Shopbots as Economic Agents

 

A truly unique aspect of shopbots is that they possess the (as yet unrealized) potential to act as economic agents themselves -- that is, they could charge buyers directly for providing a pricing information service. Thus, search costs need not be merely an exogenously determined constant -- they could be set strategically by shopbots in an effort to maximize their own profits. In this section, we suppose that a shopbot charges a price for i randomly chosen price quotes, and we examine how it can judiciously set its price schedule so as to maximize profit. We assume the shopbot has no variable production costs.

For a start, assume that the shopbot is the only means by which buyers can obtain price quotes. Then the shopbot can easily extract the entire surplus from the market by setting . According to this cost schedule, buyers do not pay an additional amount for a second quote; thus, they all request two quotes. In particular, , and the sellers charge the marginal cost r, obtaining no surplus at all. The sum of the expected cost of the item and the search cost is precisely the buyer valuation v, so the buyers buy the item, also receiving no surplus. All of the surplus (v-r) goes to the shopbot!

It is more natural to suppose that the buyers have an alternative mechanism by which they can discover prices, such as manual search. In this scenario, all of the buyers can weigh the costs and benefits of using the shopbot against those of using the alternative search mechanism to decide which mechanism to use and how much search to perform or request. In this case, the shopbot's profit is given by:

 

where represents the step function, equal to 1 for x>0 and 0 otherwise. The two step functions represent the fact that, in order to capture the Search-i segment of the market, the shopbot must both undercut the cost of using the alternate search mechanism and must price low enough so that the search cost plus the expected item price does not exceed the buyer's valuation.

Recall from the previous section that since buyers' strategy choices depend on other buyers' choices, strategy vector may have a complex time dependency. We can obtain some insights into the optimal pricing of prices, however, by considering a simple example that is analytically tractable.

Suppose that the cost of the alternative search mechanism is linear in the number of price quotes: . Furthermore, assume that the shopbot restricts itself to offering only 1 or 2 quotes. Then its task is to set the values of and so as to maximize its expected profit. Then, as shown in Sec. 3.2, rational, fully-informed buyers will evolve to an equilibrium in which the marginal benefit and cost of a second quote is balanced. Suppose that the shopbot takes a long-term view, so that it does not factor transients into its calculations, but only seeks to set a price structure to maximize Eq. 17 with set to its asymptotic equilibrium value. Finally, without loss of generality, we can set v=1 and r=0; generalized expressions for all derived quantities can be obtained by a simple rescaling.

Using the fact that, at equilibrium, the marginal benefit equals the marginal cost , the shopbot's profit can be written as follows:

 

subject to the conditions:

 

Taken together, the first condition in Eq. 19 and the rightmost expression for the shopbot's profit in Eq. 18 suggest that should always be chosen to just undercut c'. The fourth condition may be eliminated because it is redundant with the third, given that the marginal benefit and marginal cost of the second quote are equal when is in equilibrium.

Temporarily ignoring the second and third conditions, it is apparent from Eq. 18 that the shopbot's profit is maximized when is maximized. Using Eq. 14 and solving numerically for the optimal value of , we find that it occurs at . If is less than this value, the expected item prices and increase and so does their differential , but the increase in fails to compensate for the reduction in . On the other hand, if is greater than , the expected item prices and the price differential decrease more than would be compensated by the increase in .

In order for the shopbot to encourage the buyer population to evolve to , it should set the price differential to the corresponding optimal . This strategy, however, is not guaranteed to be successful, because (as illustrated in Fig. 4(b) for ) there are two values of that correspond to : the desired value and an additional unstable solution at . As discussed in Sec. 3.2, the system will evolve to a state in which if the initial value of is less than 0.465602. To cope with low initial values of , the shopbot could use a more sophisticated strategy. It could deliberately charge a very small initially, such that the unstable equilibrium for is less than the initial value of . This would cause to increase, whereupon the shopbot could gradually raise up to the desired value of .

Now consider the second condition in Eq. 19. It will be violated if , which occurs when . In this regime, the shopbot cannot charge for the second quote because buyers wishing to purchase two quotes would choose the cheaper alternate search method. Instead, the shopbot must undercut the alternate search method, which it may do by setting .

Finally, consider the third condition. Substitution of Eqs. 12 and 7 yields the constraint , where

 

This constraint comes into play when . In this regime, c' is large enough so that, with , the expected item cost plus the search cost would exceed the buyer's valuation. This would cause the buyers to opt out of the market, resulting in no profit for the shopbot. In order to decrease the total cost to the buyer, the shopbot can still set to just undercut c', but it must reduce the overall price to the buyer by manipulating so as to increase above , which in turn decreases and . The shopbot can achieve this by reducing below . In this case, the optimal value of is determined by inverting , or , and . It can be shown that, in this regime, is precisely equal to .

In addition to these three distinct ranges of c', there is a fourth scenario in which c'>1. In this case, the buyers cannot afford the alternate search mechanism. This is tantamount to the shopbot being the only search mechanism, and as discussed earlier, the shopbot extracts all of the market surplus.

The analytic results for , , , and in these four different ranges of c' are summarized in Table 5 and illustrated in Fig. 7. In Table 5 and Fig. 7, (v-r) is normalized to 1; the result for general (v-r) can be obtained by multiplying all search cost parameters (e.g., c', , and ) by this quantity.

