Example :-15.13 (N.D Vohra)Q. In a small town ,there are only two stores that handle sundry goods- ABC and XYZ.The total number of customers is equally divided between the two, because price and quality of goods sold are equal. Both stores have good reputation in the community, and they render equally good customer service. Assume that a gain of customer by ABC is a loss to XYZ and vice versa. Both stores plan to run annual pre Diwali sales during the first week of November. Sales are advertised through a local newspaper, radio and television media. With the aid of an advertising firm ,store ABC constructed the game matrix given below.(Figures in the matrix represent a gain or loss of customers.)

Determine optimal strategies and worth of such strategies for both ABC and XYZ.

StrategyOf

ABC

Strategy Of XYZ

News paper Radio Television

News paper 30 40 -80

Radio 0 15 -20

Television 90 20 50

Solution :-

The game matrix, with row minima and column maxima

MINI MAX STRATEGY FOR XYZ MAXIMIN STRATEGY FOR ABC

Evidently, saddle point does not exist. Now ,we can see that row 3dominates row 2, and column 3 dominates column 1. Deleting R₂ andC₁,the game reduces to the order 2 Ẋ 2 ,as shown in next slide.

StrategyOf

ABC

Strategy Of XYZ RowMinimalNews paper Radio Television

News paper 30 40 -80 -80Radio 0 15 -20 -20

Television 90 20 50 20ColumnMaxima

90 40 50

With usual meaning of the symbols used, we have

X = a₂₂ -a₂₁ = 50 - 20 = 30 =1/5.(a₁₁+a₂₂)-(a₁₂+a₂₁) (40+50)-(-80+20) 150

Y = a₂₂ -a₁₂ = 50 –(-80) = 130 = 13/15. (a₁₁+a₂₂)-(a₁₂+a₂₁) (40+50)-(-80+20) 150

V = (a₁₁ ₓ a₂₂) – (a₁₂ ₓ a₂₁) = (40 ₓ50) +(80 ₓ 20) = 3600 = 24 . (a₁₁+a₂₂)-(a₁₂+a₂₁) (40+50)-(-80+20) 150

Radio TelevisionNewspaper 40 -80

Television 20 50

Thus Optimal strategies are :

Thus the value of game = 24.

Newspaper Radio Television For ABC : 1/5 0 4/5

For XYZ : 0 13/15 2/15

Example 15.16 A Company is currently involved in negotiations with its union on the upcoming wage contract. With an aid of an outside mediator, the table below was constructed by the management group. The pluses are to be interpreted as proposed wage increases while a minus figure indicates that a wage reduction is proposed . The mediator informs the management groups that he has been in touch with the union and that they have constructed a table that is comparable to the table developed by the management. Both the company and the union must decide on an overall strategy before negotiations begin. The management group understands the relationship of company strategies to union strategies in the following table but lacks specific knowledge of game theory to select the best strategy (or strategies) for the firm. Assist the management on this problem. What game value and strategies are available to the opposing groups?

Conditional costs to company

Company strategies

(In Lac Rs)Union strategies

U₁ U₂ U₃ U₄

C₁ +0.25 +0.27 +0.35 -0.02C₂ +0.02 +0.16 +0.08 +0.08C₃ +0.14 +0.12 +0.15 +0.13C₄ +0.30 +0.14 +0.19 0.00

Since the company represents the “minimising “ and the union the “maximising player “,we shall recast the pay-off matrix (by taking transpose of given matrix) as follows:

This game has no saddle point.We observe that all entries in the third row of this matrix are greater than ,or equal to, the corresponding entries in the fourth row . Thus ,fourth row is dominated by the third row and hence can be deleted .

Unionstrategies

(In Lac Rs)Company strategies

C₁ C₂ C₃ C₄

U₁ +0.25 +0.20 +0.14 +0.30U₂ +0.27 +0.16 +0.12 +0.14U₃ +0.35 +0.08 +0.15 +0.19U₄ -0.02 +0.08 +0.13 0.00

Deleting it we get,

In this matrix ,the first column is dominated both by the second and the third columns; and the fourth column is dominated by the third column .Deleting the dominated columns.

Unionstrategies

(In Lac Rs)Company strategies

C₁ C₂ C₃ C₄

U₁ +0.25 +0.20 +0.14 +0.30U₂ +0.27 +0.16 +0.12 +0.14U₃ +0.35 +0.08 +0.15 +0.19

The matrix is reduced to the following:

Here the second row is dominated by the first . Deleting this row we get the following matrix of the order 2 x 2,and obtain the solution to the game analytically.

Unionstrategies

(In Lac Rs)Company strategies

C₂ C₃

U₁ +0.20 +0.14U₂ +0.16 +0.12U₃ +0.08 +0.15

If x be the probability with which the union adopts policy U₁ and Y be the probability of adoption of C₂ by the company ,we havex = 0.15-0.08 = 7 ; (0.20+0.15) – (0.08+0.14) 13y = 0.15 – 0.14 = 1 ; and

(0.20+0.15)- (0.08+0.14) 13V = 0.20 ẋ 0.15 – 0.08 ẋ 0.14 = 0.0188 = 47

(0.20 + 0.15)–(0.08 + 0.14) 0.1300 325

Unionstrategies

(In Lac Rs)Company strategies

C₂ C₃

U₁ +0.20 +0.14U₃ +0.08 +0.15

Thus , optimal strategy for the company is (0,1/13,12/13,0); for the union it is (7/13,0,6/13,0) and the game value is 47/325(representing increased wages).

Example 15.1 (N.D. Vohra)Two leading firms ,Nirmala,Textiles Ltd. And Swati Rayons Ltd.,

for years have been selling shirting which is but a small part of both firms total sales. The marketing Director of Nirmala Textiles raised the question ,”What should the firm’s strategies be in terms of advertising for the product in Question ?” The system groups of Nirmala textiles developed the following data for varying degrees of advertising:

(a) No advertising ,medium advertising and heavy advertising for both firms will result in equal market share.

(b) Nirmala Textiles with no advertising :40 % of the market with medium advertising by Swati Rayons and 28% of the market with heavy advertising by Swati Rayons.

(c) ) Nirmala Textiles with medium advertising : 70% of the market with no advertising by Swati Rayons and 45% of the market with heavy advertising by Swati Rayons.

(d) ) Nirmala Textiles using heavy advertising : 75% of the market with no advertising by Swati Rayons and 52.5% of the market with medium advertising by Swati Rayons .Based upon the above information ,answer the marketing director,s question .

The pay-off matrix from the viewpoint of Nirmala Textiles Ltd.,Showing its market sahre under several combinations of the strategies,is given below . Also ,row minima and column maxima have been obtained to see if saddle point exists.

Question Contd. ……

Swati Rayons Ltd.’s Strategy

Nirmala Textile

Ltd. Strategy

No. Advt.

b₁

Med. Advt.

b₂

Hvy. Advt.

b₃

Row Minima

No. Advt.

a₁ 50 40 28 28

Med. Advt.

a₂ 70 50 45 45

Hvy. Advt.

a₃ 75 52.5 50 50*

Column Maxima 75 52.5 50*

We observe that saddle point exists at the intersection of a₃ and b₃. Thus ,the optimal strategy for each one is to engage in heavy advertising and it will result in an even distribution of the market between the firms. Nirmala textiles Ltd’s marketing director should , therefore, resort to heavy advertising .