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Popularity signals in trial-offer markets with social influence and position bias

https://doi.org/10.1016/j.ejor.2017.10.056Get rights and content

Highlights

  • We study a trial-offer market consisting of N products.

  • Consumer’s choices are dependent on past purchases (social influence).

  • We characterize the long term equilibrium of the market share of each of the N products.

  • A sublinear social signal reduces the market share inequalities among products.

  • Benefits of using a quality ranking instead of a popularity ranking are highlighted.

Abstract

This paper considers trial-offer markets where consumer preferences are modelled by a multinomial logit with social influence and position bias. The social signal for a product is given by its current market share raised to power r (or, equivalently, the number of purchases raised to the power of r). The paper shows that, when r is strictly between 0 and 1, and a static position assignment (e.g., a quality ranking) is used, the market converges to a unique equilibrium where the market shares depend only on product quality, not their initial appeals or the early dynamics. When r is greater than 1, the market becomes unpredictable. In many cases, the market goes to a monopoly for some product: which product becomes a monopoly depends on the initial conditions of the market. These theoretical results are complemented by an agent-based simulation which indicates that convergence is fast when r is between 0 and 1, and that the quality ranking dominates the well-known popularity ranking in terms of market efficiency. These results shed a new light on the role of social influence which is often blamed for unpredictability, inequalities, and inefficiencies in markets. In contrast, this paper shows that, with a proper social signal and position assignment for the products, the market becomes predictable, and inequalities and inefficiencies can be controlled appropriately.

Introduction

The impact of social influence and product visibilities on consumer behaviour in Trial-Offer (T-O) markets1 has been explored in a variety of settings (e.g., Salganik, Dodds, Watts, 2006, Tucker, Zhang, 2011, Viglia, Furlan, Guevara, 2014). Social influence can be dispensed through different types of social signals: a market place may report the number of past purchases of a product, its consumer ratings, and/or its consumer recommendations. Recent studies Engstrom and Forsell (2014), Viglia et al. (2014) however came to the conclusion that the popularity signal (i.e., the number of past purchases or the market share) has a much stronger impact on consumer behaviour than the average consumer rating signal.2 These two experimental studies were conducted in very different settings, using the Android application platform in one case and hotel selection in the other. Consumer preferences are also influenced by product visibilities, a phenomenon that has been widely observed in internet advertisement (e.g., Craswell, Zoeter, Taylor, & Ramsey, 2008), in online stores such as Expedia, Amazon, and iTunes, as well as physical retail stores (see, e.g., Lim, Rodrigues, & Zhang, 2004).

Despite its widespread use in online settings (including for songs, albums, movies, hotels, and cell phones to name only a few), there is considerable debate in the scientific community about the benefits of social influence. Many researchers have pointed out the potential negative effects of social influence. The seminal work of Salganik et al. (2006) on the MusicLab experimental market demonstrated that social influence can introduce significant unpredictability, inequality, and inefficiency in T-O markets. These results were reproduced by many researchers (e.g., Muchnik, Aral, Taylor, 2013, van de Rijt, Kang, Restivo, Patil, 2014, Salganik, Watts, 2008, Salganik, Watts, 2009). More recently, Hu, Milner, and Wu (2015) studied a newsvendor problem with two substitutable products with the same quality in which consumer preferences are affected by past purchases. The authors showed that the market is unpredictable but can become less so if one of products has an initial advantage. Altszyler, Berbeglia, Berbeglia, and Van Hentenryck (2017) has recently studied the impact of product appeal and product quality in a trial-offer market model with social influence under a finite time horizon. The authors showed that there exists a logarithmic tradeoff between the two: the final product market share remains constant if a decrease in product quality is followed by an exponential increase in the product appeal. Other researchers have focused on understanding when these undesirable side-effects arise and where they come from. Ceyhan, Mousavi, and Saberi (2011) studied a market specified by a logit model where a constant J captures the strength of the social signal. They showed that the market behaviour (e.g., whether it is predictable) depends on the strength of the social signal. Their results did not consider product visibilities, which is another important aspect of T-O markets. Indeed, various researchers (e.g., Abeliuk, Berbeglia, Cebrian, Hentenryck, 2015, Abeliuk, Berbeglia, Cebrian, Hentenryck, 2016, Abeliuk, Berbeglia, Hentenryck, Hogg, Lerman, 2017, Hentenryck, Abeliuk, Berbeglia, Berbeglia, Maldonado, 2016, Lerman, Hogg, 2014) indicated that unpredictability and inefficiencies often depend on how products are displayed in the market. In particular, Abeliuk et al. (2015) show that social influence is always beneficial in expectation when the products are ordered by the performance ranking that maximises the purchases greedily at each step. This result was obtained using the generalised multinomial logit model proposed by Krumme, Cebrian, Pickard, and Pentland (2012) to reproduce the MusicLab experiments. Hentenryck et al. (2016) proves a similar result for the quality ranking that assigns the highest quality products to the most visible positions. In addition, they show that the market converges to a monopoly for the highest quality product. These results contrast with the MusicLab experiments which relied on the popularity ranking that dynamically assigns the most popular products to the most visible positions.

