Fig. 2(a) illustrates cyclical price wars that typically
occur when 5 pricebots use the myoptimal pricing strategy. Regardless
of the initial value of the price vector, a pattern quickly emerges in
which prices are positioned near the monopolistic price v = 1,
followed by a long episode during which pricebots successively
undercut one another by 
. During this latter phase, no two
prices differ by more than 
, and the prices fall
linearly with time. Eventually, when the lowest-priced seller is
within above the value 
, the next seller
finds it unprofitable to undercut, and instead resets its price to v
= 1. The other pricebots follow suit, until all but the
lowest-priced seller are charging v = 1. At this point, the
lowest-priced seller finds that it can maintain its market share but
increase its profit dramatically -- from 
to 
-- by raising its price to 
. No sooner
than the lowest-priced seller raises its price does the next seller to
reset its price undercut, thereby igniting the next cycle of the price
war.
Fig. 2(b) shows the sellers' profits averaged during the intervals between successive resetting of prices. The upper curve represents a linear decrease in the average profit attained by the lowest-priced seller as price decreases, whichever seller that happens to be. The lower curve represents the average profit attained by sellers that are not currently the lowest-priced; near the end of the cycle they suffer from both low market share and low margin. The expected average profit can be computed by averaging the profit given by Eqs. 5 and 6 over one price-war cycle:
which yields 
in this instance. The
simulation results match this closely: the average profit per time
step is 0.0515, which is just over twice the average profit
obtained via the game-theoretic pricing strategy.
Since prices fluctuate over time, it is of interest to compute the
probability distribution of prices. Fig. 2(a) depicts the
cumulative distribution function for myoptimal pricing. This measured
cumulative density function has exactly the same endpoints and v = 1 as those of the theoretical mixed strategy
equilibrium, but the linear shape between those endpoints (which
reflects the linear price war) is quite different from what is
displayed in Fig. 1(a).
Figure 2: (a) and (b) Price and profit dynamics, respectively, for 5 MY pricebots.
(c) Cumulative distribution of prices observed between times 10 and 100 million.
