Game Theory 1
Game Theory
Wiebe van der Hoek
Computer Science
University of Liverpool
United Kingdom
wvdh 2
Literature
� Fun and Games
� Ken Binmore
� A Course in Game Theory
� Martin Osborne en Ariel Rubinstein
wvdh 3
What is it all about?
� game = interaction� traffic
� supermarket
� employee, employer, board, union
� student / teacher
� judge and lawyers
� George en Osama
� marriage and career
wvdh 4
What is it all about?
� terminology of chess and bridge
� logic and systematics of interaction
� analysis takes you from irrational issues
� 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....
wvdh 5
Surprise and Paradox
� does it make sense� to vote for a candidate you fancy least?
� for a general, to toss a coin?
� in poker, place a maximal bid with the worst cards?
� to throw some goods away before starting to negotiate about them
� to sell your house to the second best bidder? YES
!!
YES!!
YES!!
YES!!
YES!!
wvdh 6
Strategic voting
� Boris, Horace and Maurice determine who can be a member of the Dead Poet Society
� proposal: allow Alice
� amendment: allow Bob, rather than Alice
� first vote over amendment, then over proposal
Game Theory 2
wvdh 7
Strategic voting
Bob
Nobody
Alice
Bob
Alice
Nobody
Nobody
Alice
Bob
Borice Horace Maurice
wvdh 8
Strategic voting
� first between A, B� winner Alice
� then between A, N� winner Alice
� strategic voting H:� first vote for Bob!
� result … B, N
� M anticipates: vote for A
Bob
Nobody
AliceBob
Alice
Nobody
Nobody
Alice
Bob
Borice
Horace
Maurice
wvdh 9
History
� Von Neumann and Morgenstern
� The Theory of Games and EconomicBehaviour (1944)
� ideas from economics and mathematics
� initially very optimistic, then draw-back
� revival since 1970’s
� Nash, Aumann, Shapley, Selten, Harsanyi
wvdh 10
Game Theory
� theory of decision makers� 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
wvdh 11
Abstracts from `emotions’
� suppose you’re offered £ 1.000
� you make a deal with the first person you encounter: (1.000-x,x) x = 1, 2 ...� if he accepts: (you,person) get (1.000-x,x)
� else (0,0)
� only money counts, and that is known
� both are rational: prefer y+1 over y
� what will you offer?ONE
£!!!!!!!
wvdh 12
Game Theory
� theory of decision makers� are rational
� reason strategically
� players anticipate on knowledge and expectations about behaviour of other decision makers
Game Theory 3
wvdh 13
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
� a ≥i b ⇔ ui(a) ≥ ui(b)� also called payoff-function
� (although not the same)
wvdh 14
Representation strategic games
� N = {1,2}
� A1 = {T,B}
� A2= {L,R}
� u1(T) =w1, etcz1‚z2y1‚y2
x1‚x2w1‚w2T
B
L R
wvdh 15
Representation strategic games
� Interpretation� one-shot
� simultaneous
� independent
� utilities are known
� not the choice of others
z1‚z2y1‚y2
x1‚x2w1‚w2T
B
L R
wvdh 16
Example: BoS
� N = {1,2}
� A1 = {B,S}
� A2= {B,S}
� u1, u2 see figure� B: Bach
� S: Strawinsky
� Battle of the Sexes
1,20,0
0,02‚1B
S
B S
wvdh 17
Profiles
� 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
wvdh 18
Profiles: example
� 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)
� (x-6,x’6) = ((25,22,20,12,0,22),26) would have been better for player 6, given the profiles of the others
Game Theory 4
wvdh 19
Nash equilibrium
� John Nash
� equilibrium (“solution”)
� every player is rational
� ever player plays optimally
� no use to devert individually
� not an algorithmic approach
wvdh 20
Nash equilibrium (definitie)
� 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) � `no player i can improve in a*, if the other players still play a*-i’
wvdh 21
Nash equilibrium (alternative)
� 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)}
� a* is N.eq iff ∀i∈N a*i∈ Bi(a*-i)
wvdh 22
Example: BoS (N.eq)
� N = {1,2}
� A1 = {B,S}
� A2= {B,S}
� u1, u2 see figure� B: Bach
� S: Strawinsky
� two equilibria:
� (bach,bach) and
� (strawinsky, strawinsky)
1,20,0
0,02‚1B
S
B S
wvdh 23
Example: coordination game
� Mozart or Mahler?
