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The authors have declared that no competing interests exist.

We study a trial-offer market where consumers may purchase one of two competing products. Consumer preferences are affected by the products quality, their appeal, and their popularity. While the asymptotic convergence or stationary states of these, and related dynamical systems, has been vastly studied, the literature regarding the transitory dynamics remains surprisingly sparse. To fill this gap, we derive a system of Ordinary Differential Equations, which is solved exactly to gain insight into the roles played by product qualities and appeals in the market behavior. We observe a logarithmic tradeoff between quality and appeal for medium and long-term marketing strategies: The expected market shares remain constant if a decrease in quality is followed by an exponential increase in the product appeal. However, for short time horizons, the trade-off is linear. Finally, we study the variability of the dynamics through Monte Carlo simulations and discover that low appeals may result in high levels of variability. The model results suggest effective marketing strategies for short and long time horizons and emphasize the significance of advertising early in the market life to increase sales and predictability.

Social influence is a ubiquitous phenomenon in numerous markets: Product recommendations and information about past purchases have been shown to influence consumers choices significantly whether it is for music, movie, book, technological, and other type of products. Social influence often induces a rich-get-richer phenomenon (or Matthew effect) where popular products tend to become even more popular [

In this paper, we consider a trial-offer market where consumers may purchase one of two competing products and consumer preferences are affected by the quality, appeal, and popularity of the products. While the asymptotic convergence of these, and related, dynamical systems has been widely studied (e.g., [

This paper focuses on such trial-offer markets over a finite horizon and is particularly interested in answering the following questions to design a particular marketing campaign:

Can a low quality be compensated by a greater appeal of the product?

Given the quality and appeal of a competing product, which combination of quality and appeal are necessary for a product to capture the desired market share?

How is important the timing of a marketing strategy?

Will the marketing strategy increase or decrease the predictability of the market?

The main contribution of this paper is a principled study of these questions, to move from an understanding of the market behavior to the design of successful marketing and product development strategies. First, we derive a system of Ordinary Differential Equations (ODEs) which approximates the discrete market dynamics as a continuous system. Second, we perform an assortment of Monte Carlo simulations which corroborate that the ordinary differential equations are a good approximation of the median value. Using the solution of the ODEs, we conclude that, for medium and long-term marketing strategies, the trade-off between quality and appeal is logarithmic: The expected value of the market share remains constant if a decrease of quality is followed by an exponential increase of the product appeal. However, for very short-term marketing strategies, the trade-off is linear. The Monte Carlo simulations are then used to understand the magnitude of the stochastic fluctuations (i.e., variability) in the market share for different values of the parameters. We observe that market share variability increases as products are closer in quality and when product appeal are low. We then conclude with recommendations on marketing strategies for markets with short and long time horizons and the importance of investing early in marketing campaigns.

The seminal work on the MusicLab [

Krumme et al [

The closest related work is by [

Another related paper by [

We consider a simplification of the model proposed by [

Its appeal _{i} which represents inherent customer preference for trying product

Its quality _{i} which represents the probability that a consumer purchases product

An incoming customer may try product 1 or 2. The purchase probability of product _{i} that product _{i}, i.e.,
_{i} is the number of purchases product _{1} ≠ 0 and _{2} ≠ 0.

Without loss of generality, this paper presents the results from the point of view of firm 2 whose product has the lowest quality. Hence it is only necessary to analyze how the parameters of product 2 shape its success in the market. The success is measured by the _{2} ≐ _{2}/(_{1} + _{2}).

We begin by studying the dynamics of the cultural market behavior in the transitory regime using a continuous approximation, which is modeled as a system of Ordinary Differential Equations (ODEs). The use of ODEs for solving discrete systems is well-established [

By taking the interval between two consecutive customer arrivals as the time unit, the change in purchases at time _{1}_{2} ≐ _{2}/_{1}, where _{2} denotes the quality ratio. The dynamical system can be rewritten as:

The dynamical behavior is only determined by the quality ratio _{2}, while the magnitudes of _{1} and _{2} only affect the time scale.

