Profit landscapes; a simple price war

We define the state of the system at time t, tex2html_wrap_inline708 , as the collection of broker prices tex2html_wrap_inline710 , broker interest vectors tex2html_wrap_inline612 , and subscription matrix elements tex2html_wrap_inline714 at time t. Our goal is to understand the evolution of tex2html_wrap_inline708 , given

  1. an initial configuration tex2html_wrap_inline720 ,
  2. the values of the various extrinsic (possibly time-varying) variables, comprising the category prevalences tex2html_wrap_inline722 , the costs tex2html_wrap_inline604 , tex2html_wrap_inline606 , and tex2html_wrap_inline670 , the consumer value V, and the consumer interest vectors tex2html_wrap_inline672 , and
  3. a specification of the algorithms used by each agent to dynamically modify those variables over which it has control in an attempt to maximize its utility, 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 (e.g., for the numbers just quoted, 10110). This is mainly due to the tex2html_wrap_inline742 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 the optimal--for it-- set of subscribers or subscriptions. Recall, however, that both broker and consumer must consent before a subscription may be made. 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 tex2html_wrap_inline612 and tex2html_wrap_inline710 . 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 tex2html_wrap_inline752 , and the subspace associated with broker b by tex2html_wrap_inline756 . Thus tex2html_wrap_inline758 , where tex2html_wrap_inline756 is the space of possible values of the (J+1) broker variables tex2html_wrap_inline764 .

In the reduced system, the broker utility function tex2html_wrap_inline766 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 tex2html_wrap_inline766 by setting the values of tex2html_wrap_inline774 , given the values of tex2html_wrap_inline776 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 tex2html_wrap_inline784 . For the moment, assume that broker 1 charges less than broker 2 for the good ( tex2html_wrap_inline786 ). 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

  eqnarray112

The expected profit per article for the brokers is

  eqnarray120

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 tex2html_wrap_inline792 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 tex2html_wrap_inline794 , and if the resulting expected profit per article for broker i is negative we shall override this, setting tex2html_wrap_inline798 . 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 tex2html_wrap_inline800 and tex2html_wrap_inline802 . First, note that each term in the expression for broker 1's expected utility in Eq. 2 can be maximized independently by setting tex2html_wrap_inline804 , where tex2html_wrap_inline806 represents the step function: tex2html_wrap_inline808 for x>0, and 0 otherwise. In other words, it is only worthwhile to send articles to consumer c if c's interest level tex2html_wrap_inline672 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 tex2html_wrap_inline828 in order to maximize each term of the sum over consumers in the expression for tex2html_wrap_inline830 . Putting all of this together, we have

  eqnarray146

Now consider the situation from the consumer's perspective, using Eq. 1. The consumer c will choose the optimal setting of tex2html_wrap_inline834 from among the four choices: (0,0), (0,1), (1,0) and (1,1).

First, suppose that tex2html_wrap_inline844 . If c chooses to set tex2html_wrap_inline848 , then tex2html_wrap_inline850 and therefore tex2html_wrap_inline852 , so that tex2html_wrap_inline854 . Alternatively, if c chooses to set tex2html_wrap_inline858 , then tex2html_wrap_inline860 , and so tex2html_wrap_inline862 . Which is the better choice? If tex2html_wrap_inline864 , then this quantity always exceeds tex2html_wrap_inline866 because tex2html_wrap_inline786 . In this case, tex2html_wrap_inline870 should be set to 1, and the value of tex2html_wrap_inline872 is immaterial because tex2html_wrap_inline850 substituted into Eq. 3 shows that tex2html_wrap_inline876 . However, if tex2html_wrap_inline878 , then tex2html_wrap_inline880 as well, and both tex2html_wrap_inline870 and tex2html_wrap_inline872 should be set to zero.

Now consider the other alternative: tex2html_wrap_inline886 . In this case the value of tex2html_wrap_inline870 is immaterial, tex2html_wrap_inline860 , and tex2html_wrap_inline862 . The value of tex2html_wrap_inline872 matters only if tex2html_wrap_inline896 , in which case the optimal value for tex2html_wrap_inline872 is 1 if tex2html_wrap_inline900 and 0 otherwise.

Assembling all of the above analysis, we obtain:

  eqnarray194

In the expression for tex2html_wrap_inline902 , 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 tex2html_wrap_inline904 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 brokers' prices tex2html_wrap_inline906 and tex2html_wrap_inline908 . To determine the optimal prices, it is convenient to use the fact that we are taking the limit as the number of consumers tex2html_wrap_inline784 to replace the sums in Eq. 2 with integrals over a consumer population with a distribution of interest levels given by tex2html_wrap_inline912 , with the result:

  eqnarray205

where

  eqnarray208

Suppose that tex2html_wrap_inline912 is the uniform distribution, i.e. tex2html_wrap_inline916 for all values of tex2html_wrap_inline918 . Then substitution of Eq. 4 into Eq. 6, and some integration by parts and other algebra lead to analytic solutions for the integrals tex2html_wrap_inline920 and tex2html_wrap_inline922 . In the interval tex2html_wrap_inline924 ,

  eqnarray213

and outside this interval tex2html_wrap_inline926 . Note that the step function introduces no discontinuity in tex2html_wrap_inline928 because the term in square brackets that multiplies it is equal to zero at tex2html_wrap_inline930 . The solution for tex2html_wrap_inline922 is:

  eqnarray231

in the region satisfied by the constraints tex2html_wrap_inline934 , tex2html_wrap_inline936 , and tex2html_wrap_inline938 . Outside of this region, tex2html_wrap_inline940 . Again, there is no discontinuity in this function, despite the presence of the step function.