� same preferences
� two equilibria:
� (Mozart,Mozart) and
� (Mahler,Mahler)
� N.eq right concept?
1,10,0
0,02,2Mo
Ma
Mo Ma
wvdh 24
Pareto Efficiency
� (Mozart,Mozart) and
� (Mahler,Mahler)
� N.eq right concept?
� (2,2) is (strongly) Pareto efficient:
� ¬∃x¬∃y (x,y) > (2,2)
1,10,0
0,02,2Mo
Ma
Mo Ma
Game Theory 5
wvdh 25
Pareto Optimality (definition)
� Given G = ⟨ N‚(Ai)‚(≥i)⟩� a* = (a1, a2 ,... , an) ∈ A1x A2 x ... x Anis Pareto optimal iff
� ∀i∈N ∀b* ∈ A1x A2 x ... x An (a*) ≥i (b*)
wvdh 26
Example: prisoner’s dilemma
� C: cooperate with the other, keep silent
� D: justify against the other -1,-13,-2
-2,30,0C
D
C D
wvdh 27
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
-1,-13,-2
-2,30,0C
D
C D
wvdh 28
Example: hawk-dove
� preference:� hawkish if other is dovish
� dovish if oher is hawkish
� N.eq: (Dove,Hawk)
� and (Hawk,Dove)
0,04,1
1,43,3D
H
D H
wvdh 29
Example: Matching Pennies
� 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
1,-1-1,1
-1,11,-1H
T
H T
wvdh 30
SCSG
� 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:
� with u1 and u2 we have
� u1 + u2 = 0
Game Theory 6
wvdh 31
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)
� action y* ∈ A2 is maxminimizer for 2:
� ∀y∈A2 min u2(x,y*) ≥2 min u2(x,y)x∈A1
y∈A2 y∈A2
x∈A1
wvdh 32
maxminimizer
� action x* ∈ A1 is maxminimizer for 1:
� ∀x∈A1 min u1(x*,y) ≥1 min u1(x,y)
� 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
y∈A2 y∈A2
wvdh 33
Equilibria and maxminimizers
� (x*,y*) is N.eq for G, iff:
� x* is a maxminimizer for 1;
� y* is a maxminimizer for 2
� maxxminyu1(x,y) =
� minymaxxu1(x,y) =u1(x*,y*)
wvdh 34
maxminimizers
� solves for 1
� maxxminy u1(x,y) =� max{
� min{u1(x,y)|y∈A2} � |x ∈A1} =
4,-45,-56,-63,-34,-4x6
3,-32,-24,-45,-53,-3x5
5,-57,-75,-58,-86,-6x4
3,-33,-34,-42,-25,-5x3
4,-46,-64,-45,-53,-3x2
1,-11,-13,-32,-22,-2x1
y5y4y3y2y1...
wvdh 35
maxminimizers
� solves for 1
� maxxminy u1(x,y) =� max{
� min{u1(x,y)|y∈A2} � |x ∈A1} =
4,-45,-56,-63,-34,-4x6
3,-32,-24,-45,-53,-3x5
5,-57,-75,-58,-86,-6x4
3,-33,-34,-42,-25,-5x3
4,-46,-64,-45,-53,-3x2
1,-11,-13,-32,-22,-2x1
y5y4y3y2y1...
x1: miny u(x1,y) = 1
wvdh 36
maxminimizers
� solves for 1
� maxxminy u1(x,y) =� max{
� min{u1(x,y)|y∈A2} � |x ∈A1} =
4,-45,-56,-63,-34,-4x6
3,-32,-24,-45,-53,-3x5
5,-57,-75,-58,-86,-6x4
3,-33,-34,-42,-25,-5x3
4,-46,-64,-45,-53,-3x2
1,-11,-13,-32,-22,-2x1
y5y4y3y2y1...
x1: miny u(x1,y) = 1
x2: miny u(x2,y) = 3
xn: miny u(xn,y) = 3
..: .......... .. = .. max = 5 for x* = x4
Game Theory 7
wvdh 37
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} =
4,-45,-56,-63,-34,-4x6
3,-32,-24,-45,-53,-3x5
5,-57,-75,-58,-86,-6x4
3,-33,-34,-42,-25,-5x3
4,-46,-64,-45,-53,-3x2
1,-11,-13,-32,-22,-2x1
y5y4y3y2y1...
wvdh 38
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} = -5!
