OPTIMAL BUNDLING STRATEGIES FOR COMPLEMENTS AND SUBSTITUTES WITH HEAVY-TAILED VALUATIONS

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Running title: Optimal bundling strategies Rustam Ibragimov2 Imperial College Business School and Innopolis University Artem Prokhorov The University of Sydney Business School and St. Petersburg State University Johan Walden Haas School of Business, the University of California at Berkeley ABSTRACT We develop a framework that allows one to model the optimal bundling problem of a multiproduct monopolist providing interrelated goods with an arbitrary degree of complementarity or substitutability. Using the framework, we derive characterizations of the optimal bundling strategies in the case of heavy-tailed valuations of the products by the consumers. The results show that patterns in the optimal bundling strategies are the opposites of one another, depending on the degrees of heavy-tailedness of consumers’ valuations and the degrees of complementarity and substitutability among the goods provided. For substitutes with moderately heavy-tailed valuations, the seller’s optimal bundling strategies are the same as in the case of independently priced goods with thin-tailed (log-concavely distributed) or moderately heavy-tailed valuations discussed in the previous literature. That is, the seller prefers separate provision for substitute goods with high marginal costs and provision in a single bundle for substitutes with low marginal costs. These conclusions are reversed for complements with sufficiently (extremely) heavy-tailed valuations. In such a case, seller’s optimal strategy is to provide complement goods with low marginal costs separately, and as a single bundle under high marginal costs. As discussed in the paper, the conclusions may help to explain several bundling strategies commonly observed in real-world markets. JEL Classification: C02, D42, L12, L21 KEYWORDS: Optimal bundling strategies, multiproduct monopolist, interrelated goods, substitutes, complements, heavy-tailed valuations, robustness 1 Helpful comments from Donald W. K. Andrews, Donald J. Brown, Paul Kattuman, Benoit B. Mandelbrot, Peter C. B. Phillips, Herbert E. Scarf and Vladimir Smirnov and the participants at seminars at Imperial College Business, Innopolis University, Kazan (Volga Region) Federal University, Harvard University, the University of Sydney Business School, Yale University, the 10th International Conference on Computational and Financial Econometrics (CFE 2016, Seville, Spain), the 17th Conference of the Applied Stochastic Models and Data Analysis International Society (ASMDA 2017, London, United Kingdom) and the 31st European Meeting of Statisticians (EMS 2017, Helsinki, Finland) are gratefully acknowledged. Research for this paper was supported by a grant from the Russian Science Foundation (Project No. 16-18-10432). 2 Corresponding author, [email protected]

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Introduction

1.1

Optimal bundling

The last quarter of a century has witnessed a surge of interest in the analysis of optimal bundling strategies for a multiproduct monopolist. In particular, many studies in the literature emphasized that bundling decisions of a monopoly providing two goods depend on correlations between consumers’ valuations for the products (see Adams and Yellen, 1976, McAfee, McMillan and Whinston, 1989, Schmalensee, 1984, Salinger, 1995, Banciu and Odegaard, 2016, and Chen and Ni, 2017), the degrees of complementarity and substitutability between the goods (e.g., Lewbel, 1985, and Venkatesh and Kamakura, 2003, and Armstrong, 2013) and marginal costs of the products (see, among others, Salinger, 1995, and Venkatesh and Kamakura, 2003). Most of the studies on bundling have focused, however, on prescribed distributions for consumers’ valuations in the case of two products and their packages, such as bivariate uniform or Gaussian distributions, and only a few general results are available for bundles of more than two goods and more general distribution classes (e.g., Palfrey, 1983, Bakos and Brynjolfsson, 1999, 2000, Chakraborty, 1999, Geng et al., 2005, Fang and Norman, 2006, Ibragimov and Walden, 2010, Section 2.5 in Ibragimov et al., 2015, and references therein). Further, most of the literature on bundling focuses on the case of independently priced goods, that is, such products that consumers’ valuations for their bundles are additive in valuations of the component goods, as opposed to the case of interrelated goods, e.g., substitutes or complements (see Dansby and Conrad, 1984, Lewbel, 1985, Section 3.2 in Bakos and Brynjolfsson, 1999, Venkatesh and Kamakura, 2003, Armstrong, 2013, Bhargava, 2014, and Section 4 in the present paper). Bakos and Brynjolfsson (1999) have focused on optimal bundling decisions for a multiproduct monopolist providing large bundles of independently priced goods with zero marginal costs (information goods) at profit-maximizing prices to consumers whose valuations belong to a class that includes, by Proschan’s (1965) results reviewed in Appendix A.1, valuations with symmetric log-concave densities. Among other results, Bakos and Brynjolfsson (1999) show that, in such settings, if the seller prefers bundling a certain number of goods to selling them separately and if the optimal bundle price per good is less than the mean valuation, then bundling any greater number of goods will further increase the seller’s profits, compared to the case where the additional goods are sold separately. From the results it follows that, if consumers’ valuations are log-concave, then a form of superadditivity for bundling decisions holds, that is, the benefits to the seller grow as the number of goods in the bundle increases.3 Fang and Norman (2006) show that a multiproduct monopolist providing bundles of independently priced goods to consumers with valuations with log-concave densities prefers selling them separately to any other bundling decision if the products’ marginal costs are sufficiently large; under some additional distributional assumptions, the seller prefers providing the goods as a single bundle to any other bundling decision if the marginal costs of the goods are sufficiently small.4 3 This

property is similar to the case of Vickrey auctions with two buyers (see Remark 2 in Palfrey, 1983). and Norman (2006) use Proschan’s (1965) results on majorization and peakedness properties of symmetric log-concave distributions reviewed in Appendix A.1. From Proschan’s (1965) results given by Proposition 7.1 in Appendix A1 it also follows the distributional assumptions in Bakos and Brynjolfsson (1999) are satisfied for valuations with log-concave densities symmetric about their mean. In particular, the assumptions are satisfied for valuations with a finite support [v, v] distributed as the truncation XI(|X − µ| < h), h > 0, of an arbitrary random variable X with a log-concave density symmetric about µ = (v + v)/2, where h = (v − v)/2 and I(·) is the 4 Fang

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The assumptions that consumers’ valuations have log-concave densities or a bounded support imply that the valuations’ distributions are very thin-tailed: e.g., log-concave densities (with the standard example being the Gaussian case) exhibit at most an exponential decay in the tails (see the discussion in Section B.2 in Ibragimov and Walden, 2010, Section 2.1.2 in Ibragimov et al., 2015, and Section 4 in this paper). Motivated, in part, by numerous empirical results on departures from Gaussianity and heavy-tailed distributions of key variables in economics and finance, and modeling of the effects of crises, extreme observations, outliers and large fluctuations of economic and financial indicators, the recent works by the authors and co-authors have focused on the analysis of robustness of economic and financial models to heavy-tailedness assumptions (see the review in Section 3, Ibragimov et al., 2015, and references therein). According to the results presented in Ibragimov et al. (2015), the presence of heavy-tailed distributions can either reinforce or reverse the conclusions of a number of key models in economics and finance, including, importantly, optimality of diversification, depending on the degree of heavy-tailedness. That is, the conclusions of the models are robust and continue to hold under moderate heavy-tailedness but are reversed under sufficiently, or extremely, heavy-tailed distributions. Similar conclusions continue to hold for heavy-tailed distributions characterized by a wide class of dependence structures, including models with common shocks and dependence classes defined in terms of copulas (see Ibragimov and Walden, 2011, Ibragimov and Prokhorov, 2017, and references therein). According to the analysis in Ibragimov and Walden (2010) (see also Section 2.5 in Ibragimov et al., 2015), the conclusions on (non-)robustness hold in the analysis of optimal bundling. In particular, the results for thin-tailed log-concavely distributed valuations reviewed in this section continue to hold for moderately heavy-tailed valuations. In contrast, for sufficiently heavy-tailed valuations, separate provision is optimal for products with low marginal costs (e.g., information goods) and their provision in a single bundle is optimal under high marginal costs. A somewhat similar result arises in the setting of auctions with nonlinear valuations (see Smirnov and Wait, 2011). Heavy-tailedness concepts provide a natural framework for modelling marketing strategies for goods with extreme valuations in the real world. For example, strategies involving exclusion of goods with extreme (heavy-tailed) valuations and selling them separately are often employed on the market, in particular, by cable and direct satellite broadcast television firms that have marginal costs of reproduction close to zero. The latter firms typically offer a “basic” bundle and use such strategies as pay-per-view approach for unusual special events such as boxing matches (see Bakos and Brynjolfsson, 1999). Season tickets for entertainment performances offered by sporting and cultural organizations that have sufficiently high marginal costs of production may illustrate the dual pattern in bundling. It seems plausible that most of the demand for season tickets is concentrated around a relatively small fraction of consumers that have high (and heavy-tailed) valuations and preferences for performances offered by the entertainment organizations. However, in contrast to television firms, the companies offering the tickets often choose providing them in bundles to consumers with extreme valuations and preferences for the performances. To our knowledge, all general results in the bundling literature available for an arbitrary number of goods maintain the assumption that the goods provided are independently priced, and, so far, no results on optimal bundling of an arbitrary number of interrelated goods (complements or substitutes) with non-additive preferences are available, indicator function (see also Remark 2 in An, 1998). Log-concavity of the density of consumers’ valuations also ensures that the regularity condition on the valuations’ quantiles assumed in the analysis of optimal bundling in auctions in Chakraborty (1999) is satisfied.

