Fig. 2(a) illustrates the cyclical price wars that
typically occur when all 5 sellers 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 the sellers 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 sellers 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 who resets its price
undercut, thereby igniting the next cycle of the price war.
Fig. 3(a) 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. 7 and 8 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. 4(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 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).