Profit Landscapes; A Simple Price War

We define the state of the system at time t, , as the collection of broker prices , broker interest vectors , and subscription matrix elements at time t. Our goal is to understand the evolution of , given

1. an initial configuration ,
2. the values of the various extrinsic (possibly time-varying) variables, comprising the category prevalences , the costs , , and , the consumer value V, and the consumer interest vectors , and
3. a specification of the utility-maximization algorithms used by each agent to dynamically change its own parameters, including
   1. the state information accessible to the agent (and its accuracy and timeliness), and
   2. the times or conditions when the agent updates its own state.

Even in systems of modest size (e.g., J=10 categories, B=10 brokers, C=1000 consumers), the state space can be quite large: its dimension is (J+1+C)B, or more than for the numbers just quoted. This is mainly due to the elements of the subscription matrix S. However, it is possible to reduce the dimensionality by factoring out the degrees of freedom associated with S. This is done by assuming that each broker and consumer instantly adapts to changes in its environment by selecting its optimal set of subscribers or subscriptions. Recall, however, that both broker and consumer must consent to a subscription. The conflicting opinions on what constitutes ``optimal'' may be resolved via a game-theoretic analysis. Thus the subscription matrix becomes a function of the remaining system variables and . Note that, by factoring out the subscription matrix, we have in effect factored out the consumer population, so that the ``reduced'' system's state is expressed entirely in terms of the brokers' states. We will denote the reduced state space by , and the subspace associated with broker b by . Thus , where is the space of possible values of the (J+1) broker variables .

In the reduced system, the broker utility function defines broker b's profit landscape, and the system dynamics is a co-evolutionary process in which each broker b attempts to maximize its profit by setting the values of , given the values of for all the other brokers i.

The remainder of this section is devoted to an analysis of the simplest possible multi-broker system, in which two brokers compete in a large consumer market in which there is only one type of information good. Thus B=2, J=1, and . For the moment, assume that broker 1 charges less than broker 2 for the good ( ). In this case, we can apply the definition of the system given in the last section to get an expected utility per article for consumer c of

The expected profit per article for the brokers is

From equation 2, it is readily seen that, independent of any other choices it may make, broker i can maximize its expected profit by setting its interest level equal to 1 if the quantity in parentheses is positive, and 0 otherwise. In other words, a self-interested broker will never have a negative expected profit, because it always has the option of pulling out of the market by setting its interest level to 0. In all that follows, we shall assume that , and if the resulting expected profit per article for broker i is negative we shall override this, setting . This reduces the dimensionality of the landscape, nominally 4, to 2.

Having established the optimal setting of interest levels by the brokers, now consider the subscription matrix elements and . First, note that each term in the expression for broker 1's expected utility in Eq. 2 can be maximized independently by setting , where represents the step function: for x>0, and 0 otherwise. In other words, it is only worthwhile to send articles to consumer c if c's interest level is sufficiently high that c's expected payment for interesting articles exceeds the cost of sending articles to c. Broker 2 is in a different situation. If c is already subscribed to broker 1, it will never purchase articles from broker 2 because it charges a higher price for the same good. Under such circumstances, broker 2 should not attempt to send articles to c because it will be paying for article transport with no hope of reimbursement. However, if c is not subscribed to broker 1, then broker 2 should set in order to maximize each term of the sum over consumers in the expression for . Putting all of this together, we have

Now consider the situation from the consumer's perspective, using Eq. 1. The consumer c will choose the optimal setting of from among the four possible choices: (0,0), (0,1), (1,0) and (1,1).

First, suppose that . If c chooses to set , then and therefore , so that . Alternatively, if c chooses to set , then , and so . Which is the better choice? If , then this quantity always exceeds because . In this case, should be set to 1, and the value of is immaterial because substituted into Eq. 3 shows that . However, if , then as well, and both and should be set to zero.

Now consider the other alternative: . In this case the value of is immaterial, , and . The value of matters only if , in which case the optimal value for is 1 if and 0 otherwise.

Assembling all of the above analysis, we obtain:

In the expression for , the step function on the left represents the veto power of the brokers, and the one on the right represents the veto power of the consumers. The expression for is similar, except that it is automatically zero if the consumer already subscribes to the lower-priced broker.

Having established the subscription matrix elements, and the brokers' interest levels, the only remaining decisions to be considered are the optimal brokers' prices and . Since the number of consumers , we may replace the sums in Eq. 2 with integrals over a consumer population with a distribution of interest levels given by , with the result:

where

Suppose that is the uniform distribution, i.e. for all values of . Then substitution of Eq. 4 into Eq. 6, and some integration by parts and other algebra lead to analytic solutions for the integrals and . In the interval ,

and outside this interval . (Despite the step function in the last term of Eq. 7, is not discontinuous at .) The function has a simple interpretation: it is the marginal utility per article per consumer for a monopolist broker as a function of its price . See Appendix A for further discussion of , including an analytic expression for the monopolist's optimal price and a graph of vs. P for a relevant choice of , , and V.

