Fig. 2(a) and Fig. 2(b) portray simulations
of heterogeneous play. Consider the price dynamics of
Fig 2(a), which depicts 1GT vs. 1DF. In this scenario,
the game-theoretic pricebot outperforms the derivative follower by
more often than not capturing greater market share with its lower
price. In response, the derivative follower charges relatively high
prices, although it does not oscillate precisely around the
monopolistic price 
; instead, it prices in a range slightly
below this optimum, since in doing so it more frequently finds itself
to be the lower-priced of the two pricebots. The average profits of
GT were 0.0682, while DF's average profits were less than half this
value at 0.0334.
In contrast, Fig 2(b) depicts the price dynamics of 1GT
vs. 1MY. Unlike the derivative follower, the myoptimal pricebot
outperforms the game-theoretic pricebot; specifically, the
time-averaged profits of GT were merely 0.0235, while MY achieved
0.0494. MY pricing has two notable advantages over derivative
following: (i) access to full information pertaining to both
competitors' prices and buyer demand, and (ii) the ability to change
its price discontinuously, if necessary. Accordingly,
Fig. 2(b) reveals that MY prices generally just undercut
GT prices, unless GT charges 
, in which case MY charges . We temporarily defer discussion of competition between MY and
DF pricebots.
Table 1 summarizes the results of our 1-on-1 GT, MY, and DF simulations by depicting the time-averaged profits obtained by pricebots that employed the various strategies as indicated. It is interesting to consider this profit matrix as representing the payoffs of a normal form game in which there are three possible strategies, namely MY, DF, and GT. In doing so, we observe that the strategy profiles (1MY, 1MY) and (1DF, 1DF) are both pure strategy Nash equilibria, with the latter as Pareto optimal. Moreover, regardless of the opponents' behavior, it is always preferable to choose strategy MY or DF, rather than behave as prescribed by GT: i.e., the elimination of dominated strategies eliminates the game-theoretic strategists. This outcome is not entirely surprising in view of the fact that GT is rooted in a stage game analysis, and prescribes play with no regard for historical data. In contrast, MY and DF take into account changing environmental conditions, and are thus more apt in asynchronous, repeated game settings.
Figure 2: 1-on-1 Pricebot Simulations
Table 1: 1-on-1 Profit Matrix. Within a given
cell, the left-hand profit is that received by an agent employing the
strategy corresponding to that cell's row, while the right-hand profit
is that received by an agent employing the strategy corresponding to
that cell's column.
Figure 3: 4-on-1 Pricebot Simulations: Prices vs. Time
We now turn to heterogeneous simulations of 5 pricebots. These
simulations were conducted assuming 1 individual pricebot vs. 4
pricebots playing the same strategy: e.g., 1MY vs. 4DF.
Fig 3 displays a 
matrix depicting the 9
possible combinations of 1 vs. 4 pricebots of strategies MY, DF, and
GT, with the cells in the matrix indexed by 
: e.g.,
Fig 3 
refers to the cell that depicts 1MY vs. 4DF.
We describe two of the more interesting off-diagonal entries.
