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决策学代写-MAST30022-Assignment 4

时间：2022-08-16

MAST30022 Decision Making

Assignment 4, Semester 1 2022

Due time: 4pm, Friday 26 August.

Name:

Student ID:

• To complete this assignment, you need to write your solutions into the blank answer

spaces following each question in this assignment PDF.

If you have a printer (or can access one), then you must print out the assignment

template and handwrite your solutions into the answer spaces.

If you do not have a printer but you can figure out how to annotate a PDF using

an iPad/Android tablet/Graphics tablet or using Adobe Acrobat, then annotate your

answers directly onto the assignment PDF and save a copy for submission.

Failing both of these methods, you may handwrite your answers as normal on blank

paper and then scan for submission (but note that you will thereby miss valuable

practice for the exam process). In that case, however, your document should have

the same length as the assignment template otherwise Gradescope will reject your

submission. So you will need to add as many blank pages as necessary to reach that

criterion.

Scan your assignment to a PDF file using your mobile phone (we recommend Cam-

Scanner App), then upload by going to the Assignments menu on Canvas and submit

the PDF to the GradeScope tool by first selecting your PDF file and then clicking on

‘Upload PDF’.

– it is a single pdf with all pages in correct template order and the correct way up,

and with any blank pages with additional working added only at the end of the

template pages;

– has all pages clearly readable;

– has all pages cropped to the A4 borders of the original page and is imaged from

directly above to avoid excessive ’keystoning’

These requirements are easy to meet if you use a scanning app on your phone and take

some care with your submission - please review it before submitting to double check

you have satisfied all of the above requirements.

MAST30022 Semester 2, 2022 Assignment 1 1

• The submission deadline is 4pm Melbourne time on Friday 26 August. You

have longer than of the normal one week to complete this assignment. Late submission

within 20 hours after the deadline will be penalised by 5% of the total available marks

for every hour or part thereof after the deadline. After that, the Gradescope submission

channel will be closed, and your submission will no longer be accepted. We recommend

you submit at least a day before the due date to avoid any technical delays. If there

are extenuating, eg medical circumstances, contact the Subject Coordinator, Dr Mark

Fackrell.

• There are 4 questions, of which 2 randomly chosen questions will be marked. Note

you are expected to submit answers to all questions, otherwise a mark penalty will

apply.

• Working and reasoning must be given to obtain full credit. Give clear and concise

explanations. Clarity, neatness, and style count.

∗ ∗ ∗

1

MAST30022 Semester 2, 2022 Assignment 1 2

1. Consider the game in extensive form shown in Figure 1 in which Player 1 owns vertices

a, d, e, f , and g, and Player 2 owns vertices b and c.

a

(2, −1)

P1

b c

d e f gP1 P1 P1 P1

P2 P2

(0, 3) (1, −1) (−2, 3) (1, 1) (−2, 4) (−3, 1)

Figure 1: Game in extensive form for Question 1

For vertex x ∈ {a, b, c, d, e, f, g}, use the label Lx if the left hand edge is chosen, Rx if

the right hand edge is chosen, and Cx if there is only one possible move.

(a) Redraw the game in extensive form labelling all the edges accordingly.

2

MAST30022 Semester 2, 2022 Assignment 1 3

(b) Describe the strategy set for each player, give the normal form of the game, and

find any pairs of pure strategies in equilibrium if the game is of

(i) perfect information;

3

MAST30022 Semester 2, 2022 Assignment 1 4

(ii) imperfect information and Player 2 cannot distinguish between vertices b and

c, and Player 1 cannot distinguish between vertices d and e.

4

MAST30022 Semester 2, 2022 Assignment 1 5

2. Consider the game with chance moves in extensive form shown in Figure 2 in which

Player 1 cannot distinguish between the two vertices s/he owns. Describe the strategy

set for each player and give the normal form of the game where the payoffs are expected

payoffs.

C represents Chance or Nature.

Find all pairs of Nash equilibria in pure strategies, if they exist.

