Uninformed consumers

For a variety of reasons, a consumer agent may have to rely on adaptive estimates of its valuation distribution, g. Suppose for example that consumer i knows that g is an exponential distribution, but does not know its individual parameter . Depending on the consumer's beliefs about the dynamics of the environment, it might wish to place more or less weight on recent observations. Since the mean of the distribution is , one reasonable and flexible approach to estimating is to start with an assumed prior and after each period of subscription to update the estimate according to

 

where is the mean of the N valuations received during the subscription period t, and is the consumer's ``flightiness'' factor. This factor could be set to a constant, or be time-varying; in fact, if , all observations are weighted equally, yielding the standard maximum likelihood estimator of for exponential distributions [7].

Fig. 3 presents one view of what happens when the consumers all start with exactly the right estimates of their parameters, but update their estimates in accordance with Eq. 14 with . The profit landscape of Fig. 3 was generated as follows: for each pair of f and in a large grid, a simulation was run for 200 subscription periods, and the average profit during that period was recorded on the vertical axis. Compared to Fig 1a, the landscape peaks at a lower value of f and a higher value of , and the overall profit at that peak is lower. More precisely, the peak of the landscape of Fig. 3 occurs at , as compared to the ``ideal'' peak of Fig. 1a, which occurs at . In other words, if the producer is constrained to a fixed price schedule, then over the course of 200 subscription periods it is most profitable to eliminate subscription fees entirely, compensating only partly by raising the price-per-item by a large factor. The optimal and in Fig. 3 obtained from simulation are consistent with Eq. 13: with F fixed at 0, optimization with respect to yields an optimal and a corresponding .

 
Figure: Profit landscape when consumers estimate their values, and the flightiness parameter is . The graph shows the expected average profit after 200 subscription periods when f and are held fixed. The market consists of M=1000 consumers and one seller offering N=10 articles per subscription period. As in Fig 1a, h is a uniform distribution with , , and the production cost .  

Figure 4 provides a different perspective on how consumer learning affects the producer's price-setting strategy and profits. Suppose that the producer were to fix its price parameters f and to their ideal values as given by Fig. 2. Again, we suppose that the production cost is , so the ideal values are , the corresponding ``ideal'' profit , and the fraction of subscribers m=1. As before, we suppose that the consumers all start with correct estimates of their parameters and then update their estimates in accordance with Eq. 14 with as they continue to purchase items. Fig. 4a illustrates the resultant profit (normalized by dividing by the ``ideal'' value of 0.41367) and consumer subscription rate as a function of time. Both continue to diminish over time. Interestingly, they do so at almost exactly the same rate, suggesting that the loss of profit is almost entirely attributable to the continual loss of consumers.

Fig. 4b offers some insight into the phenomenon of consumer leakage. It shows the distribution of estimated in the consumer population at iteration 0 (the dashed line, representing the original distribution ) and at iteration 40. Although the distribution of is uniform at iteration 0, it evolves to a bimodal distribution by iteration 40. As consumer i continually updates its estimate , this estimate exhibits a random walk about its correct value . If drifts above a certain critical threshold , the expected value of subscribing falls below the subscription fee, and consumer i opts out of the market. Once this occurs, consumer i has no further source of information that might demonstrate that its estimated valuation is overly pessimistic, so it remains permanently disenfranchised from the market. The threshold value therefore acts as an absorbing boundary; its value can be computed implicitly from Eq. 12: . In the scenario presented in Fig. 4b, . Consumers with below this value are still subscribing; a consumer i with above this value will never subscribe again unless the producer entices it back into the market by lowering f or sufficiently to raise the threshold above .

 
Figure 4: (a)Profit (normalized to its ideal value) and proportion of subscribed consumers m vs. time (in subscription periods) when consumers estimate their values; flightiness parameter . Both and m diminish indefinitely and nearly identically, although the rate of reduction slows with time. The market consists of M=10000 consumers and one seller offering N=10 articles per subscription period. (b) Histogram of in the consumer population at iteration 40. The dashed line indicates the initial distribution of .  

Fig. 4b shows that there is a large portion of consumers with only slightly more than . Even a slight lowering of f and/or by the producer would encourage these consumers to re-enter the market. To help illustrate this point, Fig. 5 shows a profit landscape as a function of f and conditioned on the distribution of at iterations a) 0, b) 40, and c) 200. In other words, the simulation is run with the ideal settings of f and for t iterations, and then f and are changed suddenly to different values at iteration t+1. The landscapes show the expected profit for iteration t+1. According to Fig. 4b, the profit landscape initially peaks at the ideal values , but as time passes the peak quickly develops into an ever-deepening trough that is bordered by two ridges, each of which contains a peak. The producer could do better by switching to either of the two peaks -- either by lowering or raising its price parameters appropriately. In the first scenario, the producer may lower f and/or so as to lower . This will lure previously disenfranchised consumers back into the market, whereupon they may discover that their valuation estimates were overly pessimistic. These recaptured consumers may then be willing to stay in the market even if prices rise again. In the second scenario, the seller increases profits by raising f and/or . This works in the short term because the remaining consumers tend to be those for whom the actual value is considerably less than , and therefore they are willing to pay more. However, in the long term, this reduces the threshold substantially, causing consumer leakage to occur at a much higher rate.

 
Figure: Profit landscape showing the mean profit per article per consumer as a function of f and conditioned on the distribution of a) at iteration 0, b) at iteration 40, and c) at iteration 200. Consumers are assumed to estimate their values with flightiness parameter . (Note that, compared to Fig 1a and Fig 3, this figure uses a smaller range along the axis representing .)  

Regardless of , , or other such parameters, consumer leakage will eventually lead to complete market failure for any fixed, positive settings of f and . The only way to sustain profits is to fix f=0, in which case the maximal profit is obtained at . But this is only 0.684 of the ideal profit. In the next subsection, we discuss how a producer might improve its performance above this level by using dynamic pricing and optimization, allowing it to charge finite subscription fees without suffering from unchecked erosion of profits and subscription levels.