Unit 8 Supply and demand: Markets with many buyers and sellers

8.9 How competition works: Transforming a cartel coordination game into a competitive prisoners’ dilemma

Before you start

To understand the model in this section, you will need to know about game theory and the concept of Nash equilibrium, which was introduced in Unit 4. If not, you can either skip this section, or read Sections 4.2, 4.3, and Section 4.13 before beginning work on it.

In 2020, seven airlines serving the Indonesian market were convicted of colluding to sustain high prices.

The OPEC cartel is managed by the governments of Saudi Arabia and the largest oil-producing countries, rather than by the oil companies themselves, but cartels may also exist among private companies. Sustaining a cartel without government support is difficult, however, if the number of companies involved is large. In this case, it may be in the interest of a single company not to charge the high prices agreed upon by the cartel, but instead to act like a competitor and charge a low price to capture a larger share of the market. When this happens, the cartel will fall apart, leading to lower prices for consumers.

We can use game theory to understand how competition can work to destroy a cartel. Consider a market with two firms (A and B) selling an identical good that costs $1 to produce. If both firms charge a high price ($4), total market sales of the product will be 60 units, equally split between the two firms. If both charge a low price ($2), total sales will be 72 units, also equally split.

Profit per unit is $3 at the high price, and $1 at the low price. So each firm’s profit is profit per unit times one half the market sales at the price chosen or:

  • $3 × 30 = $90 if both charge high prices
  • $1 × 36 = $36 if both charge low prices.

So the firms (that is, the owners of the firms) could benefit from working together as a cartel, agreeing on a high market price. But that does not mean that the cartel can be sustained. We have to ask: could one firm make higher profit by dropping out of the cartel? Suppose Firm A dropped out and charged $2, while Firm B continued to charge $4. Since the goods are identical, the entire market would go to the lower-price firm. Firm A would then make a profit of $72 (that is, profit per unit, $1, times the number of units sold in the whole market at the lower price, 72 units). This is less than its $90 profit within the cartel. So Firm A will decide not to withdraw from the cartel, and the same calculation will convince Firm B to stay, too. The cartel can be sustained.

Translating this into game theory terms, we have just shown that charging a high price is the best response if the other firm does the same. The pay-off matrix in Figure 8.20 shows that charging high prices is a Nash equilibrium. But charging low prices is also a Nash equilibrium: if the other firm is charging $2, then charging $4 would mean selling nothing. This price-setting game with two firms and two Nash equilibria is a coordination game, in which the owners of both firms prefer the high-price equilibrium. They can achieve it by agreeing that both will set the high price (that is, by forming a cartel), and they can sustain it because neither would benefit from dropping out. Such agreements are typically informal and secret, because colluding to sustain high prices is never popular with customers, and in many countries is illegal.

In this payoff matrix, Firm A and Firm B can set a high price or a low price. Payoffs are (Firm A’s, Firm B’s). If both set a high price, payoffs are (90, 90). If Firm A sets a high price and Firm B sets a low price, payoffs are (0, 72). If Firm A sets a low price and Firm B sets a high price, payoffs are (72, 0). If both set a low price, payoffs are (36, 36).

Figure 8.20 Pay-offs in the price-setting game with two firms, a coordination game with two Nash equilibria.

barriers to entry, entry barriers
The term barriers to entry refers to anything making it difficult for new firms to enter a market, such as intellectual property rights or economies of scale in production.

If there are barriers to entry, preventing any more firms from selling in the market, then the two firms can maintain the high market price. But think about what changes if a third firm, C, can enter the market.

Figure 8.21, shows the pay-off matrix from the viewpoint of Firm A, assuming that firms B and C behave in the same way as each other.

If all three charge the same price, the market is split equally: if the price is high, they each sell 20 units and receive $60 of profit each; if it is low, they each sell 24 units and receive $24 in profit. If they set different prices, 72 units are sold and $72 profit is shared equally between the low-price firms, while those charging high prices sell nothing.

Again, the firms would prefer a cartel, where all agree to set the high price, to the outcome where they all set a low price. But in this case, a cartel cannot be sustained. The first column of the pay-off matrix shows that if B and C are setting a high price, Firm A prefers to drop out, and sell 72 units with a profit of $72—more than its $60 profit within the cartel. The other firms, in order to survive, would then follow and the cartel would be destroyed.

In this payoff matrix, Firm A, Firm B, and Firm C can set a high price or a low price. Firms B and C set their price together. Payoffs are (Firm A’s, Firm B’s, Firm C’s). If all set a high price, payoffs are (60, 60, 60). If Firm A sets a high price and Firms B and C set a low price, payoffs are (0, 36, 36). If Firm A sets a low price and Firms B and C set a high price, payoffs are (72, 0, 0). If all set a low price, payoffs are (24, 24, 24).

Figure 8.21 Pay-offs in the price-setting game with three firms, a prisoners’ dilemma.

The increased competition with three firms transforms the price-setting game from a coordination game supporting a cartel with high prices, to a prisoners’ dilemma game in which the dominant strategy is to set a low price. A cartel cannot succeed, because each member would rather violate the price-setting agreement and cut prices. This happens because the additional firm reduces the profits available from the cartel, making defection from the cartel a more profitable strategy.

As in the examples in Unit 4, the failure to cooperate in a prisoners’ dilemma game yields lower pay-offs for all of the firms. But this may be judged to be a better outcome because of the benefits to consumers, whose pay-offs in terms of lower prices are not included in the game. In the case of cartels, public policy may seek to reduce barriers to entry so that the third firm (or more) could enter, thereby increasing competition and making it more difficult for the firms to cooperate in price setting.

Exercise 8.9 Price-setting game with three firms: Players B and C behave differently

In the price-setting game shown in Figure 8.21, suppose Firms B and C can now set different prices from each other.

  1. Use the numbers provided in this section to calculate the pay-offs for the case in which B sets a high price and C sets a low price.
  2. Draw two payoff matrices, like that in Figure 8.21, each featuring A and B’s choices, where one payoff matrix shows payoffs to all three firms when C charges a high price, and the other shows payoffs to all three firms when C charges a low price. Find the Nash equilibrium of this modified game and compare it to the Nash equilibrium of the game in Figure 8.21. (Hint: compare payoffs for A and B within each matrix, and compare payoffs for C across the matrices.)