C

P

P

P

P1 2

2 1

0.3 0.7

(0,0)

0.5 0.5 0.6 0.4

(3,0) (−1,2) (1,−1)(2,−2)(3,1)(−2,1)(1,−2)

CC

Figure 2: Game with chance moves in extensive form

You may continue your answer on the next page.

5

MAST30022 Semester 2, 2022 Assignment 1 6

6

MAST30022 Semester 2, 2022 Assignment 1 7

3. Consider the following two-person zero-sum game

V =

[ −1 3 2 1

5 5 2 3

]

.

(a) Determine the value of the game, as well as a pair of optimal strategies (x∗1,y

∗).

7

MAST30022 Semester 2, 2022 Assignment 1 8

(b) Show that x∗2 =

(

1

3

, 2

3

)

is an optimal mixed strategy for player 1, and that

x =

(

2

3

, 1

3

)

is not an optimal mixed strategy for player 1.

8

MAST30022 Semester 2, 2022 Assignment 1 9

(c) Show that the convex combination

x∗t = tx

∗

1 + (1− t)x∗2, 0 ≤ t ≤ 1

of the two optimal strategies x∗1 and x

∗

2 is also an optimal strategy for player 1.

Using this result, provide a third optimal strategy for player 1.

9

MAST30022 Semester 2, 2022 Assignment 1 10

4. Consider the following two-person constant-sum game

(4, 10) (5, 9) (7, 7)

(3, 11) (2, 12) (5, 9)

(8, 6) (9, 5) (2, 12)

(5, 9) (6, 8) (6, 8)

.

Use the linear programming method, possibly in combination with other methods

(eg. saddle points, dominance elimination, 2× 2 formulae, etc.) when necessary, to de-

termine the values and optimal strategies for both players of this two-person constant-

sum game.

You may continue your answer on the next page.

10

MAST30022 Semester 2, 2022 Assignment 1 11

You may continue your answer on the next page.

11

MAST30022 Semester 2, 2022 Assignment 1 12

12

Assignment 4, Semester 1 2022

Due time: 4pm, Friday 26 August.

Name:

Student ID:

• To complete this assignment, you need to write your solutions into the blank answer

spaces following each question in this assignment PDF.

If you have a printer (or can access one), then you must print out the assignment

template and handwrite your solutions into the answer spaces.

If you do not have a printer but you can figure out how to annotate a PDF using

an iPad/Android tablet/Graphics tablet or using Adobe Acrobat, then annotate your

answers directly onto the assignment PDF and save a copy for submission.

Failing both of these methods, you may handwrite your answers as normal on blank

paper and then scan for submission (but note that you will thereby miss valuable

practice for the exam process). In that case, however, your document should have

the same length as the assignment template otherwise Gradescope will reject your

submission. So you will need to add as many blank pages as necessary to reach that

criterion.

Scan your assignment to a PDF file using your mobile phone (we recommend Cam-

Scanner App), then upload by going to the Assignments menu on Canvas and submit

the PDF to the GradeScope tool by first selecting your PDF file and then clicking on

‘Upload PDF’.

– it is a single pdf with all pages in correct template order and the correct way up,

and with any blank pages with additional working added only at the end of the

template pages;

– has all pages clearly readable;

– has all pages cropped to the A4 borders of the original page and is imaged from

directly above to avoid excessive ’keystoning’

These requirements are easy to meet if you use a scanning app on your phone and take

some care with your submission - please review it before submitting to double check

you have satisfied all of the above requirements.

MAST30022 Semester 2, 2022 Assignment 1 1

• The submission deadline is 4pm Melbourne time on Friday 26 August. You

have longer than of the normal one week to complete this assignment. Late submission

within 20 hours after the deadline will be penalised by 5% of the total available marks

for every hour or part thereof after the deadline. After that, the Gradescope submission

channel will be closed, and your submission will no longer be accepted. We recommend

you submit at least a day before the due date to avoid any technical delays. If there

are extenuating, eg medical circumstances, contact the Subject Coordinator, Dr Mark

Fackrell.

