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 Oligopoly Models Game Theory

The fifth model of oligopoly is not so much a model as a general theory used to examine strategy, called game theory. Game theory was developed relatively recently as a tool for examining strategic behavior. In this theory, firms often try to outsmart each other, but often will find that they are moving in circles. They often over-anticipate. Game theory examines how firms or people behave when they know that others can anticipate their behavior.

There are many game theory models that apply to oligopoly behavior. For example, there are only two firms making collectable "limited edition" commemorative dinner plates for the year 2000. If both firms choose not to make very many plates, they will both make a lot of money. But if one firm makes a lot of plates, while the other chooses to not make very many, the firm that produces a lot will make even more money. If they both make a lot of plates, then the plates will not be rare and they both will gain very little profit. Here is the game theory: Firm 1 says, "If I make a little, I know that Firm 2 will want to make a lot, so I will make a lot. But if Firm 2 knows that these are my choices, it knows I will make a lot, no matter what I say.”

These strategy options can be summarized with a payoff matrix. A payoff matrix puts one player’s choices on the vertical axis and the other player’s choices on the horizontal axis and then fills in a table showing the payoff (in this case, profit) for each combination of strategies. Following is a payoff matrix for the dinner plate game.

Firm 2’s Strategy

If both firms make few plates, they will be in the top left box of the table, each making \$100. If one of the firms makes a lot of plates while the other makes a few, then the one that makes a lot will get \$120 while the other gets \$20. But if they both make a lot, they will only get \$40 each, since they will have flooded the market. In this game, there is a dominant strategy, meaning a strategy that is best regardless of the other person’s actions.

In order to properly solve the game, it is necessary to find the game-theoretic equilibrium. Here is the solution for Firm 1. If Firm 2 produces few plates, Firm 1 should produce many plates, since it will make \$120 instead of \$100. If Firm 2 produces many plates, Firm 1, according to the table, should produce many plates because it will realize a profit of \$40 instead of \$20. Thus, no matter what Firm 2 does, Firm 1 should produce many plates. Knowing this, Firm 2 is better off producing many plates, since \$40 is better than \$20. In the end, both Firm 1 and Firm 2 make many plates, even though they would both be better off if they both made a few. Clearly this is an inefficient outcome.

This game is called a prisoner’s dilemma, and there has been much written about the strategies of this game. To better illustrate the prisoner’s dilemma, there was a contest in which people could write computer programs with strategies of how to play the game and then the computer programs played each other repeatedly. The winner of the repeated game was the program with the "tit for tat" strategy. This strategy says that a firm should do whatever its opponent did the last time they met. Thus, it is possible to have an outcome in this game where both firms get the \$100 profit, but only if the same opponent is played repeatedly.

Games do not always have a dominant strategy, however. To illustrate this, take the same table, but change Firm 1’s potential profits:

In this case, Firm 1 would prefer to make few plates if Firm 2 made few plates, and would prefer to make many plates if Firm 2 made many plates. By looking at Firm 2, the game theory can be solved. If Firm 2 knows the preferences of Firm 1, then Firm 2 must compare the "both making few plates" option to the "both making many plates" option. Firm 2 is better off if both are making few plates, and so the equilibrium is that they both make few plates. This is called a Nash equilibrium, because each player’s strategy was the best choice given the other player’s strategy. If Firm 2 made few plates, then Firm 1’s best move is to make few plates. If Firm 1 made few plates, it is Firm 2’s best strategy to make few plates.

When there is some uncertainty about the strategy of another player, there is a strategy called the maximin strategy, which argues that a player should choose the strategy that maximizes the worst-case outcome that could occur when the other player chooses a strategy. This strategy was mentioned often when discussing Cold War options between the U.S. and the Soviet Union. Another way of wording this strategy is that a player should take an action that makes the best of the worst-case scenario.

All in all, game theory is an incredibly fascinating field of study, but it is very specific to the payoff structure of the market and so it offers little in the way of generalized theories of oligopolistic behavior.

For more work with the dominat firm model, try the following Active Graph Level Two exercise:

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