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APPENDIX

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An Alternative View of the Payoff Matrix n Assume total maximum profits of all oligopolists is constant at 200 units. n Alternative policies include price increases, keeping the price constant, price reductions, and an increase in advertising expenditures.

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An Alternative View of the Payoff Matrix n The matrix is essentially a duopoly, but the competitor may be all other oligopolists.

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Payoff Matrix Competitors’ Alternative Strategies Raise p p = k Reduce p More ads p107 92 88 99 p=k110 106 94 97 p116 108 100102 ads108 109 96 98 Oligopolists Alternative Strategies If the oligopolist reduces p and others leave prices unchanged, He gets 108, competitors get 92.

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n This is a zero-sum game. The payoff (total profit available = 200 units) remains the same and one player’s gains are the other player’s losses. What policy should the oligopolist choose? Play it safe? (Assume that opponents will always follow that strategy which produces the worst possible result from his own point of view.) This calls for the maximin strategy.

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The maximin strategy is to maximize the minimum gains you expect your opponents to permit you to take. Secure the result that will be least harmful, no matter what the opponent decides to do. We act pessimistically. The Maximin Strategy

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Competitors’ Alternative Strategies Raise p p = k Reduce p More ads p107 92 88 99 p=k110 106 94 97 p116 108 100102 ads108 109 96 98 Oligopolists Alternative Strategies We assume if the oligopolist increases p, the competitor will reduce the price. If we keep our prices constant… If we reduce our price… If we increase our advertising… Playing it safe, this pessimistic oligopolist reduces the price and Gets the maximum from the minimum gains the competitors allow

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The Minimax Strategy n The numbers in the matrix represent losses to the competitors (since they are the gains of the oligopolist). They wish to minimize the profits of the olligopolist, which are their own losses. n They will choose to reduce prices. Note their potential losses with each policy they can choose.

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Competitors’ Alternative Strategies Raise p p = k Reduce p More ads p107 92 88 99 p=k110 106 94 97 p116 108 100102 ads108 109 96 98 Oligopolists Alternative Strategies Note the competitors’ most favorable outcome for each Possible policy choice. Here, the competitors choose the policy of minimizing the maximum loss the oligopolist will inflict on them.

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Competitors’ Alternative Strategies Raise p p = k Reduce p More ads p107 92 88 99 p=k110 106 94 97 p116 108 100102 ads108 109 96 98 Oligopolists Alternative Strategies The outcome 100 is a “saddle point” where the maximum minimum of the oligopolist = the minimum maximum of Competitors. It is a stable equilibrium.

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n If 100 were not a stable equilibrium, things could get complicated. Various solutions are possible. n A player might adopt a “mixed” strategy in an attempt to take advantage of opponent’s mistakes or to keep them off balance. n Laws of probability can be applied in evaluating possible policy choices.

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The application of game theory assumes that businessmen do play a game of business according to a general theory of games and that they seek to maximize profits. This behavior may lead to outcomes similar to those of marginal analysis. But as we saw at the beginning, it provides a fertile field for imaginative approaches.

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GAME THEORY Day 5. Minimax and Maximin Step 1. Write down the minimum entry in each row. Which one is the largest? Maximin Step 2. Write down the maximum.

GAME THEORY Day 5. Minimax and Maximin Step 1. Write down the minimum entry in each row. Which one is the largest? Maximin Step 2. Write down the maximum.

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