Game Theory Models of Oligopoly

The Cournot duopoly model

Price wars erupt when each of the firms in the market adopts the conjecture that other firms will hold their prices constant, thus the first firm can gain by undercutting the price. An alternative belief firms can hold is that competitors have constant output. This idea, termed Cournot conjectures after the 19th century French economist who first conceived it, offers another way to look at the behavior of firms in an oligopoly,

The Cournot model is most easily understood in a duopoly, an oligopoly which consists of only two firms (thus it is the oligopoly with the smallest possible number of firms). Each of the two sellers posits that its rival's output will not change, and continues to hold such a belief even after it is proved wrong. Thus, a weakness of the model is that firms do not learn from the past.

Cournot presented the market as initially a monopoly. A single firm, call it firm A, with constant marginal and average cost (equal to C) maximizes its profit, as shown in figure 12.17. Marginal revenue equals marginal cost when output is Q_a1, and the firm charges a price of P₁. The firm makes a profit of P₁deC.

Another firm sees the profit and enters the market. Call this new firm B. It must decide how much to produce. Under the Cournot conjecture, firm B assumes that it has the remainder of the market demand, that part of the demand curve to the right of Q_a1, as its own demand. Figure 12.18 shows what this means to firm B. If firm A's output is fixed at Q_a1, then any production by B will lower market price below P₁. Its demand curve is the accented portion of the market demand, thus marginal revenue for firm B is MR_b1.

At this point, firm B just maximizes profit by setting its marginal cost, assumed to be the same as for firm A, equal to its marginal revenue. So firm B produces Q_b1 off the residual demand, and total market output is Q_1b+Q_a1, giving a market price of P₂. Under most situations Q_b1 will be smaller than Q_a1.

After firm B has entered the market, firm A finds that the price it can get is P₂, not P₁, and B is producing Q_b1. Firm A adopts the same conjecture as B did and assumes B will always produce Q_b1. Thus, A faces the demand situation pictured in figure 12.19. Firm A now faces the residual demand to the right of Q_b1, so its marginal revenue is MR_a2. To maximize profit it sets output at Q_a2, total market production is Q_a2+Q_b1, and market price is P₃.

Of course, B gets to go again, and under the Cournot conjecture assumes A's output will stay at Q_a2. They keep going around and around, until an equilibrium is reached when neither firm wants to change its output given the output of the other firm. An equilibrium has two parts:

                        - when A produces Q_a*, B maximizes profit by producing Q_b*.

Thus, neither firm has any incentive to change output.

The progression to an equilibrium had each firm changing its output - reacting - in response to the output of the rival firm. For every possible output by B we can find A's best output. This relationship is called a reaction function, and shows how A's output changes in response to Q_b. Similarly we can find a reaction function which tells us what B's output will be for every Q_a.

Reaction functions are easily understood in an algebraic model. Suppose the inverse market demand is given by P=300-Q_t, where Q_t=Q_a+Q_b is the total market quantity. The Cournot conjecture for firm A is that Q_b is fixed so the residual demand for firm A is given by P=(300-Q_b)-Q_a. With this demand curve, the marginal revenue for firm A is MR_a=(300-Q_b)-2Q_a. When it maximizes profit firm A sets its marginal cost equal to marginal revenue, so MC_a=(300-Q_b)-2Q_a. Suppose MC_a=MC_b=30. Then when A maximizes profit, 30=(300-Q_b)-2Q_a, or Q_a=(270-Q_b)/2, which is firm A's reaction function to B's output. By symmetry, firm B has as its reaction function to A's output Q_b=(270-Q_a)/2. Solving these together gives that Q_a=Q_b=90.

This answer is easily found by graphing the reaction functions. The heavy line in figure 12.20 is B's reaction function to Q_a. If Q_a is zero (B has a monopoly), Q_b=135. Conversely, B will decide to produce zero units if Q_a is 270. The symmetry of our example gives A's reaction function, the lighter line on the graph. The Cournot equilibrium is at the intersection of the two reaction functions, where each firm is producing 90 units, so market price is 120. Each firm makes a profit of (120-30)x90 or 8100.

Contrast the Cournot solution to that which would result if the market were a perfect competition or a monopoly. In perfect competition price equals minimum long run average cost. The constant costs of this example (MC=AC=30) tells us that P=300-Q_c=30, so Q_c would equal 270 in perfect competition. Obviously, profit for each of the firm is zero. For a monopoly facing this demand MR_m=300-2Q_m, and setting MR equal to MC gives Q_m=135 and P_m=165. Profit is 135x(165-30)=18225. These results are shown in figure 12.21.

There is one more interpretation of the Cournot model, called the Stackelberg model, which does not have the firms being quite as naive. Firm B could see that A reacts to Q_b, and decide to exploit that reaction. B is called the leader and finds that point on A's reaction function which maximizes firm B's profit. A is termed the follower for obvious reasons. While the derivation is left as an exercise, the best move for firm B would be to produce 135 units, at which point A will produce 67.5 units. Total market output is 202.5 so price is 97.5. Firm B makes a profit of 135x(97.5-30)=9112.5 while firm A has a profit of 67.5x(97.5-30)=4556.25. B has more profit than under the Cournot conjecture, but A has less.

