Game theory and the pay-off matrix

Overview

Game theory and why is it useful for analysing oligopoly, three elements of any game and pay-off matrix.

What you need to know

Answer each one before you reveal it.

Questions from this subtopic

Each one with the model answer underneath.

1. 1

   What is game theory and why is it useful for analysing oligopoly?

   Model answer

   Game theory is the study of decision making where the outcome for each decision maker depends on the choices of the others as well as on their own. It suits oligopoly because interdependence is precisely that situation, and unlike models that simply assume how rivals react, game theory works out which reactions are in the rival's own interest.

2. 2

   What are the three elements of any game?

   Model answer

   The players, who make the decisions; the strategies available to each player; and the pay-offs, which state the outcome each player receives for every possible combination of strategies.

3. 3

   What is a pay-off matrix?

   Model answer

   A pay-off matrix is a table setting out every combination of the players' strategies and the result each player gets from it. One player's strategies form the rows and the other's the columns, and each cell contains two figures, conventionally the row player's pay-off first and the column player's second.

4. 4

   Two firms, A and B, each choose to set a high price of £70 or a low price of £60. Market demand is P = 100 - 2Q and each firm's marginal cost is a constant £40, with output measured in thousands of units. Derive the four pay-offs.

   Model answer

   At £70, market quantity is (100 - 70)/2 = 15 thousand; if both charge £70 they split it, so each sells 7.5 thousand at a margin of £30, giving £225,000 each. At £60, market quantity is (100 - 60)/2 = 20 thousand; if both charge £60 they split it, so each sells 10 thousand at a margin of £20, giving £200,000 each. If one charges £60 while the other charges £70, the cheaper firm takes the whole market of 20 thousand at a margin of £20, earning £400,000, and the dearer firm sells nothing and earns nothing.

5. 5

   Describe in words the pay-off matrix for two firms A and B choosing High price or Low price, where both High gives each £225,000, both Low gives each £200,000, and undercutting gives the low-priced firm £400,000 and the high-priced firm nothing.

   Model answer

   The matrix has two rows for firm A, High and Low, and two columns for firm B, High and Low, with pay-offs in thousands of pounds written as A's first. The top-left cell, both High, is (225, 225). The top-right cell, A High and B Low, is (0, 400). The bottom-left cell, A Low and B High, is (400, 0). The bottom-right cell, both Low, is (200, 200).

6. 6

   What is a dominant strategy?

   Model answer

   A dominant strategy is one that gives a player a better pay-off than any of its alternatives no matter what the other players choose. Because it is best in every case, a player with a dominant strategy does not need to predict what the rival will do.

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