  
Table 2: Optimal shopbot prices and , Strategy-2 population , and shopbot profit as a function of the alternate search cost c'. Special values , and are defined in the text.

  
Figure 7: Optimal shopbot parameters as a function of alternative search cost c', with v-r normalized to 1. a) Shopbot prices and as a function of c'. b) Population of Search-2 buyers , shopbot profit , total seller profit , and buyer surplus .

Figure 8(a) displays two additional quantities of interest: the total seller profit (summed over all sellers), which by Eq. 8 is simply , and the buyer surplus , which in this case is as follows:

 

Note therefore that the total social welfare is simply . For general v and r, these quantities can be rescaled to yield a total social welfare of v-r. This is a consequence of the assumption that all buyers have equal valuations v, and the fact that the shopbots are motivated to manipulate the market to ensure that all buyers purchase the item.

  
Figure 8: Buyer surplus and total seller profit as a function of alternate search cost c' under two different assumptions: a) shopbot sets and to maximize its own profit, and b) no shopbot present.

For comparison, Fig. 8(b) depicts the buyer surplus and seller profit in the case where there is no shopbot, i.e., the buyers simply pay c' for one quote and 2 c' for two quotes. There are two regimes. Recall that if is sufficiently small, there are two solutions to , one of which is stable. Strategy-1 and Strategy-2 buyers can co-exist at the non-trivial stable equilibrium defined by provided that c' < 0.103872, the maximal value attained by (this occurs at ). Again, this assumes that the initial conditions are such that the population will evolve towards the non-trivial equilibrium rather than the one at . In the range c' < 0.103872, the total seller profit is simply , and the buyer surplus is . However, if c' exceeds the threshold 0.103872, then there is no solution such that , and the system will evolve to the only stable equilibrium: the trivial one at . Since none of the buyers compare prices, the sellers are free to behave as monopolists. All sellers set an item price of v-c', which is the maximum that the buyers will pay, given that they must pay c' to discover the sellers' price. Thus the total seller profit is v-c' and the buyer surplus is exactly 0. These curves are plotted in Fig. 8(b).

When c' is small, the buyer surplus is relatively high, and is completely unaffected by the presence or absence of the shopbot. This is because the shopbot is forced to just undercut the alternate search mechanism on both the single-quote and double-quote prices, so the buyer sees no difference in the search cost. However, the behaviors are quite different in an intermediate range of c'. Without the shopbot, the buyer surplus drops to zero above c' = 0.103872. With the shopbot, however, the surplus drops but remains positive all the way up to c'=0.706946. For larger c', the buyer surplus is zero in both cases. Overall, despite the fact that the shopbot is seeking only to maximize its own profit, it still provides a significant benefit to buyers by extending the range of c' for which they can experience a positive surplus.

In the foregoing analysis, we have assumed that the shopbot only provides one or two quotes. Of course, this was done purely for reasons of tractability -- there is no practical reason for a shopbot to constrain itself in such a way. If freed from this constraint, how might a shopbot maximize its profits? One plausible method would be to use a numerical optimization technique in which each candidate price schedule is evaluated by simulating the evolution of the buyers' strategies until an equilibrium or approximate equilibrium is reached (just as was done in Sec. 4). However, even this computationally intensive approach is likely to be insufficient, given our observation that the shopbot may need to employ a dynamic price schedule to encourage the buyers to reach the desired final equilibrium.

   Figure 9: Heuristic pricing of prices by a monopolist shopbot. a) . b) . c)

One heuristic for setting the price schedule dynamically is to set and then to set the remaining prices so that is constant for all i. In other words, at any moment in time, the shopbot sets the marginal price of quote i ( ) equal to the marginal benefit of that quote to a buyer, given the current setting of . Then a fraction of buyers switch their strategies, is updated to reflect this, and the cycle begins afresh in the next time step. Note that the algorithm causes buyers to be indifferent as to which search strategy they choose. This heuristic strongly encourages and to settle to an equilibrium. In fact, it can be over-aggressive, causing the shopbot to settle upon a price structure that is not quite optimal, but in practice it seems to yield reasonably good solutions.

Figure 9 depicts a simulation in which, initially, all parameters are exactly as in Fig. 5(a). Then, at time t=1000 (and at intervals of every 100 time steps thereafter) the shopbot pricing heuristic is put into effect. The cost of the alternate search mechanism is taken to be c' = 0.25 per quote. Immediately, the shift from a linear dependence on the search strategy i to a highly nonlinear one, and the shopbot prices and buyer strategies continue to co-evolve for a while until they finally reach an equilibrium in which the approximate the nonlinear form assumed in Eq. 16, with . In effect, the shopbot is selling the set of all five price quotes as a bundle, encouraging almost all buyers to switch to purchasing the full bundle rather than just purchasing two quotes, as they preferred to do with linear costs. This switchover to a strong preference for Search-5 is reflected in Fig. 9(a). The shopbot's profit increases dramatically as soon as the heuristic is put into effect, and increases further to roughly 0.359 by the end of the simulation run at time t=10000. Although this is not known to be an optimal solution, it still compares favorably with the maximal profit of 0.287 that could have been obtained by the shopbot had it only offered one or two quotes. (The theoretical optimum for at most two quotes is computed by multiplying the results in Table 5 by (v-r) = 0.5.)