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.

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, q_{1} and q_{2}. 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, q_{1}+q_{2}.

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

P = 600 - q_{1} - q_{2}.

For example, if each firm produces 100 units, so that q_{1}=q_{2}=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 q_{1} to be an equilibrium output, it must maximize firm
1's profit given firm 2's choice of q_{2}. Suppose that firm 1 thinks
that firm 2 will produce q_{2}. Then, firm 1 estimates that if it
produces q_{1} units of output, its profit will be:

Profit of firm 1 = P * q_{1} = (600-q_{1}-q_{2})*q_{1}

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

profit maximizing value of q_{1} = 300 - .5q_{2}

this profit-maximizing value of q_{1} is called firm 1's *best
response* to firm 2. According to this equation, firm 1's best response is a
decreasing function of q_{2}. 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 q_{2} = 300 - .5q_{1}

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:

q_{1} = q_{2} = 200.

Substituting these choices of quantity into the demand equation gives

P = 600 - q_{1} - q_{2} = 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.

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 - q_{1} - q_{2}.

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 q_{1}. This is given by the *best response* of firm
2, which we calculated above:

profit maximizing value of q_{2} = 300 - .5q_{1}

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 * q_{1} = (600-q_{1}-q_{2})*q_{1}

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 * q_{1} = (600-q_{1}-(300 - .5q_{1}))*q_{1}
= (300-.5q_{1})q_{1}

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

q_{1} = 300

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

P = 600 - q_{1} - q_{2} = 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.