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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
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DefinitionDefinition Typical Normal Form GamesTypical Normal Form Games Rational BehaviorRational Behavior Iterated DominanceIterated Dominance
Cournot CompetitionCournot Competition
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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,….)).
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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)
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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.
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(1, -1) (-1,1)
(-1, 1)
(1, -1)
Head
Tail
Head
Tail
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Players: 1, 2 Strategy sets: {Head, Tail}, {Head, Tail} Strategy profiles:
(Head, Head), (Head, Tail),
(Tail, Head), (Tail, Tail).
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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,
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(2, 1) (0, 0)
(0, 0)
(1, 2)
Football
Opera
Football
Opera
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Where Husband selections are rows wife’s are columns
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(-1, -1) (-10,0)
(0, -10)
(-3, -3)
Cooperate Defect
Cooperate
Defect
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Intuition: I would choose Defect to avoid 10 years
of prison
Note: This is the most famous example of Normal Form Games
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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).
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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)
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What is a rational behavior?
The answer depends on my beliefs of my opponent’s actions and my decisions!
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)(),( '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
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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
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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
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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
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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.
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(2, 2)
(1, 1)
(4, 0)
(1, 2)
(4, 1)
(3, 5)
L M R
D
U
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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
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(2, 2)
(4, 0)
(1, 2)
(3, 5)
L R
D
U
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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
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As a rational player, player 2 chooses L. A “Rational” game yields the result (U, L).
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ii SS 0
Step 1 Define:
Step 2 Define:
)},(),(,{ '00'01iiiiiiiiiiiii ssussuSsSsSsS
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kiki SS
1
Step k+1:define:
)},(),(,{ ''1iiiiii
kii
kii
kii
ki ssussuSsSsSsS
Step : Let
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The computation must stop after finite number of steps if the strategy sets are finite.
An example of Iterated Dominance Deletion:
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(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
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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)
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(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
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Step3: S1
2 = {B, C} (A dominated by B, D
dominated by C) S2
2 = {B, C} (D dominated by B)
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(3, 2)
(2, 1)
(2, 2)
(1, 5)
B C
B
C
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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.
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Definition: G is solvable by pure Iterated Deletion of Strict Dominance if S contains a single strategy profile.
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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:
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(1, 1)
(0, 0)
(1, 1)
(2, 1)
(0, 0)
(2, 1)
L R
T
M
B
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- 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).
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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).
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We can use the Iterated Strict Dominance Deletion to obtain a maximum profit strategy profile for the two competitive firms.
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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
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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:
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0),(
1
211 dq
qqdu
0),(
1
211 dq
qqdu
And
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c
qifqc
otherwise
q2
2
22
0
1
c
qifqc
otherwise
q1
1
22
0
2
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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:
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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.
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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
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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.
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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:
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[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 !