� Equilibrium (5,-5)
4,-45,-56,-63,-34,-4x6
3,-32,-24,-45,-53,-3x5
5,-57,-75,-58,-86,-6x4
3,-33,-34,-42,-25,-5x3
4,-46,-64,-45,-53,-3x2
1,-11,-13,-32,-22,-2x1
y5y4y3y2y1...
wvdh 39
security level vs equilibria
� consider cooperative game G
� (2,2) looks like `the optimal’ solution
� security strategy of1 is r, gives 1!
� Nash equilibria?
1,11,1
0,02,2l
r
L R
wvdh 40
security level vs equilibria
� consider game G
� (2,2) looks like `the optimal’ solution
� security strategy of 1 is r, gives 1!
� Nash equilibria?
1,11,1
0,02,2l
r
L R
wvdh 41
bimatrix games
� m x n matrix
� 1 has strategies s1and s2, 2 has t1, t2and t3
� payoff π1(si,tj) = ij� π2(si,tj) = (i-2)(j-2)
6,04,02,0
3,-12,01,1s1
s2
t1 t2 t3
wvdh 42
bimatrix games
� m x n matrix
� 1 has strategies s1and s2, 2 has t1, t2and t3
� Nash equilibrium (σ,τ):� ∀s,t π1(σ,τ) ≥ π1(s,τ) � ∀s,t π2(σ,τ) ≥ π2(σ,t)
6,04,02,0
3,-12,01,1s1
s2
t1 t2 t3
Game Theory 8
wvdh 43
domination
� strategy sd of 1 dominates sistrongly:
� ∀t π1(sd,t) > π1(si,t)� and weakly if:
� ∀t π1(sd,t) ≥ π1(si,t)� ∃t π1(sd,t) > π1(si,t)
� t1 dominates t2weakly
6,04,02,0
3,-12,01,1s1
s2
t1 t2 t3
wvdh 44
Iterated elimination
� s2 of 1 strongly dominates s1
� No further (weak) domination: all is left are Nash Equilibria
� this is not generally so
6,04,02,0
3,-12,01,1s1
t1 t2 t3
wvdh 45
Order of elimination
3,33,3BF
3,33,3BE
1,10,2AF
1,12,0AE
DC3,33,3BF
3,33,3BE
1,10,2AF
1,12,0AE
DC
3,33,3BF
3,33,3BE
1,10,2AF
1,12,0AE
DC
lost equilibrium!
wvdh 46
elimination: conclusions
� strict strategies: no problem
� with weakly dominated strategies:
� some equilibria can get lost
� order of elimination is important
wvdh 47
Example: BoS
� N = {1,2}
� A1 = {B,S}
� A2= {B,S}
� u1, u2 see figure� B: Bach
� S: Strawinsky
� Battle of the Sexes
1,20,0
0,02‚1B
S
B S
no dominant strategies
still (two) Nash equilibira
wvdh 48
Ex: coordination game
� Mozart of Mahler?
� Same preference
1,10,0
0,02,2Mo
Ma
Mo Ma
No dominant strategy
still two Nash equilibria
Game Theory 9
wvdh 49
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)
-1,-13,-2
-2,30,0C
D
C D
wvdh 50
Ex: Matching Pennies
� Head and Tail
� if different, 1 pays a Pound to 2; if the same, 2 pays a Pound to 1 1,-1-1,1
-1,11,-1H
T
H T
no dominant strategy
No pure Nash equilibrium
wvdh 51
mixed strategies
� 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
wvdh 52
security level and strategy
� maximin for 1:
� 2, via s2� note: (s2,t2) is a saddlepoint
� then 2 is also security level of player 1
0,40,39,0s3
3,22,04,6s2
7,31,20,1s1
t3t2t1
wvdh 53
security level and strategy
� maximin for 1: 2 via s3� (s3,t2) is not a saddlepoint
� security level of 1 is indeed not 2, but 22/3
� How?4,02,33,7s3
3,00,22,1s2
0,96,41,0s1
t3t2t1
wvdh 54
mixed strategies
� remove strictly dominated strategy s2
� 2 has no pure dominating strategy, but, .....
4,02,33,7s3
3,00,22,1s2
0,96,41,0s1
t3t2t1
Game Theory 10
wvdh 55
mixed strategies� 2 has no pure dominating strategy, but, .....
