Proposition 1 establishes that dynamic behavior under adjustment processes of the form (3) is quite different from more sophisticated adjustment processes, such as Nash play or fictitious play, particularly when the number of firms in the market is large. However, the latter typically require more cognitive effort. In this section, we introduce an evolutionary competition between the different adjustment processes. For this, we model our Cournot oligopoly as a population game. That is, we consider a large population of firms from which in each period groups of n firms are sampled randomly to play the one-shot n-firm Cournot oligopoly. Firms may use different adjustment processes, and they switch between these processes according to a general, monotone selection dynamic, capturing the idea that an adjustment process that performs better is more likely to spread through the population of firms. In this paper, we focus on the interaction between Nash play and a single short-memory adjustment process of the form (3). Denote by \(\rho _{t}\in \left[ 0,1\right] \) the fraction of Nash firms in the population in period t, with a fraction \(1-\rho _{t}\) using the short-memory adjustment process—from here on we will refer to the latter as F-firms. After each period, the fraction \(\rho _{t}\) is updated and the random matching procedure is repeated.
First consider the decision of a Nash firm that knows the fraction of Nash firms in the population and the production decision of the F-firms, but does not know the exact composition of firms in its market (or it has to make a production decision before observing this). This firm forms expectations over all possible mixtures resulting from independently drawing \(n-1\) other players from a large population, each of which is either a Nash firm or a F-firm. Nash firm i therefore chooses quantity \(q_{i}\) such that the objective function
$$\begin{aligned} \sum _{k=0}^{n-1}\left( {\begin{array}{c}n-1\\ k\end{array}}\right) \rho _{t}^{k}\left( 1-\rho _{t}\right) ^{n-1-k}\left[ P\left( \left( n-1-k\right) q_{t}+kq^{N}+q_{i}\right) q_{i}-C\left( q_{i}\right) \right] , \end{aligned}$$
is maximized. Here \(q^{N}\) is the (symmetric) output level of each of the other Nash firms, and \(q_{t}\) is the output level of each F-firm, which is given by (3). Assuming that F-firms respond to the industry-wide average quantity from the previous period, the quantities they set will be symmetric (provided all of them start out with the same quantity \(q_{0}\)), see Eq. (8) below. The first-order condition for an optimum is characterized by equality between marginal cost and expected marginal revenue. We assume that, given the value of \(q_{t}\), all Nash firms coordinate on the same output level \(q^{N}\). The first-order condition, with \(q_{i}=q^{N}\), reads
$$\begin{aligned}&\sum _{k=0}^{n-1}\left( {\begin{array}{c}n-1\\ k\end{array}}\right) \rho _{t}^{k}\left( 1-\rho _{t}\right) ^{n-1-k} \nonumber \\&\quad \times \left[ P\left( \left( n-1-k\right) q_{t}+\left( k+1\right) q^{N}\right) \right. \nonumber \\&\quad \left. +\,q^{N}P^{\prime }\left( \left( n-1-k\right) q_{t}+\left( k+1\right) q^{N}\right) -C^{\prime }\left( q^{N}\right) \right] =0. \end{aligned}$$
(7)
Let the solution to (7) be given by \(q^{N}=H\left( q_{t},\rho _{t}\right) \).Footnote 16 Note that if the F-firms play the Cournot-Nash equilibrium quantity \(q^{*}\), or if all firms are Nash firms, then Nash firms will produce \(q^{*}\) as well, that is \(H\left( q^{*},\rho _{t}\right) =q^{*}\), for all \(\rho _{t}\) and \(H\left( q_{t},1\right) =q^{*}\) for all \(q_{t}\). Moreover, a Nash firm that is certain it will only meet F-firms plays a best-reply to current aggregate output of these F-firms, that is \(H\left( q_{t},0\right) =R\left( \left( n-1\right) q_{t}\right) \), for all \(q_{t}\).
We assume that F-firms know the average quantity \({\overline{q}}_{t-1}\) played across the population of firms in period \(t-1\). We therefore obtain
$$\begin{aligned} q_{t}=F\left( q_{t-1},\left( n-1\right) {\overline{q}}_{t-1}\right) =F\left( q_{t-1},\left( n-1\right) \left( \rho _{t-1}H\left( q_{t-1},\rho _{t-1}\right) +\left( 1-\rho _{t-1}\right) q_{t-1}\right) \right) , \end{aligned}$$
(8)
with the output of a Nash firm in period t given by \(q_{t}^{N}=H\left( q_{t},\rho _{t}\right) \).
