Games and Decisions
Problem 1
Matrix
| 0.79 | 0 | 0 |
| 0 | 0.04 | 0 |
| 0 | 0 | 0.06 |
p1=0.79 p2=0.04 p3=0.06
Objective function
P=p1+p2+p3
Solution
To calculate the solution, Storm application was used.
Objective function value
P=42,93249
Strategies
X1=Y1=1.2658/42.93249=0.02948 X2=Y2=25/42.93249=0.58231 X3=Y3=16,6667/42.93249=0.38821 Value of the game = 1/42.93249 = 0,02330
Interpretation
Saddam Husain has three residences where he can spend following night. Iraq opposition knows about all of his residences. They want to kill Husain, but it is supposed to look like an accident, so they can destroy just one of his residences. If the Husain is in residence number one and the Iraq opposition will destroy this residence, the chance that Husain will be killed is 77%. If opposition will destroy residence 2 and Husain will be in this residence, the chance that he will be killed is 4%. For the third residence, the percentage is 6%. So in which residence should Husain spend the night?
Residence 1: 2,95% Residence 2: 58,23% Residence 3: 38,82%
The strategy for Iraq Opposition is the same as Husain's strategy. The best chance to kill him is to destroy residence 2.
Problem 2
Object valus
s1=279, s2=106, s3=104
Disposable amounts of investors
i1=20, i2=10
| Strategies of Investor 2 | |||||||
|---|---|---|---|---|---|---|---|
| *) | 100 | 010 | 001 | ||||
| 200 | 259.0 | 0.0 | 259.0 | 96.0 | 259.0 | 94.0 | |
| 110 | 230.5 | 134.5 | 314.0 | 48.0 | 365.0 | 94.0 | |
| 101 | 228.5 | 134.5 | 363.0 | 96.0. | 316.0 | 47.0 | |
| 020 | 86.0 | 269.0 | 86.0 | 0.0 | 86.0 | 94.0 | |
| 011 | 190.0 | 269.0 | 142.0 | 48.0 | 143.0 | 47.0 | |
| 002 | 84.0 | 269.0 | 84.0 | 96.0 | 84.0 | 0.0 | |
*) Strategies of Investor 1
There are no pure strategy equilibria in this game.
Problem 3
Object values
s1=279, s2=106, s3=104
Disposable amounts of investors
i1=20, i2=10
| Strategies of Investor 2 | |||||||
|---|---|---|---|---|---|---|---|
| *) | 100 | 010 | 001 | ||||
| 200 | 259.0 | 0.0 | 259.0 | 96.0 | 259.0 | 94.0 | |
| 110 | 230.5 | 134.5 | 314.0 | 48.0 | 365.0 | 94.0 | |
| 101 | 228.5 | 134.5 | 363.0 | 96.0 | 316.0 | 47.0 | |
| 020 | 86.0 | 269.0 | 86.0 | 0.0 | 86.0 | 94.0 | |
| 011 | 190.0 | 269.0 | 142.0 | 48.0 | 143.0 | 47.0 | |
| 002 | 84.0 | 269.0 | 84.0 | 96.0 | 84.0 | 0.0 | |
*) Strategies of Investor 1
| Strategies of Investor 2 | |||||||
|---|---|---|---|---|---|---|---|
| *) | 100 | 010 | 001 | ||||
| 110 | 230.5 | 134.5 | 314.0 | 48.0 | 365.0 | 94.0 | |
| 101 | 228.5 | 134.5 | 363.0 | 96.0 | 316.0 | 47.0 | |
| 011 | 190.0 | 269.0 | 142.0 | 48.0 | 143.0 | 47.0 | |
*) Strategies of Investor 1
| Strategies of Investor 2 | ||||
|---|---|---|---|---|
| *) | 100 | 010 | 001 | |
| 110 | 365 | 362 | 459 | |
| 101 | 363 | 459 | 363 | |
| 011 | 459 | 190 | 190 | |
*) Strategies of Investor 1
System of conditions
v(1) = 230.5 a1 + a2 = 459 v(2) = 134.5 a1 ≥ 230.5 v(1,2) = 459 a2 ≥ 134.5
Superadditive effect
v(1,2) - v(1) - v(2) = 94
Optimal division of the total pay-offs
a1 = 230.5 + 94/2 = 277.5 a2 = 134.5 + 94/2 = 181.5
The investors are supposed to choose strategies 1,0,1 and 0,1,0 (respectively 0,1,1 and 1,0,0 or 1,1,0 and 0,0,1), with additional financial transfers.
