Games Oligopolists Play

When a firm’s decision significantly affects the profits of other firms in the industry, the firms are in a situation of interdependence.

In our duopoly example and in real life, each oligopolistic firm realizes both that its profit depends on what its competitor does and that its competitor’s profit depends on what it does. That is, the two firms are in a situation of interdependence, where each firm’s decision significantly affects the profit of the other firm (or firms, in the case of more than two).

The study of behavior in situations of interdependence is known as game theory.

In effect, the two firms are playing a “game” in which the profit of each player depends not only on its own actions but on those of the other player (or players). In order to understand more fully how oligopolists behave, economists, along with mathematicians, developed the area of study of such games, known as game theory. It has many applications, not just to economics but also to military strategy, politics, and other social sciences.

Let’s see how game theory helps us understand oligopoly.

The Prisoners’ Dilemma

The reward received by a player in a game, such as the profit earned by an oligopolist, is that player’s payoff.

Game theory deals with any situation in which the reward to any one player—the payoff—depends not only on his or her own actions but also on those of other players in the game. In the case of oligopolistic firms, the payoff is simply the firm’s profit.

A payoff matrix shows how the payoff to each of the participants in a two-player game depends on the actions of both. Such a matrix helps us analyze situations of interdependence.

When there are only two players, as in a duopoly, the interdependence between the players can be represented with a payoff matrix like that shown in Figure 14-1. Each row corresponds to an action by one player (in this case, ADM); each column corresponds to an action by the other (in this case, Ajinomoto). For simplicity, let’s assume that ADM can pick only one of two alternatives: produce 30 million pounds of lysine or produce 40 million pounds. Ajinomoto has the same pair of choices.

A Payoff Matrix Two firms, ADM and Ajinomoto, must decide how much lysine to produce. The profits of the two firms are interdependent: each firm’s profit depends not only on its own decision but also on the other’s decision. Each row represents an action by ADM, each column, one by Ajinomoto. Both firms will be better off if they both choose the lower output, but it is in each firm’s individual interest to choose the higher output.

The matrix contains four boxes, each divided by a diagonal line. Each box shows the payoff to the two firms that results from a pair of choices; the number below the diagonal shows ADM’s profits, the number above the diagonal shows Ajinomoto’s profits.

These payoffs show what we concluded from our earlier analysis: the combined profit of the two firms is maximized if they each produce 30 million pounds. Either firm can, however, increase its own profits by producing 40 million pounds while the other produces only 30 million pounds. But if both produce the larger quantity, both will have lower profits than if they had both held their output down.

Prisoners’ dilemma is a game based on two premises: (1) Each player has an incentive to choose an action that benefits itself at the other player’s expense (2) When both players act in this way, both are worse off than if they had acted cooperatively.

The particular situation shown here is a version of a famous—and seemingly paradoxical—case of interdependence that appears in many contexts. Known as the prisoners’ dilemma, it is a type of game in which the payoff matrix implies the following:

The original illustration of the prisoners’ dilemma occurred in a fictional story about two accomplices in crime—let’s call them Thelma and Louise—who have been caught by the police. The police have enough evidence to put them behind bars for 5 years. They also know that the pair have committed a more serious crime, one that carries a 20-year sentence; unfortunately, they don’t have enough evidence to convict the women on that charge. To do so, they would need each of the prisoners to implicate the other in the second crime.

So the police put the miscreants in separate cells and say the following to each: “Here’s the deal: if neither of you confesses, you know that we’ll send you to jail for 5 years. If you confess and implicate your partner, and she doesn’t do the same, we’ll reduce your sentence from 5 years to 2. But if your partner confesses and you don’t, you’ll get the maximum 20 years. And if both of you confess, we’ll give you both 15 years.”

Figure 14-2 shows the payoffs that face the prisoners, depending on the decision of each to remain silent or to confess. (Usually the payoff matrix reflects the players’ payoffs, and higher payoffs are better than lower payoffs. This case is an exception: a higher number of years in prison is bad, not good!) Let’s assume that the prisoners have no way to communicate and that they have not sworn an oath not to harm each other or anything of that sort. So each acts in her own self-interest. What will they do?