This paper seeks to expand our understanding of social influence in T-O markets and explores the role of the social signal in conjunction with product visibilities. Our starting point is the generalised multinomial logic model of Krumme et al. (2012), which we extend to vary the strength of the social signal. More precisely, this paper considers a T-O market where the probability of purchasing product i at time t is given by pi(ϕt)=vσ(i)qi(ϕit)rj=1nvσ(j)qj(ϕjt)rwhere σ is a bijection from n products to n positions, vkR is the visibility of position k, qiR is the inherent quality of product i, ϕit is the market share of product i at time t, and r > 0 is the strength of the social signal. As should be clear from the discussion above, prior work on T-O markets with product visibilities (e.g., Abeliuk, Berbeglia, Cebrian, Hentenryck, 2015, Abeliuk, Berbeglia, Maldonado, Hentenryck, 2016, Hentenryck, Abeliuk, Berbeglia, Berbeglia, Maldonado, 2016, Lerman, Hogg, 2014) focused on the case of a linear social signal (r=1). The primary objective of this paper is to understand what happens to the T-O market when r < 1.

The paper contains both theoretical and simulation results and its contributions can be summarised as follows:

  • 1.

    When r < 1 and a static ranking is used, the market converges to a unique equilibrium, which we characterise analytically. In the equilibrium, the market shares depend only on the product qualities qi and no monopoly occurs. Moreover, a product of higher quality receives a larger market share than a product of lower quality, introducing a notion of fairness in the market and reducing the inequalities introduced by a linear social signal.

  • 2.

    When r > 1 and a static ranking is used, the equilibria can be characterised similarly. However, contrary to the case r < 1, the equilibria that are not monopolies can be shown to be unstable under certain conditions. As a result, the market will typically go to a monopoly for some product: which product wins the entire market share depends on the initial condition and the early dynamics.

  • 3.

    Agent-based simulations show that the market converges quickly towards an equilibrium when using sublinear social signals and the quality ranking. They also show that the quality ranking outperforms the popularity ranking in maximising the efficiency of the market. The popularity ranking is also shown to have some significant drawbacks in some settings.

These theoretical results indicate that, when the social influence signal is a sublinear function of the market share and a static ranking of the products (e.g., the quality ranking) is used, the market is entirely predictable, depends only on the product quality, and does not lead to a monopoly. This contrasts with the case of r=1 where the market is entirely predictable but goes to a monopoly for the product of highest quality (assuming the quality ranking) (Hentenryck et al., 2016) and the case of r > 1 where the market becomes unpredictable (even with a static ranking). As a result, sublinear social signals provide a way to balance market efficiency and the inequalities introduced by social influence. In particular, with sublinear social signals and a static ranking, markets do not exhibit a Matthew effect where the winner takes all, and remain predictable.

