Simultaneous vs. sequential games

In this section we consider Nash equilibrium for the case in which a continuum of infinite actions is available to each player. To be specific, we analyze the behavior of two firms competing with each other in terms of quantities produced. Both firms would like to maximize their profit, but they influence each other since their choices of quantities have an impact on the price at which the product is sold on the market. This price is common to both firms, as we assume that they produce a perfectly identical product and there is no possibility of differentiating prices. This kind of competition is called Cournot competition; the case in which firms compete on prices is called Bertrand competition, but it will not be analyzed here. We start with the case of simultaneous moves, which leads to the Cournot–Nash equilibrium.¹⁷ Then we will consider sequential moves.

To clarify these concepts it is useful to tackle a simple model, in which we assume that each firm has a cost structure involving only a variable cost:

Here, TC_i denotes total cost for firm i, c_i is the variable cost, and q_i is the amount produced by firm i = 1, 2. The total amount available on the market is Q = q₁ + q₂ and it is going to influence price. In Section 2.11.5 we introduced the concepts of demand and inverse demand functions, relating price and demand. There, we assumed the simplest relationship, i.e., a linear one. By a similar token, we assume here that there is a linear relationship between total quantity produced and price:

Incidentally, this stylized model assumes implicitly that all produced items are sold on the market. Then, the profit for firm i is

Note that the profit of each firm is influenced by the decisions of the competitor. Assuming that the two firms make their decisions simultaneously, it is natural to wonder what the Nash equilibrium will be. Note that we assume complete information and common knowledge, in the sense that each player has all of the above information and knows that the other player has such information. We can find the equilibrium by finding the best response function R_i(q_j) for each firm, i.e., a function giving the profit-maximizing quantity  for firm i, for each possible value of q_j set by firm j. Enforcing the stationarity condition for the profit of firm 1, we find

Fig. 14.3 Finding a Nash equilibrium in Cournot competition.

By the same token, for firm 2 we have

To solve the problem, we should find where the two response functions intersect; in other words, we should solve the system of equations

where we use the superscript “c” to denote Cournot equilibrium. In our case, we are lucky, since response functions are linear. In particular, both response functions are downward-sloping lines, as illustrated in Fig. 14.3. Hence, to find the Nash equilibrium we simply solve the system of linear equations

which yields

The resulting equilibrium price turns out to be

and the profit of each firm is

It is interesting to note that if a firm manages to reduce its cost, it will increase its produced quantity and profit as well. We leave this check as an exercise. It is also worth noting that if the firms have the same production technology (i.e., c₁ = c₂), then we have a symmetric solution  as expected.

So far, we have assumed that the two competing firms play simultaneously. In the supply chain management problem of Section 14.3, since there are two different types of actions, it is more natural to assume that one of the two players moves first. Hence, we may also wonder what happens in the quantity game of this section if we assume that firm 1, the leader, sets its quantity q₁ before firm 2, the follower. Unlike the simultaneous game, firm 2 knows the decision of firm 1 before making its decision; thus, firm 2 has perfect information. The analysis of the resulting sequential game leads to von Stackelberg equilibrium. Firm 1 makes its decision knowing the best response function for firm 2, as given in Eq. (14.10). Hence, the leader’s problem is

where the superscript “s” refers to von Stackelberg competition. Applying the stationarity condition yields

We see that firm 1 produces more in this sequential game than in the Cournot game. If we plug this value into the best response function R₂(q₁), we obtain

We see that the output of firm 2 is a fraction of that of the Cournot game, plus a term that is positive if firm 1 is less efficient than firm 2. Now it would be interesting to compare the profits for the two firms under this kind of game. This is easy to do when marginal production costs are the same; we illustrate the idea with a toy numerical example.

Example 14.9 Two firms have the same marginal production cost, c₁ = c₂ = 5, and the market is characterized by the price/quantity function

In this example we compare three cases:

1. The two firms collude and work together as a cartel. We may also consider the two firms as two branches of a monopolist firm. Note that if the two marginal costs were different, one of the two branches would be just shut down (assuming infinite production capacity, as we did so far).
2. The firms do not cooperate and move simultaneously (Cournot game).
3. The firms do not cooperate and move sequentially (von Stackelberg game).

In the first case, we just need to work with the aggregate output Q. The monopolist solves the problem

where superscript “m” indicates that we are referring to the monopolist case. We solve the problem by applying the stationarity condition

which yields the following market price and profit:

In the second case, the solution given by (14.11) is symmetric:

The overall output and price are

respectively. The profit for each firm is

Note that the total overall profit is

In fact, the monopolist would restrict output to increase price, resulting in a larger overall profit than with the Cournot competition. So, collusion results in a larger profit than competition, which is no surprise.

Let us consider now the von Stackelberg sequential game. Using (14.14) and (14.15), we see that

Table 14.6 Battle of the sexes, alternative version.

from which we see that, with respect to the simultaneous game, the output of firm 1 is increased whereas the output of firm 2 is decreased. The total output and price are

respectively. The price is lower than in both previous cases, and the distribution of profit is now quite asymmetric:

The overall profit for the sequential game is lower than for the simultaneous one; however, the leader has a definite advantage and its profit is larger in the sequential game.

The toy example above shows that the privilege of moving first may yield an advantage to the leader. Given the structure of the game, it is easy to see that the leader of the sequential game cannot do worse than in the simultaneous game; in fact, she could produce the same amount as in the Cournot game, anyway. However, this need not apply in general. In particular, when there are asymmetries in information or things are random, the choice of the leader, or its outcome when there is uncertainty, could provide the follower with useful information. The following example shows that being the first to move is not always desirable.

Example 14.10 (Battle of the sexes, alternative versions) Let us consider again the battle of the sexes of Example 14.5, where now we assume that Juliet has the privilege of moving first. Given the payoffs in Table 14.2, she knows that, whatever her choice, Romeo will play the move that allows him to enjoy her company. Hence, she will play shopping for sure. In this case, Juliet does not face the uncertainty due to the presence of two Nash equilibria in Table 14.2 and is certainly happy to move first. The situation is quite different for the payoffs in Table 14.6. In this case, Romeo is indifferent between going to cinema or restaurant. What he really dreads is an evening with Morticia. It is easy to see that this game has no Nash equilibrium, as one of the two players has always an incentive to deviate. An equilibrium can be found if we admit mixed strategies, in which players select a strategy according to a probability distribution, related to the uncertainty about the move of the competitor. We do not consider mixed strategies here, but the important point in this case is that no player would like to move first.

Fig. 14.4 An example of the Braess’ paradox.

We noted that the first version of the battle of the sexes is a stylized coordination game for two firms that should adopt a common standard; in this second version, one firm wants to adopt the same standard as the competitor, whereas the other firm would like to select a different one.