A Proof of \tau-inequality


In market modeling, one often treats buyers as a homogeneous group. In this paper we consider buyers with heterogeneous preferences and products available in many variants. Such a framework allows us to successfully model various market phenomena. In particular, we investigate how is the vendor’s behavior influenced by the amount of available information and by the presence of correlations in the system.
Keywords: Market model, supply-demand law, correlations, matching problem.

Market Model with Heterogeneous Buyers

[10pt] Matúš Medo and Yi-Cheng Zhang

[10pt] Physics Department, University of Fribourg, CH-1700 Fribourg

1 Introduction

The standard economics textbooks make the supply-demand law as one of the pillars of the modern economic theory. However, many people, especially economists (see for example [1]), gradually realize that the most important factor is missing in the traditional supply-demand law. The study of complex systems [2, 3, 4] has already led to novel approaches to market phenomena. In a previous work [5], one of us introduced a simple framework to treat both quality and information capability, yielding a generalized supply demand law. However, in the previous paper, product is simply characterized by a single scalar variable: quality. In the modern economy we face a much more complex world, where products have many attributes and consumers have heterogeneous tastes [6]. These preferences cannot be simply represented as price and quality alone. We therefore generalize the previous work to allow multiple variants of each product as well as many different tastes among consumers.

Thus the producers face a dilemma: whether to target the average taste by producing a single or a few variants to leverage the economy of scale, or to match precisely each consumer’s taste [7]. We shall see that the answer depends on the information level that the producers may access: whether they know, and how well they know the consumers’ preferences. In addition, producers face also the nonlinear production costs. All these factors have to compromise to yield a combined result that gives various degrees of product diversity. With our approach, the supply demand problem of producers with the capability of producing variations and consumers’ diverse tastes becomes a matching problem [8, 9], where many mathematical and statistical mechanic tools are available to handle the complexity of the combinatorial problem.

In this paper we build a market model and investigate its behavior under various circumstances. In the first part of the paper we do not consider correlations between preferences of the parties included in the system. While unrealistic, this assumption allows us to discover basic properties of the model and outline the way of reasoning which can be used also in later, more realistic considerations. In the second part of the paper we discuss correlations and the ways how they can be introduced to the system. The last part of the paper deals with the consequences of the correlations for the model.

2 General framework: one vendor with many buyers

Let’s start with a market where only one vendor and buyers are present. The vendor can produce different variants of a product (e.g. many different shoes). With regard to the market, he has to decide which variants it is optimal to produce. We assume that all buyers satisfied with the offer buy one item, others stay out of the trading. Buyers in the market we label with lowercase Latin letters (). The different variants the vendor can produce we label with Greek letters (). The price of variant we label as . We assume that every variant can be produced in as many pieces as it is needed and as fast as it is needed.

The simple structure sketched above offers us enough space to model basic features of real markets. To establish a mathematical model for the market we have to introduce some assumptions about participants’ preferences and their consequences on the trading process. To keep complexity of the model at minimum we assume that buyer’s opinion about a variant can by represented by one scalar quantity, which we call cost and label it with ; we assume . The smaller is the cost , the bigger is the probability that buyer is satisfied with variant when asked. Preferences of the vendor are easier to introduce; they are represented by costs which he suffers during production and sale of a particular variant. The cost for variant we label and after a proper rescaling of monetary units . To simplify our considerations, we arrange the variants in order of cost: .

We stress a conceptual difference between vendor’s and buyer’s costs. The seller’s cost is strictly monetary—it represents a real amount of money (although in arbitrary units). In contrast, the buyer’s cost has no tangible interpretation, it simply represents something as airy as happiness with the given variant.

The vendor is able to produce different variants. However, when he is producing more variants, his expenses grows due to need of an additional investment. The vendor’s tendency to produce only few different variants can be modeled e. g. by a nonlinearity of expenses (doubled production of one single variant does not require doubled expenses). We adopt another approach; we assume that to initiate the production of a variant, the vendor has to pay additional charge (this represents initial costs).

