Lingo 做线性规划 - Game Thoery

Reference: <An Introduction to Management Science Quantitative Approaches to Decision Making, Revised 13th Edition>

Instance

Suppose that two companies are the only manufacturers of a particular product; they compete against each other for market share. In planning a marketing strategy for the coming year, each company will select one of three strategies designed to take market share from the other company. The three strategies, which are assumed to be the same for both companies, are as follows:

Strategy 1: Increase advertising.

Strategy 2: Provide quantity discounts.

Strategy 3: Extend warranty.

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

Doing so, Company A identifies the minimum payoff for each of its strategies, which is the minimum value in each row of the payoff table. These row minimums are 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 in market share of at least 2% is guaranteed.

Considering the entries in the Column Maximum row, Company B can be 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 that Company A cannot gain more than 2% in 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 has changed.

Because these values are not equal, a pure strategy solution does not exist. In this case, it is not optimal for each company to be predictable and select a pure strategy regardless of what the other company does. The optimal solution is for both players to adopt a mixed strategy.

With a mixed strategy, each player selects its strategy according to a probability distribution.Weighting each payoff by its probability and summing provides the expected value of the increase in market share for Company A.

Company A will 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 that Company B will select a strategy that provides the minimum expected gain for Company A. Thus, Company B will 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 will be the minimum expected gain. This strategy will minimize Company A’s expected gain in market share. Company A will select its optimal mixed strategy using a maximin strategy, which will maximize the minimum expected gain. This objective is written as follows:

Define GAINA to be the optimal expected gain in market share for Company A.

Now consider the game from the point of view of Company B. Company B will select one of its strategies based on the following 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 market share for Company B for each Company A strategy is provided in Table 5.10.

Company A will 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 will be the expected loss, which will maximize Company B’s expected loss in market share. Company B will select its optimal mixed strategy using a minimax strategy to minimize the maximum expected loss. This objective is written as follows:

Define LOSSB to be the optimal expected loss in market share for Company B.