Literature Game Theory - University of Liverpool...

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

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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....

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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

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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

£!!!!!!!

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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

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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)

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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

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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

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Nash equilibrium

� 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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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’

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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)

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Example: BoS (N.eq)

� N = {1,2}

� A1 = {B,S}

� A2= {B,S}


� 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*)

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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

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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

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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

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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


� 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


� 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...



xn: miny u(xn,y) = 3

..: .......... .. = .. max = 5 for x* = x4

Game Theory 7

wvdh 37

maxminimizers

� solves for 1


� min{u1(x,y)|y∈A2} � |x ∈A1} = 5

� solves for 2


� 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


� min{u1(x,y)|y∈A2} � |x ∈A1} = 5

� solves for 2


� 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!


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}


� 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


� 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


� 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



� 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



� 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


� 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


� 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


� 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)



� 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)



� 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


� strategic voting H:� first vote Bob!

� solution… B, N

Bob

Nobody

AliceBob

Alice

Nobody

Nobody

Alice

Bob

Borice

Horace

Maurice

wvdh 105

Strategic voting



� 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


� 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


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


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 ≤ !

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