The remaining of this paper is organised as follows. Section 2 describes the related work. Section 3 introduces T-O markets and the generalised multinomial logit model for consumer preferences considered here. Section 4 reviews some necessary mathematical preliminaries, including the fact that T-O markets can be modelled as Robbins–Monro algorithms. Section 5 derives the equilibria for the market as a function of the social signal and also presents the convergence results. Section 6 reports the results from the agent-based simulation. Section 7 discusses some additional results on sublinear signals. Section 8 discusses the results and concludes the paper.

Section snippets

Related work

The research presented in this paper was motivated by the seminal work of Salganik et al. (2006). They study an experimental market called the MusicLab, where participants were presented a list of unknown songs from unknown bands, each song being described by its name and band. The participants were partitioned into two groups exposed to two different experimental conditions: the independent condition and the social influence condition. In the independent group, participants were shown the

The trial-offer model

The paper builds on the work by Krumme et al. (2012) who propose a framework in which consumer choices are captured by a multinomial logit model whose product utilities depend on the product appeal, position bias, and a social influence signal representing past purchases. A marketplace consists of a set N of n items. Each item i ∈ N is characterised by two values:

  • 1.

    its appeal ai > 0 which represents the inherent preference of trying item i;

  • 2.

    its quality qi > 0 which represents the conditional

Trial-offer markets as Robbins–Monro algorithms

This section establishes some basic definitions and concepts that are useful in the rest of the paper. In particular, it shows that T-O markets can be modelled as Robbins–Monro algorithms and states some useful results. The results in this section are well-known in stochastic approximation. The section starts with a brief introduction of Ordinary Differential Equations (ODE) and some stability criteria (e.g., see Hirsch, Smale, Devaney, 2012, Jordan, Smith, 1999).

Equilibria of trial-offer markets

This section characterises the equilibria and the asymptotic behaviour of the continuous dynamics dϕtdt=p(ϕt)ϕt,(ϕtΔn1),which is associated with the RMA (8). For simplicity, we remove the visibilities by stating q¯j=vjqj. We are interested in the case where f(x)=xr with (r > 0, r ≠ 1), since the case r=1 has been settled in earlier work. Let Q be the set of positive market shares, this is, Q={iN:ϕi0}, clearly Q ≠ ∅ since i=1nϕi=1.

Theorem 5.1

Let f(x)=xr,r>0, and r ≠ 1. Any equilibria ϕ for Eq. (10)

Agent-based simulation results

We now report results from an agent-based simulation to highlight and complement the theoretical analysis. The agent-based simulation uses the setting from Abeliuk et al. (2015), which used a dataset to emulate an environment similar to the MusicLab. The setting consists of 50 songs with the values of qualities and appeals specified in Appendix D. As mentioned in the introduction, the MusicLab is a trial-offer market where participants can try a song and then decide to download it. The

The benefits of social influence

A linear social signal has been shown to be beneficial to the market efficiency, i.e. it maximises the expected number of downloads. This result was proved by Abeliuk et al. (2015) for the performance ranking and by Hentenryck et al. (2016) for any static ranking such as the quality ranking. Unfortunately, sublinear social signals are not always beneficial to the market in that sense, as one can see in Example 7.1. Consider, once again, the quality ranking and assume that q1 ≥ ⋅⋅⋅ ≥ qn. When

Discussion and conclusion

This paper studied the role of social influence in trial-offer markets where customer preferences are modelled by a generalisation of a multinomial logit. In this model, both position bias and social influence impact the products tried by consumers.

The main result of the paper is to show that trial-offer markets, when the ranking of the products is fixed, converge to a unique equilibrium for sublinear social signals of the form ϕir, where ϕi represents the cumulative market share of product i.

Acknowledgements

Thanks are due to three anonymous referees for their valuable comments.

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