The system dynamics does not depend explicitly on the appeals (_{1} and _{2}), whose effects are incorporated only in the initial conditions. Thus, from a dynamical standpoint, the effects of the appeals are equivalent to those of the initial purchases. This provides a natural interpretation of the role of the appeals in terms of the equivalent initial purchases (see, e.g., [

Eqs _{2} and _{2} ≠ 1, _{2} > 1 (i.e., _{2} > _{1}), _{2} = 1 is a stable fixed point and _{2} = 0 is unstable ([_{2}(_{2} = 1: As time goes to infinity, the increasing number of purchases makes the _{2} ≈ _{2}. When _{2} < 1 (i.e., _{2} < _{1}), _{2} = 0 becomes the stable state and the system evolves towards _{2} = 0. As shown by [

Using the chain rule, Eqs _{2} as a function of _{2} with the system parameters for a given _{T} ≐ _{1} + _{2}. By writing _{2} as a function of _{2} and _{T}, we can compute _{2} as a function of _{2} and _{2} for different values of _{1} and _{T}.

We analyze the case in which product 1 has a better quality than product 2, i.e., _{2} = _{2}/_{1} < 1. Although product 1 is the winner (_{2} ≈ 0) in the stationary solution, at finite times, there are some combinations of parameters in which product 2 is the winner (i.e., _{2} ≈ 1), even for cases with large _{T} values. Low _{2} values can be compensated with _{2} values that are much higher than _{1}. It can be seen in _{T} values (i.e., large number of customers), the values for _{2} shows an approximate linear behavior between _{2} and log(_{2}) (see the green dividing stripes in the right column). In other words, to compensate for an increment of quality for product 1, it is necessary to either increment _{2} at the same rate or improve _{2} by an exponential amount.

To study the trade-off between _{2} and _{2}, we can analyze the functional form of the _{2} = 0.5 level curves. These curves are displayed in _{2} ≈ _{2} and _{2} and ln(_{2}), as observed in

For short time horizons, if both appeals are large (_{1} ≪ _{1} and _{2} ≪ _{2}), the purchase probabilities described in _{1}), we can choose to increase our product quality or appeal linearly by the exact same rate (since _{2} = _{2}/_{1}). In this case, the curve that separates the winning and losing regions (level set of _{2} = 0.5) is
_{2} decreases inversely with _{2} and not with its logarithm. Thus, the relationship between the appeal and quality of different products is linear. This approximation holds as long as _{1} ≪ _{1} and _{2} ≪ _{2}. Hence large appeal values result in longer time lags during which _{2} does not evolve and

In the previous section, we used a deterministic model to gain some insight into the dependencies between system parameters for finite times. We now employ Monte Carlo simulations to study the effects of the stochastic fluctuations on the market share and their dependency on the system parameters. Additionally, we examine how well the ODE model approximates the real dynamic system.

_{2} versus _{T} of the discrete system and their ODE approximations for different values of _{2} ∼ 1). In particular, for the case of _{1} = _{2} = 1 and _{2} = 0.8, product 2 obtains more purchases than product 1 in 20% of the simulations at _{T} = 1000, despite the fact that the expected market share of product 2 at this _{T} is _{2} = 0.28). Obviously, the system possesses a positive feedback mechanism in which the purchase of a product increases its future purchase probability. A stochastic fluctuation in favor of one product increases its purchase probability, which amplifies this initial fluctuation. However, when the appeals are larger, a substantial reduction in variability is observed. This is consistent with the fact that large appeals produce longer time lags in which the purchase probabilities do not depend on the number of purchases (see Section 3.2). This drastically reduces the positive feedback effect and diminishes the system variability. Larger appeal values result in longer time lags, which allow the system to reach high _{T} values under the long-term approximation. Furthermore, as _{T} increases, a small stochastic fluctuation has proportionally less influence on _{T}, producing negligible variations in the purchase probabilities. Thus, stochastic fluctuations at short times are the major contributor to the final variability. _{T} increases.

The Monte Carlo simulations involved 10^{4} realizations of the system for nine different set of parameters (grey lines). In all cases we used _{1} = 1, the appeal used for both products were the same (_{1} = _{2}) and both products start with zero purchases, _{1}(_{2}(_{T}_{2}) space, we plot each simulation with straight lines that link consecutive dots to follow trajectories easily.

_{2}. Low expected values of _{2} tend to be associated with less variability. The higher the purchase probability difference between the products, the more skewed and narrow the initial distribution is. This is discussed in more detail in the Section 4.1. In the case of _{2} = 0.2, when moving from _{1} = _{2} = 10 to _{1} = _{2} = 100, the variability increases slightly, which is the opposite behavior compared to the _{2} = 0.8 case. Although the increase in the time lag tends to decrease the variability, it also delays the increase in _{2}. The distribution of _{2} thus stays closer to the values of _{2} = 0.5, where the system is more susceptible to fluctuations.