� q = (1/2,0,1/2) dominates t2strongly!
� π2(s1,q) = � (1/2)�0 + (1/2)�9 =4.5 > 4
� π2(s3,q) = � (1/2)�7 + (1/2)�0 =3.5 > 3
4,02,33,7s3
3,00,22,1s2
0,96,41,0s1
t3t2t1
wvdh 56
mixed strategies� 2 has no pure dominating strategy, but, .....
� q = (1/2,0,1/2) dominates t2strongly!
� π2(s1,q) = � (1/2)�0 + (1/2)�9 =4.5 > 4
� π2(s3,q) = � (1/2)�7 + (1/2)�0 =3.5 > 3
4,02,33,7s3
3,00,22,1s2
0,96,41,0s1
t3t2t1
wvdh 57
mixed strategies
� after iterated elimination
� what is security level of 1?
� suppose 1 plays mixed strategy (1-r,0,r)
4,02,33,7s3
3,00,22,1s2
0,96,41,0s1
t3t2t1s
wvdh 58
mixed strategies
� 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
4,02,33,7s3
3,00,22,1s2
0,96,41,0s1
t3t2t1s
wvdh 59
mixed strategies
� E1(r)= 1(1-r) + 3r = 1 + 2r
� E2(r)= 6(1-r) + 2r = 6 – 4r
� E3(r)= 0(1-r) + 4r = 4r
� m(r) = min{E1,E2, E3}
0
1
2
3
4
5
6
1r →
1/2 5/63/4
max for r = 5/6payoff is E1(r) = 22/3
wvdh 60
Ex: Matching Pennies
� Head and Tail
� if different, 1 pays a Pound to 2; if the same, 2 pays a Pound to 1 1,-1-1,1
-1,11,-1H
T
H T
no dominant strategy
No Nash equilibrium
Game Theory 11
wvdh 61
Ex: Matching Pennies
� Suppose r plays (0.5,0.5), and
� c plays (0.5,0.5)
1,-1-1,1
-1,11,-1H
T
H T
wvdh 62
Ex: Matching Pennies
� 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 +� 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
1,-1-1,1
-1,11,-1H
T
H T
wvdh 63
Ex: Matching Pennies
� 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 +� 0.5 ⋅ 0.5 ⋅ 1 + 0.5 ⋅ 0.5 ⋅ -1 = 0
� no other Nash: � 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
1,-1-1,1
-1,11,-1H
T
H T
Game Theory: Part II: Extensive Games
Wiebe van der Hoek
Computer Science
University of Liverpool
United Kingdom
wvdh 65
Rules of the Game
root
terminal
When?
wvdh 66
Rules of the Game
When?
r
l
l
r
L
RR
R
MLLWhat?
I
I
II II
II
Who?w w
w
w
w
l
lHow much?
Game Theory 12
wvdh 67
Example: tictactoe
ox
I
ox
o
II
ox
o
x
I
ox
o
x o
oxo
II
oxo xo
II
oxo xo
x o
II
oxo x
I
oxo xo
x
I
oxo xo
x o
xI
I
II
o o
x
ox
o x
o
x
o
xo
o
x
o o
x
o
x
I
II
II
o
I
I
wo
xo xo
x o
x o
d
o
II I
wvdh 68
Strategies
� A pure strategie for player p specifies for every decision note of p what he will do there
� If all players choose such a strategy, the outcome of the game is determined
wvdh 69
Strategies
� A pure strategie for player p specifies for every decision note of p what he will do there
� Strategies for I:r
l
l
r
L
RR
R
MLL
I
I
II II
IIw w
w
w
w
l
lllllllll lrlrlrlr rlrlrlrl rrrrrrrr
b
a
c d
e
wvdh 70
Strategies
� A pure strategie for player p specifies for every decision note of p what he will do there
� Strategies for II:
LLL, LLR, LML, LMR, LRL, LRR
RLL, RLR, RML, RMR, RRL, RRR
r
l
l
r
L
RR
R
MLL
I
I
II II
IIw w
w
w
w
l
l
b
a
c d
e
wvdh 71
Strategy profile
r
l
l
r
L
RR
R
MLL
I
I
II II
IIw w
w
w
w
l
l
b
a
c d
e
Example:
[lr,RMR]
wvdh 72
extensive to strategic
1
1
2A B
C D
E F
a b
c
d
dd
dd
cb
caAE
AF
BE
BF
DC
dd
cb
caAE
AF
B
DC
reduced strategic form
Game Theory 13
wvdh 73
Equilibria: example
� Nash equilibria?