The Prisoners’ Dilemma Each of two prisoners, held in separate cells, is offered a deal by the police—a light sentence if she confesses and implicates her accomplice but her accomplice does not do the same, a heavy sentence if she does not confess but her accomplice does, and so on. It is in the joint interest of both prisoners not to confess; it is in each one’s individual interest to confess.

An action is a dominant strategy when it is a player’s best action regardless of the action taken by the other player.

The answer is clear: both will confess. Look at it first from Thelma’s point of view: she is better off confessing, regardless of what Louise does. If Louise doesn’t confess, Thelma’s confession reduces her own sentence from 5 years to 2. If Louise does confess, Thelma’s confession reduces her sentence from 20 to 15 years. Either way, it’s clearly in Thelma’s interest to confess. And because she faces the same incentives, it’s clearly in Louise’s interest to confess, too. To confess in this situation is a type of action that economists call a dominant strategy. An action is a dominant strategy when it is the player’s best action regardless of the action taken by the other player.

It’s important to note that not all games have a dominant strategy—it depends on the structure of payoffs in the game. But in the case of Thelma and Louise, it is clearly in the interest of the police to structure the payoffs so that confessing is a dominant strategy for each person. So as long as the two prisoners have no way to make an enforceable agreement that neither will confess (something they can’t do if they can’t communicate, and the police certainly won’t allow them to do so because the police want to compel each one to confess), Thelma and Louise will each act in a way that hurts the other.

PITFALLS: PLAYING FAIR IN THE PRISONERS’ DILEMMA

PITFALLS

PLAYING FAIR IN THE PRISONERS’ DILEMMA
One common reaction to the prisoners’ dilemma is to assert that it isn’t really rational for either prisoner to confess. Thelma wouldn’t confess because she’d be afraid Louise would beat her up, or Thelma would feel guilty because Louise wouldn’t do that to her.
But this kind of answer is, well, cheating—it amounts to changing the payoffs in the payoff matrix. To understand the dilemma, you have to play fair and imagine prisoners who care only about the length of their sentences.
Luckily, when it comes to oligopoly, it’s a lot easier to believe that the firms care only about their profits. There is no indication that anyone at ADM felt either fear of or affection for Ajinomoto, or vice versa; it was strictly about business.

So if each prisoner acts rationally in her own interest, both will confess. Yet if neither of them had confessed, both would have received a much lighter sentence! In a prisoners’ dilemma, each player has a clear incentive to act in a way that hurts the other player—but when both make that choice, it leaves both of them worse off. When Thelma and Louise both confess, they reach an equilibrium of the game. We have used the concept of equilibrium many times; it is an outcome in which no individual or firm has any incentive to change his or her action.

A Nash equilibrium, also known as a noncooperative equilibrium, results when each player in a game chooses the action that maximizes his or her payoff given the actions of other players, ignoring the effects of his or her action on the payoffs received by those other players.

In game theory, this kind of equilibrium, in which each player takes the action that is best for her given the actions taken by other players, and vice versa, is known as a Nash equilibrium, after the mathematician and Nobel laureate John Nash. (Nash’s life was chronicled in the best-selling biography A Beautiful Mind, which was made into a movie.) Because the players in a Nash equilibrium do not take into account the effect of their actions on others, this is also known as a noncooperative equilibrium.

Now look back at Figure 14-1: ADM and Ajinomoto are in the same situation as Thelma and Louise. Each firm is better off producing the higher output, regardless of what the other firm does. Yet if both produce 40 million pounds, both are worse off than if they had followed their agreement and produced only 30 million pounds. In both cases, then, the pursuit of individual self-interest—the effort to maximize profits or to minimize jail time—has the perverse effect of hurting both players.

Prisoners’ dilemmas appear in many situations. The upcoming For Inquiring Minds describes an example from the days of the Cold War. Clearly, the players in any prisoners’ dilemma would be better off if they had some way of enforcing cooperative behavior—if Thelma and Louise had both sworn to a code of silence or if ADM and Ajinomoto had signed an enforceable agreement not to produce more than 30 million pounds of lysine.