As mentioned in Section 3, when _{2} > _{1}, there are regions in the parameter space (_{2}, _{2}) in which product 2 obtains a substantial market share at finite times. _{2} = 1 to _{2} = 5) produce large effects sustained over time, which again can be explained in terms of a cumulative advantage process.

The Monte Carlo simulations involved 10^{4} realizations of the system for nine different sets of parameters (grey lines). In all cases, _{1} = 1 and both products start with zero purchases, i.e., _{1}(_{2}(_{T}_{2}) space, we plot each simulation with straight lines that link consecutive dots to follow trajectories easily.

By Lemma 1 in [_{1} ≪ _{1} and _{2} ≪ _{2}, the next purchase probabilities for products 1 and 2 are given by

The probability of having exactly _{T} purchases is a binomial distribution:
_{1} ≪ _{1} _{2} ≪ _{2} and _{1} = _{2}_{2}, which coincides with the case where the expected market shares of the products are equal. This second factor also explains the reduction of variability when _{2} is low, which was observed in

Under the short-term approximation, when appeals are large, the mean _{2} value is _{T} increases, the non-linearities of the purchase probabilities can produce a discrepancy between the ODE solution and the mean value of the simulation. _{1} = _{2} = 1 and _{2} = 0.8, there is a difference between them. This is consistent with the fact that, when the _{T} ranges and hence the mean value and the ODE match during the times when the system is most susceptible to stochastic fluctuations. Note also that the median of the simulations seems to be in strong agreement with the ODE and to outperform the mean. However, for very short times, it fails to reproduce the ODE solution, which is expected since the median is restricted to the possible discrete values of the simulations. For example, if the ODE gives a _{2} = 0.2 for _{T} = 2 there are only three possible combinations of (_{1}, _{2}) at this time; {(2, 0), (1, 1), (0, 2)}. These correspond to only three possible _{2} values, {1, 0.5, 0}, which would not match the ODE solution.

This paper studied the transitory dynamics for a simple trial-offer market model, originally proposed in [_{i}), past purchases (_{i}), and product quality (_{i}). In this model, consumers arrive sequentially, select a product to try, and then decide whether to purchase the sampled product. Although the asymptotic convergence or stationary states of these and related dynamical systems has been studied in depth, the short-term dynamics remained relatively unexplored. More precisely, the asymptotic convergence of the trial-offer market studied here was solved by [

To analyze this finite-time behavior, the paper modeled the discrete market dynamics as a system of Ordinary Differential Equations (ODEs) and restricted attention to two products for simplicity. From an ODE nondimensionalization, the paper showed that: (1) The dynamical behavior is only determined by the quality ratio (_{2}/_{1}); and (2) The effect of the appeals is equivalent to those of the initial purchases. This provides a natural interpretation of the role of the appeals in terms of the equivalent initial purchases.

Moreover, to analyze possible marketing strategies, the paper considered a firm whose product (product 2) is of a lower quality than the product of its competitor (product 1). The paper solved the ODEs for different parameter sets and quantified the market share at different time points. Although product 1 is the winner asymptotically, the paper showed that, at finite times, there are parameter regions in which product 2 obtains a higher market share, even in the case of long time horizons. In other words,

The analytic solution of the ODEs also allowed us to identify the impact of the system parameters on the market shares. In particular, the paper identified two limit cases for which we can propose different market strategies: (1) the long-term strategy, in which the appeals are negligible in comparison to the total purchases (_{1}, _{2} ≪ _{T}) and (2) The immediate reward strategy, in which the time horizon is short enough that the number of purchases are negligible in comparison to the appeals. For the long-term strategy, the paper showed that the trade-off between the appeals and qualities lies in a logarithmic scale for the appeals and in a linear scale for the qualities.

Finally, the paper ran Monte Carlo simulations to study the effect of the stochastic fluctuations on the market share. The results indicate a good agreement between the ODEs solutions and the median and mean of the simulations for a variety of parameter settings. The results also showed that the system exhibits large variability when both appeals are small and their qualities are similar. These observations have once again important implications on potential marketing strategies. Indeed, since the system has an implicit positive feedback, in which the purchase of a product increases its future purchase probability,

In summary, the paper provides evidence that (1) early investments in advertising are highly beneficial for short time horizons and for improving market predictibility; and (2) quality improvements are desirable for long-term strategies.

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We would like to thank David Schnoerr for numerous helpful discussions.