In section 3, all of the pricing strategies converged towards
the game-theoretic equilibrium price vector,
regardless of their sophistication or naivete.
In section 4, we only made a few changes to the buyers,
and found that large-amplitude price wars were possible
unless the sellers were derivative followers, in which
case the game-theoretic analysis held. How can this behavior
be explained?
As has been discussed in [1, 2, 3], the
profit landscape is a useful construct for understanding
price-war dynamics. A seller's profit landscape is its expected
profit as a function of all of the sellers' prices (and any
other parameters they may control), assuming that buyers
can react instantly to any changes in the sellers' parameters.
Cyclical price wars are possible when profit landscapes contain
multiple peaks and sharp cliffs.
A typical pair of profit landscapes for a two-seller system with
a population of buyers as defined in section 3 is illustrated in
Figs. 10 and 11.
Analysis confirms that, for buyers of this type, profit landscapes
are quite generally single-peaked and devoid of cliffs.
Figure 10: Profit landscape for seller 1 in two-seller market with
quality-sensitive buyers. The sellers' qualities are 
.
Figure 11: Profit landscape for seller 2 in two-seller market with
quality-sensitive buyers. The sellers' qualities are .
Figure 12: Profit landscape for seller 1 in two-seller market with price-sensitive buyers. The sellers' qualities are .
For the buyer population defined in section 4, the landscapes have
the following analytic form for two sellers:
Figure 12 shows an example of the profit
landscape for seller 1, given . Note the sharp
cliff at 
and the two peaks, features that were shown to
give rise to price wars in [4].
To see how this gives rise to a price war, we can introduce the
function 
, the value of 
that optimizes 
,
for each possible 
, and the analogous function 
.
These functions can be expressed analytically as:
A simple graphical construction involving these two functions yields the
price dynamics, starting from any initial price vector 
,
as illustrated in Fig. 13.
Figure 13: Graphical construction of price war for two myopic sellers,
.
Now we are in a position to understand the dynamical behavior observed
in sections 3 and 4. In section 3, the
profit landscape only contained one peak, and no cliffs.
Any reasonably decent pricing strategy should eventually find
its way to the top of this peak, although the rates at which
they do vary considerably, from instant in the case of the
myopic sellers to very slowly for the trial-and-error strategists.
In section 4, the 5-dimensional profit landscape is multi-peaked
and full of cliffs. The derivative-following algorithm only does
a local search. Therefore, once it finds its way to the top of a
peak, it can only jitter around on the top of that peak, and has
no way to find other peaks. In our case, the derivative followers
found their way to the game-theoretic peak. In fact, there is a more
optimal peak (in the myopic sense) for seller s=1, but in order
to discover it the seller would have to make a large discontinuous
jump. All of the other pricing strategies, including the very naive
trial-and-error strategy, permit large jumps. Even though the trial-and-error
strategy only permits large jumps on rare occasions, eventually a
large random jump will take the system from the game-theoretic
peak to a location somewhere near a more (myopically) optimal one,
triggering a price war.