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even in the case of thin-tailed valuations.

1.2

Objectives and main results

The paper’s goals are three-fold. First, we provide a general approach to the analysis of optimal bundling in many settings, including non-linear preferences, heavy-tailedness and dependence in consumers’ valuations, using the powerful methods of majorization theory (see Marshall et al., 2011). The majorization relation is a formalization of the concept of diversity in the components of vectors. Over the past decades, majorization theory, which focuses on the study of this relation and functions that preserve it (so-called Schur-convex functions) has found applications in disciplines ranging from statistics, probability theory, and economics to mathematical genetics, linear algebra, and geometry (see Marshall et al., 2011, Ibragimov et al., 2015, and the references therein). In many cases, the study of optimal bundling reduces to the analysis of majorization properties of tail probabilities of linear combinations of random variables, for which a number of results are available in the literature in probability, statistics, econometrics, economics, finance and related fields (see Appendix A.1, Ibragimov et al., 2015, and the references therein). Second, we develop a general framework that allows one to model the optimal bundling problem of a multiproduct monopolist providing interrelated goods with an arbitrary degree of complementarity or substitutability. In the framework, consumers’ valuations for bundles are assumed to be convex or concave functions of sums of their valuations or tastes for component goods, for which sub- (the case of substitutes) or super-additivity (the case of complements) with respect to stand-alone valuations hold. Sub- and super-additivity of preferences for bundles of interrelated goods can be conveniently modelled using power functions. In such a case, the study of optimal bundling of interrelated goods can be reduced to the analysis of Schur-convexity of tail probabilities of linear combinations of random variables with respect to majorization orderings for vectors of powers of the linear combinations’ weights. This allows applications of several recent results in majorization theory (op. cit.). Third, substantially extending the analysis in Ibragimov and Walden (2010) (see also Section 2.5 in Ibragimov et al., 2015), we provide a rather complete study of the interplay of the degree of heavy-tailedness in consumers’ valuations and that of the products’ substitutability or complementarity in the analysis of optimal bundling strategies. Using general results on majorization properties of tail probabilities of weighted sums of random variables, we obtain characterizations of optimal bundling strategies for a monopolist who provides goods with an arbitrary degree of complementarity or substitutability to consumers with heavy-tailed tastes for profit-maximizing prices (Theorems 5.1-5.4). Our study shows that patterns in the optimal bundling strategies are the opposites of one another, depending on the degrees of heavy-tailedness of consumers’ valuations and the degrees of complementarity and substitutability among the goods provided. Our results imply that, for substitutes with moderately heavy-tailed valuations, the patterns in the seller’s optimal bundling strategies are the same as in the case of independently priced goods with thin-tailed (log-concavely distributed) or moderately heavy-tailed valuations discussed in the previous section (see Theorem 5.1). That is, the seller prefers separate provision for substitute goods with high marginal costs and provision in a single bundle for

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substitutes with low marginal costs. These conclusions are reversed for complements with sufficiently (extremely) heavy-tailed valuations (see Theorem 5.2). In such a case, the seller’s optimal strategy is to provide complement goods with low marginal costs separately (as may be illustrated by “basic” bundles and pay-per-view provision for unusual special events in the cable TV industry), and as a single bundle under high marginal costs (as may be illustrated by season tickets for entertainment performances offered by sporting and cultural organizations).

1.3

Organization of the paper

The paper is organized as follows. Section 2 discusses the framework for modeling bundling of interrelated goods developed in the paper. Section 4 reviews the classes of heavy-tailed distributions we consider. Section 5 states and discusses the main results. Section 6 concludes. Appendix A.1 reviews the key concepts and results in majorization theory used in the analysis of optimal bundling in the paper. Technical proofs are provided in the Appendix A.2.

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Optimal Bundling of Complements and Substitutes

Throughout the paper, we consider a setting with a risk-neutral single seller providing m goods. Let M = {1, 2, ..., m} be the set of goods sold on the market. Let 2M stand for the set of all subsets of M. As in Palfrey (1983), Ibragimov and Walden (2010) and Section 2.5 in Ibragimov et al. (2015), the seller’s bundling decisions B are defined as partitions of the set of items M into a set of subsets, {B1 , ..., Bl } = B, where l is the cardinality of B; the subsets Bs ∈ 2M , s = 1, ..., l, are referred to as bundles. That is, Bs 6= ∅ for s = 1, ..., l; Bs ∩ Bt = for s 6= t, s, t = 1, ..., l; and ∪ls=1 Bs = M (see Palfrey, 1983, Bakos and Brynjolfsson, 1999, Fang and Norman, 2006, Ibragimov and Walden, 2010, and Section 2.5 in Ibragimov, 2015). It is assumed that the seller can offer one (and only one) partition B for sale on the market (this is referred to as pure bundling, see Adams and Yellen, 1976).5 We denote by B = {{1}, {2}, ..., {m}} and B = {1, 2, ..., m} the bundling decisions corresponding, respectively, to the cases where the goods are sold separately (that is, unbundled sales) and as a single bundle M. For a bundle B ∈ 2M , we write card(B) for a number of elements in B and denote by πB the seller’s profit resulting from selling the bundle. For a bundling decision B = {B1 , ..., Bl }, we write ΠB for the seller’s total profit Pl resulting from following B, that is, ΠB = s=1 πBs . The risk-neutral seller prefers a bundling decision B1 to a bundling decision B2 if EΠB1 ≥ EΠB2 (resp., if EΠB1 > EΠB2 ), where E denotes the expectation operator. Consumers’ preferences over the bundles B ∈ 2M are determined by their valuations (reservation prices) v(B) for the bundles and, in particular, by their valuations v({i}) for goods i ∈ M (when the goods are sold separately) which are referred to as stand-alone valuations.6 5 The

analysis of mixed bundling, in which consumers can choose among all bundling decisions available (see Adams and Yellen, 1976, and McAfee et. al., 1989) is beyond the scope of this paper. 6 The terms “valuations” and “reservation prices” are traditionally used as synonyms in the bundling literature.

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In the literature on bundling it is typically assumed that consumers’ valuations v(B) for bundles B ∈ 2M , and, in particular, their valuations v({i}) for individual goods i ∈ M are nonnegative: v(B) ≥ 0, B ∈ 2M ; v({i}) ≥ 0, i ∈ M. In the paper, the stand-alone valuations v({i}), i ∈ M, and the valuations v(B) for bundles B ∈ 2M are allowed to be negative. This corresponds to the situation where the goods provided and their bundles may have negative value to some consumers (e.g., articles exposing certain political views, advertisements, etc. in the case of information goods, see Bakos and Brynjolfsson, 1999; this agrees with the widespread attitudes that “most of the Web is boring”, “most news is worthless,” or “most TV is boring” that are, nevertheless, consistent with being an avid Web surfer, newspaper reader, or TV junkie having an ISP, a newspaper subscription or a subscription to cable TV due to the suppliers’ bundling their services, see Geng et al., 2005). If the valuations for bundles satisfy v(B1 ∪ B2 ) ≥ max(v(B1 ), v(B2 )), for all B1 , B2 ∈ 2M , and, in particular, v(B) ≥ maxi∈B v({i}), so that a consumer can discard any part(s) of a bundle at no cost, it is said that the goods in M and their bundles satisfy the free disposal condition. Although it is typically assumed in the bundling and, more generally, economics literature, in a large part due to technical convenience, the assumption of free disposal is increasingly being questioned in the case of information goods and access services where it often does not hold (see, e.g., the discussion in Essegaier et al., 2002, Chellappa and Shivendu, 2010, and references therein). The assumptions in the paper allow for no free disposal of products and their bundles. If consumers’ valuations for a bundle of goods are additive in those of component goods: v(B) =

P

i∈B

v({i}),

then the products provided by the monopolist are said to be independently priced (see Venkatesh and Kamakura, 2003). Obviously, under independent pricing, v(B1 ∪ B2 ) = v(B1 ) + v(B2 ) for all B1 , B2 ∈ 2M , and, under the assumption of valuations’ non-negativity, this implies the free disposal condition. The natural analogues of the above property for interrelated goods are subadditivity v(B) ≤