Now let’s assume that the vendor offered most favorable variants (thus , ) to customers and the number of units sold of variant is . The total vendor’s profit is


Here the last term comes for the initial costs of produced variants, is the profit for one sold unit of variant . Due to the monetary rescaling used to confine to the range , units for profit, initial costs and prices are arbitrary.

It is natural to assume that when buyer is asked about interest to buy variant , the decision is based on the cost . We formalize this by the assumption that the probability of acceptance is a function of the variant cost; this function we call acceptance function. Obviously, is a decreasing function of the cost . Moreover, we assume . This means that if a buyer considers a variant to be the perfect one, she surely buys it.

When we offer a random variant to one buyer, the acceptance probability is


Here is the probability distribution of cost (i.e. it defines what “to offer a random variant” really means). The probability of accepting a random proposal is an important parameter of the model. From our everyday life we know that largely we do not agree to such an offer. For this reason we assume in our calculations.

One example of a reasonable choice for the acceptance function is (see fig. 1)


with . This choice is especially convenient due to its simplicity. If we now assume the uniform distribution of the buyer’s costs, for , parameter of the acceptance function (3) is just the probability of accepting a random offer introduced in the previous paragraph.

Figure 1: One particularly simple choice for the buyers’ acceptance function .

In the rest of this paper we assume that the prices of all variants are the same and equal to , . This relieves us from many technicalities, and helps to highlight important features of the model. Nevertheless, generalization to various prices is straightforward.

3 No correlations in costs

We begin our investigation with the simplest case of the presented model—the market without correlations, where all costs and are mutually independent. We model this by costs uniformly distributed in the range . To keep variants ordered, we first draw their costs and then we renumber all variants achieve . It follows that after averaging over realizations, the formula holds.

3.1 A vendor without knowledge of buyers’ preferences

If a vendor wants to discover which variants are most acceptable for buyers, in a market without correlations each buyer has to be asked for preferences. This cannot be done in big markets, thus it is natural to investigate the case with no information about buyers’ preferences on the vendor’s side. In sec. 3.4 we show that even an expensive global opinion survey brings only a negligible contribution to the vendor’s income.

Without any information about preferences, the vendor is not able to discover which variants are most favored by buyers. Therefore the best strategy is to offer variants that are most favorable from his point of view. Let’s label the number of variants the vendor is willing to offer as . We assume that all these variants are available to buyers simultaneously, similarly to different types of shoes available in a shoe shop. Every buyer goes through the offered variants and decides whether some of them are suitable or not.

From the buyer’s point of view, the vendor makes random proposals; the probability of accepting one particular offer is thus by definition equal to . The probability that one particular buyer accepts one of proposed variants is complementary to the probability of denying all offered variants. Thus we have


where the approximation used is valid for , i.e. for very choosy consumers (than is a small quantity) and a small number of offered variants. Now the average number of items sold by the vendor to all buyers is . Since no correlations are present, the average number of items sold of variant is , it is a decreasing function of .

The quantity of vendor’s interest is the total profit introduced in (1). Its expected value can be found using , , and . We obtain


Here the subscript reminds that we are dealing with an “uninformed” vendor. This function is sketched in fig. 2 for three different choices of initial cost . The optimal number of variants the vendor should offer maximizes his profit. One can easily show that when , for all . Thus the condition , which can be rewritten as , characterizes a market where the optimal vendor’s strategy is to stop the production and stay idle.

Figure 2: Expected profit of the vendor without informations, , drawn against for small value of initial costs (solid line), medium initial costs (dashed line) and high initial costs (dotted line). In the last case the condition is fulfilled and the optimal vendor’s strategy is to stop the production.

Since for every product numerous numerous variations can be made, the total number of variants the vendor can offer, , is large. Thus we are allowed to assume that the optimal number of offered variants satisfies the condition and solve the maximization condition approximately. We obtain


where is the optimal expected profit, . The used approximations are valid when and .

Figure 3: The optimal number of offered variants and the optimal profit as functions of initial cost for , , and . Numerical results (empty circles) are averages over 1 000 realizations, analytical results (solid lines) come from eq. (6). For the optimal profit arbitrary units are used.

