First, we focus on the sellers and their efforts to set prices so as to maximize profits. Here we consider three basic types of pricebot algorithms that span a wide range of requirements on the availability of market information and computational power, and observe the price dynamics that ensue when various combinations of these algorithms are run against one another.
The first such algorithm, GT, is based on a game-theoretic
computation. It can be shown that, unless the price quantum is
very coarse, there is no pure-strategy Nash
equilibrium [10] (i.e., there is no
stable set of equilibrium prices). However, there is a mixed-strategy
Nash equilibrium in which each seller chooses prices
randomly from a distribution f(p) that can be computed
from the buyer parameters w and 
and the number
of sellers S. The GT algorithm simply computes f(p)
and periodically chooses a different random price
from that distribution. An implicit assumption in the
game-theoretic derivation is that
no seller observes another seller's
price before it sets its own.
The second algorithm, MY, is referred to as the myoptimal (``myopically optimal'') or ``best-response Cournot'' algorithm. Like GT, it requires perfect knowledge of the buyer population and the number of competing sellers. In addition, it requires knowledge of the current prices of all competitors. Using this information, it sets the price to the value that will maximize profits in the short term -- up until the moment when some other seller changes its price. We have experimented somewhat with variations in which the optimization is imperfect, either due to imperfect search or imperfect information.
The third algorithm, DF, or derivative-follower, is the least informationally and computationally intensive of the algorithms. It does not require any knowledge or assumptions about buyers or competitors. It simply experiments with incremental increases (or decreases) in price, continuing to move its price in the same direction until the observed profitability level falls, at which point the direction of movement is reversed.
Figure 1 summarizes the wide range of dynamical behaviors that result from various mixtures of pricebots. It is organized as a 3-by-3 matrix depicting the 9 possible combinations of 1 vs. 4 pricebots of strategy types GT, MY and DF. Table 1 summarizes the corresponding time-averaged profits for each of the 9 mixtures.
Figure 1: Prices vs. time for 4-on-1 pricebot simulations. Open circles
represent the ``one'' pricebot that may employ a different strategy than
the other four. First row: One GT agent vs.
4 GT, 4 MY and 4 DF agents. Second row: One MY agent vs. 4 GT, 4 MY and 4 DF agents.
Third row: One DF agent vs. 4 GT, 4 MY, and 4 DF agents.
Table 1: 4-on-1 profit matrix. Within each cell, the profit of the
1 agent is given on the left; the average profits of each of the
4 agents of identical type are given on the right.
In all of these simulations, each of S=5 sellers held its price fixed for a short random interval, and then (asynchronously) used its pricing algorithm to generate a new price. The buyer strategies were held fixed: 25% checked just one price, 25% checked two prices, and the remaining 50% compared all prices and selected the seller with the lowest price. The buyers' valuations were distributed uniformly between 0 and 1. The marginal production cost r turns out to have no qualitative effect on the results; it was set to zero.
The pure combinations (5 GT, 5 MY, and 5 DF sellers) lie along the
diagonals of Table 1 and Figure 1. First,
consider the 5 GT sellers. Since GT does not observe its opponent's
choice of price prior to making its own, and since an undercutter can
grab a significant portion (80%) of the market share, there is a
strong incentive to price low. The prices for 5 GT pricebots are
strongly clustered just above the threshold price 
. Below this price, undercutting is not worthwhile because the
profit margins are so low that it is better to charge the monopolist
price 
and accept the low 5% market share consisting of
1/S of the s=1 buyers. By pricing just above this 
, a GT
pricebot is playing a game of ``chicken'' -- a higher price yields a
higher profit, but increases the chances of being undercut. Due to the
generally low prices, each GT pricebot earns relatively low profits of
0.0129 on average, as compared to the theoretical maximum of 0.05 on
average
that could be obtained by a
collusive cartel in which each seller charges the monopolistic price
of 0.5.
Now consider the 5 MY sellers. As seen in the central cell in
Figure 1, they undercut one another until the price falls to
, at which point undercutting becomes less profitable
than charging the monopolistic price of 0.5. Thus prices suddenly jump
up to this level. However, as soon as this occurs, undercutting
once again becomes attractive, and the price war cycle begins anew.
We have observed similar cyclic behavior in other models of markets
in which myoptimal agents are
present [18, 24], some of
which will be illustrated in the next section of this paper. Although
the MY sellers fall into endless price wars, their average prices are
higher than those of GT sellers. This is reflected in their much
higher average profit: 0.0337 vs. 0.0129 in this example.
Now consider the 5 DF sellers. Interestingly, although they are the least informed, they maintain the highest prices and therefore the highest profits. Their average profit of 0.0387 is not very much less than the optimal value of 0.05 that could be attained by a cartel.
Can a less-informed and less computationally intensive algorithm such as DF really fare better than MY and GT? The off-diagonal elements of Fig. 1 and Table 1 illustrate what happens when different types of pricebot algorithms are mixed together. In particular, consider the rightmost column of the figure and the table. Note that, when a single GT or MY agent is pitted against 4 DFs, it fares much better than does a DF. GT consistently undercuts the 4 DFs because it is biased towards low prices; MY is even more effective because it undercuts the 4 DFs by the minimal amount necessary to grab the 80% market share. Thus, if agents were permitted to select pricing strategies on the basis of expected long-term payoff, a society of 5 DFs would be unstable. The first agent to reconsider its strategy choice would switch to MY, as would each successive agent until all were converted to MY. The situation is analogous to the Prisoner's Dilemma: self-interest compels all of the sellers to defect to the MY strategy, even though this leads to a lower profit than would be obtained if they were to adhere to the DF strategy.
The bottom row of Fig. 1 and Table 1 illustrates the inferiority of the DF algorithm from another perspective. Here we find that, when a lone DF pricebot is introduced into a society of 4 GTs or 4 MYs, it is exploited: the GT's and MY's fare better with DF than they do with an agent of their own kind, and the DF pricebot fares worse than the other agents.
Can we conclude from this that MY will be the algorithm of choice for pricebots? Certainly not. If detailed buyer information is unavailable, then a simpler strategy like DF may be the only choice. In practice, the detailed buyer information required by MY and GT is unlikely to be obtained very easily. At best, one might be able to design an approximate version of MY that employs adaptive learning techniques to set prices. Additionally, even if reasonable approximations to the MY policy can be made, it still cannot be regarded as the preferred strategy because we have only considered three strategies out of an infinite spectrum. Is there another pricing algorithm that can beat MY? Certainly. We have explored two ways of improving MY: making it faster and making it less myopic. We now discuss each of these in turn.