Problem 4
Oligopolists' strategy spaces
X1=<0, 1.79> X2=<0, 2.04> X3=<0, 1.06>
Costs functions
c1(X1) = x1/2 + 3 c2(X2) = 3*x2/4 + 2 c3(X3) = 5*x3/2 + 1
Price function
f(x1 + x2 + x3) = 6 - (x1 - x2 - x3)/2
Profit functions
M1(x1 ,x2 ,x3)=x1*(6 - (x1+x2+x3)/2) - (x1/2+3) M2(x1 ,x2 ,x3)=x2*(6 - (x1+x2+x3)/2) - (3*x2/4+2) M3(x1 ,x2 ,x3)=x3*(6 - (x1+x2+x3)/2) - (5*x3/2+1)
Solution (Using Excel Solver)
Final profits of all possible coalitions (using minimax method):
| Profit | X1 | X2 | X3 | |
|---|---|---|---|---|
| V(1) | 2,46845 | 1,79 | 2,04 | 1,06 |
| V(2) | 3,7222 | 1,79 | 2,04 | 1,06 |
| V(3) | 0,1183 | 1,79 | 2,04 | 1,06 |
| V(1,2) | 6,19065 | 1,79 | 2,04 | 1,06 |
| V(2,3) | 3,963013 | 1,79 | 2,04 | 0,56 |
| V(1,3) | 2,6552 | 1,79 | 2,04 | 0,69 |
| V(1,2,3) | 7,22055 | 1,79 | 2,04 | 0,00 |
Problem 5
Oligopolists' strategy spaces
X1=<0, 1.79> X2=<0, 2.04> X3=<0, 1.06>
Costs functions
c1(X1) = x1/2 + 3 c2(X2) = 3*x2/4 + 2 c3(X3) = 5*x3/2 + 1
Price function
f(x1 + x2 + x3) = 6 - (x1 - x2 - x3)/2
Profit functions
M1(x1 ,x2 ,x3)=x1*(6 - (x1+x2+x3)/2) - (x1/2+3) M2(x1 ,x2 ,x3)=x2*(6 - (x1+x2+x3)/2) - (3*x2/4+2) M3(x1 ,x2 ,x3)=x3*(6 - (x1+x2+x3)/2) - (5*x3/2+1)
Solution (Using Excel Solver)
Final profits of all possible coalitions (using minimax method)
| Profit | X1 | X2 | X3 | |
|---|---|---|---|---|
| V(1) | 2,46845 | 1,79 | 2,04 | 1,06 |
| V(2) | 3,7222 | 1,79 | 2,04 | 1,06 |
| V(3) | 0,1183 | 1,79 | 2,04 | 1,06 |
| V(1,2) | 6,19065 | 1,79 | 2,04 | 1,06 |
| V(2,3) | 3,963013 | 1,79 | 2,04 | 0,56 |
| V(1,3) | 2,6552 | 1,79 | 2,04 | 0,69 |
| V(1,2,3) | 7,22055 | 1,79 | 2,04 | 0,00 |
Profit distribution
a1+a2+a3=7,22055 a1+a2≥6,19065 a1+a3≥2,6552 a2+a3≥3,963013 a1≥2,46845 a2≥3,7222 a3≥0,1183
Results (using Excel Solver)
a1=2,46845 a2=3,7222 a3=1,0299
The core of oligopoly is not empty.
Comment: Cooperation is most profitable for oligopoly number 3, because they get approximately 8 times more then from the common profit that he would get if he stayed alone. Oligopoly number 1 and 2 gets same profit with and without cooperation.
Problem 6
Oligopolists' strategy spaces
X1=<0, 1.79> X2=<0, 2.04> X3=<0, 1.06>
Costs functions
c1(X1) = x1/2 + 3 c2(X2) = 3*x2/4 + 2 c3(X3) = 5*x3/2 + 1
Price function
f(x1 + x2 + x3) = 6 - (x1 - x2 - x3)/2
Profit functions
M1(x1, x2, x3)=x1*(6 - (x1+x2+x3)/2) - (x1/2+3) M2(x1, x2, x3)=x2*(6 - (x1+x2+x3)/2) - (3*x2/4+2) M3(x1, x2, x3)=x3*(6 - (x1+x2+x3)/2) - (5*x3/2+1)
Solution (Using Excel Solver)
Final profits of all possible coalitions (using minimax method):
| Profit | X1 | X2 | X3 | |
|---|---|---|---|---|
| V(1) | 2,46845 | 1,79 | 2,04 | 1,06 |
| V(2) | 3,7222 | 1,79 | 2,04 | 1,06 |
| V(3) | 0,1183 | 1,79 | 2,04 | 1,06 |
| V(1,2) | 6,19065 | 1,79 | 2,04 | 1,06 |
| V(2,3) | 3,963013 | 1,79 | 2,04 | 0,56 |
| V(1,3) | 2,6552 | 1,79 | 2,04 | 0,69 |
| V(1,2,3) | 7,22055 | 1,79 | 2,04 | 0,00 |
The Shapley Value
Oligopolist 1:v(1) = 2,46 v(1,2) - v(2) = 6,19 - 3,72 = 2,46 v(1,3) - v(3) = 2,65 - 0,11 = 2,54 v(1,2,3) - v(2,3) = 7,22 - 3,96 = 3,26
The mean value of these contributions: 10,72 / 4 = 6,28
Oligopolist 2:v(2) = 3,72 v(1,2) - v(1) = 6,19 - 2,46 = 3,73 v(2,3 - v(3) = 3,96 - 0,11 = 3,85 v(1,2,3) - v(1,3) = 7,22 - 2,65 = 4,57
The mean value of these contributions: 15,87 / 4 = 3,96
| Profit | X1 | X2 | X3 | |
|---|---|---|---|---|
| V(1) | 2,46845 | 1,79 | 2,04 | 1,06 |
| V(2) | 3,7222 | 1,79 | 2,04 | 1,06 |
| V(3) | 0,1183 | 1,79 | 2,04 | 1,06 |
| V(1,2) | 6,19065 | 1,79 | 2,04 | 1,06 |
| V(2,3) | 3,963013 | 1,79 | 2,04 | 0,56 |
| V(1,3) | 2,6552 | 1,79 | 2,04 | 0,69 |
| V(1,2,3) | 7,22055 | 1,79 | 2,04 | 0,00 |
v(3) = 0,11 v(1,3) - v(1) = 2,65 - 2,46 = 0,19 v(2,3) - v(2) = 3,96 - 3,72 = 0,24 v(1,2,3) - v(1,2) = 7,22 - 6,19 = 1,03
The mean value of these contributions: 1,57 / 4 = 0,39
The Shapley Value (vector): 6,28; 3,96; 0,39