In fig. 3, these results are shown to match a numerical treatment of the problem. In the figure we see how the initial cost influences diversity of the vendor’s production: decreasing increases differentiation of the vendor’s supply in full agreement with expectations. We can examine this feature in detail if we plot the optimal number of offered variants against for one particular realization of the model as it is shown in fig. 4 (thickness of the lines is proportional to the number of buyers of a variant).

Figure 4: Differentiation of the vendor’s production for various initial costs : single realization of the model (no averaging present) with , , . Vertical axis has no significant meaning, it serves purely to distinguish different variants.

3.2 Improvement of the vendor’s profit by a sequential offering of variants

So far we dealt with a very passive approach of the vendor. While offering variants to the market, he had no influence on the sale. In consequence, due to the absence of correlations in the system, every offered variant had the same average number of items sold. In a big market this is a natural approach. While the use of advertising can promote some variants, its treatment exceeds our scope.

In a small market a personal offering is possible. The vendor can promote favorable variants to increase the profit simply by offering the most favorable variant first. If a buyer is not interested, the second most favorable variant follows, etc. The average sale of the first variant is , for the second variant it is and in general we have . Hence the expected total sale is

This is equal to the expected total sale of the uninformed vendor in the previous section. We can conclude that the vendor’s profit improvement (if any) does not come from an increased total sale but rather from an increased sale of the variants that are more profitable for the vendor.

Now we investigate the optimal number of variants to offer in this case, . Since decreases with , at some moment it is not profitable to offer one more variant and the vendor’s profit is maximized. The corresponding equation can be solved with respect to , leading to . When the total number of possible variants is big, and the term can be neglected. The approximate solution is then


This optimal number of variants to offer is smaller than given by eq. (6). We can also notice that when is small, using approximation we are left with . This is an intriguing property—by the two different approaches we obtained the same result. To compare in markets with different sizes, we plot it as a function of in fig. 5. As can be seen, in a big market () eq. (7) fits well a numerical simulation of the system.

Figure 5: Successive offering: numerical and analytical results for the vendor using stopping condition described in the text in the markets with various sizes (on horizontal axis we have ). All numerical results are obtained as average of 10 000 realizations with , ; solid line represents eq. 7.

One can examine also the increase of the vendor’s profit caused by the change of the sale method. Using previous results, the approximate formula can be obtained. We see that when the total number of variants is big, sequential offering results in a small growth of the vendor’s profit. Nevertheless, in a system with a limited offer (small ) or with very choosy buyers (very small ), the improvement can be substantial.

Here we should notice, that the stopping condition “income greater than expenses” introduced above can be hard to use in practice. It is because is a random quantity and can drop to the disadvantageous region even when is big enough to cover the expenses. Thus for the vendor it is not enough to simply check profitability of the sale of one particular variant . Rather he has to take into account sales of all previously offered variants. This is especially important in systems with a small number of buyers where relative fluctuations are bigger. This effect is shown in figure 5 where numerical results for the vendor blindly using the stopping condition are shown for various market sizes. Clearly as increases, numerical results approach the analytical result (7).

3.3 Competition of two vendors

In real markets we seldom find a monopolist vendor; competition and partition of the market is a natural phenomenon. To investigate the model behavior in such a case we introduce the second vendor to the market. We assume that the vendors differ by initial costs, which are and . Again we do not consider the influence of advertisements and reputation, albeit they are vital in a market competition.

The course of the solution is similar to the one leading to eq. (5). We label the number of variants offered by vendor 1 as , the number of variants offered by vendor 2 as , and we assume that there is no overlap between offered variants. The aggregate sale of two buyers is where

With our assumptions about the equal status of the vendors, every offered variant has the same average sale. Therefore both vendors gain the share proportional to the number of variants they offer. Thus vendor 1 takes of the total sale and vice versa. When , we can simplify the expected profits to the form


Both parties maximize their profits by adjusting and . The corresponding system cannot be solved analytically but its numerical treatment is straightforward. The result is shown in fig. 6 where we have fixed the initial cost to investigate how and for both vendors vary with .

Figure 6: The optimal number of variants to offer (left) and the optimal profit (right) for vendor 1 (solid line) and for vendor 2 (dashed line) against . The initial cost of the second vendor is , , and .