But if B is so smart, why isn't A? When both firms try to be leaders, each assuming the other is willing to be a follower, each produces 135 units, total market output is 270, price is 30, and profit is zero, which is the same result as the Bertrand equilibrium.
 

Strategic behavior and an introduction to the theory of games

The difference between the Cournot equilibrium, and the Stackelberg result offers a convenient way to begin an approach to oligopoly called strategic behavior and game theory. Game theory studies how individuals and organizations will react when interests of the players of the game are codependent. In noncooperative games, the interests of the players conflict, while in cooperative games their interests coincide. Oligopoly markets can often be construed as noncooperative games. Each firm in the market makes a move - how much to produce. The rules of the game - the demand and cost curves - determine how much profit each firm gets. The profit is called the "payoff" and is the reward for playing the game.

Suppose we have the duopoly market talked about in the previous section, and the leaders of each firm have read this book. Outright collusion is illegal so they cannot act like a monopoly, but both want to avoid the zero profit result of Stackelberg behavior. So they agree to not be leaders. Instead, they will each use the Cournot conjecture.

Each firm can follow the agreement, or cheat and try to be a leader. The payoff matrix shows what the profit each firm would make depending on whether it cheats (acts like a leader), and if the other firm cheats:
 

Notice, both firms are better off acting with Cournot conjectures than if both act like leaders. But, if firm A believes B will adhere to the agreement, A will do better to try to lead. B has the same incentive if it figures A will be a follower.

With this payoff matrix there is a dominant strategy. A dominant strategy is one for which a single choice is the best one for a player whatever choice his rival makes. We can see that A has a dominant strategy by finding his best choice for each choice of B.  Showing the dominant strategy is left to an technical exercise.

Sometimes a game doesn't have a dominant strategy.  Look at the following payoff matrix.

Notice, if A is Cournot, B will want to be Stackelberg.  But if A is Stackelberg, B will want to be Cournot.  symmetry gives A the same desires as B.

Without a dominant strategy, we need a rule for the players to follow. One common strategy is a maximin strategy which has the player choosing that strategy which maximizes his minimum gain. Looking at the payoff matrix, A gains at least 5100 if he chooses Cournot conjectures, but could see 4000 if he chooses Stackelberg conjectures if B chooses the same way. His maximin strategy is to behave Cournot. B sees the same. If she chooses Stackelberg, it is possible she will see 4000. But if she chooses to behave as Cournot suggested, the worst she will gain is 5100. Her maximin strategy is also to be Cournot. Under the maximin rule, the Cournot equilibrium is the outcome.

An example of a game with a dominant strategy is called the Prisoner's Dilemma. It shows the tradeoffs two partners-in-crime face after being caught by the police. the interesting result is the dominant strategy leads to an  outcome which is not the best for the thief. If neither confesses, they get a small penalty - say a year in jail. But if one confesses and the other doesn't, the one that confesses gets only probation while the other gets 10 years. If they both confess, so neither must testify against the other, they both get 5 years.

Their best choice is for both to remain silent, and get at most one year in jail. But, if the first crook expects the second to remain silent, the first crook will want to confess. The second has the same incentive. The prisoner's dilemma is whether the partner in crime can be trusted to remain silent.

Suppose we set up the payoff matrix for the prisoner's dilemma.
 
 
 

NOTE: P = years in prison

Look at the choices for prisoner A. No matter what B does, A is better off by confessing. If B confesses, A lowers his sentence to 5 years by confessing as well. And if B doesn't confess, A gets off with probation by confessing, instead of one year in prison. His dominant strategy is to confess - he is always better off. In a symmetric game like this one, B has the same choices. With little honor among thieves, they both end up spending 5 years in prison.

It is often thought that firms in oligopoly markets face payoff matrices with elements of the prisoner's dilemma. Take, for example, the possibility of collusion. In a collusive agreement the firms try to act like a monopoly, maximizing joint profits, under some formula for dividing the gains. But if both firms cheat they lose money.

Let us look at an example payoff matrix:

P = profit

The dominant strategy for each firm is to cheat. Whatever firm A chooses, B does better by cheating, and similarly for B. Thus, we see that both firms should end up cheating on any agreement to collude.

OPEC is a cartel of oil producing countries which has sought to maintain the price of oil on the international market by restricting the quantity supplied. Often the collusion is successful. For a long time Saudi Arabia, the leading oil exporting country in the world, has tried to keep the price of oil at about $20 per barrel. But sometimes the cartel falls apart. Member countries unilaterally expand output, driving the price lower. This cheating brings short term gains, especially if there is a lag before the market price reacts. In fact, since Saudi Arabia is so intent on a steady oil price, it sometimes cuts its output to prevent the price from falling.

But the Saudis carry a "big stick" as well. Exporting almost twice as much as Iran, the second largest (in terms of oil exports) member of the cartel, they can flood the market, drive down the price of oil, and punish members for cheating. In fact, one reason OPEC may be so successful is that Saudi Arabia acts as an enforcer.