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1 General Game Playing Game Theory Lecture 7: Game Theory Games in Normal Form Equilibria Dominance Mixed Strategies

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1General Game Playing Game Theory

Lecture 7:Game Theory

Games in Normal Form

Equilibria

Dominance

Mixed Strategies

2General Game Playing Game Theory

Example (1)

3General Game Playing Game Theory

Example (2)

4General Game Playing Game Theory

Example (3)

5General Game Playing Game Theory

Competition and Cooperation

The “pathological” assumption says that opponents choose the joint move that is most harmful for us.

This is usually too pessimistic for non-zerosum games or games with n > 2 players. A rational player chooses the move that's best for him rather than the one that's worst for us.

6General Game Playing Game Theory

Example

Game Theory gives the answer

7General Game Playing Game Theory

Strategies

Game model:

S – set of statesA1, ..., An – n sets of actions, one for each playerl1, ..., ln – where li A⊆ i S, the legality relationsg1, ..., gn – where gi S ⊆ ℕ, the goal relations

A strategy xi for player i maps every state to a legal move for i

xi : S → Ai ( such that (xi (S),S) ∈ li )

(Note that even for Chess the number of different strategies is finite. They outnumber the atoms in the universe, though ...)

8General Game Playing Game Theory

Strategies for Agent-Battle

Example strategy for Ag1:

{At(Ag1,1,1), At(Ag

2,5,1), At(Ag

3,5,5)} → Go(East)

{At(Ag1,1,1), At(Ag

2,5,1), At(Ag

3,4,5)} → Go(North)

⋮{At(Ag

1,1,4), At(Ag

2,3,3), At(Ag

3,3,5)} → Go(North)

Similar for Ag2, Ag

3

9General Game Playing Game Theory

Towards the Normal Form of Games

Each n-tuple of strategies directly determines the outcome of a match.

Example:

Start with 7 coins. Players A and B take turn in removing one or two coins. Whoever takes the last coin wins.

Strategy Player A:

{7 # 2, 6 # 2, 5 # 2, 4 # 1, 3 # 1, 2 # 2, 1 # 1}

Strategy Player B:

{7 # 2, 6 # 1, 5 # 2, 4 # 1, 3 # 2, 2 # 2, 1 # 1}

Outcome: (0, 100)

10General Game Playing Game Theory

Games in Normal Form

An n-player game in normal form is an n+1-tuple

Г = (X1, ..., Xn ,u)

where Xi is the set of strategies for player i and

u = (u1, ..., un): Xi → ℕi

are the utilities of the players for each n-tuple of strategies.

n

i=1

11General Game Playing Game Theory

Roshambo

Rock Scissors Paper

Rock

50 0 100

50 100 0

Scissors

100 50 0

0 50 100

Paper

0 100 50

100 0 50

2-player games are often depicted as matrices

12General Game Playing Game Theory

2-Finger-Morra

1 Finger 2 Fingers

1 Finger

30 90

70 10

2 Fingers

90 0

10 100

13General Game Playing Game Theory

Battle of the Sexes

Ballgame Opera

Ballgame

3 2

4 2

Opera

1 4

1 3

14General Game Playing Game Theory

Prisoner's Dilemma

Cooperate Defect

Cooperate

3 4

3 1

Defect

1 2

4 2

15General Game Playing Game Theory

Equilibria

Let Г = (X1, ..., Xn,u) be an n-player game.

(x1*, ..., xn*) ∈ X1 ... Xn equilibrium

if for all i = 1, ..., n and all xi ∈ Xi

ui(x1*, ..., xi-1*, xi, xi+1*, ..., xn*) ≤ ui(x1*, ..., xn*)

An equilibrium is a tuple of optimal strategies: No player has a reason to deviate from his strategy, given the opponent's strategies.

16General Game Playing Game Theory

Full Cooperation

a b

a

4 1

4 2

b

2 3

3 1

17General Game Playing Game Theory

Battle of the Sexes

Ballgame Opera

Ballgame

3 2

4 2

Opera

1 4

1 3

(Note that the outcome for both players is bad if they choose to play different equilibria.)

18General Game Playing Game Theory

Cooperation

a b

a

4 2

4 2

b

1 3

1 3

(Note that the concept of an equilibrium doesn't suffice to achieve the best possible outcome for both players.)

19General Game Playing Game Theory

Prisoner's Dilemma

Cooperate Defect

Cooperate

3 4

3 1

Defect

1 2

4 2

(Note that the outcome which is better for both players isn't even an equilibrium!)

20General Game Playing Game Theory

Dominance

A strategy x ∈ Xi dominates a strategy y ∈ Xi if

ui(x1, ..., xi-1, x, xi+1, ..., xn) ≥ ui(x1, ..., xi-1, y, xi+1, ..., xn)

for all (x1, ..., xi-1, xi+1, ..., xn) ∈ X1 ... Xi-1 Xi+1 ... Xn.

A strategy x ∈ Xi strongly dominates a strategy y ∈ Xi if

x dominates y and y does not dominate x.

Assume that opponents are rational: They don't choose a

strongly dominated strategy.