We see that at vendor 1 stops the production for he cannot stand the competition of vendor 2. By putting in the equations and we obtain the expression for the value when this price-out occurs


It is in a good agreement with the values found by a numerical simulation of the model. Important feature of this result is that it depends on the initial price of the competitive vendor—decreasing the production costs can expel others from the market.

One can notice that when vendor 1 tries to increase the profit by deliberately increasing (with the intention to increase the sale), the term prevents the success of this strategy. As a result, the vendors have to adapt to each other. In mathematic terms, . At the same time, the sum of profits is not maximized at and . It is more profitable to remove the less efficient producer (the one with the higher value of initial costs). This is an analogy of a real market where ruining (or taking over) of a competitor can improve company profit.

3.4 An informed vendor

Now we would like to investigate the artificial case of the market where the vendor knows costs of all buyers. This knowledge can be used to increase the optimal profit. We start with a simpler question: if the vendor offers only one variant, how much the sale can be increased by a good choice of the variant?

The probability that buyer is agreeable to buy variant is . Since costs are random and independent, only the average acceptance probability plays a role and the number of users willing to buy this variant, , is thus binomially distributed with the mean and the variance . When the number of buyers is big, we can pass to a continuous approximation and assume the normal distribution of


The biggest value from the set () we label as . This is the number of potential buyers for the most accepted variant and the vendor does the best when by offering this variant. The probability density (often called extremal distribution) is


The multiplication by appears because we do not care which one of all variants is “the most accepted” one and the error function term represents the probability that the remaining variants are less accepted.

Since we are interested in big values of , we expect that the difference is big in comparison with . Therefore we use the approximation , which is valid for . When the error function value is close to one, we can also use the approximation () to obtain

This form is too complicated to obtain an analytical result for . Instead we compute the most probable value

Here the first term represents the average value of the sale and the additional term represents the gain arising from the additional vendor’s knowledge. To get a better notion about the sale growth we use the relative sale growth


To simplify the formula, the assumption has been used. A comparison of this result with a numerical simulation of the model is shown in fig. 7. As can be seen, a good agreement is obtained.

Figure 7: The relative growth of the vendor’s sale is drawn against the total number of buyers . Solid lines represent the analytical result, outcomes from numerical simulations are shown as symbols.

When , all vendor’s information is indeed useless and the average sale improvement is negligible. The inequality leads to the condition


Thus when the number of variants is not large enough, buyers’ opinions in the uncorrelated market cannot be used to increase the vendor’s sale and profit.

From the previous results we can draw useful implications about the vendor with perfect information, offering more than only one variant. When the total number of variants is big, the number of variants offered by the vendor is small in comparison with . Therefore the average sale of all offered variants is increased at most by given by eq. (12) and the same applies to the total sale. However, the vendor is interested mainly in his profit. When we take into account different costs of variants, the resulting growth of the income due to the informations is even smaller than because the variant with the highest sale can have a high cost for the vendor. Thus condition (13) is has more general consequences. It specifies the circumstances when even the perfect information about buyers’ preferences do not help the vendor to achieve a significant improvement of his profit.

4 Correlations in the system

Now we would like to add one important flavor to the model—correlations. They arise from conformity of people’s tastes (buyer-buyer correlations) and from the fact that high quality preferred by buyers results in high costs on the vendor’s side (buyer-vendor anticorrelations). To approach the behavior of a real market, we investigate how these correlations influence our results obtained so far. Before doing so, we briefly discuss correlations from a general point of view.

4.1 Measures of correlations

A correlation is the degree to which two or more quantities are associated. We shall discuss different ways how to measure correlations and how to introduce them to the system. In particular, we would like to measure the correlation between two lists (vectors) of costs: and (two buyers) or and (a buyer and the vendor). All lists of our interest have length and contain real numbers between and . A common choice for the correlation measure is Pearson’s correlation coefficient . For lists and it is defined as


This measure is sensitive to non-linear transformations of values in lists and . In addition, since it originates in the least-square fitting of the data by a straight line, it measures only a linear correlation. For this reasons, in this work we use another correlation measure, Kendall’s tau. For lists and it is given by the formula


and it ranges from (exactly the same ordering of lists and ) to (reverse ordering of lists); uncorrelated lists have . Notably, Kendall’s tau is insensitive to all monotonic mappings of the data. This is the strongest property we can expect from a correlation measure—more general transformations, nonmonotonic mappings, can sweep out any structure present in the data.