The restriction tex2html_wrap_inline786 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 tex2html_wrap_inline906 and tex2html_wrap_inline908 :

  eqnarray254

where tex2html_wrap_inline948 is given by:

  equation257

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, all of the individual decisions that would be made by self-interested consumers and brokers about subscriptions and interest levels are taken into account.

The profit landscape tex2html_wrap_inline950 is illustrated for the case tex2html_wrap_inline952 , V = 1 in Fig. 2. Note that there are two distinct humps, the one on the right corresponding to tex2html_wrap_inline956 in Eq. 10 and the one on the left corresponding to tex2html_wrap_inline958 . The hump on the right corresponds to a situation in which broker 1 is cheaper ( tex2html_wrap_inline786 ). The hump on the left is more interesting. It 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 is charging so little that it cannot afford to keep marginal customers--i.e., customers with low interest levels tex2html_wrap_inline672 --as subscribers. (Recall a broker pays tex2html_wrap_inline606 for each article it sends to each of its subscribers, but receives payment only for those articles a subscriber is interested in). For these marginal customers, the only alternative is to subscribe to broker 1. Broker 1 is in effect making money off broker 2's rejects.

   figure263
Figure: Profit landscape tex2html_wrap_inline966 for broker 1 when tex2html_wrap_inline952 , V = 1.

We can use the profit landscape to help us understand the behavior of an idealized system in which the brokers use the profit landscape itself to periodically update their parameters so as to maximize their profitability. The updates are assumed to be asynchronous, and the entire agent population is assumed to instantaneously respond to any changes made by a broker, with the exception of the other broker's prices. 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 tex2html_wrap_inline972 that gives the value of tex2html_wrap_inline906 that maximizes tex2html_wrap_inline976 for each possible tex2html_wrap_inline908 (footnote). Figure 3 shows a contour plot of tex2html_wrap_inline950 on which tex2html_wrap_inline972 is overlaid as a heavy solid line. (As before, tex2html_wrap_inline952 , V = 1.) For tex2html_wrap_inline994 , tex2html_wrap_inline972 is given by the solution to a cubic equation involving cube roots of square roots of tex2html_wrap_inline908 ; in this region it looks fairly linear. The ``vertical'' segment at tex2html_wrap_inline1000 is a discontinuity as the optimal price jumps from the tex2html_wrap_inline922 peak to the tex2html_wrap_inline920 peak. In the region tex2html_wrap_inline1006 , tex2html_wrap_inline1008 , where tex2html_wrap_inline1010 is a price quantum -- the minimal amount by which one price can exceed another. For tex2html_wrap_inline1012 , tex2html_wrap_inline1014 . This is the value of tex2html_wrap_inline906 that maximizes tex2html_wrap_inline1018 , i.e. it is the price that would be established by a monopolist. It is never worth increasing tex2html_wrap_inline906 beyond 0.589511 because the loss in sales volume would exceed the gain in profit.

   figure275
Figure: Contour map of landscape, with overlaid optimal price function tex2html_wrap_inline972 for tex2html_wrap_inline1024 .

   figure284
Figure: Iterative graphical construction of price-war time series, using functions tex2html_wrap_inline972 and tex2html_wrap_inline1028 . See text.

If broker 2 also uses a myoptimal strategy, then by symmetry its price-setting function is identical to that of broker 1 with tex2html_wrap_inline906 and tex2html_wrap_inline908 interchanged. The profit landscape tex2html_wrap_inline1034 and optimal price tex2html_wrap_inline1028 are likewise identical under an interchange of tex2html_wrap_inline906 and tex2html_wrap_inline908 .

Now the evolution of both tex2html_wrap_inline906 and tex2html_wrap_inline908 can be obtained simply by alternate application of the two price optimization functions. I.e., first broker 1 sets its price tex2html_wrap_inline1046 , then broker 2 sets its price tex2html_wrap_inline1048 , and so forth. The time series may be traced graphically on a plot of both tex2html_wrap_inline972 and tex2html_wrap_inline1028 together, as shown in Fig. 4. Assume any initial price vector tex2html_wrap_inline1054 , and suppose broker 1 is the first to move. Then the graphical construction starts by holding tex2html_wrap_inline908 constant while moving horizontally to the curve for tex2html_wrap_inline972 . Then, tex2html_wrap_inline906 is held constant while moving vertically to the curve tex2html_wrap_inline1028 . Alternate horizontal moves to tex2html_wrap_inline972 and vertical moves to tex2html_wrap_inline1028 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 tex2html_wrap_inline1068 . 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 tex2html_wrap_inline922 in its profit landscape. Raising the price to tex2html_wrap_inline1072 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.

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