Now consider the opposite extreme, in which the buyers
are completely price sensitive: provided that a certain
minimal quality level 
is met, b seeks
the least expensive seller. This limit is
obtained by setting 
for all b. We shall
again assume that 
is distributed uniformly
in the interval (0,1). We make the further assumption
that the price ceiling and quality floor are perfectly
correlated: buyers that require high quality
are more tolerant of paying a higher price for that
quality. This can be achieved (for example) by setting
.
Just as we did for quality-sensitive buyers, we
assume that every buyer has access to perfect, completely
up-to-date information about the sellers' prices and
qualities, and that each seller s's quality 
is
immutable, so that the sellers are only free to set their prices.
As before, we first compute the game-theoretic prices,
and then examine the collective behavior of the system
when all sellers employ the price-updating algorithms
described previously. Again, we can without loss of
generality order the sellers such that s=1 is the seller
with the highest quality, s=2 is the seller with
second-highest quality, etc.
Then, as computed in the Appendix,
Evaluating this numerically for the set of five sellers
studied in the previous section, we find that the game-theoretic
price vector 
= (0.9, 0.545, 0.35, 0.25, 0.1875).
Figure 6 shows the results of a typical
simulation run of this system in the case where all five sellers
are myoptimal. All parameters are identical to those used in the
simulation represented in Fig. 2, except
for the buyer parameters, which are exactly as described in the first
paragraph of the current section.
In contrast to what was found in section 3, the
system of myoptimal sellers does not reach the game-theoretic
equilibrium price -- in fact it fails spectacularly to reach any
equilibrium at all. The system regularly passes through
a price vector that is extremely close to the game-theoretic value:
= (0.900, 0.523, 0.350, 0.252, 0.185) at times
t = 565 + 560 n, where 
. However, on the next
time step, the highest quality seller s=1 realizes that it can make more
profit by dropping its price from 0.900 to 0.523, the price currently
charged by s=2. Although its margin drops from 0.7 per unit to 0.323
per unit (the production cost is 
), it steals the
market formerly held by s=2, resulting in an increase in sales volume
from 0.1 B to 0.477 B. However, this gain is short-lived because
s=2 immediately retaliates by settings its price just slightly below
that of s=1. Now s=1 is reduced back to its market share of 0.1 B,
and is making much less per unit than it was at the original price of 0.900.
Thus s=1 retaliates by matching s=2's lowered price. The war between
s=1 and s=2 continues, with the other three sellers staying at fixed
prices that are essentially equally to the game-theoretic values.
Eventually, the two warring sellers suddenly find it worthwhile to
horn in on s=3's market. The top three sellers continue in a three-way
price war, until the price becomes so depressed that s=1 finally opts
out and sets its price back up to 0.900. Once this price pressure is off,
the other sellers all set their prices up to their game-theoretic values,
but as soon as this occurs then s=1 is tempted to instigate a new price
war.
Figure 6: Simulation of 5 myoptimal sellers in a cyclic price war.
All buyers are price-sensitive.
Figure 7: Simulation of 5 computationally-limited myoptimal
sellers in a series of price wars.
All buyers are price-sensitive.
Figure 8: Simulation of 5 sellers employing trial-and-error pricing
strategy. All buyers are price-sensitive.
Figure 9: Simulation of 5 derivative followers, which reach price
equilibrium computed via game theory. All buyers are price-sensitive.
A simulation of computationally-limited myoptimal
sellers, shown in Fig. 7,
displays very similar behavior. The price wars are
somewhat faster than in the pure myoptimal case because,
while the sellers easily find that undercutting is (myopically)
favorable, they usually undercut by more than is absolutely
necessary. There is some additional jitter introduced into
the price-war period because s=1 and the other higher-quality
sellers may take a little longer than pure myoptimals to
realize that they should opt out of a price war.
As shown in Fig. 9, derivative followers quickly
gravitate towards the game-theoretic prices, and do not engage
in price-war behavior. During the interval from time 10,000 to time 50,000,
the average prices are measured to be
(0.89556, 0.53141, 0.35784, 0.25377, 0.18556),
and tend to stay within approximately 0.03 of these values.
It might be tempting to conclude at this point that myoptimal
and nearly-myoptimal sellers are too clever for their own good,
and that the individual ignorance of the derivative followers
is responsible for overall societal bliss (at least on the
part of the sellers, not necessarily the consumers!)
However, this is wrong on at least two counts. First, even
if individual ignorance could be shown to lead to good societal
behavior, this would not necessarily be a useful result
because in an open, massively distributed agent economy
we are very unlikely to be able to legislate an agent's degree
of intelligence. In other experiments, we have found
that a single myoptimal agent introduced into a society of
derivative followers can take tremendous advantage of its fellow
agents. Thus there is every incentive to create an intelligent
agent rather than a stupid one. Second, it is untrue that
simplistic strategies lead to stable behavior, as we are about
to see.
Figure 8 illustrates a typical simulation
run with 5 sellers employing the trial-and-error pricing strategy.
Despite a large amount of jitter due to the sellers' continual random
explorations, two longer-scale trends are evident: roughly metastable
periods, during which the prices are roughly equal to the game-theoretic
values, and price-war episodes (see in particular the period between
time 32,500 through time 40,000). This example shows that price wars
are not an artifact of having an unrealistic amount of knowledge and
computational power; they can occur even when an incredibly simplistic
pricing strategy is used.