P

i∈B

v({i}),

v(B1 ∪B2 ) ≤ v(B1 )+v(B2 ), in the case of substitutability for individual goods and their bundles, and superadditivity P v(B) ≥ i∈B v({i}), v(B1 ∪ B2 ) ≥ v(B1 ) + v(B2 ), in the case of complementarity (see Dansby and Conrad, 1984, Lewbel, 1985, Venkatesh and Kamakura, 2003, Armstrong, 2013, and Bhargava, 2014). In our main results presented in the next section, Xi , i ∈ M, denote i.i.d. random variables (r.v.’s) that determine their valuations for the goods and their bundles. These r.v.’s may be viewed as representing the distribution of consumers’ tastes or preferences for goods i ∈ M. For convenience and in order to provide intuition for the settings and the results in the paper, the r.v.’s will be referred as consumers’ tastes (preferences) for the goods i ∈ M provided in what follows. We suppose that consumers’ valuations v(B) for a bundle B of goods produced by the monopolist is a function of their tastes and preferences for the component goods in the bundle. More precisely, we model the setting with
P interrelated goods by assuming that consumers’ valuations for bundles B ∈ 2M are given by v(B) = f i∈B Xi , where f : R → R is an increasing function with f (0) = 0. The valuations for goods i ∈ M in the case where they are sold separately are thus v({i}) = f (Xi ), i ∈ M. Further, for B1 , B2 ∈ 2M , and B = B1 ∪ B2 , one has
P P v(B) = f i∈B1 Xi + i∈B2 Xi . The case of (partial) substitutability is modeled using functions f (x) that are concave in x ≥ 0 and are convex

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in x < 0.7 In such a case, the valuations v(B) are subadditive for non-negative valuations for the components and sub-bundles of the bundles B ∈ 2M . The valuations v(B) are superadditive for non-positive valuations for the component goods and sub-bundles of B ∈ 2M (see Marshall et al., pp. 650-651). More precisely, for B1 , B2 ∈ 2M and B = B1 ∪ B2 , one has v(B) ≤ v(B1 ) + v(B2 ) if v(B1 ), v(B2 ) ≥ 0, and v(B) ≥ v(B1 ) + v(B2 ) if v(B1 ), v(B2 ) ≤ 0. P In particular, for any bundle B ∈ 2M , one has v(B) ≤ i∈B v({i}) for v({i}) = f (Xi ) ≥ 0, i ∈ B, and v(B) ≥ P i∈B v({i}) for v({i}) = f (Xi ) ≤ 0, i ∈ B. In a similar fashion, (partial) complementarity is modeled using functions f (x) that are convex in x ≥ 0 and are concave in x < 0. In such a case, the valuations v(B) are superadditive for non-negative valuations for the components and sub-bundles of the bundles B ∈ 2M . The valuations v(B) are subadditive for non-positive valuations for the component goods and sub-bundles of B ∈ 2M . That is, for B1 , B2 ∈ 2M and B = B1 ∪ B2 , one has v(B) ≥ v(B1 ) + v(B2 ) if v(B1 ), v(B2 ) ≥ 0, and v(B) ≤ v(B1 ) + v(B2 ) if v(B1 ), v(B2 ) ≤ 0. For any bundle B ∈ 2M , P P one has v(B) ≥ i∈B v({i}) for v({i}) = f (Xi ) ≥ 0, i ∈ B, and v(B) ≤ i∈B v({i}) for v({i}) = f (Xi ) ≤ 0, i ∈ B. In what follows, we assume that f (x) = gr (x) or f (x) = hr (x), where, for r > 0, gr (x) = xr I(x ≥ 0), hr (x) = x|x|r−1 , x ∈ R, and I(·) stands for the indicator function. The corresponding valuations v(B) for bundles B ∈ 2M , are given by v(gr , B) = gr

X




(1)


Xi .

(2)

Xi

i∈B

and v(hr , B) = hr

X i∈B

The valuations for goods i ∈ M in the case where they are sold separately are thus v(gr , {i}) = gr (Xi ) and v(hr , {i}) = hr (Xi ), i ∈ M. A number of conclusions in the paper allow extensions to the case of general functions f modeling consumers’ valuations for bundles, and will be presented elsewhere. The above functions gr (x) and hr (x) defining valuations for bundles B ∈ 2M model (partial) substitutability for r ≤ 1 and (partial) complementarity for r ≥ 1. The valuations v(gr , B) are nonnegative: v(gr , B) ≥ 0 for all B ∈ 2M . P The case r = 1 with consumers’ valuations for bundles v(h1 , B) = i∈B Xi models the case of independently priced goods. Clearly, in the case r = 1, one has v(h1 , {i}) = h1 (Xi ) = Xi , i ∈ M.

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Heavy-Tailedness in Economics and Finance

As discussed in the introduction, most of the previous literature has focused on the analysis of optimal bundling under valuations with thin-tailed distributions, such as those with log-concave densities or with a bounded support (see the next section for the definition and a review of properties of log-concave distributions). At the same time, it has been documented in numerous studies in economics, finance and marketing that key variables in these fields have distributions that exhibit departures from Gaussianity and have heavy tails (see, among 7 These properties of valuations v(B) for bundles B ∈ 2M are consistent with the assumption typically imposed on the value function of gains and losses in mental accounting and prospect theory (e.g., Kahneman and Tversky, 1979, and Thaler, 1985).

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others, the discussion and reviews in Embrechts et al., 1997, Cont, 2001, Rachev et al., 2005, Gabaix, 2009, Ibragimov and Walden, 2010, Ibragimov et al., 2015, and references therein). This paper belongs to the large stream of literature that has focused on applications and analysis of heavytailedness phenomena in the above fields. This literature dates back to Mandelbrot (1963) (see also the papers in Mandelbrot, 1997, and Fama, 1965), who pioneered the study of heavy-tailed distributions in social sciences. In settings that involve a heavy-tailed r.v. X (representing consumers’ valuations or preferences; a risk, return or a foreign exchange rate; income or wealth, etc.) it is usually assumed that the distribution of X follows a power law, so that P (|X| > x) ∼ x−ζ , ζ > 0.

(3)

Here, f (x) ∼ g(x) means that c1 g(x) ≤ f (x) ≤ c2 g(x) for some constants c1 > 0, c2 > 0 for large x. The parameter ζ in (3) is referred to as the tail index (or the tail exponent) of the distribution of the r.v. X. It characterizes the heaviness (the rate of decay) of the tails of heavy-tailed power law distribution (3). The more the probability mass in the tails, the smaller are the tail index parameters, and vice versa. Heavy-tailedness (i.e., the tail index ζ) of the variable X governs the likelihood of observing extreme values and large fluctuations in the variable. The smaller values of the tail index ζ correspond to a higher degree of heavy-tailedness in X and, thus, to a larger likelihood of observing extreme values and large fluctuations in realizations of this variable. The tail index may be regarded as being infinite: ζ = ∞ for thin-tailed distributions like Gaussian or exponential ones. An r.v. X with a distribution that satisfies (3) has finite moments E|X|p of order p < ζ. However, the moments are infinite for p ≥ ζ. A r.v. X is said to be thin-tailed, if its moments of all orders are finite: E|X|p < ∞ for all p > 0. It is heavy-tailed, if it follows a power law distribution (3). Distributions with log-concave densities, for which several general results in the optimal bundling literature exist, have finite moments of all orders and are therefore thin-tailed (see the discussion in the previous section). An important wide class of heavy-tailed distributions satisfying (3) is generated by scale mixtures of normal and other thin-tailed variables. Scale mixtures of normals include, for instance, the Student−t distributions with arbitrary degrees of freedom, the double exponential distribution, the logistic distribution as well as symmetric stable distributions that are closed under convolutions and portfolio formation (see the next section). As discussed in many works in the literature reviewed in Ibragimov and Walden (2010) and Ibragimov et al. (2015), in many aspects, under extreme observations and pronounced heterogeneity, heavy-tailed frameworks outperform those based on thin-tailed distributions such as log-concave or superexponential (e.g., normal) ones. Many studies in economics and finance argue that the tail indices ζ in heavy-tailed models (3) typically lie in the interval ζ ∈ (2, 4) for financial returns and foreign exchange rates in developed markets, implying finite variances and infinite fourth moments (see, among others, Loretan and Phillips (1994), Gabaix et al., 2006, Gabaix, 2009, the review in Ibragimov et al., 2015, and references therein). Importantly, foreign exchange rates and financial returns in emerging markets typically exhibit more pronounced heavy-tailedness compared to their counterparts in developed countries and often have tail indices ζ < 2 and, thus, infinite variances (see Ibragimov et al., 2013, Section 3.2 in

