Assume that each buyer b has 
and 
; that is, it is
extremely quality-sensitive, seeking
the highest-quality seller for which the price does
not exceed 
. Assume that the number of buyers
, and that is
distributed uniformly between 0 and 1. Furthermore,
assume that every buyer has access to perfect, completely
up-to-date information about the sellers' prices and
qualities. Finally, assume that each
seller s's quality 
is immutable, so that the sellers are
only free to set their prices. This allows the
cost for seller s to be abbreviated as a
fixed constant 
.
Now we can compare the behavior of the system under
several different assumptions about the profit-maximization
strategies employed by the sellers. Without loss of generality,
we can 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 seller 1 will attract
all buyers for which 
, and in
general (taking 
) seller s will attract all buyers for
which 
. Thus the profit
for seller s will be
provided that 
and
for all s' < s.
(If the first condition were not satisfied, then seller s would
lose money on each sale because it would be charging less than
the marginal cost. If the second condition were not satisfied,
because s would be undersold by a higher-quality seller.)
First, suppose that the sellers behave game-theoretically.
Seller s=1 is free to set its price at will because any
buyer that is willing to pay its price 
(i.e. 
)
will prefer it since no other seller offers higher quality.
Taking the derivative of Eq. 2 with
respect to and setting equal to zero easily yields
the conclusion that the optimal price
. Once s=1 has set
its price, s=2 can determine its price using the same
technique, and so on. In general, we find that
, or
.
Recursive substitution yields a simple closed-form expression:
The same analysis holds for myoptimal sellers. The highest
quality seller is completely unaffected by the prices
charged by its competitors. It can act as though it is
the only seller in the market, compute its optimal price,
and ignore the rest of the sellers because none can compete
on quality. Once s=1 has established its price, the seller
with the second highest quality concedes the premium buyers
to s=1, and acts as the highest quality seller in the
remaining market, and so on recursively, resulting in exactly
the same equilibrium prices as are given in Eq. 3.
As an example, we now examine the behavior of a system with
5 sellers with fixed qualities 
, 
, 
,
, and 
. The cost of producing a unit
of quality Q is taken to be a simple linear function: c(Q) = 0.1 + 0.1Q.
If the 5 sellers behave in a game-theoretic manner and the number
of buyers is infinite, then the
resultant prices can be computed from Eq. 3:
, 
, 
,
, and 
.
Now suppose that the sellers use the myoptimal strategy. In other
words, when it is a seller's turn to reevaluate its price, it
does an exhaustive search over all possible candidate
prices as follows. For each candidate price, it uses its knowledge
of its competitors prices and qualities and its knowledge of
the individual parameters of each of the buyers to compute the
expected profit for that candidate price. (Equivalently, an oracle
could perform this computation on the seller's behalf.)
It chooses the candidate
price that maximizes its expected price, assuming that no competitor
will alter its price in response.
Figure 2: Simulation of 5 myoptimal sellers. All buyers are quality-sensitive.
Figure 2 illustrates a typical simulation run for a
population of 5 myoptimal sellers and 1000 buyers.
After just a few time steps,
the simulated system reaches an equilibrium in which the prices
are very close to the game-theoretic values:
, 
,
, 
, and 
.
The small discrepancies can be attributed to the fact that
the number of buyers in the simulated system is finite
rather than infinite.
Computationally-limited myoptimal sellers have access to the
same unlimited information
that myoptimals do, but are more limited in computational capability.
They differ from myoptimals only in that they do not perform
an exhaustive search over all possible prices. Instead, they
randomly generate a small set of candidate prices, use
their perfect knowledge of all competitors and individual
buyers (or an oracle)
to compute the expected profit for each candidate price,
and select the best price.
In our implementation, computationally-limited myoptimals
consider the current price and 10 other randomly generated
candidate prices. With probability
0.9, the candidate prices are generated from a gaussian distribution
with standard deviation 0.02 centered about the current price. With
probability 0.1, the proposed price is chosen from a uniform distribution
in the interval (0,1). A typical simulation run is illustrated
in Fig. 3. After 5000 time steps, the equilibrium prices
were (0.627156 0.384965 0.252813 0.189860 0.163891) -- again,
reasonably close to the game-theoretic values.
Figure 3: Simulation of 5 computationally-limited myoptimal sellers.
All buyers are quality-sensitive.
It is useful to consider the opposite extreme, in which sellers
are uninformed about their competitors and the buyer population.
In this case, the sellers must use some sort of trial and
error, perhaps coupled with memory and/or learning. One
extremely simple approach is to use the Trial-and-Error
strategy: when it is time to re-evaluate price, with a
small small-jumping probability (0.05), generate a new price
by adding a small random increment drawn from a zero-mean
gaussian distribution with a small standard deviation (0.02).
Even less frequently, with some very small big-jumping
probability (0.001), generate a new price by drawing it from
a uniform distribution in the interval (0,1).
If the price has just undergone a small or big jump, the profit
that accrues until the next opportunity for a price adjustment
is measured. The profit per unit time is compared to what it was
prior to the jump. If it is higher, then the new price is retained.
If it is not, then the price reverts to what it was before.
A typical simulation run is shown in Fig. 4.
Again, the prices tend towards an approximate equilibrium,
although it is impossible for them to settle completely due
to the nature of the algorithm. Averaging over the last 1000
time steps, the approximate equilibrium price vector was
(0.6271 0.3889 0.2518 0.1849 0.1434), which is
fairly close to the computed game-theoretic equilibrium.
A second algorithm that can be used in situations where
sellers have no direct knowledge of competitors or buyers
is the derivative-following algorithm. A derivative
follower starts by measuring its profitability and then
perturbing its price up or down by a random amount.
If, when the seller next
re-evaluates its price, it finds that the profit per unit
time has increased, it will modify the price in the same
direction as before; otherwise it will reverse the direction
of the price change. We have found it helpful to use
a random step size; this avoids entrapment at ``false''
local maxima in the profit curve that exist in systems with
a finite number of buyers.
A typical simulation run with step size chosen uniformly
between 0 and 0.02 is shown in Fig. 5.
An average over the last 1000 time steps yields a price vector
(0.6292 0.4028 0.2689 0.2050 0.1701), which is again fairly
close to the computed game-theoretic value.
Figure 4: Simulation of 5 sellers, each of which employs the trial-and-error
pricing policy. All buyers are quality-sensitive.
Figure 5: Simulation of five derivative-following sellers.
All buyers are quality-sensitive.
