Game Theory and Business Strategy

Competition among Oligopolists


When players do not have dominant or dominated strategies, we must find equilibria through cell-by-cell inspection or best response analysis. This best response analysis applies not only to games for which we can draw game diagrams, but also games with pure strategies that are continuous variables (See your text, Dixit and Skeath, chapter 4, section 9). For example, firm output decisions (how much to produce) are often continuous variables. In this exercise, you will apply best response analysis to such output decisions in both a simultaneous and sequential setting.


Cournot Competition

An often-studied simultaneous game is the Cournot quantity-competition model of oligopoly. Augustin Cournot developed one of the first models of competition between firms in 1835. Consider two firms, firm 1 and firm 2, producing identical products so that they are forced to charge identical prices. In Cournot's model, the sole strategic choice of the firms is the amount they choose to produce, q1 and q2. Once the firms select their quantities, the resulting industry price is whatever price is required to "clear the market" which is the price at which consumers are willing to buy the total production, q1+q2.

Suppose that the firms have no marginal costs of production. market demand is given by:

P = 600 - q1 - q2.

For example, if each firm produces 100 units, so that q1=q2=100, the resulting price is 400.

If the firms must make their output decisions simultaneously, how much will each firm produce? In order to be at equilibrium, each firm must be producing a best response to the choice of the other firm. Consider the decision of firm 1. In order for q1 to be an equilibrium output, it must maximize firm 1's profit given firm 2's choice of q2. Suppose that firm 1 thinks that firm 2 will produce q2. Then, firm 1 estimates that if it produces q1 units of output, its profit will be:

Profit of firm 1 = P * q1 = (600-q1-q2)*q1

To maximize the above profit, we can use calculus to deduce that firm 1's optimal quantity is given by

profit maximizing value of q1 = 300 - .5q2

this profit-maximizing value of q1 is called firm 1's best response to firm 2. According to this equation, firm 1's best response is a decreasing function of q2. That means that if firm 1 expects firm 2 to increase output, it will reduce its own output. In a similar manner, we can calculate firm 2's best response to firm 1's choice of quantity:

profit maximizing value of q2 = 300 - .5q1

Now we know what each firm will produce given the output choice of the other firm. In order to be at equilibrium, the choices of the two firms must, simultaneously, be best responses to each other. Otherwise, a firm will always have incentive to change its output.

Only one pair of outputs simultaneously solves both best reply equations. The solution turns out to be:

q1 = q2 = 200.

Substituting these choices of quantity into the demand equation gives

P = 600 - q1 - q2 = 600-200-200 = 200.

So the equilibrium profits for each firm are: 200*200 = 40,000.

This shows that finding the equilibrium of games in which we cannot write down a "game box" is also possible. Since each firm has many strategies (an infinite number, in fact), we cannot write down each strategy and look for equilibria in a game box. However, the same principle applies. In order to be at equilibrium, a firm must be playing the best response to the choice of the other firm. In the next section, we consider the effects of transforming this simultaneous game into a sequential one.

 

 


Stackelberg Competition

The Cournot model has often been criticized as unrealistic because it is rare that firms would select quantities simultaneously, without any ability to observe the choices of their competitors. Heinrich von Stackelberg, in his 1934 criticism of the Cournot model, pointed out that the results would be quite different if the two firms chose quantities sequentially. considered the same model as Cournot competition, except that one firm gets to act before the other. A natural question is which position is better - to go first or second? Would you prefer to have to select your quantity first, or would you prefer if your competitor selected her quantity first, and you, only upon observing this decision, had to make yours?

The intuitive answer is often "I would prefer to go second, since I can see what the first firm chose." However, as we will see, the first mover has the advantage. 

Consider a two-period model. In the first period, firm 1 selects its output, and in the second period, firm 2 observes the choice of firm 1 and then selects its own output. Firm 1 is better off, and the intuition is as follows: What advantage would you gain by being second? You would observe what the first firm chose. Then, you would select the quantity that maximizes your profit given what the first firm chose. But, if you're so smart, isn't your competitor smart as well? Can't your competitor also calculate, ahead of time, what you will do? Obviously, the first firm can predict the reaction of the second firm. Then, who has more power? The second firm is simply reacting  to the choice of firm 1. The first firm, then, by choosing its quantity is maximizing its own profit, taking this reaction  into account.

Using the same demand curve from the previous section,

P = 600 - q1 - q2.

Again, there are no marginal costs. Firm 1 selects its quantity in period 1, and firm 2 in period 2. In order to find the equilibrium of this sequential game, we use rollback.

Start with the decision of firm 2. What will firm 2 do in the second period if firm 1 chose q1. This is given by the best response of firm 2, which we calculated above:

profit maximizing value of q2 = 300 - .5q1

Now, what will firm 1 do in the first period. Firm 1 will anticipate firm 2's response. Firm 1 maximizes its profit given by:

Profit of firm 1 = P * q1 = (600-q1-q2)*q1

But, firm 1 knows exactly what firm 2 will pick: its profit-maximizing value. Substituting firm 2's best response into firm 1's profit:

Profit of firm 1 = P * q1 = (600-q1-(300 - .5q1))*q1 = (300-.5q1)q1 

From which we may find that the profit-maximizing output for firm 1 is:

q1 = 300

If firm 1 produces 300, firm 2 produces 150, and the market price is:

P = 600 - q1 - q2 = 600-300-150 = 150

And the resulting profits are:

Profit of firm 1 = 300*150 = 45,000

Profit of firm 2 = 150*150 = 22,500

So firm 1 earns a greater profit than firm 2. Also, firm 1 earns a greater profit than it did in the simultaneous (Cournot) game. Moving first is an advantage!

Summarizing:

In a sequential model, firm 1 is able to commit to a quantity higher than the equilibrium of the simultaneous game. By so doing, firm 1 forces firm 2 to cut back on its own production. This commitment yields the first firm a market share of 2/3 compared to the 1/2 it had in the simultaneous move game.