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In oligopoly markets, there is a tension between cooperation and self-interest. If all the firms limit their output, the price is high, but then firms have an incentive to expand output. The techniques of game theory are used to solve for the equilibrium of an oligopoly market. “Duopoly” example: Jack and Jill choose how many gallons of water to pump and sell in town. To keep things simple, assume zero costs. The demand schedule gives us the price the buyers are willing to pay, as a function of the total combined output of Jack and Jill.

If the market structure were perfectly competitive, the market supply curve would be based on marginal costs, so the price would be zero and the quantity would be 120. This is the socially efficient outcome. If the market structure were a monopoly, the price would be 60 and the quantity would be 60. This outcome maximizes industry profits, and there is a deadweight loss. If Jack and Jill could form a cartel, they would collude with each other to act in unison, like a monopoly. Each seller would produce 30 gallons and share the monopoly profits. Would they come to this arrangement if they had to choose their quantities separately?

Suppose Jack expects Jill to produce 30 gallons. If he produces 30, both receive profits of $1800. But, if he produces 40, the total quantity supplied to the market is 70 gallons, so the market price is $50. Therefore, Jack’s profit would be $2000, and Jill’s profit would be $1500. By increasing output and expanding market share, total industry profits fall but Jack is better off. Since Jill is in the same situation, both sellers have an incentive to produce more than their share of the monopoly output. The cartel solution is not stable. Without the ability to commit to the cartel, self-interest pushes the price below the monopoly level.

What is the duopoly solution? Suppose Jack and Jill each produce 40 gallons, so that the market price is $40 and each receives profits of $1600. Can either seller do better by choosing a different quantity? If Jack were to produce 30 gallons, the price would be $50 and his profits would only be $1500. If Jack were to produce 50 gallons, the price would be $30 and his profits would only be $1500. Jack and Jill each producing 40 gallons is the Nash equilibrium of this oligopoly game. It is stable–neither seller would want to change behavior. Put another way, if each seller expected the other to choose 40, their best response is to choose 40, thereby confirming the expectations.

For general games, a Nash equilibrium is defined to be a combination of strategies (one for each player), for which no player has an alternative strategy that yields a higher payoff, given the strategies chosen by the other players. This is a notion of joint rationality. Each player is best responding to the other players.

Beyond the example: what can we say, in general, about the Nash equilibrium of “quantity competition” oligopoly games? With duopoly (like monopoly), there is an output effect and a price effect. Output effect: Selling one more gallon allows the seller to receive the market price for that gallon. Price effect: Selling one more gallon causes the market price to fall for all of the gallons the seller produces. The difference between monopoly and duopoly is that the price effect is...