• There are 4 questions, of which 2 randomly chosen questions will be marked. Note

you are expected to submit answers to all questions, otherwise a mark penalty will

apply.

• Working and reasoning must be given to obtain full credit. Give clear and concise

explanations. Clarity, neatness, and style count.

∗ ∗ ∗

1

MAST30022 Semester 2, 2022 Assignment 1 2

1. Consider the game in extensive form shown in Figure 1 in which Player 1 owns vertices

a, d, e, f , and g, and Player 2 owns vertices b and c.

a

(2, −1)

P1

b c

d e f gP1 P1 P1 P1

P2 P2

(0, 3) (1, −1) (−2, 3) (1, 1) (−2, 4) (−3, 1)

Figure 1: Game in extensive form for Question 1

For vertex x ∈ {a, b, c, d, e, f, g}, use the label Lx if the left hand edge is chosen, Rx if

the right hand edge is chosen, and Cx if there is only one possible move.

(a) Redraw the game in extensive form labelling all the edges accordingly.

2

MAST30022 Semester 2, 2022 Assignment 1 3

(b) Describe the strategy set for each player, give the normal form of the game, and

find any pairs of pure strategies in equilibrium if the game is of

(i) perfect information;

3

MAST30022 Semester 2, 2022 Assignment 1 4

(ii) imperfect information and Player 2 cannot distinguish between vertices b and

c, and Player 1 cannot distinguish between vertices d and e.

4

MAST30022 Semester 2, 2022 Assignment 1 5

2. Consider the game with chance moves in extensive form shown in Figure 2 in which

Player 1 cannot distinguish between the two vertices s/he owns. Describe the strategy

set for each player and give the normal form of the game where the payoffs are expected

payoffs.

C represents Chance or Nature.

Find all pairs of Nash equilibria in pure strategies, if they exist.

C

P

P

P

P1 2

2 1

0.3 0.7

(0,0)

0.5 0.5 0.6 0.4

(3,0) (−1,2) (1,−1)(2,−2)(3,1)(−2,1)(1,−2)

CC

Figure 2: Game with chance moves in extensive form

You may continue your answer on the next page.

5

MAST30022 Semester 2, 2022 Assignment 1 6

6

MAST30022 Semester 2, 2022 Assignment 1 7

3. Consider the following two-person zero-sum game

V =

[ −1 3 2 1

5 5 2 3

]

.

(a) Determine the value of the game, as well as a pair of optimal strategies (x∗1,y

∗).

7

MAST30022 Semester 2, 2022 Assignment 1 8

(b) Show that x∗2 =

(

1

3

, 2

3

)

is an optimal mixed strategy for player 1, and that

x =

(

2

3

, 1

3

)

is not an optimal mixed strategy for player 1.

8

MAST30022 Semester 2, 2022 Assignment 1 9

(c) Show that the convex combination

x∗t = tx

∗

1 + (1− t)x∗2, 0 ≤ t ≤ 1

of the two optimal strategies x∗1 and x

∗

2 is also an optimal strategy for player 1.

Using this result, provide a third optimal strategy for player 1.

9

MAST30022 Semester 2, 2022 Assignment 1 10

4. Consider the following two-person constant-sum game

(4, 10) (5, 9) (7, 7)

(3, 11) (2, 12) (5, 9)

(8, 6) (9, 5) (2, 12)

(5, 9) (6, 8) (6, 8)

.

Use the linear programming method, possibly in combination with other methods

(eg. saddle points, dominance elimination, 2× 2 formulae, etc.) when necessary, to de-

termine the values and optimal strategies for both players of this two-person constant-

sum game.

You may continue your answer on the next page.

10

MAST30022 Semester 2, 2022 Assignment 1 11

You may continue your answer on the next page.

11

MAST30022 Semester 2, 2022 Assignment 1 12

12