!worldview! FOR INQUIRING MINDS: Prisoners of the Arms Race

Between World War II and the late 1980s, the United States and the Soviet Union were locked in a seemingly endless struggle that never broke out into open war. During this Cold War, both countries spent huge sums on arms, sums that were a significant drain on the U.S. economy and eventually proved a crippling burden for the Soviet Union, whose underlying economic base was much weaker. Yet neither country was ever able to achieve a decisive military advantage.

As many have pointed out, both nations would have been better off if they had both spent less on arms. Yet the arms race continued for 40 years.

Why? As political scientists were quick to notice, one way to explain the arms race was to suppose that the two countries were locked in a classic prisoners’ dilemma. Each government would have liked to achieve decisive military superiority, and each feared military inferiority. But both would have preferred a stalemate with low military spending to one with high spending.

Caught in the prisoners’ dilemma: heavy military spending hastened the collapse of the Soviet Union.
©ITAR-TASS/Sovfoto

However, each government rationally chose to engage in high spending. If its rival did not spend heavily, its own high spending would lead to military superiority; not spending heavily would lead to inferiority if the other government continued its arms buildup. So the countries were trapped.

The answer to this trap could have been an agreement not to spend as much; indeed, the two sides tried repeatedly to negotiate limits on certain weapons. But these agreements weren’t very effective. In the end the issue was resolved as heavy military spending hastened the collapse of the Soviet Union in 1991.

Unfortunately, the logic of an arms race did not disappear. A nuclear arms race developed between Pakistan and India, two neighboring countries with a history of mutual antagonism. In 1998 both countries confirmed the unrelenting logic of the prisoners’ dilemma by publicly testing nuclear weapons in a tit-for-tat sequence.

However, by 2013 a glimmer of hope emerged, as the prime ministers of these South Asian nuclear rivals began a series of meetings aimed at making “a new beginning.”

But in the United States an agreement setting the output levels of two oligopolists isn’t just unenforceable, it’s illegal. So it seems that a noncooperative equilibrium is the only possible outcome. Or is it?

Overcoming the Prisoners’ Dilemma: Repeated Interaction and Tacit Collusion

A firm engages in strategic behavior when it attempts to influence the future behavior of other firms.

Thelma and Louise in their cells are playing what is known as a one-shot game—that is, they play the game with each other only once. They get to choose once and for all whether to confess or hang tough, and that’s it. However, most of the games that oligopolists play aren’t one-shot; instead, they expect to play the game repeatedly with the same rivals. An oligopolist usually expects to be in business for many years, and it knows that its decision today about whether to cheat is likely to affect the way other firms treat it in the future. So a smart oligopolist doesn’t just decide what to do based on the effect on profit in the short run. Instead, it engages in strategic behavior, taking account of the effects of the action it chooses today on the future actions of other players in the game. And under some conditions oligopolists that behave strategically can manage to behave as if they had a formal agreement to collude.

Suppose that ADM and Ajinomoto expect to be in the lysine business for many years and therefore expect to play the game of cheat versus collude shown in Figure 14-1 many times. Would they really betray each other time and again?

A strategy of tit for tat involves playing cooperatively at first, then doing whatever the other player did in the previous period.

Probably not. Suppose that ADM considers two strategies. In one strategy it always cheats, producing 40 million pounds of lysine each year, regardless of what Ajinomoto does. In the other strategy, it starts with good behavior, producing only 30 million pounds in the first year, and watches to see what its rival does. If Ajinomoto also keeps its production down, ADM will stay cooperative, producing 30 million pounds again for the next year. But if Ajinomoto produces 40 million pounds, ADM will take the gloves off and also produce 40 million pounds the next year. This latter strategy—start by behaving cooperatively, but thereafter do whatever the other player did in the previous period—is generally known as tit for tat.

Tit for tat is a form of strategic behavior, which we have just defined as behavior intended to influence the future actions of other players. Tit for tat offers a reward to the other player for cooperative behavior—if you behave cooperatively, so will I. It also provides a punishment for cheating—if you cheat, don’t expect me to be nice in the future.