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Ibragimov et al., 2015, Gu, 2016, Ibragimov and Gu, 2016, and references therein). Heavy-tailed distributions provide a framework in modeling extreme rare events like crises and natural disasters. In this regard, the results in the literature point out that tail indices may be considerably smaller than one, implying infinite first moments, for returns from technological innovations (Silverberg and and Verspagen, 2007), losses from operational risks (see Neˇslehova et al., 2006) and those from natural disasters (see Ibragimov et al., 2009, Section 2.5.1 in Ibragimov et al., 2015, and references therein). Extreme events and heavy-tailedness phenomena are also especially pronounced in markets characterized by the nobody knows, the winner takes all, and the success breeds success principles with high uncertainty in individual demands and in the success or failure of new products, such as markets for technological innovations and information goods and creative (e.g., motion picture, music, and book publishing) industries (see, among others, the discussion and reviews in Shapiro and Varian, 1999, Brynjolfsson et al., 2003, de Vany, 2004, Frank and Cook, 1995, Anderson, 2006, Eliashberg et al., 2006, Ghose and Sundararajan, 2006, and Gaffeo et al., 2008). As discussed in Ibragimov and Walden (2010), empirical results in the economics and marketing literature, including the above works, provide estimates ζ = 1.2 for book demand on Amazon.com; ζ ∈ (0.9, 1.5) for book sales and revenues in Italy; ζ ∈ (1, 2) for box office revenues, rates of return and profits for motion pictures in North America, and ζ ∈ (2.1, 2.6) for their losses; ζ < 1 for motion picture revenues in the United Kingdom, and pronounced heavy-tailedness for a number of other variables important in marketing. As in Ibragimov and Walden (2010) and Ibragimov et al. (2015), in order to illustrate the main ideas and in order to simplify the presentation of the main results in this paper, we focus on modeling heavy-tailedness using the framework of independent stable distributions, that is, distributions satisfying power-law relation (3) with ζ < 2 (see the next section). As discussed in the above works, the stable distributions provide a fairly restricted class, but are extremely convenient to analyze. They provide natural extensions of the Gaussian law since they are the only possible limits for appropriately normalized and centered sums of i.i.d. r.v.’s. This property is useful in representing heavy-tailed marketing, economic and financial data as cumulative outcomes of market agents’ decisions in response to information they possess. In addition, stable distributions are flexible to accommodate both heavy-tailedness and skewness in data. Importantly, similar to Section 5.2 in Ibragimov and Walden (2010) and Ibragimov et al. (2015) all the results in the paper can be easily extended to wide classes of dependent valuations, including those with distributions given by convolutions of multivariate α−symmetric ones. The class of α−symmetric distributions contains, as subclasses, heavy-tailed models with common shocks with common shocks affecting all units in population (such as macroeconomic or political ones) which are of high importance in economics (see, the discussion in Andrews, 2005) and also spherical distributions corresponding to the case α = 2. Spherical distributions, in turn, include such examples as Kotz type, multinormal and multivariate spherically symmetric α−stable distributions. Multivariate t−distributions with k degrees of freedom are examples of spherical distributions with finite moments of any order smaller than k : In the case k > 2, in particular, the distributions generate r.v.’s with common shocks dependence structure and finite variances and higher moments. These distributions were used in a number of works to model heavy-tailedness phenomena with finite moments up to some order (see, among others, Praetz, 1972, Blattberg and 8

Gonedes, 1974, and Glasserman et al., 2002). Similarly, the results of Ibragimov and Prokhorov (2016, 2017) can be used to generalize the results in the paper to case of heavy-tailed marginals with a wide range of dependence structures characterized by general copula families. The proof of the above generalizations are straightforward and so is omitted.

4

Classes of Distributions

In this section, we introduce certain classes of distributions for which we will be deriving the main results. These classes are those considered in Ibragimov and Walden (2010) and Ibragimov et al. (2015) (the notation and the definitions of the distributional classes in this paper are somewhat different from those in the above works). In what follows, a univariate density f (x), x ∈ R, is referred to as symmetric (about zero) if f (x) = f (−x) for all x > 0. An absolutely continuous distribution or an r.v. X with the density X is said to be symmetric if f (x) is symmetric (about zero). We say that a r.v. X with the density f : R → R and the convex distribution support Ω = {x ∈ R : f (x) > 0} is log-concavely distributed if log f (x) is concave in x ∈ Ω, that is, if for all x1 , x2 ∈ Ω, and any λ ∈ [0, 1], f (λx1 + (1 − λ)x2 ) ≥ (f (x1 ))λ (f (x2 ))1−λ .

(4)

(see An, 1998, Bagnoli and Bergstrom, 2005, Section 18.B in Marshall and Olkin, 2011, and Section 2.1.2 in Ibragimov et al., 2015). A distribution is said to be log-concave if its density f satisfies (4). Examples of log-concave distributions include (see, for instance, Marshall and Olkin, 2011, p. 764-765, and the discussion in Section 2.1.2 in Ibragimov et al., 2015) the normal distribution N (µ, σ 2 ), the uniform density U(θ1 , θ2 ), the exponential density, the logistic distribution, the Gamma distribution Γ(α, β) with the shape parameter α ≥ 1, the Beta distribution B(a, b) with a ≥ 1 and b ≥ 1; and the Weibull distribution W(γ, α) with the shape parameter α ≥ 1. The class of log-concave distributions is closed under convolutions. It is important to emphasize that if a r.v. X is log-concavely distributed, then its density is extremely thin-tailed. More precisely, the density of X has at most an exponential tail, that is, f (x) = O(exp(−λx)) for some λ > 0, as x → ∞ and all the power moments E|X|γ , γ > 0, of the r.v. exist (see Corollary 1 in An, 1998). Karlin (1968), An (1998), Bagnoli and Bergstrom (2005) and Marshall and Olkin (2011) provide surveys of many other properties of log-concave and related distributions. Throughout the paper, LC denotes the class of symmetric log-concave distributions. For 0 < α ≤ 2, σ > 0, β ∈ [−1, 1] and µ ∈ R, we denote by Sα (σ, β, µ) the stable distribution with the characteristic exponent (index of stability) α, the scale parameter σ, the symmetry index (skewness parameter) β and the location parameter µ. That is, Sα (σ, β, µ) is the distribution of a r.v. X with the characteristic function ( exp {iµx − σ α |x|α (1 − iβsign(x)tan(πα/2))} , α 6= 1, E(eixX ) = exp {iµx − σ|x|(1 + (2/π)iβsign(x)ln|x|} , α = 1,

9

x ∈ R, where i2 = −1 and sign(x) is the sign of x defined by sign(x) = 1 if x > 0, sign(0) = 0 and sign(x) = −1 otherwise. In what follows, we write X ∼ Sα (σ, β, µ), if the r.v. X has the stable distribution Sα (σ, β, µ). A closed form expression for the density f (x) of the distribution Sα (σ, β, µ) is available in the following cases (and only in those cases): α = 2 (Gaussian distributions); α = 1 and β = 0 (Cauchy distributions)8 ; α = 1/2 and β = ±1 (L´evy distributions)9 . Degenerate distributions correspond to the limiting case α = 0. The index of stability α characterizes the heaviness (the rate of decay) of the tails of stable distributions Sα (σ, β, µ). In particular, if X ∼ Sα (σ, β, µ), 0 < α < 2, then the distribution of X satisfies power law (3) with the tail index ζ equal to the index of stability α : ζ = α. This implies that the p−th absolute moments E|X|p of a r.v. X ∼ Sα (σ, β, µ), α ∈ (0, 2) are finite if p < α and are infinite otherwise. The symmetry index β characterizes the skewness of the distribution. The stable distributions with β = 0 are symmetric about the location parameter µ. The stable distributions with β = ±1 and α ∈ (0, 1) (and only they) are one-sided, the support of these distributions is the semi-axis [µ, ∞) for β = 1 and is (−∞, µ] (in particular, the L´evy distribution with µ = 0 is concentrated on the positive semi-axis for β = 1 and on the negative semi-axis for β = −1). In the case α > 1 the location parameter µ is the mean of the distribution Sα (σ, β, µ). The scale parameter σ is a generalization of the concept of standard deviation; it coincides with the latter in the special case of Gaussian distributions (α = 2). Stable distributions are closed under convolutions and portfolio formation. In particular, for i.i.d. r.v.’s Xi ∼ Sα (σ, β, µ) with µ = 0, α 6= 1 or β = 0, α = 1 (such distributions are called strictly stable) and any portfolio weights wi ≥ 0, i = 1, ..., n, one has n X

wi Xi =

i=1

d

n X

wiα

1/α

X1

(5)