4.2 Lists with a given correlation degree

Now we would like to construct a set of lists that have mutual values of Kendall’s tau equal to . Such a set would represent lists of buyers’ preferences in an equally dispersed society. Since buyers’ tastes are to a certain extent similar, we expect positive correlations with . Nevertheless, in the following discussion we do not confine ourself to this region.

First we address a different question. Let’s assume that between lists and there is , between lists and there is . Does it imply any constraints on ? The answer is yes. It can be shown (see Appendix A) that fulfills the inequality


which is an analogy of the triangular inequality for side lengths of a triangle. From (16) we can draw various simple conclusions. First, if we want to construct three lists which have pairwisely , it is possible only for .1 Thus it is impossible to have more than two lists which are perfectly anticorrelated. Another simple result is that when , inevitably .

Now the question is whether we are able to create the whole system of lists which all have pairwisely Kendall’s tau equal to . The answer depends on the magnitude of . It can be shown (see appendix C) that the upper bound for is


As can be seen in fig. 8, this quantity grows slowly with the list length . Therefore to model a market with a large number of equally correlated buyers we would need an enormous number of possible variants.

Figure 8: The upper bound as a function of for two different lengths of lists . In both cases the upper bound is the same over a large part of the region and drops to when .

4.3 Generation of correlated lists

In the previous paragraphs we found that the society with a fixed mutual correlation degree of buyers is limited in its size. Therefore to introduce correlations to the presented market model we need a different approach. While copulas represent a general tool (see e.g. [10, 11]), they are useful mainly for numerical simulations and offer only small possibilities for analytical results. Here we adopt a simpler way to generate correlated lists.

Let’s consider the lists of variables


where , and are independent random variables uniformly distributed in the range . Here indicates correlation/anticorrelation between and and is the binding parameter controlling strength of the correlation: leads to uncorrelated lists, to perfectly correlated () or anticorrelated () lists. In all cases, values , lie in the range .

For the lists defined above, it can be shown that (see appendix B)


where . Plots of and are shown in fig. 9. Since buyers’ lists are prepared using the same formula, the average value of their correlation is non-negative. Notably, for any value of we can find suitable and that produce lists with the expected correlation equal to .

Lists created using eq. (18) do not have fixed mutual correlation, its actual value fluctuates around the mean value given by (19). According to the law of large numbers, is normally distributed. In appendix B it is shown that the variance of is proportional to . Such fluctuations are negligible for long lists. We can conclude eq. (18) present a way to create a system with the desired amount of correlation for any , and .

Yet there is a hitch in the proposed construction of correlated lists. The parameter influences the distribution of costs: for or they are distributed uniformly, for has a tent shape. This is an implausible property: the changes of the cost distributions can drive or distract the phenomena we are interested in. To fix this problem we propose the following two solutions.

First, to obtain correlated lists we can use the formulae


where is a random quantity distributed uniformly in the range . The complicated form of the has a simple meaning. The vendor’s costs grow uniformly with and buyers’ costs are connected to the vendor’s by the parameter . The term proportional to introduces a noise to the system, resulting in differences between buyers’ and vendor’s lists. Finally, the term represents the average value of buyers’ costs. It is easy to check that given by (20) is confined to the range for every and . The overall distribution of costs is uniform in the range and thus we avoid the problems of eq. (18). Moreover, this construction is simple enough to tract the proposed model analytically.

Using the techniques shown in appendix B we can find Kendall’s tau in this case. In the limit one obtains


where again . The form of is identical with (19) found before for a different construction of correlated lists.

As we will see later, eq. (20) is not appropriate to produce anticorrelated lists. Hence we present one more approach here—less accessible to analytical computation but more robust. The normal distribution is stable with respect to addition of random variables and this motivates us to make the following choice


where are drawn from the standard normal distribution. It can be shown2 that in this case


The course of is shown in fig. 9.

