Lingo do linear programming-Game Thoery

Reference: <an Introduction to Management science quantitative approaches to decision Making, revised 13th edition>

Instance

Suppose that companies is the only manufacturers of a particular product; They compete against each other for the market share. In planning a marketing strategy for the coming year, each company would select one of the three strategies designed to take MA Rket share from the other company. The three strategies, which is assumed to being the same for both companies, is as follows:

Strategy 1:increase Advertising.

Strategy 2:provide Quantity discounts.

Strategy 3:extend warranty.

A payoff table showing the percentage gain in the the market share for company A for each combination of strategies is shown I N Table 5.5.

Doing So, company A identifies the minimum payoff for each of its strategies, which are the minimum value in each row of th e payoff table. These row minimums is shown in Table 5.6.

After comparing the row minimum values, company A selects the strategy that provides the maximum of the row minimum values . This is called a maximin strategy. Thus, company A selects strategy A1 as its optimal strategy; An increase on market share of least 2% is guaranteed.

Considering the entries in the Column Maximum row, company B can is guaranteed a decrease in market share of no more than 2% by selecting the strategy B3. This is called a minimax strategy. Thus, company B selects B3 as its optimal strategy. Company B has guaranteed, A cannot gain more than 2% in the market share.

Let us continue with the Two-company Market-share game and consider a slight modification in the payoff table as shown in Table 5.8. Only one payoff have changed.

Because These values is not equal, a pure strategy solution does not exist. In this case, it isn't optimal for each company to be predictable and select a pure strategy regardless of the Company does. The optimal solution is for both players to adopt a mixed strategy.

with a mixed strategy, each player selects their strategy according to a probability distribution. Weighting each payoff by it probability and summing provides the expected value of the increase in market share for Compa NY A.

Company A would select one of its three strategies based on the following probabilities:
PA1 = The probability that company A selects strategy A1
PA2 = The probability that company A selects strategy A2
PA3 = The probability that company A selects strategy A3

Given the probabilities PA1, PA2, and PA3 and the expected gain expressions in Table 5.9, game theory assumes B would select a strategy that provides the minimum expected gain for company A. Thus, company B would select B1, B2, or B3 based on
Min {eg (B1), eg (B2), eg (b3)}
When company B selects its strategy, the value of the game would be the minimum expected gain. This strategy would minimize company A's expected gain in the market share. Company A would select its optimal mixed strategy using A Maximin strategy, which would maximize the minimum expected gain. This objective is written as follows:

Define Gaina to is the optimal expected gain in market share for company A.

Now consider the game from the point of view of company B. Company B would select one of its strategies based on the follow ing probabilities:
PB1 = The probability that company B selects strategy B1
PB2 = The probability that company B selects strategy B2
PB3 = The probability that company B selects strategy B3

The expression for the expected loss-in-the-share for company-B for each company A strategy are provided in Table 5.10.

Company A would select A1, A2, or A3 based on
Max {El (A1), El (A2), El (A3)}
When company A selects its strategy, the value of the game would be the expected loss, which would maximize company B ' s EXPE CTED loss in the market share. Company B would select its optimal mixed strategy using a MINIMAX strategy to minimize the maximum expected loss. This objective is written as follows:

Define LOSSB to is the optimal expected loss in market share for company B.

Lingo do linear programming-Game Thoery