Price Competition in Attraction Choice Models with Linear Cost and Random Coefficients


Student thesis: Doctoral Thesis

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Awarding Institution
  • Houmin YAN (Supervisor)
  • Shouyang Wang (External person) (External Supervisor)
Award date20 Jul 2018


Retailers selling a set of differentiated products face the problem of setting appropriate prices such that the total profit is maximized. There is a tradeoff between the price and market share. Specifically, raising up the price reduces the corresponding market share in general, and therefore it may fail to boost the total sales and the profit. Meanwhile, other retailers' pricing strategies also affect the pricing decision, thus makes the problem complicated. In this study, under various attraction choice models, we consider this price decision problem in a competitive setting. The adopted choice models provide a behavioral basis for the dependence structure between the product's attributes and its market share. Therefore, they have gained wide applications in both the academics and industries. The research question is investigating the existence conditions and structure properties of the Nash equilibrium for this competition problem, and particularly, how the random coefficients and the cost structures affect the equilibrium.

We start with the basic attraction choice models including the multinomial logit model (MNL), multiplicative competitive interaction model (MCI), and linear attraction model. After discussing the sufficient conditions for the existence of a Nash equilibrium, we provide closed-form solutions of the equilibrium market shares and prices under these basic attraction choice model. We further develop structure properties to elaborate the underlying mechanism and the economics insight of the equilibrium. In particular, the equilibrium market shares depend on the linear combination of the performance and cost of each product under the MNL and linear attraction models. Whereas the equilibrium market shares depend on the ratio of the performance and cost under the MCI model.

We then turn to two generalized attraction choice models: the nested attraction model and random coefficient attraction model. These two models allow us to drill down complicated dependence relationships among the products.
For the nested attraction model, we discuss the conditions under which a closed-form solution of the equilibrium can be developed. In order to illustrate how the dependence structures among the products affect the equilibrium, we compare the equilibrium under nested models with the equilibrium under basic models. Specifically, for any retailer, with fixed performances and costs of the selling products, the higher the independency among products is, the larger the equilibrium market share is. For the random coefficient attraction model, we develop the existence condition for a Nash equilibrium: if the reciprocal of the market share function is convex in product's price, then a Nash equilibrium exists. After that, for both the monopoly and duopoly scenarios, we further carry out structural analysis to demonstrate the effect of the random coefficients on the equilibrium price and profit. Specifically, when facing uncertain coefficients, we suggest retailers who sell "high-end product" cut off the price, and retailers who sell "low-end product" raise up the price in order to achieve the maximum profit. Finally, we conduct an empirical study on China TV market data to calibrate our theoretical results.

This thesis contributes to literature in the following aspects. First, to the best of our knowledge, this is the first study focusing on the closed-form equilibrium solutions of the price competition problem under various attraction choice models. In particular, similar to the Lambert W function, we define specified function forms that represent the equilibrium solutions as well as a bridge to the structure properties. Second, we generalize the results in existing literature. Specifically, we extend the equilibrium conditions for nested logit model to nested attraction choice model, and provide an alternative proof as well as tools for a general structure analysis. Third, this is the first study that theoretically investigates the price competition game for attraction models with random coefficients. For this general model, most existing researches are focused on developing estimation methods or conducting empirical studies. Nevertheless, we focus on the Nash equilibrium of the price competition problem and illustrate the effect of random coefficients on the equilibrium price and profit. Finally, we conduct an empirical study on China TV market data that has not been addressed in other related works before. Our empirical analysis calibrates the theoretical results and also deepens the understanding of China TV market.

    Research areas

  • Attraction choice model, Price competition, Nash equilibrium, Nested structure, Random coefficient, China TV market