As our market model assumes costs confined to the range , the costs given by (22) have to be transformed using the cumulative distribution function of the standard normal distribution . In this way we obtain


Since this transformation is monotonic, it does not affect the value of and we can use eq. (23) for transformed lists of costs.

Figure 9: The dependence of on for the proposed constructions of correlated lists.

5 A market with correlations

When we discussed the market without correlations, the probability distribution of the variant cost was independent of . Consequently, the probability of accepting variant


was also independent of (we labeled ). As a result, when we change the acceptance function while preserving the quantity , the derived results remain unchanged.

In the presence of correlations we witness a very different picture: the detailed shape of the acceptance function is important. To keep the algebra as simple as possible, from now on we adopt the simplest choice for : the step function . This means that a buyer accepts a proposed variant only when its cost is smaller than .

In the following we first deal with the market where costs are given by eq. (20) for it is more accessible to analytical treatment. Then we shortly present analytical results for the market where to introduce correlations, eq. (22) is used.

5.1 An uninformed vendor in a market with correlations

Here we assume cost correlations created using eq. (20). When the vendor has no information about the preferences of buyers, similarly to sec. 3.1 the best strategy is to produce vendor’s most favorable variants. First we focus on the case of positive correlations; in (20) we set and . Using (25) and the chosen step acceptance function , the probability that one buyer accepts variant is


Since we expect the total number of variants to be very large and rather small, the second region makes the major contribution and thus we simplify eq. (26) to .

We assume that the vendor is simultaneously offering his most favorable variants. The probability that one buyer denies all offered variants is


Since is big, we use the approximation to evaluate this expression analytically, leading to


Here we used also which is valid when is small. When this is not the case, denying probability approaches zero and thus accepting probability is virtually one regardless to the approximation used.

With respect to (20), the sum of expected vendor’s costs can be written as


Here the first term represents fixed costs for producing different variants, is the probability that to one buyer variant is sold. Since the probability that of the successful trade is , from the condition we can deduce


This corresponds to the portioning of the probability among variants according to their probability of acceptance.

Now we can use (28), (29) and (30) to write down the expected profit of the vendor offering his topmost variants . It’s not possible to carry out the maximization of this expression analytically—numerical techniques have to be used to find and . Results are shown in 10 as lines together with outcomes from a numerical simulation of the model; a good agreement is found for . Results confirm that positive correlations between buyers and the vendor increase the vendor’s profit. This pattern is most obvious in the case when the vendor can offer only the most favorable variant and still every buyer buys it.

Figure 10: The optimal number of variants to produce (left) and the optimal profit drawn against for two different values of the initial cost . Lines show analytical results derived above, symbols represent numerical simulations (averages of 1 000 realizations), model parameters are set to , , . The decay of both quantities for is in agreement with eq. (31).

In the numerical results shown in fig. 10 we can notice one striking feature. When , changes rapidly and falls to zero quickly. Such a behavior is rather surprising for one do not expect abrupt changes in the region when there were none in the opposite region . The reason for this behavior is simple—when , vendor’s most preferred variants have cost too high to be accepted by a buyer. This effect can be quantified. When buyers’ costs are generated by (20), the inequality holds. Due to the acceptance function only the variants with cost smaller than are accepted. Therefore only variants with can be possibly accepted, where


Thus with negative correlations in the market, the vendor is able to sell the most favorable variant (the one with ) only if . When , the vendor sells no variants . Since grows steeply with (already with one obtain ), the vendor offering his top variants has to offer too many of them and he suffers both big initial costs and big costs . As a result the vendor is pushed out of the market.

Without detailed investigation we can infer the system behavior when the step acceptance function is replaced by a different choice. In the limit of careless buyers with , the influence of correlations vanishes and both and do not depend on and the model simplifies to the case investigated in sec. 3.1. Thus as gradually changes from the step function to a constant function, the dependence on gets weaker. In particular, if the largest cost for which is (for the step function ), in eq. (31) is replaced by . As a consequence, decreases and the steep decline of in fig. 10 shifts to a lower value of .