21General Game Playing Game Theory

Removing Strongly Dominated Strategies

a b

a

4 1

4 2

b

2 3

3 1

22General Game Playing Game Theory

Iterated Dominance

a b c d e

a 10 7 6 9 8

b 10 4 6 9 5

c 9 7 9 8 8

d 2 6 4 3 7

Let a zero-sum game be given by

23General Game Playing Game Theory

Iterated Dominance (2)

a b c d e

a 10 7 6 9 8

b 10 4 6 9 5

c 9 7 9 8 8

d 2 6 4 3 7

24General Game Playing Game Theory

Iterated Dominance (3)

a b c d e

a 10 7 6 9 8

c 9 7 9 8 8

25General Game Playing Game Theory

Iterated Dominance (4)

b c

a 7 6

c 7 9

26General Game Playing Game Theory

(60,50) Player 1

(40,40) (60,50) (20,60) Player 2

(75,25) (40,40) (50,30) (80,40) (40,40) (60,50) (35,60) (20,60) (10,50)

Game Tree Search with Dominance

27General Game Playing Game Theory

(40,40) ≤ 40? ≤ 35?

(75,25) (40,40) (50,30) (80,40) (40,40) (60,50) (35,60) (20,60) (10,50)

The - - Principle does not Apply

28General Game Playing Game Theory

The Need to Randomize: Roshambo

Rock Scissors Paper

Rock

50 0 100

50 100 0

Scissors

100 50 0

0 50 100

Paper

0 100 50

100 0 50

This game has no equilibrium

29General Game Playing Game Theory

2-Finger-Morra

1 Finger 2 Fingers

1 Finger

30 90

70 10

2 Fingers

90 0

10 100

This game, too, has no equilibrium

30General Game Playing Game Theory

Mixed Strategies

Let (X1, ..., Xn, u) be an n-player game, then its mixed extension is

Г = (P1, ..., Pn, (e1, ..., en))

where for each i=1, ..., n

Pi = {pi: pi probability measure over Xi}

and for each (p1, ..., pn) P∈ 1 ... Pn

ei(p1, ..., pn) = ∑ ... ∑ ui(x1, ..., xn) * p1(x1) * ... * pn(xn)x1∈X1 xn∈Xn

31General Game Playing Game Theory

Existence of Equilibria

Nash's Theorem.

Every mixed extension of an n-player game

has at least one equilibrium.

32General Game Playing Game Theory

Roshambo

Rock Scissors Paper

Rock

50 0 100

50 100 0

Scissors

100 50 0

0 50 100

Paper

0 100 50

100 0 50

The unique equilibrium is

13

,13

,13 ,13 ,

13

,13

33General Game Playing Game Theory

2-Finger-Morra

1 Finger 2 Fingers

1 Finger

30 90

70 10

2 Fingers

90 0

10 100

The unique equilibrium is

(p1*, p2

*) =

with e1(p1*, p2

*) = 46 and e2(p1*, p2

*) = 54

35, 2

5 , 35, 2

5

34General Game Playing Game Theory

Computing the Equilibrium

Solution known for 2-player, zero-sum games:

Let aij be the goal value for player 1 for the pair (i, j) of pure strategies.

For player 1, minimize under conditions (j)

si 0 (

i)

For player 2, maximize under conditions (i)

tj 0 (

j)

Normalization gives the final answer.

∑i

s i

∑j

t j

∑i

aij⋅s i≥1

∑j

aij⋅t j≤1

35General Game Playing Game Theory

Example: 2-Finger-Morraa

11 = 70 a

12 = 10 a

21 = 10 a

22 = 100

minimize s1 + s

2 under conditions 70⋅s

1 + 10⋅s

2 1

10⋅s1 + 100⋅s

2 1

s1 0 s

2 0

maximize t1 + t

2 under conditions 70⋅t

1 + 10⋅t

2 1

10⋅t1 + 100⋅t

2 1

t1 0 t

2 0

Solutions: s1 = 0.01304 s

2 = 0.00870

t1 = 0.01304 t

2 = 0.00870

Normalization gives p1* = = (0.6, 0.4)

p2* = = (0.6, 0.4)

s1

s1s2

, s2

s1s2

t1

t1t 2

, t2

t1t 2

36General Game Playing Game Theory

Then p1 = strongly dominates p1' = (0,1,0).

Hence, for all (pa', pb', pc') ∈ P1 with pb' > 0 there exists a dominating strategy (pa, 0, pc) ∈ P1.

Iterated Row Dominance for Mixed Strategies

a b c

a 10 0 8

b 6 4 4

c 2 8 7

Let a zero-sum game be given by

12,0 , 1

2

37General Game Playing Game Theory

Iterated Row Dominance for Mixed Strategies (2)

a b c

a 10 0 8

b 6 4 4

c 2 8 7

Now p2 = dominates p2' = (0,0,1). 12, 1

2,0

38General Game Playing Game Theory

Iterated Row Dominance for Mixed Strategies (3)

a b c

a 10 0 8

c 2 8 7

The unique equilibrium is 38,0 , 5

8 , 12, 1

2,0 .

39General Game Playing Game Theory

Challenges

From a game theoretic point of view, modeling simultaneous moves as a sequence of our move followed by the joint moves of our opponents is incorrect.

How to modify the node expansion?

How to compute equilibria and mixed strategies?

How to model (and coin against) “stupid” opponents, e.g. who always choose Rock in Roshambo?

40General Game Playing Game Theory

The Floor is Yours!