The payoff to ADM of each of these strategies would depend on which strategy Ajinomoto chooses. Consider the four possibilities, shown in Figure 14-3:

How Repeated Interaction Can Support Collusion A strategy of tit for tat involves playing cooperatively at first, then following the other player’s move. This rewards good behavior and punishes bad behavior. If the other player cheats, playing tit for tat will lead to only a short-term loss in comparison to playing always cheat. But if the other player plays tit for tat, also playing tit for tat leads to a long-term gain. So a firm that expects other firms to play tit for tat may well choose to do the same, leading to successful tacit collusion.

  1. If ADM plays tit for tat and so does Ajinomoto, both firms will make a profit of $180 million each year.

  2. If ADM plays always cheat but Ajinomoto plays tit for tat, ADM makes a profit of $200 million the first year but only $160 million per year thereafter.

  3. If ADM plays tit for tat but Ajinomoto plays always cheat, ADM makes a profit of only $150 million in the first year but $160 million per year thereafter.

  4. If ADM plays always cheat and Ajinomoto does the same, both firms will make a profit of $160 million each year.

Which strategy is better? In the first year, ADM does better playing always cheat, whatever its rival’s strategy: it assures itself that it will get either $200 million or $160 million (which of the two payoffs it actually receives depends on whether Ajinomoto plays tit for tat or always cheat). This is better than what it would get in the first year if it played tit for tat: either $180 million or $150 million. But by the second year, a strategy of always cheat gains ADM only $160 million per year for the second and all subsequent years, regardless of Ajinomoto’s actions.

Over time, the total amount gained by ADM by playing always cheat is less than the amount it would gain by playing tit for tat: for the second and all subsequent years, it would never get any less than $160 million and would get as much as $180 million if Ajinomoto played tit for tat as well. Which strategy, always cheat or tit for tat, is more profitable depends on two things: how many years ADM expects to play the game and what strategy its rival follows.

If ADM expects the lysine business to end in the near future, it is in effect playing a one-shot game. So it might as well cheat and grab what it can. Even if ADM expects to remain in the lysine business for many years (therefore to find itself repeatedly playing this game with Ajinomoto) and, for some reason, expects Ajinomoto always to cheat, it should also always cheat. That is, ADM should follow the old rule “Do unto others before they do unto you.”

But if ADM expects to be in the business for a long time and thinks Ajinomoto is likely to play tit for tat, it will make more profits over the long run by playing tit for tat, too. It could have made some extra short-term profits by cheating at the beginning, but this would provoke Ajinomoto into cheating, too, and would, in the end, mean lower profits.

When firms limit production and raise prices in a way that raises one anothers’ profits, even though they have not made any formal agreement, they are engaged in tacit collusion.

The lesson of this story is that when oligopolists expect to compete with one another over an extended period of time, each individual firm will often conclude that it is in its own best interest to be helpful to the other firms in the industry. So it will restrict its output in a way that raises the profits of the other firms, expecting them to return the favor. Despite the fact that firms have no way of making an enforceable agreement to limit output and raise prices (and are in legal jeopardy if they even discuss prices), they manage to act “as if” they had such an agreement. When this happens, we say that firms engage in tacit collusion.

!worldview! ECONOMICS in Action: The Rise and Fall and Rise of OPEC

The Rise and Fall and Rise of OPEC

Call it the cartel that does not need to meet in secret. The Organization of Petroleum Exporting Countries, usually referred to as OPEC, includes 12 national governments (Algeria, Angola, Ecuador, Iran, Iraq, Kuwait, Libya, Nigeria, Qatar, Saudi Arabia, the United Arab Emirates, and Venezuela), and it controls 40% of the world’s oil exports and 80% of its proven reserves. Two other oil-exporting countries, Norway and Mexico, are not formally part of the cartel but act as if they were. (Russia, also an important oil exporter, has not yet become part of the club.)

It is in OPEC’s interest to keep oil prices high and output low.
©Laura Gangi Pond/thinkstock

Unlike corporations, which are often legally prohibited by governments from reaching agreements about production and prices, national governments can talk about whatever they feel like. OPEC members routinely meet to try to set targets for production.

