GAMES OF STRATEGY

Games of Strategy

Games of Strategy

Chapter 9 - U1

The expected utility for Jack is dependent on his luck, which will be accumulated through applying the formula:

(U good * P good) + (U bad * P bad)

Therefore in the coming year the expected utility for Jacks would be:

Expected U = (v250,000 * 0.5) + (v90,000 * 0.5)

Expected U = (500 * 0.5) + (300 * 0.5)

Expected U = 300 + 250

Expected U = 400

The above equation can be used to find similar levels of Jack's Utility and Income yields:

Utility = U, Income = I, & U = vI

400 = vI

I = 4002

I = 160,000

The guaranteed amount of Income and Jack's expected Utility would be same when expected Income from investment is $ 160,000.

Luck-outcome pairs of Jack & Janet from four possible ways:

The table below demonstrates the Pairs for luck-outcome & 0.5 Probability of occurrence for each possibility:

Pairs for Luck-outcome (Janet & Jack)

Probability

1

Janet good, Jack good

0.5 * 0.5 = 0.25

2

Janet bad, Jack good

0.5 * 0.5 = 0.25

3

Janet good, Jack bad

0.5 * 0.5 = 0.25

4

Janet bad, Jack bad

0.5 * 0.5 = 0.25

Janet & Jacks Expected Utility under the above arrangement

The following table demonstrates the pooled earnings of Janet & Jack, which are equally split:

1

0.25 * v500,000/2

= 125

2

0.25 * v340,000/2

= 103

3

0.25 * v340,000/2

= 103

4

0.25 * v180,000/2

= 75

Expected Utility for Janet & Jack = 125 + 103 + 103 + 75 = 406

The definite amount of Income that yields the same Utility as Janet's or Jack's expected Income from investing can be found using the equation for Utility:

Utility = U, Income = I, & U = vI

406 = vI

I = 4062

I = 164,836

The definite amount of similar Income and expected Utility for Janet and Jack would be same when expected Income from investment is $ 164,836.

Luck-outcome pairs of Jack, Janet & Chrissy from four possible ways:

The table below demonstrates the Pairs for luck-outcome & 0.5 Probability of occurrence for each possibility:

Pairs for Luck-outcome (Chrissy, Janet & Jack)

Probability

1

Chrissy good, Janet good, Jack good

0.5 * 0.5 * 0.5 = 0.125

2

Chrissy good, Janet good, Jack bad

0.5 * 0.5 * 0.5 = 0.125

3

Chrissy good, Janet bad, Jack good

0.5 * 0.5 * 0.5 = 0.125

4

Chrissy bad, Janet good, Jack good

0.5 * 0.5 * 0.5 = 0.125

5

Chrissy good, Janet bad, Jack bad

0.5 * 0.5 * 0.5 = 0.125

6

Chrissy bad, Janet bad, Jack good

0.5 * 0.5 * 0.5 = 0.125

7

Chrissy bad, Janet good, Jack bad

0.5 * 0.5 * 0.5 = 0.125

8

Chrissy bad, Janet bad, Jack bad

0.5 * 0.5 * 0.5 = 0.125

Chrissy, Janet, & Jacks Expected Utility under the above arrangement

The following table demonstrates the pooled earnings of Chrissy, Janet, & Jack, which are equally split:

1

0.125 * v750,000/3

= 62.5

2

0.125 * v590,000/3

= 55.43

3

0.125 * v590,000/3

= 55.43

4

0.125 * v590,000/3

= 55.43

5

0.125 * v430,000/3

= 47.32

6

0.125 * v430,000/3

= 47.32

7

0.125 * v430,000/3

= 47.32

8

0.125 * v270,000/3

= 37.5

Expected Utility for Chrissy, Janet & Jack = 62.5 + 55.43 + 55.43 + 55.43 + 47.32 + 47.32 + 47.32 + 37.5 = 408.25

The definite amount of Income that yields the same Utility as Chrissy, Janet's or Jack's expected Income from ...