� via strategic form:1
2A B
L R
0,0 2,1
1,2
1,21,2
2,10,0A
B
L R
wvdh 74
Strategy profile
r
l
l
r
L
RR
R
MLL
I
I
II II
IIw w
w
w
w
l
l
b
a
c d
e
LL
w
w
w
w
RR RRRLMRMLLRLLRLMRMLLR
wwwwwwwwwwwrr
wwwwwlwwwwlrl
llllllwwwwwlr
llllllwwwwwll
RL G
strategic form of G extensive form of G
a strategy profile: [rr,RLL]
wvdh 75
Backward InductionI
w
II
w l
II
wwII
I
wl
G
w
I
w
II
wII
I
wl
G
l
I
w
II
wwII
w
G
l
I
w
II
ww
G
l
I
w
G
l
G I
w
wvdh 76
backward induction
I
w
II
w l
II
wwII
I
wl
G
wvdh 77
I can win
LLL
w
w
w
w
LRR RRRRRLRMRRMLRLRRLLLRLLMRLMLLLR
wwwwwwwwwwwrr
wwwwwlwwwwlrl
llllllwwwwwlr
llllllwwwwwll
again: rr is winning strategy,since that row only contains a w
wvdh 78
extensive games: definitions
� extensive games: G = ⟨ N‚H‚P,(≥i)⟩� N: set of players
� H histories: ∅, (ak)k=1..K (may be infinite)� closed under prefixes
� terminals Z: no successor or infinite
� P: H\Z → N player who is to move
� ≥i: preference relation on Z
Game Theory 14
wvdh 79
extensive games: definitions
� N: set of players
� H histories: ∅, (ak)k=1..K (may be infinite)� 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
wvdh 80
Subgame perfect solutions
� extensive games: Γ = ⟨ N‚H‚P,(≥i)⟩� 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)⟩� all continuations of h
wvdh 81
Subgames
� history h
� subgame Γ(h)1
1
2A B
C D
E F
a b
c
d
wvdh 82
subgame perfect N.-eq
� let Γ = ⟨N‚H‚P,(≥i)⟩ extensive� s* is N.-eq if ∀i∀si O(s-i*,si*) ≥i O(s-i*,si)
� s* is subgame perfect N.-eq if� ∀i∀h∈H\Z (P(h)=i ⇒� Oh(s-i*|h,si *|h) ≥i|hO(s-i*|h,si))� for all strategies si for i in Γ(h)
� s*|h is N.-eq for all Γ(h)
wvdh 83
Equilibria: Example
� Nash equilibria?1
2A B
L R
0,0 2,1
1,2
wvdh 84
equilibria (ctd)
� so: (A,R) and (B,L)
� interpretation (B,L):
� given that 2 plays Lafter A, 1 better choose B
� intuitive?
� what is optimal for 1?
1
2A B
L R
0,0 2,1
1,2
1
2A B
L R
0,0 2,1
1,2
Game Theory 15
wvdh 85
equilibria (ctd)
� 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
1
2A B
L R
0,0 2,1
1,2
1
2A B
L R
0,0 2,1
1,2
wvdh 86
equilibria (ctd)
� 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!
