程序代写代做代考 game C Normal Form Representation of a Game

Normal Form Representation of a Game
In the normal-form representation of a game, each player simultaneously chooses a strategy and the combination of strategies chosen by players determines a payo↵ for each player.
Three ingredients
Set of Players
Set of Strategies Available to Each Player
A Payo↵ Function that assigns a payo↵ to each player for every possible strategy profile
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Normal Form, Strict Dominance and Nash Equilibrium

Normal Form Representation
Prisoners’ Dilemma
P1 (Row Player)
P2 (Column Player)
D
C
D
2,2
10,0
C
0,10
8,8
D refers to ”Defect” and C refers to ”Cooperate” Set of Players P1 and P2
Set of Strategies Si = {D,C} for i = 1,2
Saltuk Ozerturk (SMU)
Normal Form, Strict Dominance and Nash Equilibrium

Normal Form Representation
A payo↵ function that assigns a payo↵ to each player for every possible strategy profile
P2 P1
D
C
D
2,2
10,0
C
0,10
8,8
u1(D,D) = 2
u2(D,D) = 2
u1(D,C) = 10
u2(D,C) = 0
u1(C,D) = 0
u2(C,D) = 10
u1(C,C) = 8
u2(C,C) = 8
Saltuk Ozerturk (SMU)
Normal Form, Strict Dominance and Nash Equilibrium

Strict Dominance
What is so special about the Prisoners’ Dilemma? P2
P1
(C,C) most socially desirable outcome.
Can players achieve (C,C) as an ”equilibrium” outcome when they choose actions to maximize own payo↵?
Based on a simple ”rationality requirement” the answer is no.
Saltuk Ozerturk (SMU)
D
C
D
2,2
10,0
C
0,10
8,8
Normal Form, Strict Dominance and Nash Equilibrium

Strict Dominance
Rationality Requirement: A rational player never uses a strictly dominated strategy
D
C
D
2,2
10,0
C
0,10
8,8
u1(D,C) = 10 > u1(C,C) = 8 u1(D,D) = 2 > u1(C,D) = 0
Regardless of what P2 does, playing D is strictly better than playing C for P1.We say that D strictly dominates C for P1.
Saltuk Ozerturk (SMU)
Normal Form, Strict Dominance and Nash Equilibrium

Strict Dominance
Rationality Requirement: A rational player never uses a strictly dominated strategy
D
C
D
2,2
10,0
C
0,10
8,8
u1(D,C) = 10 > u1(C,C) = 8 u1(D,D) = 2 > u1(C,D) = 0
Regardless of what P2 does, playing D is strictly better than playing C for P1.We say that D strictly dominates C for P1.
D strictly dominates C for P2 as well.
Saltuk Ozerturk (SMU)
Normal Form, Strict Dominance and Nash Equilibrium

Definition
Definition of Strict Dominance in a 2-Player Game
Consider two actions x and y for P1.
Let a2 2 S2 denote a generic action in action set S2 for P2. We say that x strictly dominates y for P1 if
u1(x,a2) > u1(y,a2) for every a2 2 S2. and denote the strict dominance relationship with
x y. Saltuk Ozerturk (SMU)
Normal Form, Strict Dominance and Nash Equilibrium

Strict Dominance
L
C
R
T
8,4
6,2
7,3
M
10,6
3,5
0,4
B
4,8
5,6
5,7
Observe that L C because
u2(T,L) = 4 > u2(T,C) = 2 u2(M,L) = 6 > u2(M,C) = 5 u2(B,L) = 8 > u2(B,C) = 6.
Similarly, verify that L R. Since L strictly dominates every other strategy, we call L as the strictly dominant strategy for P2.
Saltuk Ozerturk (SMU)
Normal Form, Strict Dominance and Nash Equilibrium

Strict Dominance
Another example
L
C
R
T
8,4
3,5
4,3
M
10,6
6,7
5,4
B
4,4
5,6
-2,4
Saltuk Ozerturk (SMU)
Normal Form, Strict Dominance and Nash Equilibrium

Strict Dominance
Another example
L
C
R
T
8,4
3,5
4,3
M
10,6
6,7
5,4
B
4,4
5,6
-2,4
Verify formally that C R and C L. Hence C is a strictly dominant strategy for P2.
Verify formally that M T and M B. Hence M is a strictly dominant strategy for P1.
We refer to (M,C) a strictly dominant strategy equilibrium.
Saltuk Ozerturk (SMU)
Normal Form, Strict Dominance and Nash Equilibrium