i=1

(see Section 2.1.2 in Ibragimov et al., 2015, and references therein). For a detailed review of properties of stable distributions the reader is referred to, e.g., the monographs by Zolotarev (1986) and Uchaikin and Zolotarev (1999). For 0 < r < 2, we denote by CS(r) the class of distributions which are convolutions of symmetric stable distributions Sα (σ, 0, 0) with characteristic exponents10 α ∈ (r, 2] and σ > 0. That is, CS(r) consists of distributions of r.v.’s X such that, for some k ≥ 1, X = Y1 + ... + Yk , where Yi , i = 1, ..., k, are independent r.v.’s such that Yi ∼ Sαi (σi , 0, 0), αi ∈ (r, 2], σi > 0, i = 1, ..., k. Further, CSLC stands for the class of convolutions of distributions from the classes LC and CS(1). That is, CSLC is the class of convolutions of symmetric distributions which are either log-concave or stable with characteristic exponents greater than one11 . In other words, CSLC consists of distributions of r.v.’s X such that X = Y1 + Y2 , where Y1 and Y2 are independent r.v.’s with distributions belonging to LC or CS(1). 8 The

densities of Cauchy distributions are f (x) = σ/(π(σ 2 + (x − µ)2 )). distributions have densities f (x) = (σ/(2π))1/2 exp(−σ/(2x))x−3/2 , x ≥ 0; f (x) = 0, x < 0, where σ > 0, and their shifted

9 L´ evy

versions. 10 Here and below, CS stands for “convolutions of stable”; the overline indicates that convolutions of stable distributions with indices of stability greater than the threshold value r are taken. 11 CSLC is the abbreviation of “convolutions of stable and log-concave”.

10

Finally, for 0 < r ≤ 2, we denote by CS(r) the class of distributions which are convolutions of symmetric stable distributions Sα (σ, 0, 0) with indices of stability12 α ∈ (0, r) and σ > 0. That is, CS(r) consists of distributions of r.v.’s X such that, for some k ≥ 1, X = Y1 + ... + Yk , where Yi , i = 1, ..., k, are independent r.v.’s such that Yi ∼ Sαi (σi , 0, 0), αi ∈ (0, r), σi > 0, i = 1, ..., k. All the classes LC, CSLC, CS(r) and CS(r) are closed under convolutions. In particular, the class CSLC coincides with the class of distributions of r.v.’s X such that, for some k ≥ 1, X = Y1 + ... + Yk ,

(6)

where Yi , i = 1, ..., k, are independent r.v.’s with distributions belonging to CS(1) or LC. Similar to property (5), a linear combination of independent stable r.v.’s with the same characteristic exponent α also has a stable distribution with the same α. However, in general, this does not hold true in the case of convolutions of stable distributions with different indices of stability. Therefore, the class CS(r) of convolutions of symmetric stable distributions with different indices of stability α ∈ (r, 2] is wider than the class of all symmetric stable distributions Sα (σ, 0, 0) with α ∈ (r, 2] and σ > 0. Similarly, the class CS(r) is wider than the class of all symmetric stable distributions Sα (σ, 0, 0) with α ∈ (0, r) and σ > 0. Clearly, CS(1) ⊂ CSLC and LC ⊂ CSLC. It should also be noted that the class CSLC is wider than the class of (two-fold) convolutions of log-concave distributions with stable distributions Sα (σ, 0, 0) with α ∈ (1, 2] and σ > 0. By definition, for 0 < r1 < r2 ≤ 2, the following inclusions hold: CS(r2 ) ⊂ CS(r1 ) and CS(r1 ) ⊂ CS(r2 ). In what follows, we write X ∼ LC (resp., X ∼ CSLC, X ∼ CS(r) or X ∼ CS(r)) if the distribution of the r.v. X belongs to the class LC (resp., CSLC, CS(r) or CS(r)). We also denote R+ = [0, ∞). Heavy-tailedness properties of stable r.v.’s discussed above imply that the distributions of r.v.’s X ∼ CS(r) are moderately heavy-tailed in the sense that they follow power laws (3) with tail indices ζ < r. This implies that, as x → ∞, their tails P (|X| > x) decay to zero faster than those of power law distributions (3) with the tail index ζ = r. Thus, their power moments of order r are finite: E|X|r < ∞. Similarly, the distributions in CS(1) follow power laws (3) with tail indices ζ > 1. The same is true for distributions of r.v.’s X ∼ CSLC if their convolution representation (6) includes at least one r.v. Yi from the class CS(1). Distributions of r.v.’s X in the classes CS(1) and CSLC have tails that decay to zero faster than those of power laws (3) with the tail index ζ = 1, and their absolute first moments are finite: E|X| < ∞. In contrast, the distributions of r.v.’s X ∼ CS(r) are moderately heavy-tailed in the sense that they follow power laws (3) with tail indices ζ < e. As x → ∞, their tails P (|X| > x) decay to zero slower than those of power law distributions (3) with the tail index ζ = r, and, thus, their power moments of order r their absolute first moments are infinite: E|X|r = ∞. In particular, the distributions of r.v.’s X ∼ CS(1) follow power laws (3) with tail indices ζ < 1, and have infinite absolute first moments E|X| = ∞. In some sense, symmetric (about µ = 0) Cauchy distributions S1 (σ, 0, 0) with the tail index ζ = 1 are at the 12 The

underline indicates considering stable distributions with indices of stability less than the threshold value r.

11

dividing boundary between the classes CS(1) and CS(1) (and between the classes CS(1) and CSLC). Similarly, for r ∈ (0, 2), symmetric stable distributions Sr (σ, 0, 0) with the characteristic exponent and the tail index in (3) equal to r : α = ζ = r are at the dividing boundary between the classes CS(r) and CS(r). Further, symmetric normal distributions S2 (σ, 0, 0) are at the dividing boundary between the class LC of log-concave distributions and the class CS(2) of convolutions of symmetric stable distributions with indices of stability13 α < 2.

5

Main Results

As before, we let ci , i ∈ M, be the marginal costs of goods in M and suppose that the seller can provide bundles B of goods in M for prices per good p ∈ [0, pmax ], where pmax is the (regulatory) maximum price, with the convention that pmax can be infinite. For a bundle of goods B ∈ 2M , denote by pB the profit-maximizing price per good for the bundle, P so that the seller’s expected profit from selling B (at the price pB per good) is πB = (kpB − i∈B ci )P (v(B) ≥ kpB ), where k = card(B). The following Theorems 5.1 and 5.2 characterize the optimal bundling strategies for a multiproduct monopolist providing goods with an arbitrary degree of complementarity or substitutability to consumers with heavy-tailed valuations and preferences for the goods and their bundles. In the theorems, it is assumed that consumers’ valuations
P v(B) for bundles B ∈ 2M of goods i ∈ M are given by the functions v(B) = v(gr , B) = gr i∈B Xi and
P r v(B) = v(hr , B) = hr i∈B Xi , r ∈ (0, 2], defined in (1) and (2), where for r > 0, gr (x) = x I(x ≥ 0), hr (x) = x|x|r−1 , x ∈ R, and Xi are i.i.d. r.v.’s that may be viewed, as discussed in Section (2), as representing consumers’ preferences or tastes for the goods provided. As discussed in Section (2), so defined valuations v(B) for bundles represent (partial) substitutability of the goods provided for r ≤ 1 and their (partial) complementarity for r ≥ 1. In the case, where the valuations v(B) = v(gr , B) follow (1) with the functions gr , they are nonnegative: P v(B) = v(gr , B) ≥ 0 for all B ∈ 2M . The case r = 1 with consumers’ valuations for bundles v(h1 , B) = i∈B Xi models the case of independently priced goods. In this case, the stand-alone valuations for goods i ∈ M are given by the r.v.’s Xi : v(h1 , {i}) = h1 (Xi ) = Xi , i ∈ M. Theorem 5.1 extends Theorem 4.2 in Ibragimov and Walden (2010) and Theorem 2.5.2 in Ibragimov et al. (2015) to the case of non-additive valuations and interrelated goods. It concerns the case where consumers’ valuations and preferences for the goods and their bundles provided are moderately heavy-tailed and the goods are independently priced or are substitutes (or are complements with not very high degree of complementarity). According to the theorem, in such settings, the seller’s optimal bundling strategies are the same as in the case of independently priced goods with extremely thin-tailed log-concavely distributed (see Bakos and Brynjolfsson, 1999, and Fang and Norman, 2006) or moderately heavy-tailed valuations (see Ibragimov and Walden, 2010, and Section 2.5 in Ibragimov et al., 2015). Theorem 5.1 Let µ ∈ R, r ∈ (0, 2), and let consumers’ valuations v(B) for bundles B ∈ 2M of goods from M be 13 More precisely, the symmetric Cauchy distributions are the only ones that belong to all the classes CS(r) with r > 1 and all the classes CS(r) with r < 1. Symmetric stable distributions Sr (σ, 0, 0) are the only ones that belong to all the classes CS(r0 ) with r0 > r and all the classes CS(r0 ) with r0 < r. Symmetric normal distributions are the only distributions belonging to the class LC and all the classes CS(r) with r ∈ (0, 2).

