Consider the following game in which two firms decide how much of a homogeneous

Question

Consider the following game in which two firms decide how much of a homogeneous good to produce. The annual profit payoffs for each firm are stated in the cell of the game matrix, and Firm A's payoffs appear first in the payoff pairs:

Firm B - low output

Firm B - high output

Firm A - low output

300, 250

200, 100

Firm A - high output

200, 75

75, 100

Suppose the technology decision between A and B will be made simultaneously. Answer the following questions:

(a)What is (are) the dominant strategy (strategies) in this game? Explain interms of the payoffs.

(b)  Identify the Nash equilibrium (equilibria), if there is (are) any, for this simultaneous decision. Explain.

Firm B - low output

Firm B - high output

Firm A - low output

300, 250

200, 100

Firm A - high output

200, 75

75, 100

Explanation / Answer

Game theory involves two players. They have more than one option to decide. Pay off from each options adopted by two players are available. They have to select a strategy which will maximize their own return. But for optimizing their decision, they have to consider the action of his rival.

In this problem, two players are firm A and firm B. They have two strategies low output and high output. The strategies of firm a are measured in rows and for firm B in columns. They have to select a strategy which will maximize their payy off. Each cell has two pay offs. First one is for Firm A and second one is for firm B.

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(a) Dominant strategy is a strategy which will always give higher payoffs in comparison with pay off of other strategies. Consider first strategy of firm 1. If it adopts strategy of low output, then firm 2 can also adopt either strategy of low output or high output. In that case pay off of firm 1 will be 300 or 200.

Alteratively if firm 1 adopts high output then pay offs are 200 or 75. 200 is earned if firm B also go for low productivity. It is 75 if firm B adopts high productivity.

Now compare two payoffs side by side. Note that firm A has higher pay off in low output [300,200] in comparison with the pay off of high output [200,75]. So whatever strategy firm B adopts, Firm A will always go for low production. So low production strategy of firm A dominates high production strategy.

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Same result is not observed for firm B. Pay off from low production strategy of firm B is [ 250,75]. Pay off from high production strategy are [100,100]. Now compare the two. If Firm A go for low production, then firm B will select low production. It will give pay off 250. Similarly when firm A decides for high production, then firm will also decide for high production. It will maximize its pay off. Amount is 100. Thus no strategy dominates for firm B.

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(b) Now you have to ascertain ash equilibrium. It is a combination of strategy of two players which will optimize their pay offs. Deviation from this combination, will not be beneficial for any of them. So the combination will be stable.

In this problem, firm A due to dominant strategy of low output will always go for it. It will not be affected by the decision of firm B. It is row 1. So Firm B has to optimize its payoff when Firm A has selected low output. Pay off s are 250 if low output strategy is selected and 100 if high output is selected. So firm B will select low output strategy.

Thus Nash equilibrium is low output strategy for both firm A and firm B. Pay off are 300 for A and 250 for B.

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