In article <AsGa7.22166$m8.7847311@news1.rdc1.md.home.com>, Earl Cox <earldcox1@home.com> wrote:> I refuse to get drawn into a discussion on this matter. But let me say that > these four rules, posted by Will, come from an actual product pricing model > which contained about 120 rules.
So it seems that I'm correct in thinking that the system of four rules is ridiculous, if the actual system needs 120 rules. Presumably, however, the original poster, and Cox, think that the four rules below are are good illustration of the general technique, even if a bit oversimplified:> 1. The price should be high > 2. The price must be low > 3. The price must be around 2 times cost > 4. If the competition price is not very high, then the price should be > near the competition price > > Mathematical definitions terms like "high", "low", "near the competition > price" etc. are part of this fuzzy system, which yields a suggested > price.
So let's compare this to a probabilistic / decision theory treatment. First, we'd need to define the goal. This is to maximize profit, which can be broken down into short-term profit and long-term profit. The latter can be discounted according to whatever is thought to be the appropriate interest rate. Short-term profit can in turn be represented as profit = price*sales - cost*sales - overhead Let's assume that overhead and cost (per item) are fixed, and sales depends on price, which we can set as we please. If we know the function relating sales to price, this is a mathematical problem, in which there is no role for fuzziness or probability. Of course we don't know exactly how sales would change in response to price. The relevant information for this would include past experience of sales changes in response to price changes, comments from customers, and beliefs about how competitors would respond. We could assess a probability distribution over price->sales functions by imagining several possible prices, and for each coming up with probability distributions for (a) how many customers would buy at this price, if they have no better alternative, (b) what price the competitors would charge in response, perhaps using whatever uncertain knowledge we have about their production costs, and (c) what fraction of the customers would be aware of our product and of the competitors' products, if these were the prices. This last will of course depend heavily on our assessment of what fraction of customers are currently aware of our product and the competitors' products, which will also be uncertain, but to which we can give a probability distribution. If we set a high price, however, customers will be motivated to research alternatives, perhaps becoming aware of competitors that they aren't currently aware of. This last point links to the question of how the price we set now could affect long-term sales. Of course, we can change the price later on, so we're talking here about long-term changes in the behaviour of customers and competitors, not simple questions of elasticity of demand. In particular, setting the price high might motivate customers to look for alternatives, which will then affect their future purchases, even if we later lower the price. Similarly, a high price might motivate investment in production equipment by competitors. These assessments are likely to be difficult and uncertain, but fortunately much of the work in assessing the short-term effects can be reused - eg, if we think that most customers are already aware of all the companies offering the product, we don't have to worry that setting a high price would encourage them to research alternative suppliers. Now, compare this process with the four fuzzy rules above (presuming the other 116 rules are similar in flavour). How, for instance, does "the price must be around 2 times cost" relate to the above? The answer is that it doesn't, because there isn't any reason for price to be 2 times the cost. This rule may come from somebody's unanalysed experience in some particular situation, where 2 times cost was perhaps a good price, but there's no reason to think this will be the right price in other situations. And why must the price be low? Presumably this derives from some combination of a belief that a high price will reduce sales in the short term, and that it may reduce customer "loyalty" in the long term, but surely nothing good can come of merging these two very distinct considerations (which we can see are actually inversely related, since a high level of customer knowledge makes the bad effect of a high price greater in the short term, but less in the long term). The fuzzy rules seem designed to just capture current prejudices, not produce a higher level of real understanding. The advantages of a real analysis of the problem, rather than fuzzy rules, is even greater when you consider the larger context. Might it be desirable to set a high price while increasing the advertising budget? Would investment in low-cost production equipment pay off, or might one find that being able to lower the price produced little increase in sales anyway, so that the return on the investiment would come only from lower cost, not increased sales? The probabilistic analysis above would help in answering these questions, but I don't see how the fuzzy rules would. Finally, note that there really is nothing fuzzy about this problem. It all comes down to dollars, which are pretty definite. There may be fuzzy aspects to the statements one gets from the experts, but since one can - and should - ask them what they really mean, modelling the fuzziness of their original statements is an absurd way of trying to solve the problem. All the usual bafflegab one hears about how the concept of "tall" is fuzzy is totally irrelevant. Radford Neal ---------------------------------------------------------------------------- Radford M. Neal radford@cs.utoronto.ca Dept. of Statistics and Dept. of Computer Science radford@utstat.utoronto.ca University of Toronto http://www.cs.utoronto.ca/~radford ----------------------------------------------------------------------------