12

given by v(B) = v(gr , B) in (2) or by v(B) = v(hr , B) in (2). Suppose that Xi , i ∈ M, in (1)-(2)are i.i.d. r.v.’s such that Xi ∼ Sα (σ, β, µ), i ∈ M, for some σ > 0, β ∈ [−1, 1] and α ∈ (r, 2], or Xi − µ ∼ CS(r), i ∈ M. The risk-neutral seller strictly prefers B to any other bundling decision (that is, the goods are sold as a single bundle), if ci = c, i ∈ M, and p < µ. The risk-neutral seller strictly prefers B to any other bundling decision (that is, the goods are sold separately), if ci ≥ µ, i ∈ M, or if ci = c, i ∈ M, and p > µ. The following Theorem 5.2 extends Theorem 4.3 in Ibragimov and Walden (2010) and Theorem 2.5.3 in Ibragimov et al. (2015) to the case of non-linear valuations and interrelated products. According to the theorem, the patterns in the solutions to the seller’s optimal bundling problem in Theorem 5.1 in this paper, Theorem 4.3 in Ibragimov and Walden (2010) and Theorem 2.5.3 in Ibragimov et al. (2015) are reversed if consumers’ tastes are sufficiently (extremely) heavy-tailed and the goods are independently priced or are complements (or are substitutes with not very high degree of substitutability). Theorem 5.2 Let µ ∈ R, r ∈ (0, 2], pmax < ∞, and let the valuations for bundles B ∈ 2M of goods from M be given by v(B) = v(gr , B) in (1) or by v(B) = v(hr , B) in (2). Suppose that Xi , i ∈ M, in (1)-(2) are i.i.d. r.v.’s such that Xi ∼ Sα (σ, β, µ), i ∈ M, for some σ > 0, β ∈ [−1, 1] and α ∈ (0, r), or Xi − µ ∼ CS(r), i ∈ M. The risk-neutral seller strictly prefers B to any other bundling decision (that is, the goods are sold separately), if ci = c, i ∈ M, and p < µ. The risk-neutral seller strictly prefers B to any other bundling decision (that is, the goods are sold as a single bundle), if ci ≥ µ, i ∈ M, or if ci = c, i ∈ M, and p > µ. The results in Ibragimov and Walden (2010) and Section 2.5 in Ibragimov et al. (2015) provide the analogues of Theorems 5.1 and 5.2 in the case of independently priced goods (r = 1) and additive valuations v(B) = v(h1 , B) = P sumi∈B v(h1 , {i}) = i∈B Xi for bundles B ∈ 2M of goods i ∈ M provided. For completeness, the results for the independently priced case are provided in the following Theorems 5.3 and 5.4. P Theorem 5.3 Let µ ∈ R, and let the valuations for bundles B ∈ 2M be given by v(B) = v(h1 , B) = i∈B v(h1 , {i}) P = i∈B Xi . Suppose that the stand-alone valuations v(h1 , {i}) = Xi , i ∈ M, for goods in M are i.i.d. r.v.’s such that Xi ∼ Sα (σ, β, µ), i ∈ M, for some σ > 0, β ∈ [−1, 1] and α ∈ (1, 2], or Xi − µ ∼ CSLC, i ∈ M. Then the conclusion of Theorem 5.1 holds. Theorem 5.4 Let µ ∈ R, pmax < ∞, and let the valuations for bundles B ∈ 2M be given by v(B) = v(h1 , B) = P P i∈B v(h1 , {i}) = i∈B Xi . Suppose that the stand-alone valuations v(h1 , {i}) = Xi , i ∈ M, for goods in M are i.i.d. r.v.’s such that Xi ∼ Sα (σ, β, µ), i ∈ M, for some σ > 0, β ∈ [−1, 1] and α ∈ (0, 1), or Xi − µ ∼ CS(1), i ∈ M. Then the conclusion of Theorem 5.2 holds. As discussed in Ibragimov and Walden (2010) and Section 2.5 in Ibragimov et al. (2015), similar to the argument based on variance in Bakos and Brynjolfsson (1999), the underlying intuition for Theorems 5.1 and 5.3 is that for moderately heavy-tailed distributions of valuations and the marginal costs of goods on the right of the mean valuation, bundling decreases profits since it reduces concentration (peakedness) of the valuation per good and thereby decreases the fraction of buyers with valuations for bundles greater than their total marginal costs (this is 13

implied by the results in Appendix A1). For the identical marginal costs of goods less than the mean valuation, bundling is likely to have the opposite effect on the profit. On the other hand, similar to the discussion in Ibragimov and Walden (2010) and Section 2.5 in Ibragimov et al. (2015), the results in Theorems 5.2 and 5.4 are driven by the fact that, in the case of extremely heavytailed valuations, concentration and peakedness of the valuations per good in bundles decreases with their size (see Appendix A1). Therefore, bundling of goods in the case of extremely heavy-tailed valuations and marginal costs of goods higher than the mean reservation price increases the fraction of buyers with valuations for bundles greater than their total marginal costs and thereby leads to an increase in the monopolist’s profit. This effect is reversed in the case of the identical marginal costs on the left of the mean valuation. It is worth noting that the assumptions of Theorem 5.2 with r ≥ 1 (and those of Theorem 5.4) are satisfied, in particular, for positive stable r.v.’s (consumers’ preferences or tastes for the goods provided that determine positive stand-alone valuations and valuations for bundles) Xi ∼ Sα (σ, 1, µ), i ∈ M, where σ > 0 and α ∈ (0, 1), including the L´evy distributions S1/2 (σ, 1, µ). In such settings, the free disposal condition holds. Furthermore, from the proof of Theorems 5.1-5.4 it follows that the first parts (second parts) of conclusions in the theorems hold as well in the case of arbitrary marginal costs ci if the price per good pB in each bundle B ∈ 2M is less than (greater than) µ. One should also note here that the conditions pmax < ∞ in Theorems 5.2 and 5.4 are necessary since otherwise the monopolist would set an infinite price for each bundle of goods under extremely heavy-tailed distributions of the r.v.’s (consumers preferences or tastes for the goods provided) Xi considered in the theorems. Theorems 5.2 and 5.4 shed new light on marketing strategies of excluding from the bundle the goods for which extreme (both positive and negative) valuations are more likely, and selling them separately. Such strategies are often observed on the market, in particular, in the bundling decisions of cable and direct satellite broadcast television firms that have marginal costs of reproduction close to zero. The latter firms typically offer a “basic” bundle and use such strategies as pay-per-view for unusual special events such as boxing matches (see Bakos and Brynjolfsson, 1999). The high valuations for the special events are concentrated among a small fraction of consumers and thus are likely to be very heavy-tailed. Therefore, the optimal bundling strategies for the special events are likely to be the opposites of those for thin-tailed distributions of valuations and thus, in contrast to the basic bundles, the events are likely to be provided on pay-per-view basis. Season tickets for entertainment performances offered by sporting and cultural organizations that have sufficiently high marginal costs of production might illustrate the dual pattern in bundling. Most of the demand for season tickets is plausibly concentrated around a relatively small fraction of consumers that have high valuations for performances offered by the entertainment organization. The optimal strategy is to offer tickets to such consumers as a bundle, as predicted by the results in this section for heavy-tailed preferences under the free disposal assumption or symmetric heavy-tailed valuations in the case of sufficiently large marginal costs. This strategy is the opposite of separate provision of the most of tickets to performances to consumers who are likely not to have very extreme valuations.

