Please answer this game theory question with explanations ta

Please answer this game theory question with explanations, tables, and solutions.

Q4. Suppose a decision-maker faced with FOUR alternative actions and four states of nature develops the following payoff table: STATES OF NATURE PAYOFF MATRIX ALTERNATIVEA2 A, Ag 36 38 100 80 20 120 60 80 140 30 70 80 25 70 35 80 ACTIONS (a) Suppose that the state of nature cannot be observed before the decision-maker needs to take action. Suppose nothing is known about the chances or probability of occurrence of the four states (i) What decision is recommended using the Hurwicz criterion for 1 (the maxim ax criterion)? (ii) What decision is recommended using the Hurwicz criterion for =0 (the maximin criterion)? (iii) For what range of e [0,1] would you recommend action A1 over A2? (b) Suppose that the state of nature cannot be observed before the decision-maker needs to take action. If nothing is known about the chances or probability of occurrence of the four states, what is the recommended decision using the LaPlace criterion? Construct the regret table and determine the recommended decision using Savage\'s minimax regret criterion. (c)

Solution

First we need to calculate H(AI) for each row i

H(Ai)=alpha(row i max)+(1-alpha)(row i min)

H(A1)=1(140)+0(20)=140

H(A2)=1(120)+0(30)=120

H(A3)=1(100)+0(35)=100

H(A4)=1(80)+0(80)=80

Maximax is to Find Maximum of H(Ai) calculated above =140 hence A1 to be chosen.

Maximin is to find minimum of H(Ai)=80 hence A4 to be chosen

Similar for alpha=0

H(A1)=0(140)+1(20)=20

H(A2)=0(120)+1(30)=30

H(A3)=0(100)+1(35)=35

H(A4)=0(80)+1(80)=80

In Maximax H(A4)=80 that will be chosen hence A4 to choose

In Maximin H(A4)=20 A1 to choose.

to be indifferent between A1 & A2 we will use @

@(140)+(1-@)20=@120+(1-@)30

@20=(1-@)10

@20=10-@10

@30=10

@=1/3

For Value of alpha (@)>1/3 we will choose A1 over A2

For B)

Laplace criterion  

Find average of oucomes for each action

Avg A1=221/4=55.25

Avg A2=258/4=64.5

Avg A3=265/4=66.25

Avg A4=320/4=80

As 80 is the maximum payoff hence A4 will be chosen using Laplace criterion