# 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)   2,468451,792,041,06
V(2)   3,72221,792,041,06
V(3)   0,11831,792,041,06
V(1,2)   6,190651,792,041,06
V(2,3)   3,9630131,792,040,56
V(1,3)   2,65521,792,040,69
V(1,2,3)   7,220551,792,040,00

## 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)2,468451,792,041,06
V(2)3,72221,792,041,06
V(3)0,11831,792,041,06
V(1,2)6,190651,792,041,06
V(2,3)3,9630131,792,040,56
V(1,3)2,65521,792,040,69
V(1,2,3)7,220551,792,040,00

### 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) 2,468451,792,041,06
V(2) 3,72221,792,041,06
V(3) 0,11831,792,041,06
V(1,2) 6,190651,792,041,06
V(2,3) 3,9630131,792,040,56
V(1,3) 2,65521,792,040,69
V(1,2,3) 7,220551,792,040,00

### 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) 2,468451,792,041,06
V(2) 3,72221,792,041,06
V(3) 0,11831,792,041,06
V(1,2) 6,190651,792,041,06
V(2,3) 3,9630131,792,040,56
V(1,3) 2,65521,792,040,69
V(1,2,3) 7,220551,792,040,00
Oligopolist 3:
```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