Strict Dominance
By definition, a strictly dominant strategy is the unique ”best response”.
Observe that
BR1(L) = M BR1(C) = M BR1(R) = M BR2(T) = C BR2(M) = C BR2(B) = C
Saltuk Ozerturk (SMU)
L
C
R
T
8,4
3,5
4,3
M
10,6
6,7
5,4
B
4,4
5,6
-2,4
Normal Form, Strict Dominance and Nash Equilibrium

Nash Equilibrium
L
M
R
U
8,4
6,5
4,5
D
10,6
6,7
5,4
In the above example, D weakly dominates U. We denote this as D % U.
Also observe that M L and M % R.
We refer to (D,M) as a weakly dominant strategy equilibrium.
Saltuk Ozerturk (SMU)
Normal Form, Strict Dominance and Nash Equilibrium

Nash Equilibrium
Many games are not dominance solvable in the sense that players do not have a strictly or weakly dominant strategy.
Hence, we require a stronger solution concept that produces tighter predictions.
Saltuk Ozerturk (SMU)
Normal Form, Strict Dominance and Nash Equilibrium

Nash Equilibrium
Definition of Nash Equilibrium (NE) in a 2-Player Game
We say that a strategy profile (x⇤,y⇤) is a Nash Equilibrium if x⇤ 2 BR1(y⇤)
and
y⇤ 2 BR2(x⇤).
We only require x⇤ to be a best response to y⇤ not the unique
best response.
Similarly we only require y⇤ to be a best response to x⇤ not the best response.
Saltuk Ozerturk (SMU)
Normal Form, Strict Dominance and Nash Equilibrium

Nash Equilibrium
L
R
U
2,1
2,1
D
0,0
2,0
(U,L) is a NE because U = BR1(L) and L 2 BR2(U). Can you find other NE?
Saltuk Ozerturk (SMU)
Normal Form, Strict Dominance and Nash Equilibrium

Nash Equilibrium
Observation: If (x⇤,y⇤) is a strictly dominant strategy equilibrium, then it is also a unique NE
Proof: If (x⇤,y⇤) is a strictly dominant strategy equilibrium, then x⇤ is strictly dominant for P1. This implies that
u1(x⇤,y⇤) > u1(a1,y⇤) for every a1 2 S1 ) x⇤ = BR1(y⇤). Similarly, y⇤ is strictly dominant for P2. This implies that
u2(x⇤,y⇤) > u1(x⇤,a2) for every a2 2 S2 ) y⇤ = BR2(x⇤). But then if x⇤ = BR1(y⇤) and y⇤ = BR2(x⇤), we conclude
that (x⇤, y⇤) is a unique NE. Saltuk Ozerturk (SMU)
Normal Form, Strict Dominance and Nash Equilibrium

Nash Equilibrium
L
C
R
T
8,4
3,5
4,3
M
10,6
6,7
5,4
B
4,4
5,6
-2,4
Verify that the strictly dominant strategy equilibrium (M,C) is the unique NE.
Saltuk Ozerturk (SMU)
Normal Form, Strict Dominance and Nash Equilibrium

Nash Equilibrium
L
M
R
U
8,4
6,5
4,5
D
10,6
6,7
5,4
Verify that the weakly dominant strategy equilibrium (U,M) is also a NE but it is not a unique NE.
Saltuk Ozerturk (SMU)
Normal Form, Strict Dominance and Nash Equilibrium

Nash Equilibrium
Two research firms, Firm 1 and Firm 2 simultaneously choose how much time to spend on research to develop a new drug. Firm 1 chooses x1 0 and Firm 2 chooses x2 0.
Saltuk Ozerturk (SMU)
Normal Form, Strict Dominance and Nash Equilibrium

Nash Equilibrium
The two firms’ payo↵ functions are given by u1(x1, x2) =
u2(x1, x2) =
Saltuk Ozerturk (SMU)
10x1
if x1 > x2
5x1
if x1 = x2
x1
if x1 < x2 10x2 if x2 > x1
5x2
if x1 = x2
x1
if x1 < x2 Normal Form, Strict Dominance and Nash Equilibrium Nash Equilibrium ui(xi,xj) = Is the strategy pair (x1 = 5, x2 = 5) a NE? No because u1(5,5) = 0 < u1(6,5) = 4 To verify that a strategy profile is not a NE, it is sucient to find a profitable devitiation from that profile for any of the two players. Saltuk Ozerturk (SMU) 10xi ifxi >xj
5xi
ifxi =xj
xi
ifxi xj
5xi
ifxi =xj
xi
ifxi xj
5xi
ifxi =xj
xi
ifxi xj
5xi
ifxi =xj
xi
ifxi