1
2A B
L R
0,0 2,1
1,2
1
2A B
L R
0,0 2,1
1,2
wvdh 87
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)
wvdh 88
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)
n
C
i o
k
F 5,1
2,20,0
k
F C
i o
2
k
F C
i o
2
10110
1511112
112
F C
i o
k
3
7120
12121
9122
F C
i o
3
k
5100
101017
10
2
F C
i o
k
3
5010
100117
012
F C
i o
k
3
2020
7021
4022
F C
i o
3
k
0000
50012
00
2
F C
i o
k
3
4220
92216
222
F C
i o
k
3
2200
7201
4202
F C
i o
3
k
7110
12211
9212
F C
i o
3
k
k
F C
i o
2
k
F C
1
i o
k
F C
i o
2
k
F C
i o
2
10110
1511112
112
F C
i o
k
3
7120
12121
9122
F C
i o
3
k
5100
101017
10
2
F C
i o
k
3
5010
100117
012
F C
i o
k
3
2020
7021
4022
F C
i o
3
k
0000
50012
00
2
F C
i o
k
3
4220
92216
222
F C
i o
k
3
2200
7201
4202
F C
i o
3
k
7110
12211
9212
F C
i o
3
k
k
F C
i o
2
k
F C
1
i o
Game Theory 16
k
F C
i o
2
k
F C
i o
2
10110
1511112
112
F C
i o
k
3
7120
12121
9122
F C
i o
3
k
5100
101017
10
2
F C
i o
k
3
5010
100117
012
F C
i o
k
3
2020
7021
4022
F C
i o
3
k
0000
50012
00
2
F C
i o
k
3
4220
92216
222
F C
i o
k
3
2200
7201
4202
F C
i o
3
k
7110
12211
9212
F C
i o
3
k
k
F C
i o
2
k
F C
1
i o
k
F C
i o
2
k
F C
i o
2
10110
1511112
112
F C
i o
k
3
7120
12121
9122
F C
i o
3
k
5100
101017
10
2
F C
i o
k
3
5010
100117
012
F C
i o
k
3
2020
7021
4022
F C
i o
3
k
0000
50012
00
2
F C
i o
k
3
4220
92216
222
F C
i o
k
3
2200
7201
4202
F C
i o
3
k
7110
12211
9212
F C
i o
3
k
k
F C
i o
2
k
F C
1
i o
k
F C
i o
2
k
F C
i o
2
10110
1511112
112
F C
i o
k
3
7120
12121
9122
F C
i o
3
k
5100
101017
10
2
F C
i o
k
3
5010
100117
012
F C
i o
k
3
2020
7021
4022
F C
i o
3
k
0000
50012
00
2
F C
i o
k
3
4220
92216
222
F C
i o
k
3
2200
7201
4202
F C
i o
3
k
7110
12211
9212
F C
i o
3
k
k
F C
i o
2
k
F C
1
i o
k
F C
i o
2
k
F C
i o
2
10110
1511112
112
F C
i o
k
3
7120
12121
9122
F C
i o
3
k
5100
101017
10
2
F C
i o
k
3
5010
100117
012
F C
i o
k
3
2020
7021
4022
F C
i o
3
k
0000
50012
00
2
F C
i o
k
3
4220
92216
222
F C
i o
k
3
2200
7201
4202
F C
i o
3
k
7110
12211
9212
F C
i o
3
k
k
F C
i o
2
k
F C
1
i o
k
F C
i o
2
k
F C
i o
2
10110
1511112
112
F C
i o
k
3
7120
12121
9122
F C
i o
3
k
5100
101017
10
2
F C
i o
k
3
5010
100117
012
F C
i o
k
3
2020
7021
4022
F C
i o
3
k
0000
50012
00
2
F C
i o
k
3
4220
92216
222
F C
i o
k
3
2200
7201
4202
F C
i o
3
k
7110
12211
9212
F C
i o
3
k
k
F C
i o
2
k
F C
1
i o
wvdh 96
shop-chain game
� subgame perfect equilibrium:
� 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
Game Theory 17
wvdh 97
Backward Induction
1 12
1,1 2,2 3,3
r r r
d d d
0,0
wvdh 98
Backward Induction
1 12
1,1 2,2 3,3
r r r
d d d
0,0
wvdh 99
Backward Induction
1 12
1,1 2,2 3,3
r r r
d d d
0,0
wvdh 100
Backward Induction
1 12
1,1 2,2 3,3
r r r
d d d
0,0
wvdh 101
Centipede
� 1 and 2 divide n marbles; they choose in turn, if somebody picks two, the game is over
11 12 22
2,0 1,2 3,1 2,3 4,2 3,4
3,3e e e e e e
t t t t t t
wvdh 102
Centipede
� Intuitively correct?
11 12 22
2,0 1,2 3,1 2,3 4,2 3,4
3,3e e e e e e
t t t t t t
Game Theory 18
wvdh 103
Strategic voting
� Boris, Horace and Maurice determine who can be a member of the Dead Poet Society
� proposal: allow Alice
� counterprop: allow Bob, rather than Alice
� first vote over counterprop, then over proposal
wvdh 104
Strategic voting
� first betwee A, B� winner Alice
� then between A, N� winner Alice
� strategic voting H:� first vote Bob!