The evolutionary competition between adjustment processes is driven by the profits they generate. Taking into account that a Nash firm meets between 0 and \(n-1\) other Nash firms, expected profits for a Nash firm are given by
$$\begin{aligned} \Pi _{N}\left( q^{N},q,\rho \right)= & {} \sum _{k=0}^{n-1}\left( {\begin{array}{c}n-1\\ k\end{array}}\right) \rho ^{k}\left( 1-\rho \right) ^{n-1-k}\nonumber \\&\times \left[ P\left( \left( k+1\right) q^{N}+\left( n-1-k\right) q\right) q^{N}-C\left( q^{N}\right) \right] , \end{aligned}$$
(9)
where \(q^{N}\) and q are the (symmetric) quantities set by Nash firms and F-firms, respectively. Expected profits \(\Pi _{F}\left( q^{N},q,\rho \right) \) for an F-firm can be determined in a similar manner. If the population of firms and the number of groups of n firms drawn from that population are large enough, average profits will be approximated quite well by these expected profits, which we will use as a proxy for average profits from now on. In addition, because the information requirements for Nash play are substantially higher than those for short-memory adjustment processes, we allow for differences in information or deliberation costs \(\kappa _{N},\kappa _{F}\ge 0\) required to implement these types of behavior. Performance of Nash and F-firms is then evaluated according to \(V_{i} =\Pi _{i}-\kappa _{i}\) where \(i=N,F\).
The fraction \(\rho _{t}\) of Nash firms evolves endogenously according to a dynamic which is an increasing function of the performance differential between the two adjustment processes, that is
$$\begin{aligned} \rho _{t}=G\left( V_{N,t-1}-V_{F,t-1}\right) =G\left( \Pi _{N,t-1} -\Pi _{F,t-1}-\kappa \right) , \end{aligned}$$
(10)
where \(\kappa \equiv \kappa _{N}-\kappa _{F}\) is the difference in deliberation costs, which we—given the information requirements for Nash play in a heterogeneous environment—assume to be nonnegative.Footnote 17 The map \(G: {\mathbb {R}} \rightarrow \left[ 0,1\right] \) is a continuously differentiable, monotonically increasing function with \(G\left( 0\right) =\frac{1}{2}\), \(\lim _{x\rightarrow -\infty }G\left( x\right) =0\) and \(\lim _{x\rightarrow \infty }G\left( x\right) =1\). One possible choice for \(G\left( \cdot \right) \) that satisfies these properties is the discrete choice model, \(G\left( x\right) =\left[ 1+\exp \left( -\beta x\right) \right] ^{-1}\), see Anderson et al. [4]. This model is based on stochastic choice of firms, who observe performance of the different adjustment processes and tend to choose the better performing process with a higher probability. This model is very popular in heterogeneous agent models (see, e.g., Brock and Hommes [10]) and in the literature on quantal response equilibria (see, e.g., McKelvey and Palfrey [33]), and we will use this specification in Sect. 4. It is straightforward to generalize this approach to allow for other (and more than two) adjustment processes, or to let it depend upon performance of these processes from earlier periods.