5.2 An uninformed vendor in a different market with correlations

Now we switch to the market costs drawn using eq. (22) and transformed to the range by eq. (24). As we already mentioned, this case is not allowable for an analytical treatment—hence we present only numerical results in fig. 11. They agree with our expectations: when , for the vendor it’s sufficient to produce only one variant. As the positive correlations diminish, the optimal number of offered variants grows and the profit shrinks. A closer investigation of the vendor’s behavior in this case exceeds the scope of this paper and remains as a future challenge.

Figure 11: The optimal number of variants and the optimal profit of the uninformed vendor in the market with correlations given by (22) and (24). Numerical results are averages of 1 000 repetitions, , , .

6 Another trading model

In previous sections we presented a way how to deal with the trading process. Here we shortly present a different model which arises from the same playground as our previous reasonings but highlight slightly different aspects of the market phenomenon.

Let’s have a market with buyers and different variants that the vendor can produce. Preferences of the interested parties are again represented by the scalar costs , (, ) uniformly distributed in the range (thus again we have no correlations in the system).

In a market, a vendor is aware that when some buyer is not satisfied with the offer, she can choose a different vendor. Therefore every vendor tries to induce as small cost as possible to the customers. This can be done by offering of the variants highly preferred by many buyers. We can visualize the process by sorting preference lists of all interested parties. Now when some variant is near the top of a buyer’s list, its cost is small and it is favorable for this buyer.

We have to specify the criterion for the “variant preferred by many buyers”. First, it can be the variant that is not too deep in nobody’s list. Thus, if we label the position of variant in the list of buyer as and as , the vendor chooses the variant that has the smallest . The selection process is visualized in fig. 12. Now the question is: how far buyers have to go down their lists? In other words: if we label , what is ?

Figure 12: The layout of the introduced trading model. Each column represents a sorted list of variants’ costs (the most preferred at top). The vendor is willing to go down his list by , in consequence buyers are forced go down by some value . Questionmarks signalize that after sorting of all lists we do not know standing of variants in the lists.

Since we have buyers in the market, the probability of a particular value is approximately given by the formula


Here the first term denotes the probability that there is no such a variant which is among topmost for every buyer and among topmost for the vendor.3 The second term responds to the fact that there is some variant which is exactly on th place in the list of a buyer (we do not care who it is, thus the multiplication by appears), among topmost variants in lists of other buyers and among topmost variants in the vendor’s list.

In eq. (32) we can use the approximation which is valid when . To calculate we replace the summation by the integration in the range which yields


Here we dropped terms vanishing in the limit (for there is a big number of variants that the vendor can produce).

Since there are no correlations in the system, when the vendor offers his topmost variants, every variant has the same probability to be chosen by a buyer. Thus the vendor has to go down his list on average by . On average this corresponds to the cost


A buyer is with the probability the one that has the sold variant on the th place of his list. With the complementary probability he has this variant somewhere between the 1st and th place. Thus we have


When the number of buyers is large, this approaches . A comparison of results (34) and (35) with numerical simulations is shown in fig. 13; a good agreement is found. A small discrepancy for can be corrected using the result which we develop in the following section.

Figure 13: Numerical and analytical results for plotted against for various values of (, numerical results are averages of 1 000 repetitions). The analytical result for is shown by the broken line.

To discover the scaling behavior of , one can follow a shorter path. The probability that one particular offered variant is among topmost in one buyer’s list is . For all buyers simultaneously the probability is . Since the vendor offers items, the probability that at least one of them is above the line is approximately . If this equals to , the trading is successfull. Thus we obtain . This result scales with and in the same way as the previous outcome of the detailed derivation.

At the end we have to mention that this model of transactions between the vendor and buyers is not relevant for a high number of buyers because decreases very slowly in this case. In other words: when is high, the probability that there is a variant that is not worst in any buyer list approaches zero.

6.1 A vendor producing more than one variant

From the previous discussion we know that in a big market vendor can not insist on selling only one variant. Therefore we would like to investigate a more relaxed case where the vendor offers simultaneously different variants. Then every buyer can choose the one the most suitable for him or her. We would like to investigate, how much suffer buyers in this case. To do so, we label the most favorable variant from the vendor’s offer for one particular buyer and compute the average value of this quantity.

