d
1,1
1
r
r
d
d
2,2
3,3
Backward Induction
r
1
0,0
d
1,1
wvdh
r
2
d
1,1
r
1
r
d
d
2,2
3,3
1,1
t
2,0
wvdh
Game Theory
1
e
2,2
3,3
0,0
98
r
1
r
d
d
2,2
3,3
0,0
wvdh
100
Centipede
1 and 2 divide n marbles; they choose in
turn, if somebody picks two, the game is
over
2
d
2
d
99
e
d
r
1
0,0
Centipede
1
r
Backward Induction
wvdh
1
r
wvdh
97
Backward Induction
1
2
2
e
1
e
2
e
e
Intuitively correct?
1
3,3
t
t
t
t
t
t
1,2
3,1
2,3
4,2
3,4
2,0
101
e
wvdh
2
1
e
2
e
1
e
2
e
t
t
t
t
t
1,2
3,1
2,3
4,2
3,4
e
3,3
102
17
Strategic voting
Strategic voting
Maurice
Boris, Horace and Maurice determine
who can be a member of the Dead Poet
Society
first betwee A, B
then between A, N
proposal: allow Alice
counterprop: allow Bob, rather than Alice
first vote over counterprop, then over
proposal
wvdh
103
Strategic voting
winner Alice
Horace
winner Alice
Nobody
Alice
Bob
strategic voting H:
Bob
Alice
Nobody
first vote Bob!
solution… B, N
Borice
Alice
Nobody
Bob
wvdh
104
Strategic voting
Maurice
first between A, B
then between A, N
strategic voting H:
winner Alice
winner Alice
Horace
M anticipates: vote for
A
first between A, B
then between A, N
Nobody
Alice
Bob
first vote Bob!
solution…
B, N
Maurice
Bob
Alice
Nobody
Alice
Nobody
Bob
wvdh
winner Alice
Horace
winnner Alice
Nobody
Alice
Bob
utility B H M
a
3 2 2
Borice
105
Strategic voting: extensive
Bob
Alice
Nobody
b
1 1 3
n
2 3 1
Borice
Alice
Nobody
Bob
wvdh
106
Strategic voting: extensive
1
u
a
b
n
B
3
1
2
H
2
1
3
M
2
3
1
a
a
a
a
a
n
a n n n
n a n n
a a n a
b
a
b
(aaa,aaa,xyz) is Nash
a
b
b
a
a
a
b
n
a
n
n
n
n
a
n
n
n
n
b
n b b b
b n b b
n n b n
a
b
n
b
a
a
a
n
b
b
b
n
3
1
2
Game Theory
a b b b
b a b b
a a b a
a
a
a
a
a
wvdh
a
a
b
2
1
3
3
1
2
107
wvdh
a b b b
b a b b
a a b a
b
a
b
a
b
b
2
a n n n
n a n n
a a n a
3
n
a
n
n
n
n
a
n
n
n
n
b
b
n b b b
b n b b
b n b n
b
n
b
n
b
b
b
n
a
2
3
1
a
a
n
a
a
b
2
1
3
2
3
1
108
18
Strategic voting: extensive
1
(aab,aab,nnb) is not Nash!
a
a
a
H can do better: bxb
a
a
a
a
a
a
n
Pirates on an island
a
a
b
a b b b
b a b b
a a b a
b
a
b
a
b
b
2
n
a
n
n
n
n
a
n
n
n
n
b
b
3
a n n n
n a n n
a a n a
n b b b
b n b b
a n b n
b
n
b
n
b
b
b
n
a
3
1
2
Five pirates p1, .... , p5 are on an island
There is also a bag of 100 diamonds
And hence, a need to distribute them
2
3
1
2
1
3
wvdh
wvdh
109
Five Pirates: procedure
Pirates on an island
player i proposes a division Di over pi, ..., p5
with a majority for Di: so be it done
no majority for Di: pi gets shot, we move on to pi+1
Assumptions: Any pirate
Now
you are p1. What will D1 be?
wvdh
wvdh
majority rule: x wins
binary protocol: chair decides!
X
Y
Z Y
wvdh
Game Theory
X
1: x > z > y;
2: y > x > z;
3: z > y > x
40% type 1, 30% type 2, 30% type 3
majority rule: x wins
binary protocol: chair decides!
X
Z
112
Voting agenda paradox
1: x > z > y;
2: y > x > z;
3: z > y > x
40% type 1, 30% type 2, 30% type 3
X
values his life higher than 100 diamonds
values 1 diamond higher than another’s life
votes in favour of a proposal iff others are worse
111
Voting agenda paradox
110
Z
Y Z
Y
Y
Y
X
Z
X Z
X
X
113
Y
Z Y
wvdh
X
Z
X
Z
Y Z
Y
Y
Y
Z
X Z
X
114
19
Pareto dominated paradox
1: x > y > b > a
2: a > x > y > b
3: b > a > x > y
x
x
x
Pareto dominated paradox
a
b
a
y b
y
a
b
y b
x
y
wvdh
x
115
Borda protocol
1:
2:
3:
4:
5:
6:
7:
x>c>b>a
a>x>c>b
b>a>x>c
x>c>b>a
a>x>c>b
b>a>x>c
x>c>b>a
x
a
b
y b
a
y
a
b
y b
y
wvdh
116
Arrow's theorem
allocate points: 4, 3, 2, 1.
but for all, x > y !!
1: x > y > b > a
2: a > x > y > b
3: b > a > x > y
Σ: x : 22, a : 17, b: 16, c: 15
If x withdraws
withdraws::
m agents, each with preference ≤i over D
Wanted:
G(≤1, ...... , ≤m , D) = ≤
c: 15, b: 14, a:13 !!!
wvdh
117
wvdh
118
Arrow's theorem
1
2
3
4
5
completeness: x ≤ y or y ≤ x
transitivity:
if x ≤ y ≤ z, then x ≤ z
≤i
unrestricted domain: all ≤ satifsy 1 and 2
<i
Pareto:
if ∀ i, x ≤i y, then x ≤ y
independece
of
irrelevant
choices
≤'
≤
i
if ≤i is as ≤i’ regarding x and y,
then ≤ = ≤‘ regarding x and y
6
no dictator:
no i completely determines ≤
It is impossible to generate such a ≤ !
wvdh
Game Theory
i
119
20