Representation of Games 14.12 Game Theory Muhamet Yildiz 1 Game: - - PDF document

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Representation of Games 14.12 Game Theory Muhamet Yildiz 1 Game: - - PDF document

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Lecture 3 Representation of Games 14.12 Game Theory Muhamet Yildiz 1 Game: Ingredients Who are the players (decision makers)? What moves are available to each player and when? What does each player know at the time of each of his

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SLIDE 1

Lecture 3 Representation of Games

14.12 Game Theory Muhamet Yildiz

1

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SLIDE 2

Game: Ingredients

  • Who are the players (decision makers)?
  • What moves are available to each player

and when?

  • What does each player know at the time of

each of his decisions?

  • What are the outcomes and payoffs at the

end?

2

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SLIDE 3

Road Map

  • 1. Extensive form representation
  • 2. Strategy
  • 3. Normal form representation
  • 4. Mixed strategy

3

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SLIDE 4

Extensive-form representation

Definition: A tree is a set of nodes connected with directed arcs such that

1.

There is an initial node; 2. For each other node, there is one incoming arc; 3. each node can be reached through a unique path.

4

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SLIDE 5

A tree

/ ... -- ... -,

r

I

Non-terminal

I \

rm nodes

/

ll..lIe

___ ina _ I_ N _

  • _

d_ es

....

I

,

\ , \ I I I I , -,

,

5

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SLIDE 6

Extensive form - definition

Definition: A game consists of

  • a set of

players

  • a tree
  • an allocation of

each non-terminal node to a player

  • an informational partition (to be made precise)
  • a payoff for each player at each terminal node.

6

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SLIDE 7

Information set

An information set is a collection of nodes such that

  • 1. The same player is to move at each of

these nodes;

  • 2. The same moves are available at each of

these nodes. An informational partition is an allocation

  • f each non-terminal node of

the tree to an information set.

7

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SLIDE 8

A game

1 L R (2,2) 2 I

r

u

1 (0,0) 1 A p p (1,3) (3, 1) (3,3) (1, I) 8

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SLIDE 9

Another game

1

x

T B

2

L R R L

9

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SLIDE 10

The Same Game

x

1 T

B L R

10

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SLIDE 11

What is wrong?

1

x

T B

Up

L R R L

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SLIDE 12

What is wrong?

1

x

T B

3

2 L R R L

12

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SLIDE 13

What is wrong?

3 A B 13

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SLIDE 14

Strategy

A strategy of a player is a complete contingent-plan, determining which action he will take at each information set he is to move (including the information sets that will not be reached according to this strategy).

14

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SLIDE 15

Matching pennies with perfect information

2's Strategies: HH = Head if 1 plays Head, 1 Head if 1 plays Tail; HT = Head if I plays Head, Head Tail Tail if 1 plays Tail; 2 TH = Tail if 1 plays Head, 2 Head if 1 plays Tail; head tail head tail TT = Tail if 1 plays Head, Tail if 1 plays Tail. (-1,1) (1,-1) (1,-1) (-1,1) 15

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SLIDE 16

Matching pennies with perfect information

2 1

HH HT TH TT

Q Q Q Q

Head (-1,1) -1,1)

(1,-1) (1,-1)

Q Q Q Q

Tail

(1,-1) -1,1) (1,-1) (-1,1)

Head Tai 2 2 head head tail (-1 ,1) (1,-1) (1,-1) (-1,1) 16

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SLIDE 17

N

  • rmal-form representation

Definition (Normal form): A game is any list

G = (Sp ... ' Sn; up "

, ,uJ

where, for each i E N = {1,2, ... ,

n} ,

  • S; is the set of

all strategies available to i,

  • u

: SI x·· · X Sn ---t 9t is the VNM utility function of i

player i.

Assumption: G is "common knowledge".

Definition: A player i is rational iff he tries to maximize the expected value of

U ; given his beliefs.

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SLIDE 18

Chicken

~

.~

::.....

~.

:;::

, ~-
  • :
~,

(-1,-1)

(1 ,0)

(0,1)

Chicken

) (0,1) (1,0) (-1,-1) (1/2,1/2

18

Image by MIT OpenCourseWare.

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SLIDE 19

Matching pennies

Head Tail Head (-1,1) (1,-1) Tail (1,-1) (-1,1)

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SLIDE 20

Extensive v. Normal Forms

  • Extensive to Normal:
  • Find the set of

strategies for each player

  • Every strategy profile s leads to an outcome

z( s), a terminal history

  • Utility from s is u(z(s))
  • Normal to Extensive: many possibilities

20

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SLIDE 21

Matching pennies with perfect information

2 1

HH HT TH TT

Q Q Q Q

Head (-1,1) -1,1)

(1,-1) (1,-1)

Q Q Q Q

Tail

(1,-1) -1,1) (1,-1) (-1,1)

Head Tai 2 2 head head tail (-1 ,1) (1,-1) (1,-1) (-1,1) 21

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SLIDE 22

Matching pennies with imperfect information

1 2 I

Head Tail

Head Tail

Head (-1,1) (1,-1)

head tail

Tail (1,-1) (-1,1)

(-1,1) (1,-1) (1,-1) (-1,1) 22

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SLIDE 23

A game

1 A 2 a 1 a

,-,-,-

~

(1,-5)

D d (4,4) (5,2) (3,3)

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SLIDE 24

A game with nature

(5,0)

Left 1 Head 112 Right

(2,2)

Nature 0

(3,3)

112 Left Tail

2

Right

(0, -5) 24

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SLIDE 25

Mixed Strategy

Definition: A mixed strategy of a player is a probability distribution over the set of his strategies. Pure strategies: Si = {Sil ,Si2" .. ,Sik} A mixed strategy: cri: Si --* [0,1] S.t. cri(Sij) + cri(Si2) + ... + crlsik) = 1.

If

the other players play S_i =(Sj, ... , Si_j,si+j"",sn), then the expected utility of playing cri is crlSij)Ui(Sij,SJ + crlsi2) UlSi2,SJ + ... + cri(Sik) UlSik,SJ.

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SLIDE 26

MIT OpenCourseWare http://ocw.mit.edu

14.12 Economic Applications of Game Theory

Fall 2012 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.