Normal Form Games, Normal Form Games, Rationality and IteratedRationality and Iterated Deletion of Dominated StrategiesDeletion of Dominated Strategies

Instructor: Professor Piotr Gmytrasiewicz

Presented By: BIN WU

Date:11/20/2002

DefinitionDefinition Typical Normal Form GamesTypical Normal Form Games Rational BehaviorRational Behavior Iterated DominanceIterated Dominance

Cournot CompetitionCournot Competition

A Normal Form Game is a game of complete information in which there is a list of n players, numbered 1, 2, … n. Each player has a strategy set, Si,

and a utility function

 

 

  In such a game each player simultaneously selects a

move si Si and receives Ui((s1, s2,….)).

A list of players

D={1,2,….n} A list of finite strategy sets

{S1, S2,…Sn} Set of strategy profiles

S=S1 S2 … Sn

Payoff functions

ui: S1S2 .. Sn R (i =1, 2 .. n)

Normal form games with two players and finite strategy

sets can be represented in normal form, a matrix where

the rows each stand for an element of S1 and the column

for an element of S2.

Each cell of the matrix contains an ordered pair which states the payoffs for each player. That is, the cell i, j contains (u1(si, sj), u2(si, sj)) where si is the i-th element of

S1 and sj is the j-th element of S2.

(1, -1) (-1,1)

 (-1, 1)

 (1, -1)

Head

Tail

Head

Tail

Players: 1, 2 Strategy sets: {Head, Tail}, {Head, Tail} Strategy profiles:

(Head, Head), (Head, Tail),

(Tail, Head), (Tail, Tail).

 

Payoff functions:

o u1(Head, Head) = 1, u1(Head, Tail) = -1,

u1(Tail, Head) = -1, u1(Tail, Tail) = 1

o u2(Tail, Head) = 1, u2(Tail, Tail) = -1

u2(Head, Head) = -1, u2(Head, Tail) = 1,

(2, 1) (0, 0)

 (0, 0)

 (1, 2)

Football

Opera

Football

Opera

Where Husband selections are rows wife’s are columns

(-1, -1) (-10,0)

 (0, -10)

 (-3, -3)

Cooperate Defect

Cooperate

Defect

  Intuition: I would choose Defect to avoid 10 years

of prison

Note: This is the most famous example of Normal Form Games

Two firms each chooses output level qi to

maximize his profit, the price of a single product is determined by the total output of the two firms,

i.e., p(q1+q2) and each firm suffers the cost ci(qi).

Players list: D= {1, 2} Strategy sets: S1 = S2 = R+

Utility functions:

u1(q1, q2) = q1 p(q1, q2)-c1(q1)

u2(q1, q2) = q2 p(q1, q2)-c2(q2)

 

What is a rational behavior?

The answer depends on my beliefs of my opponent’s actions and my decisions!

 

  

)(),( 'maxiii

sii

imizesi sssus

i

Definition Player i performs a rational strategy si with beliefs i if

where s-i denotes a profile of strategy choices of all other

players

For example in the prisoner’s dilemma, suppose I am player 1 and if my beliefs of my opponent’s behaviors are  

1 (Cooperate) = 0.5

1 (Defect) = 0.5

 

If I choose to cooperate, my expected payoff will be:  u1(Cooperate, Cooperate) 1Cooperate)

+ u1(Cooperate, Defect) 1 (Defect)

= -1 0.5 + (-10) 0.5 = -5.5

If I choose to defect, then my expected payoff will be: u1(Defect, Cooperate) 1(Cooperate)

+ u1(Defect, Defect) 1 (Defect)

= 0 0.5 + (-3) 0.5 = -1.5

 Thus the rational behavior of mine would be to Defect based on my belief functions

Definition: Strategy si is strictly dominated

for player i if there is some si’ Si such that

 ui(si’, s-i) > ui(si, s-i)

For all s-i S-i .

 Based on above definition, a rational player i

should not choose si no matter what his beliefs

are.

  (2, 2)

  (1, 1) 

  (4, 0)

  (1, 2)

  (4, 1)

  (3, 5)

L M R

D

U

 

If player 1 and player 2 are both rational players and they both know that the other is.  Player 2 should never choose action M because M is dominated.    Player 1 knows that player 2 is rational

  (2, 2)

  (4, 0)

  (1, 2)

  (3, 5)

L R

D

U

Player 1 never chose action D because D is dominated.Player 2 knows that player 1 is rational  

    (2, 2)

  (4, 0)

L R

U

As a rational player, player 2 chooses L.  A “Rational” game yields the result (U, L). 

ii SS 0

Step 1 Define:

Step 2 Define:

)},(),(,{ '00'01iiiiiiiiiiiii ssussuSsSsSsS

kiki SS

1

Step k+1:define:

)},(),(,{ ''1iiiiii

kii

kii

kii

ki ssussuSsSsSsS

Step : Let

  

The computation must stop after finite number of steps if the strategy sets are finite. 

