Fig. 3(a) shows the price dynamics that result when 5
derivative followers are pitted against one another. Recall that
derivative following pricebots do not base their pricing decisions on
any information that pertains to other agents in the system --
neither pricebots' price-setting tendencies nor buyers' preferences.
Nonetheless, their behavior tends towards what is in effect a
collusive state in which all pricebots charge nearly the
monopolistic price.
This is tacit collusion as
defined, for example, in Tirole [30], and so-called because
the agents do not communicate at all so there is consequently nothing
illegal about their collusive behavior. By exhibiting such behavior,
derivative followers accumulate greater wealth than myoptimal or
game-theoretic pricebots. According to Fig. 3(b),
pricebots 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 value is not far off from the
absolute collusive limit of (1/S)(v - c) = 0.10.
Figure 3: (a) and (b) Price and profit dynamics, respectively, for 5 DF pricebots.
(c) Cumulative distribution of prices observed between times 10 and 100 million.
How do derivative followers manage to collude? Like myoptimal pricebots, DFs are capable of engaging in price wars; such dynamics are visible in Fig. 3(a). These price wars, however, are easily quelled, making upward trends more likely than downward trends. Suppose X and Y are the two lowest-priced pricebots engaged in a mini-price war. Assume X's price is initially above Y's, but that X soon undercuts Y. This yields profits for seller X obtained from the entire population of type B buyers while it is lower-priced, and from its share of type A buyers all throughout. Now suppose Y undercuts X, but soon after X again undercuts Y. This yields profits for seller X once again obtained from the entire population of type B buyers during the period in which it is lower-priced, and from its share of type A buyers all throughout. In other words, given equivalent rates of price adjustment for both pricebots, market share remains fixed during mini-price wars of this kind. Thus, the only variable in computing profits is price, leaving pricebots with the incentive to increases prices more often than not. The tendency of a society of DF pricebots to reach and maintain high prices is reflected in the cumulative distribution function, shown in Fig. 3(c).