14

6

Concluding Remarks

This paper contributes to the literature on bundling and heavy-tailedness and robustness in economics in several ways. First, we develop a framework that allows one to model the optimal bundling problem of a multiproduct monopolist providing large bundles of interrelated goods with an arbitrary degree of complementarity or substitutability. Second, we derive characterizations of optimal bundling strategies for the seller in this setup in the case of heavy-tailed valuations and preferences for the goods. Third, our analysis provides a unified approach to the study of optimal bundling problem in many settings, including non-linear preferences, heavy-tailedness and dependence in consumers’ valuations, using the powerful methods of majorization theory. In particular, the approach developed in this paper reveals that the analysis of optimal bundling strategies in different settings is based on the same probabilistic concepts and results. We show that the solutions to the optimal bundling problem with moderately heavy-tailed valuations are the opposites of those in the case of extreme heavy-tailedness. The paper provides a rather complete study of the interplay of the degree of heavy-tailedness in consumers’ valuations and that of the products’ substitutability or complementarity in the analysis of optimal bundling strategies. We obtain characterizations of optimal bundling strategies for a monopolist who provides goods with an arbitrary degree of complementarity or substitutability to consumers with heavy-tailed valuations and preferences. The patterns in the optimal bundling strategies are the opposites of one another, depending on the degrees of heavy-tailedness of consumers’ valuations and the degrees of complementarity and substitutability among the goods provided. From the results in the paper it follows that, for substitutes with moderately heavy-tailed valuations, the patterns in the seller’s optimal bundling strategies are the same as in the case of independently priced goods with thin-tailed (log-concavely distributed) or moderately heavy-tailed valuations discussed in the previous literature. That is, the seller prefers separate provision for substitute goods with high marginal costs and provision in a single bundle for substitutes with low marginal costs. These conclusions are reversed for complements with sufficiently (extremely) heavy-tailed valuations. In such a case, the seller’s optimal strategy is to provide complement goods with low marginal costs separately, and as a single bundle under high marginal costs. In particular, our analysis imply that many of the results available in the literature for independently priced goods with very thin-tailed valuations (such as those with log-concave densities or those with a bounded support) are reversed in the case of valuations with extremely heavy-tailed distributions. However, the results for very thin-tailed valuations continue to hold under the assumption that distributions of the valuations are not extremely heavy-tailed. In other words, the optimal bundling strategies analyzed in the literature in the case of very thintailed valuations are robust14 to heavy-tailedness assumptions for consumers’ valuations as long as the distributions entering the assumptions are not too heavy-tailed. However, they are reversed in the case of assumptions that involve extremely heavy-tailed distributions. 14 According to the well-established parlance in this literature, robustness is understood to mean sensitivity to distributional assumptions. In the paper, the use of the terms “robust” and “robustness” accords with this tradition.

15

The main intuition behind the analysis of optimal bundling decisions under very thin-tailed valuations in the literature (see the discussion in Palfrey, 1983, Schmalensee, 1984, Salinger, 1995, Bakos and Brynjolfsson, 1999, and Fang and Norman, 2006) is that, in such settings, consumers’ valuations per good for a bundle typically have a lower variance relative to the valuations for individual goods15 . The intuition that drives our results on bundling under heavy-tailed valuations is closely related to this. Namely, the results on peakedness and majorization reviewed in Appendix A.1 (see also Ibragimov and Walden, 2010, Ibragimov et al., 2015, and references therein) imply, essentially, that, in the case of moderately heavy-tailedness, the consumers’ valuations per good for bundles always have less spread relative to the valuations for component products, as measured by their peakedness. Contrary to this, under extreme heavy-tailedness, the spread of valuations per product for bundles, as measured by peakedness, is always greater than that of valuations for components.

7

Appendices

APPENDIX A1. MAJORIZATION AND PEAKEDNESS PROPERTIES OF HEAVY-TAILED DISTRIBUTIONS As discussed in Ibragimov and Walden (2010) and Section 2.5 in Ibragimov et al. (2015), similar to the analysis of robustness of many important models in economics, finance, econometrics, statistics, risk management and insurance to heavy-tailedness assumptions (see Ibragimov et al., 2015, and references therein), powerful tool for the study of optimal bundling under general assumptions on consumers’ valuations are provided by majorization theory. Pk Pk A vector v ∈ Rn is said to be majorized by a vector w ∈ Rn , written v ≺ w, if i=1 v[i] ≤ i=1 w[i] , Pn Pn k = 1, ..., n − 1, and i=1 v[i] = i=1 w[i] , where v[1] ≥ . . . ≥ v[n] and w[1] ≥ . . . ≥ w[n] denote components of v and w in decreasing order. The relation v ≺ w implies that the components of the vector v are less diverse than those of w (see Marshall et al., 2010). In this context, it is easy to see that the following relations hold: n X

wi /n, ...,

i=1

for all wi ≥ 0 such that

Pn

i=1

n X

n  X  wi /n ≺ (w1 , ..., wn ) ≺ wi , 0, ..., 0 , w ∈ Rn+ ,

i=1

(7)

i=1

wi = 1, and

(1/(n + 1), ..., 1/(n + 1), 1/(n + 1)) ≺ (1/n, ..., 1/n, 0), n ≥ 1.

(8)

Majorization ordering naturally relates to formalization of income inequality (see Ch. 1 and Section 13.F in Marshall and Olkin, 2011) and the analysis of portfolio diversification (see Section 2.1 in Ibragimov et al., 2015, and Chapter 4 in Ibragimov and Prokhorov, 2017). For instance, consider two vectors of portfolio weights v, w ∈ In = Pn {w = (w1 , ..., wn ) ∈ R+ : i=1 wi = 1}. Further, denote w = (1/n, 1/n, ..., 1/n) ∈ In and w = (1, 0, ..., 0) ∈ In . If v ≺ w, it is natural to think about the portfolio of risks or returns with weights v as being more diversified than that with weights w. That is, for example, the portfolio with equal weights w is the most diversified and the portfolio with weights w consisting of one risk is the least diversified among all the portfolios with weights w ∈ In (in this 15 Further intuition behind the power of bundling is that, for thin-tailed distributions, it reduces uncertainty about consumers’ valuations and leads to a decrease in extreme values of the distribution of valuations per good, thereby reducing buyer diversity and increasing the predictive power of the selling strategy (see Schmalensee, 1984, and Bakos and Brynjolfsson, 1999).

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regard, the notion of one portfolio being more or less diversified than another one is, in some sense, the opposite of that for vectors of weights for the portfolio). A function φ : A → R defined on A ⊆ Rn is called Schur-convex (resp., Schur-concave) on A if (v ≺ w) =⇒ (φ(v) ≤ φ(w)) (resp. (v ≺ w) =⇒ (φ(v) ≥ φ(w)) for all v, w ∈ A. If, in addition, φ(v) < φ(w) (resp., φ(v) > φ(w)) whenever v ≺ w and v is not a permutation of w, then φ is said to be strictly Schur-convex (resp., strictly Schurconcave) on A. Pn 2 Examples of strictly Schur-convex functions φ : Rn+ → R are given by φ(w1 , ..., wn ) = i=1 wi and, more Pn generally, by φp (w1 , ..., wn ) = i=1 wip for p > 1. The functions φp (w1 , ..., wn ) are strictly Schur-concave for p < 1 (see Proposition 3.C.1.a in Marshall et al., 2010). price per good pB ) is The following concept of peakedness of r.v.’s was introduced by Birnbaum (1948). Definition 7.1 (Birnbaum, 1948, see also Proschan, 1965, and Marshall and Olkin, 2011, p. 490). A r.v. X is more peaked about µ ∈ R than is Y if P (|X − µ| > x) ≤ P (|Y − µ| > x) for all x ≥ 0. If these inequalities are strict whenever the two probabilities are not both 0 or both 1, then the r.v. X is strictly more peaked about µ than is Y. A r.v. X is said to be (strictly) less peaked about µ than is Y if Y is (strictly) more peaked about µ than is X. In the case µ = 0, we simply say that the r.v. X is (strictly) more peaked than Y. Roughly speaking, a r.v. X is more peaked about µ ∈ R than is Y, if the distribution of X is more concentrated about µ than is that of Y. Proschan (1965) obtained the following well-known result concerning majorization and peakedness properties of tail probabilities of linear combinations of log-concavely distributed r.v.’s: Proposition 7.1 (Proschan, 1965)16 . If X1 , ..., Xn are i.i.d. r.v.’s such that Xi ∼ LC, i = 1, ..., n, then the function
Pn n ψ(a, x) = P i=1 ai Xi > x is strictly Schur-convex in a = (a1 , ..., an ) ∈ R+ for x > 0 and is strictly Schur-concave in a = (a1 , ..., an ) ∈ Rn+ for x < 0. Clearly, from Proposition 7.1 it follows that, under its assumptions, Pn

i=1 bi Xi

Pn

i=1

ai Xi is strictly more peaked than

if a ≺ b and a is not a permutation of b.

Theorems 7.1-7.4 in this appendix provide analogues of Proposition 7.1 for heavy-tailed r.v.’s obtained in Ibragimov (2007, 2009) (see Theorems 3.1-3.4 in Ibragimov, 2007, Theorems C.3-C.5 in Ibragimov, 2009). The majorization results for heavy-tailed distributions provide the foundation for the analysis of robustness of key economic and financial models to heavy-tailedness in Ibragimov et al. (2015) (see Section 2.1 and Theorems 2.1.1 and 2.1.2 therein). 16 This proposition is formulated as Theorem 12.J.1 in Marshall and Olkin (2011). Proschan’s (1979) work is also presented, in a rearranged form, in Section 11 of Chapter 7 in Karlin (1968). Ibragimov et al. (2015) and references therein provide many applications of majorization properties of log-concave distributions and their analogues for heavy-tailed case (see Appendix A1 in this paper and Appendix B2 in Ibragimov and Walden (2010).