The probability that one particular value occurs is


In this formula the product represents the probability that all variants offered by the vendor are in the buyer’s list lower than , the first term represents the probability that one of offered variants is on th place in the buyer’s list. Now we can derive


Consequently, the average cost suffered by a buyer is given by . We see that does not appear in . This means that the problem with the improper behavior of the model in big markets do not appear in this variation. Since in the calculation we did not make any approximations, no numerical simulation is needed to check the result.

7 Conclusion

The aim of this paper is to explore the modeling of a market with inhomogeneous buyers and a vendor producing multiple variants. We see that the outcomes depend on whether one or both sides have adequate information about the other side or not. In standard microeconomics, Pigou [12] has introduced the concept of price or demand elasticity. Vendors, knowing the buyers’ reserve prices to pay and thus pricing individually, can reap significant profit—this is usually called the first-degree price differentiation. Our analysis can be considered as a generalization in this direction. We show that if individual tastes are taken into account, there is much complexity in the system; treating individual tastes with a large number of buyers presents a considerable mathematical challenge. Our models point out a convenient way to tackle this type of problems and we expect that many real economy-motivated problems can be analyzed in a similar way. Vendors and buyers have many ways to improve their welfare.

In this study we have proposed two simple market models. While accessible to analytical solutions, they exhibit many features of real markets. In particular, diversification of the vendor’s production and market competition are used as examples. The diversification is presented as an interplay between the vendor’s pursuit to follow the buyers’ tastes and the costs growing with the number of produced variants. We also show that in a market with many buyers without preferences correlations, the knowledge of these preferences doesn’t increase the vendor’s profit. When correlations are introduced to the system, many technical complications arise. Nevertheless, the results are consistent with the expectations: a positive correlation between the buyers’ and vendor’s costs improves the vendor’s profit. Also, when interests of the two parties diverge (the correlation are negative), the vendor is able to make only a small or even no profit. In addition, in sec. 6 a similarly aimed model based on the well-known matching problem is investigated.

As many other directions can be explored further, we do not consider this topic exhausted. First of all, in a correlated market the vendor strategies and the influence of information deserve attention. Furthermore, while in the present work we investigated the influence of tastes on the market, the product quality and price were excluded from the analysis. Eventually, the framework established herein can be used to raise the law of supply and demand from a microscopical point of view.

8 Acknowledgment

We acknowledge the partial support from Swiss National Science Foundation (project 205120-113842) as well as STIPCO (European exchange program).

Appendix A Proof of -inequality

Let’s have three lists consisting of mutually different real numbers. Kendall’s for lists and can be written as where is the number of pairs that satisfy , is the same with a negative result of the product, and is the total number of different pairs . For the given values and it follows that


We would like to find bounds for when and are given. First we reorder lists so that lists is sorted in the descending order and for it is . Such a rearrangement does not affect the values of and thus the values of Kendall’s tau between lists remain also unchanged.

Since now all differences are positive, from we can deduce that there are positive differences and negative differences. Similarly, differences are positive and are negative. The values of and depend on the relative ordering of lists and . The biggest possible value of occurs when positive differences are aligned with positive differences (see fig. 14). By contrast, the smallest value of (and thus the smallest value of ) occurs when positive differences are aligned with negative differences .

Figure 14: An illustration of the proof. The first case (first three lines) has the biggest possible value of , the second case has the smallest possible value of .

From fig. 14 we see that and fulfill the inequalities

Using and (38) we obtain

These two inequalities summed together and divided by yield the desired inequality (16).

Appendix B Expected values of

For lists created using (18) we can rearrange (15) as follows

Moreover, can be rewritten as

Here is the probability that both and are positive and so forth, the formulae , are used. According to (18) we write

where lie in the range and are equally distributed with the density . Now we have ()

If , the first case applies to all possible values of , is possible only if . Finally, using

with it follows that

The quantity can be derived in the same way.

The variance of can be found by a direct computation of . We have