An example of Iterated Dominance Deletion:   

  (5, 2)

  (2, 6)

  (1, 4)

  (0, 4)

  (0, 0)

  (3, 2)

  (2, 1)

  (1, 1)

  (7, 0)

  (2, 2)

  (1, 5)

  (5, 1)

  (9, 5)

  (1, 3)

  (0, 2)

  (4, 8)

A B C D

A

B

C

D

 

Solution with Iterated Dominance Deletion:Step1:  S1

0 = {A, B, C, D}

S20 = {A, B, C, D}

Step 2: S1

1 = {A, B, C, D}

S21 = {B, C, D} (A dominated by D)

  (2, 6)

  (1, 4)

  (0, 4)

  (3, 2)

  (2, 1)

  (1, 1)

  (2, 2)

  (1, 5)

  (5, 1)

  (1, 3)

  (0, 2)

  (4, 8)

B DC

B

C

D

A

Step3: S1

2 = {B, C} (A dominated by B, D

dominated by C) S2

2 = {B, C} (D dominated by B)

  (3, 2)

  (2, 1)

  (2, 2)

  (1, 5)

B C

B

C

Step 4: S1

3 = {B} (C dominated by B)

S23 = {B} (C dominated by B)

The resulting strategy profile is (B, B). Luckily, this problem is solvable with IDD. 

 

Definition: G is solvable by pure Iterated Deletion of Strict Dominance if S contains a single strategy profile.

  

Why not weak dominance deletion? If a game is solvable by strict dominance deletion, a consistent strategy profile is generated regardless of the order you eliminate strategies; however, weak dominance deletion may yield different results if you choose different orders. See the following example:

  (1, 1)

  (0, 0)

  (1, 1)

  (2, 1)

  (0, 0)

  (2, 1)

L R

T

M

B

-         if we first delete T then L, the final output of

utilities will be nothing other than (2, 1)

-  if we first delete B then R, the final utilities will be (1, 1).

Two firms each chooses output level qi to

maximize his profit, the price of a single product is determined by the total output of the two firms, i.e., p(q1+q2) and each firm suffers the cost ci(qi).

We can use the Iterated Strict Dominance Deletion to obtain a maximum profit strategy profile for the two competitive firms. 

 

Assume the market price is determined by the following function:

  

Assume the cost per product is a constant c for both firms

) (2 1q q p

The profits for firm 1 and firm 2 are   1211211 ),( cqqqqqqu

2212212 ),( cqqqqqqu  

To achieve the maximum profit, each firm must satisfy the first-order derivative condition: 

0),(

1

211 dq

qqdu

0),(

1

211 dq

qqdu

And

c

qifqc

otherwise

q2

2

22

0

1

c

qifqc

otherwise

q1

1

22

0

2

we denote q1 and q2 computed above as the “best response function” of the opponent’s output level: q1 = BR(q2) and q2

= BR(q1).

 Now we perform the Iterated deletion:

 

Step1: both firms can set any output level: S1

0 = S20 = R+

Step2: S1

1 = S21 = [0, (-c)/2]

This is because each firm knows that his opponent has an output equal to or greater than 0, each firm must select a strategy within this range.

Step3:

Let’s denote 0 as q- and (-c)/2 as q+, since each firm knows the other’s output is in the range [q-, q+], he must narrow his strategy set to [BR(q+), BR(q-)]—any strategy outside of this range will for sure be strictly dominated by one inside. Thus

S12 = S2

2 = [BR(q+), BR(q-)]

 Step k:

Iterate until S1k and S2

k converge to a same point, q.

 The strategy profile (q, q) is the solution generated by the Iterated Deletion of Strict Dominance.

Two companies both produce personal computers, let = $5000 (a price for the first available PC on the market), =0.5 (free if the total output reaches 10000), c = $895 (the cost is really cheap). Let’s randomly choose s1

0= 100 and s20=200 (because the

next step will guarantee the strategy sets to fall in the range [q-, q+]). The Iterated Deletion of Strict Dominance yields the following result:

[4005.00, 2102.50][3053.75, 2578.13][2815.94, 2697.03]

[2756.48, 2726.76][2741.62, 2734.19][2737.91, 2736.05][2736.98, 2736.51][2736.74, 2736.63][2736.69, 2736.66][2736.67, 2736.66][2736.67, 2736.67][2736.67, 2736.67]

 Well, to produce 2737 PCs each will be the best choice !