The solution for is:

in the region satisfied by the constraints , , and . Beyond this region, . Again, contains no real discontinuity, despite the step function.

The restriction can be removed by exploiting the symmetry arising from the fact that there is no inherent difference between the two brokers. We obtain the profit landscapes for brokers 1 and 2 as a function of the prices and :

where is given by:

Each broker's profit landscape describes the dependence of its expected profitability as a function of the price vector (all of the brokers' prices, including its own). For any given price vector, the myriad self-interested decisions of the consumers and brokers about subscriptions and interest levels are taken into account.

The profit landscape is illustrated for the case , V = 1 in Fig. 2. Note that there are two distinct humps, the one on the right corresponding to in Eq. 10 and the one on the left corresponding to . The ``cheap'' hump on the right corresponds to a situation in which broker 1 is cheaper ( ). The ``expensive'' hump on the left corresponds to the case in which broker 1 is more expensive than broker 2, but is still able to find customers. This comes about when broker 2 charges so little that it cannot afford to keep marginal customers (those with low interest levels ) as subscribers. (Recall that a broker pays for each article it sends to each of its subscribers, but receives payment only for those articles a subscriber is interested in.) Broker 1 can make money by serving the marginal customers that were rejected by the lower-priced broker 2.

Suppose that brokers use the profit landscape itself to periodically update their parameters so as to maximize their profitability. Assume further that the updates are asynchronous, and that the entire agent population adjusts its subscription matrix elements in a selfishly optimal way. Such an update strategy is guaranteed to produce the optimal profit in the very short term -- up until the moment when the next broker updates its parameters. Thus we call such a strategy ``myopically optimal'', or ``myoptimal'' for short.

If broker 1 is myoptimal, it could derive from its profit landscape a function that gives the value of that maximizes for each possible . Figure 3 shows a contour plot of on which is overlaid as a heavy solid line. (As before, , V = 1.) For , is given by the solution to a cubic equation involving cube roots of square roots of ; in this region it looks fairly linear. The ``vertical'' segment at is a discontinuity as the optimal price jumps from the peak to the peak. In the region , , where is a price quantum -- the minimal amount by which one price can exceed another. For , . This is the value of that maximizes , i.e. it is the price that would be established by a monopolist. Any further increase in would cut consumer demand by too much.

If broker 2 also uses a myoptimal strategy, then by symmetry its price-setting function is identical to that of broker 1 with and interchanged. The profit landscape and optimal price are likewise identical under an interchange of and .

  
Figure 2: Double-peaked profit landscape for broker 1 when , V = 1.

  
Figure 3: Contour map of profit landscape, with overlaid optimal price function for .

  
Figure 4: Iterative graphical construction of price-war time series, using functions and . See text.

Now the evolution of both and can be obtained simply by alternate application of the two price optimization functions. I.e., first broker 1 sets its price , then broker 2 sets its price , and so forth. The time series may be traced graphically on a plot of both and together, as shown in Fig. 4. Assume any initial price vector , and suppose broker 1 is the first to move. Then the graphical construction starts by holding constant while moving horizontally to the curve for . Then, is held constant while moving vertically to the curve . Alternate horizontal moves to and vertical moves to always lead to a price war during which the brokers successively undercut each other, corresponding to zig-zagging between the diagonal segments of the curves. The horizontal or vertical offset between the diagonals, equal to the amount by which a broker's price drops on every other iteration, is . Eventually, the price gets driven down to 0.388709, at which point the other broker (say broker 1) opts out of the price war, switching to the high-priced peak in its profit landscape. Raising the price to breaks the price war, but unfortunately, as Fig. 4 shows, it triggers the immediate start of another one. The brokers are caught in a never-ending (and, as we shall see, disastrous) limit cycle of price wars punctuated by abrupt resets.

Classic models of price wars, including those introduced by Cournot and Bertrand [Tirole, 1988], typically have the feature that prices are driven down to a stable value (e.g. the marginal cost in Bertrand's model). However, limit-cycle price wars have been observed previously in a simple model introduced by Edgeworth, in which it assumed that no single firm is able to satisfy the entire aggregate consumer demand [Shubik, 1980]. On constructing the profit landscape for Edgeworth's model, we find that it has two peaks that are qualitatively similar to those of Figure 2. In our case, the ``expensive'' hump arises because the low-priced broker may reject some consumers; this can be regarded as a sort of self-induced capacity constraint.