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According to the following Theorem 7.1, the majorization properties of convex combinations of r.v.’s in the classes CS(r) are of the same type as in Proposition 7.1 with respect to the comparisons between the powers of components of the vectors of weights of the combinations. Theorem 7.1 (see Theorem 3.3 in Ibragimov, 2007, and Theorem C.5 in Ibragimov, 2009) Let r ∈ (0, 2). If X1 , ..., Xn are i.i.d. r.v.’s such that Xi ∼ Sα (σ, β, 0), i = 1, ..., n, for some σ > 0, β ∈ [−1, 1] and α ∈ (r, 2], or Xi ∼ CS(r), i = 1, ..., n, then the function ψ(a, x), a ∈ Rn+ in Proposition 7.1 is strictly Schur-convex in (ar1 , ..., arn ) for x > 0 and is strictly Schur-concave in (ar1 , ..., arn ) for x < 0. As follows from Theorem 7.2 below, the majorization and peakedness properties of the tail probabilities ψ(a, x) in Theorem 7.1 are reversed in the case of r.v.’s from the classes CS(r). Theorem 7.2 (see Theorem 3.4 in Ibragimov, 2007, and Theorem C.5 in Ibragimov, 2009) Let r ∈ (0, 2]. If X1 , ..., Xn are i.i.d. r.v.’s such that Xi ∼ Sα (σ, β, 0), i = 1, ..., n, for some σ > 0, β ∈ [−1, 1] and α ∈ (0, r), or Xi ∼ CS(r), i = 1, ..., n, then the function ψ(a, x), a ∈ Rn+ in Proposition 7.1 is strictly Schur-concave in (ar1 , ..., arn ) for x > 0 and is strictly Schur-convex in (ar1 , ..., arn ) for x < 0. According to Theorem 7.3 below, peakedness and majorization properties of linear combinations of r.v.’s with not too heavy-tailed distributions, as modelled, e.g., by convolutions of log-concave distributions and symmetric stable distributions with characteristic exponents greater than one, are the same as in the case of log-concave distributions in Proschan (1965). Theorem 7.3 (see Theorem 3.1 in Ibragimov, 2007, Theorem C.3 in Ibragimov, 2009, and Theorem 2.1.1 in Ibragimov et al., 2015) Proposition 7.1 holds if X1 , ..., Xn are i.i.d. r.v.’s such that Xi ∼ Sα (σ, β, 0), i = 1, ..., n, for some σ > 0, β ∈ [−1, 1] and α ∈ (1, 2], or Xi ∼ CSLC, i = 1, ..., n. As follows from Theorem 7.4, peakedness properties given by Proposition 7.1 and Theorem 7.3 above are reversed in the case of r.v.’s with extremely heavy-tailed distributions, as modelled by convolutions of stable distributions with indices of stability less than one. Theorem 7.4 (see Theorem 3.2 in Ibragimov, 2007, Theorem C.4 in Ibragimov, 2009, and Theorem 2.1.2 in Ibragimov et al., 2015) If X1 , ..., Xn are i.i.d. r.v.’s such that Xi ∼ Sα (σ, β, 0), i = 1, ..., n, for some σ > 0, β ∈ [−1, 1] and α ∈ (0, 1), or Xi ∼ CS(1), i = 1, ..., n, then the function ψ(a, x) in Proposition 7.1 is strictly Schur-concave in (a1 , ..., an ) ∈ Rn+ for x > 0 and is strictly Schur-convex in (a1 , ..., an ) ∈ Rn+ for x < 0. APPENDIX A2. PROOFS Proofs of Theorems 5.1-5.4. Let r ∈ (0, 2] and let ci , i ∈ M, be arbitrary marginal costs of goods in M. Let
P the valuations v(B) for bundles B ∈ 2M be given by v(B) = v(gr ; B) = gr i∈B Xi or by v(B) = v(hr ; B) =
P hr i∈B Xi . Further, let µ ∈ R and pmax < ∞. Suppose that the tastes Xi , i ∈ M, are i.i.d. r.v.’s such that 18

Xi ∼ Sα (σ, β, µ), i ∈ M, for some σ > 0, β ∈ [−1, 1] and α ∈ (0, r), or Xi − µ ∼ CS(r), i ∈ M. We will show that the seller’s profit maximizing bundling decision is B if the prices per good pB < µ for all bundles B ∈ 2M , and is B if pB > µ for all B ∈ 2M . For a bundle B ∈ 2M , the profit maximizing price per good in the bundle is
P pB = arg maxp∈[0,pmax ] p − (1/k) i∈B ci P (v(B) ≥ kp) and the seller’s profit per good resulting from selling the
P bundle B (at the price per good pB ) is E(πB ) = k pB − i∈B ci P (v(B) ≥ kpB ), where k = card(B) is the number of goods in B. For i ∈ M, let pi be the price of good i in the case where the goods are sold separately (that is, in the case of the bundling decision B) and let πi be the monopolist’s profit from selling the good, namely, pi = pBi and πi = πBi with Bi = {i}. As in the setup of the optimal bundling problem in Section 5, in the case with ci = c for all i ∈ M, we write p = pM for the price per good in the case where all the m goods are sold as a single bundle B = M (that is, in the case of the bundling decision B) and p for the price of each good under unbundled sales (that is, p = pB with B = {i}, i ∈ M ). Suppose that pB < µ for all B ∈ 2M . Then from Theorem 7.4 and relations (8) it follows that, for any bundle
P B ∈ 2M with the number of goods card(B) = k ≥ 2, E(πB ) = kpB − i∈B ci P (v(B) ≥ kpB ) = kpB −
P

P P P 1/r 1/r ) < ≤ i∈B ci P ( i∈B Xi ≥ (kpB ) i∈B pB − ci P Xi ≥ (pB ) i∈B E(πi ). This implies that for any bundling decision B = {B1 , ..., Bl } such that card(Bs ) = ks , s = 1, ..., l, and kt ≥ 2 for at least one t ∈ {1, ..., l}, E(ΠB ) =

l X

E(πBs ) <

s=1

l X X

E(πi ) =

m X

s=1 i∈Bs

E(πi ) = E(ΠB ).

(9)

i=1

Suppose now that pB > µ for all B ∈ 2M . Then using again Theorem 7.4 and relations (8) we get that, for
P P any bundle B ∈ 2M with card(B) = k ≤ m − 1, E(πB ) = kpB − i∈B ci P ( i∈B Xi ≥ (kpB )1/r ) < kpB −
Pm P 1/r ). Therefore, for any bundling decision B = {B1 , ..., Bl } such that card(Bs ) = ks , i∈B ci P ( i=1 Xi ≥ (mpB ) s = 1, ..., l, and kt ≤ m − 1 for at least one t ∈ {1, ..., l}, E(ΠB ) =

l X

E(πBs ) <

ks pBs − (1/m)

s=1

m X i=1

ks pBs −

s=1

s=1 l X

l X

m X


ci P (

X

m
X Xi ≥ (mpB )1/r ) = ci P ( i=1

i∈Bs

Xi ≥ (mpB )1/r ) ≤

l X

(ks /m)E(ΠB ) = E(ΠB ).

(10)

s=1

i=1

From (9) and (10) we get that the profit maximizing bundling decision is B if pB > µ for all B ∈ 2M and is B if pB < µ for all B ∈ 2M . Clearly, the condition that pB > µ for all B ∈ 2M is satisfied if ci ≥ µ for all i ∈ M. Furthermore, in the case of identical marginal costs ci = c, i ∈ M, the condition that pB > µ for all B ∈ 2M holds if p > µ. Indeed, suppose that this not the case and there exists a bundle B ∈ 2M with card(B) = k > 1 and pB ≤ µ. Then, as Pk above, we get kE(π1 ) = k(p − c)P (X1 ≥ (p)1/r ) < k(p − c)P ( i=1 Xi ≥ (kp)1/r ) ≤ E(πB ). On the other hand, Pk E(πB ) = k(pB −c)P ( i=1 Xi ≥ (kp)1/r ) < k(pB −c)P (X1 ≥ (pB )1/r ) ≤ kE(π1 ), which is a contradiction. Similarly, we get that if ci = c, i ∈ M, then p < µ implies that pB < µ for all B ∈ 2M . This completes the proof of Theorem 5.2. Theorem 5.4 follows from Theorem 5.2 with r = 1. Theorems 5.1 and 5.3 could be proven in a similar way, with the use of Theorems 7.1 and 7.3 instead of Theorem 7.2. The proof is complete.

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