� solution… B, N
Bob
Nobody
AliceBob
Alice
Nobody
Nobody
Alice
Bob
Borice
Horace
Maurice
wvdh 105
Strategic voting
� first between A, B� winner Alice
� then between A, N� winner Alice
� strategic voting H:� first vote Bob!
� solution…
� B, N
� M anticipates: vote forA
Bob
Nobody
AliceBob
Alice
Nobody
Nobody
Alice
Bob
Borice
Horace
Maurice
wvdh 106
Strategic voting
� first between A, B� winner Alice
� then between A, N� winnner Alice
Bob
Nobody
AliceBob
Alice
Nobody
Nobody
Alice
Bob
Borice
Horace
Maurice
132n
311b
223a
MHButility
wvdh 107
Strategic voting: extensive
132n
311b
223a
MHBuaab
aaa
aba
baa
bbb
bba
bab
abb
a b
nnb
nnn
nbn
bnn
bbb
bbn
bnb
nbb
aan
aaa
ana
naa
nnn
nna
nan
ann
a nb
312
213
231
wvdh 108
Strategic voting: extensive
aab
aaa
aba
baa
bbb
bba
bab
abb
a b
nnb
nnn
nbb
bnn
bbb
bbn
bnb
nbb
aan
aaa
ana
naa
nnn
nna
nan
ann
a nb
312
213
231
1
2 3
(aaa,aaa,xyz) is Nash
Game Theory 19
wvdh 109
Strategic voting: extensive
aab
aaa
aba
baa
bbb
bba
bab
abb
a b
nnb
nnn
nba
bnn
bbb
bbn
bnb
nbb
aan
aaa
ana
naa
nnn
nna
nan
ann
a nb
312
213
231
1
2 3
H can do better: bxb
(aab,aab,nnb) is not Nash!
wvdh 110
Pirates on an island
� Five pirates p1, .... , p5 are on an island
� There is also a bag of 100 diamonds
� And hence, a need to distribute them
wvdh 111
Five Pirates: procedure
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
Now you are p1. What will D1 be? wvdh 112
Pirates on an island
� Assumptions: Any pirate
� 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
wvdh 113
Voting agenda paradox
� 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
XX YY
XX YY ZZZZ
XX
XX YY YY
ZZ
ZZ XX XX
YY
YY
ZZ
ZZ
binary protocol: chair decides!binary protocol: chair decides!
wvdh 114
Voting agenda paradox
� 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
XX YY
XX YY ZZZZ
XX
XX YY YY
ZZ
ZZ XX XX
YY
YY
ZZ
ZZ
binary protocol: chair decides!binary protocol: chair decides!
Game Theory 20
wvdh 115
Pareto dominated paradox� 1: x > y > b > a
� 2: a > x > y > b
� 3: b > a > x > y
xx bb
xx bb yyyy
aa bb
aa bb yyyy
xx aa
wvdh 116
Pareto dominated paradox
� 1: x > y > b > a
� 2: a > x > y > b
� 3: b > a > x > y
xx bb
xx bb yyyy
aa bb
aa bb yyyy
xx aa
but for all, x > y !!but for all, x > y !!
wvdh 117
Borda protocol
� allocate points: 4, 3, 2, 1.� 1: x > c > b > a
� 2: a > x > c > b
� 3: b > a > x > c
� 4: x > c > b > a
� 5: a > x > c > b
� 6: b > a > x > c
� 7: x > c > b > a
Σ: Σ: x : 22, a : 17, b: 16, c: 15x : 22, a : 17, b: 16, c: 15
If x withdrIf x withdrawsaws::
c: 15, b: 14, a:13 !!!c: 15, b: 14, a:13 !!!
wvdh 118
Arrow's theorem
� m agents, each with preference ≤i over D
� Wanted: � G(≤1, ...... , ≤m , D) = ≤
wvdh 119
Arrow's theorem
1 completeness: x ≤ y or y ≤ x
2 transitivity: if x ≤ y ≤ z, then x ≤ z
3 unrestricted domain: all ≤ satifsy 1 and 2
4 Pareto: if ∀ i, x ≤i y, then x ≤ y
5 independece of irrelevant choices if ≤i is as ≤i’ regarding x and y,
then ≤ = ≤‘ regarding x and y
6 no dictator: no i completely determines ≤
ii≤≤
ii<<
ii≤≤ ''
ii≤≤
It is impossible to generate suchIt is impossible to generate suchaa ≤ !