These nations are not particularly friendly with one another. Indeed, OPEC members Iraq and Iran fought a spectacularly bloody war with each other in the 1980s. And, in 1990, Iraq invaded another member, Kuwait. (A mainly American force based in yet another OPEC member, Saudi Arabia, drove the Iraqis out of Kuwait.)

Yet the members of OPEC are effectively players in a game with repeated interactions. In any given year it is in their combined interest to keep output low and prices high. But it is also in the interest of any one producer to cheat and produce more than the agreed-upon quota—unless that producer believes that his actions will bring future retaliation.

So how successful is the cartel? Well, it’s had its ups and downs. Analysts have estimated that of 12 announced quota reductions, OPEC was able to successfully defend its price floor 80% of the time.

Crude Oil Prices, 1949-2013 (in Constant 2012 Dollars)
Sources: Energy Information Administration; FRED

Figure 14-4 shows the price of oil in constant dollars (that is, the value of a barrel of oil in terms of other goods) since 1949. OPEC first demonstrated its muscle in 1974: in the aftermath of a war in the Middle East, several OPEC producers limited their output—and they liked the results so much that they decided to continue the practice. Following a second wave of turmoil in the aftermath of Iran’s 1979 revolution, prices shot still higher.

By the mid-1980s, however, there was a growing glut of oil on world markets, and cheating by cash-short OPEC members became widespread. The result, in 1985, was that producers who had tried to play by the rules—especially Saudi Arabia, the largest producer—got fed up, and collusion collapsed.

The cartel started acting effectively at the end of the 1990s, thanks largely to the efforts of Mexico’s oil minister, who orchestrated output reductions, and Saudi Arabia’s assumption of the role of “swing producer.” As the key decision maker and the largest OPEC producer by far, Saudi Arabia allowed other members to produce as much as they could, and then adjusted its own output to meet the overall limit, thereby easing friction among members. These actions helped raise the price of oil from less than $10 per barrel in 1998 to a range of $20 to $30 per barrel in 2003.

Since 2008, OPEC has experienced the steepest roller-coaster ride of oil prices in its history. By 2008, prices had soared to over $145 per barrel. But at the end of 2008, one year into the recession of 2007-2009, prices dropped sharply to $32 per barrel. In response, Saudi Arabia cut its output by 20%, as other members reduced their output by 5%. By early 2009, prices began to rebound. Political instability in several member countries caused prices to skyrocket in 2011, as Saudi Arabia increased its output to prevent a global oil shortage.

More recently, as Iraq and Iran have signaled their intention to raise output, as the United States produces increasing amounts of oil from its shale formations, and as oil production from Brazil and Canada increases, some OPEC watchers are predicting that the cartel’s future cohesion may very well be in jeopardy.

Quick Review

  • Economists use game theory to study firms’ behavior when there is interdependence between their payoffs. The game can be represented with a payoff matrix. Depending on the payoffs, a player may or may not have a dominant strategy.

  • When each firm has an incentive to cheat, but both are worse off if both cheat, the situation is known as a prisoners’ dilemma.

  • Players who don’t take their interdependence into account arrive at a Nash, or noncooperative, equilibrium. But if a game is played repeatedly, players may engage in strategic behavior, sacrificing short-run profit to influence future behavior.

  • In repeated prisoners’ dilemma games, tit for tat is often a good strategy, leading to successful tacit collusion.

14-3

  1. Question 14.4

    Find the Nash (noncooperative) equilibrium actions for the following payoff matrix. Which actions maximize the total payoff of Nikita and Margaret? Why is it unlikely that they will choose those actions without some communication?

  2. Question 14.5

    Which of the following factors make it more likely that oligopolists will play noncooperatively? Which make it more likely that they will engage in tacit collusion? Explain.

    1. Each oligopolist expects several new firms to enter the market in the future.

    2. It is very difficult for a firm to detect whether another firm has raised output.

    3. The firms have coexisted while maintaining high prices for a long time.

Solutions appear at back of book.