The dynamics of the quantities and fractions are governed by Eqs. (8) and (10). The steady state of this dynamic system is \(\left( q^{*},\rho _{\kappa }\right) \), where \(q^{*}\) is the Cournot-Nash equilibrium quantity, and \(\rho _{\kappa }=G\left( -\kappa \right) \) is the fraction of Nash firms at the steady state. Because market profits are the same in equilibrium, this fraction depends only on the difference in deliberation costs. We have the following stability result:
Proposition 2
Let \(P^{\prime }\left( Q^{*}\right) +q^{*}P^{\prime \prime }\left( Q^{*}\right) <0\). Then the equilibrium \(\left( q^{*},\rho _{\kappa }\right) \) of the model with evolutionary competition between Nash play and the short-memory adjustment process (3) is locally stable if:
$$\begin{aligned} \frac{\left( 1-\rho _{\kappa }\right) \left( n-1\right) }{1-\rho _{\kappa }\left( n-1\right) R^{\prime }\left( Q_{-i}^{*}\right) }<-\frac{1+F_{q}^{*}}{F_{Q}^{*}}, \end{aligned}$$
(11)
and unstable if
$$\begin{aligned} \frac{\left( 1-\rho _{\kappa }\right) \left( n-1\right) }{1-\rho _{\kappa }\left( n-1\right) R^{\prime }\left( Q_{-i}^{*}\right) }>-\frac{1+F_{q}^{*}}{F_{Q}^{*}}. \end{aligned}$$
(12)
Note that it follows from condition (11) that for a sufficiently large fraction of Nash firms the Cournot-Nash equilibrium will be stable. On the other hand, from rearranging condition (12), we find that a sufficient condition for instability is
$$\begin{aligned} \frac{n-\rho _{\kappa }\left( n-1\right) \left[ 1+R^{\prime }\left( Q_{-i}^{*}\right) \right] }{1-\rho _{\kappa }\left( n-1\right) R^{\prime }\left( Q_{-i}^{*}\right) }>1-\frac{1+F_{q}^{*}}{F_{Q}^{*}} \end{aligned}$$
(13)
Note that the right-hand sides of conditions (6) and (13) are the same, but that the left-hand side of (13) is smaller than n [the left-hand side of (6)], provided \(-1\le R^{\prime }\left( Q_{-i}^{*}\right) \le 0\). Introducing Nash firms in an environment with F-firms therefore has a stabilizing effect.
In the next section, we will see that instability is still possible and that the model with interaction between Nash play and a short-memory adjustment process may actually give rise to complicated and unpredictable dynamics. Before we go into that, however, a remark on the evolutionary process (10) is in order, since it does not include the well-known replicator dynamics. These replicator dynamics—developed by evolutionary biologists (see [26] and [44]), but also applied to many evolutionary economic models—can be derived from a model of imitation, see, e.g., Gale et al. [24] or Schlag [41]. For our case, the standard replicator dynamics is given as:
$$\begin{aligned} \rho _{t}=\frac{\rho _{t-1}V_{N,t-1}}{\rho _{t-1}V_{N,t-1}+\left( 1-\rho _{t-1}\right) V_{F,t-1}}. \end{aligned}$$
(14)
For \(\kappa >0\) the model consisting of (8) and (14) has two equilibria, \(\left( q^{*},0\right) \) and \(\left( q^{*},1\right) \), both of which are unstable if the market with only F-firms is unstable. Because in equilibrium Nash firms and F-firms do not coexist, the standard replicator dynamics does not seem to be a suitable model to study the stabilizing effect of an increase in the fraction of Nash firms. This issue can be addressed by introducing noisy decision making in the replicator dynamics (see, e.g., [20] and [22, 24]), which gives rise to
$$\begin{aligned} \rho _{t}=\delta +\left( 1-2\delta \right) \frac{\rho _{t-1}V_{N,t-1}}{\rho _{t-1}V_{N,t-1}+\left( 1-\rho _{t-1}\right) V_{F,t-1}}. \end{aligned}$$
(15)
Here each period a fraction \(2\delta \) of the population chooses between the adjustment processes randomly (with equal probability) and independent of past performance. For this specification of the replicator dynamics, there will be a unique equilibrium \(\left( q^{*},\rho _{\delta }\right) \), with \(\rho _{\delta }\in \left( 0,1\right) \). As \(\delta \) decreases (or as \(\kappa \) increases), \(\rho _{\delta }\) decreases and for \(\rho _{\delta }\) small enough (and n high enough) the equilibrium will be unstable. The local stability properties for the model with the noisy replicator dynamics will therefore be similar to that of the model we study here, although the global dynamics is typically different, see the discussion in the next section. Note that the economic interpretation of the replicator dynamics is also different from that of models of the form (10), such as the discrete choice model. The former relies upon (pairwise) imitation which implies that if one adjustment process performs better than the other, but is initially used by only a small fraction of the population (that is, \(\rho _{t}\) is close to 0 or 1), it may take quite some periods for that adjustment process to be used by almost all firms. In contrast, for models of the form (10) almost the full population may switch to the better performing adjustment process in only one period. Since in oligopolistic markets firms may arguably perform some kind of (possibly restricted) optimization, instead of simply imitating another firm, we have a slight preference for models of the form (10) as a description of how firms choose adjustment processes.