[Paper Review] Pricing under a multinomial logit model with non linear network effects

Felipe Maldonado, Gerardo Berbeglia+1 more|arXiv (Cornell University)|May 7, 2020

Consumer Market Behavior and Pricing15 references4 citations

TL;DR

This paper proposes a multinomial logit model with non-linear network effects to study pricing strategies in digital markets, where consumer choices are influenced by past transactions and reputation scores. It proves that network effects increase expected revenue and market share, showing that competitive pricing leads to a unique pure Nash equilibrium while collaborative pricing maximizes total revenue, with consumer utility rising as network strength increases.

ABSTRACT

We study the problem of pricing under a Multinomial Logit model where we incorporate network effects over the consumer's decisions. We analyse both cases, when sellers compete or collaborate. In particular, we pay special attention to the overall expected revenue and how the behaviour of the no purchase option is affected under variations of a network effect parameter. Where for example we prove that the market share for the no purchase option, is decreasing in terms of the value of the network effect, meaning that stronger communication among costumers increases the expected amount of sales. We also analyse how the customer's utility is altered when network effects are incorporated into the market, comparing the cases where both competitive and monopolistic prices are displayed. We use tools from stochastic approximation algorithms to prove that the probability of purchasing the available products converges to a unique stationary distribution. We model that the sellers can use this stationary distribution to establish their strategies. Finding that under those settings, a pure Nash Equilibrium represents the pricing strategies in the case of competition, and an optimal (that maximises the total revenue) fixed price characterise the case of collaboration.

Motivation & Objective

• To model consumer choice in digital markets where network effects—based on past transaction history and reputation—affect purchasing decisions.
• To analyze how network effects influence seller pricing strategies under both competitive and collaborative market structures.
• To study the impact of network effects on market share, expected revenue, and consumer utility.
• To establish theoretical convergence of market shares to a stationary distribution using stochastic approximation.
• To compare monopolistic and competitive pricing outcomes, including their effects on consumer welfare.

Proposed method

• Extends the multinomial logit (MNL) model by incorporating non-linear network effects through a time-varying score function derived from past consumption.
• Models consumer utility as a function of intrinsic product utility, price, and a network effect parameter $ r \in (0,1) $, where $ r $ scales the influence of past transaction history.
• Applies stochastic approximation techniques to prove almost sure convergence of purchase probabilities to a unique stationary distribution.
• Derives analytical expressions for equilibrium market shares under both competitive and monopolistic pricing, showing dependence on $ \frac{g_i - \beta p_i}{1 - r} $.
• Uses Gumbel-distributed error terms to model consumer choice randomness, enabling closed-form expressions for expected utilities.
• Employs numerical experiments with synthetic data to validate theoretical results and illustrate the impact of $ r $ on pricing and utility.

Experimental results

Research questions

• RQ1 How do non-linear network effects alter the market share distribution in a multinomial logit model with multiple sellers?
• RQ2 What is the impact of the network effect parameter $ r $ on expected revenue under monopolistic and competitive pricing?
• RQ3 Does a pure Nash equilibrium exist in the competitive pricing game, and how does it compare to the monopolistic optimal pricing?
• RQ4 How does the inclusion of network effects on the no-purchase option affect consumer behavior and overall market efficiency?
• RQ5 To what extent does increasing the network effect parameter $ r $ improve consumer utility and market outcomes?

Key findings

• The market share of the no-purchase option decreases monotonically as the network effect parameter $ r $ increases, indicating that stronger network effects reduce consumer inaction.
• Expected total revenue increases with $ r $, provided $ r $ is sufficiently large, showing that network effects can boost seller profitability.
• Under competition, a unique pure Nash Equilibrium exists for pricing strategies, and it is strictly lower than the monopolistic price for all products.
• Consumer expected utility increases with $ r $, as demonstrated numerically across multiple values of $ r \in \{0.2, 0.4, 0.6, 0.8\} $, with utility rising from 2.831 to 12.974 under competitive pricing.
• The product with the highest intrinsic utility ($ g_1 = 0.993 $) consistently has the highest market share across all $ r $ values, and its expected utility grows with $ r $.
• The model shows that the competitive equilibrium yields higher consumer utility than the monopolistic price, even though the latter maximizes total revenue.