Fig. 2(b) shows the price dynamics that result when 5 derivative followers are pitted against one another. Recall that derivative followers do not base their pricing decisions on any information that pertains to other agents in the system -- neither sellers' price-setting tendencies nor buyers' preferences. Nonetheless, their behavior tends towards what is in effect a collusive state in which all sellers charge nearly the monopolistic price. This is tacit collusion as defined, for example, in Tirole Tirole88, so-called because the agents do not communicate at all and there is consequently nothing illegal about their collusive behavior. Note that DF sellers accumulate greater profits than myoptimal or game-theoretic sellers. According to Fig. 3(b), sellers that are currently lowest-priced can expect an average profit of 0.30 to 0.35, while the others can expect roughly the game-theoretic profit of 0.025. Averaging over the last 90 million time steps (to eliminate transient effects), we find that the average profit per seller is 0.0841. This is near the absolute collusive limit of (1/S)(v - c) = 0.10, which would be obtained if all sellers were to fix their prices at 1.
How do derivative followers manage to collude? Like myoptimal sellers, DF sellers are capable of engaging in price wars; such dynamics are visible in Fig. 2(b). However, these price wars tend to involve only two sellers, and the positive feedback that drives them depends critically on both the sequence of price increments and the timing of the asynchronous moves by the sellers. Downward trends are therefore very easily disrupted. For example, if A's price is currently above B's, but A reduces its price by an amount insufficient to undercut B, then A's profits decrease, so that A raises its price in subsequent time steps. Soon after A breaks the downward cycle, B discovers that it can improve profits by increasing its price, and does so. Simulations clearly show that upward trends in price are much faster and more certain than downward trends. The tendency of a society of DF sellers to reach and maintain high prices is reflected in the cumulative distribution function, shown in Fig. 4(b).
It is also of interest to study the interplay among GT, MY, and DF sellers. Typically, we find that, when a myoptimal seller is introduced into a population of DF or GT sellers, it substantially outplays them, and their profits decline significantly.
