Summary of the utility landscape model

As a first step in the development of general foresight algorithms, we first consider the simplest possible case of two competing sellers, who alternately take turns adjusting their prices. We assume that the products offered by the two sellers are somewhat similar, leading to some potential for price competition between them, but there is also a degree of product differentiation, so that the cost and utility functions for the two sellers are in general asymmetric. There are several different economic models in which such asymmetries can come about. The primary model that we work with is a price-quality model that is described in detail in (Sairamesh and Kephart, 1998). In this model, products offered by different sellers are distinguished by different values of a ``quality'' parameter, with higher-quality products being perceived as more valuable by the consumers. The consumers are modeled as trying to obtain the lowest-priced product at each time step, subject to threshold-type constraints on both quality and price, i.e., each consumer has a maximum allowable price and a minimum allowable quality.

The other model that we have studied is an information-filtering model described in detail in (Kephart et al., 1998). In this model there are two competing sellers of news articles in somewhat overlapping categories. The partial overlap of the categories leads to a potential for direct price competition; however, the fact that they are not identical introduces an element of product differentiation similar to the quality differentiation in the price-quality model. This product differentiation leads to an asymmetry in the optimal pricing strategies for the two sellers. At each time step, the consumers decide to subscribe to one of the two sellers, based on price and on the consumer's particular interest categories.

In both of the models mentioned above, the consumers deterministically and instantaneously choose one of the two sellers at every time step, based on the prices of the two sellers. This means that the only relevant variables in the state space description are the prices of the two sellers at each time step. The two sellers alternately take turns adjusting their prices at each time step, and then depending on the particular prices set, the resulting consumer behavior determines the amount of ``profit'' or ``utility'' obtained by each seller. The simulation can iterate forever, and there may or may not be a discounting factor for the present value of future rewards.

An example utility function that we study, taken from the price-quality model, is as follows: Let and represent the prices charged by seller 1 and seller 2 respectively. Let and represent their respective quality parameters, with . Let c(Q) represent the cost to a seller of producing an item of quality Q. Then assuming the particular model of consumer behavior described in (Sairamesh and Kephart, 1998), one can show analytically that in the limit of infinitely many consumers, the instantaneous profits per consumer and obtained by seller 1 and seller 2 respectively are given by:

A plot of the profit landscape for seller 1 as a function of prices and is given in figure 1, for the following parameter settings: , , and c(Q) = 0.1 (1+Q). We can see in this figure that the myopic optimal price for seller 1 as a function of seller 2's price, , is obtained for each value of by sweeping across all values of and choosing the value that gives the highest profit. We can see that for small values of , the peak profit is obtained at , whereas for larger values of , there is eventually a discontinuous shift to the other peak, which follows along the parabolic-shaped ridge in the landscape. An analytic expression for the myopic optimal price for seller 1 as a function of is as follows (defining and ):

Similarly, the myopic optimal price for seller 2 as a function of the price set by seller 1, , is given by:

  
Figure 1: Sample profitability landscape for seller 1 in price-quality model, as a function of seller 1 price and seller 2 price .

We also note in passing that there are similar utility landscapes for each of the sellers in the information-filtering model (Kephart et al., 1998). In both models, it is the existence of multiple, disconnected peaks in the landscapes, with relative heights that can change depending on the other seller's price, that leads to price wars when the sellers behave myopically.

Regarding the information set that is made available to the sellers, we have made a simplifying assumption as a first step that the players have essentially perfect information. They can model the consumer behavior perfectly, and they also have perfect knowledge of each other's costs and utility functions. Hence our model is thus a two-player perfect-information deterministic game that is very similar to games like chess. The main differences are that the utilities in our model are not strictly zero-sum, and that there are no terminating or absorbing nodes in our model's state space. Also in our model, payoffs are given to the players at every time step, whereas in games such as chess, payoffs are only given at the terminating nodes.

As a final simplification, we constrain the prices set by the two sellers to lie in a range from some minimum to maximum allowable price. The prices are also discretized, so that one can create lookup tables for the seller utility functions . Furthermore, the optimal pricing policies for each seller as a function of the other seller's price, and , can also be represented in the form of table lookups.