Title: Weighted Linear Discrete Choice
Domain: economics

Question: Classical discrete choice models, such as the multinomial logit (MNL), struggle to explain certain empirical phenomena like the "red bus, blue bus" paradox and the persistence of market share for superior products against numerous inferior alternatives. Your task is to design a tractable probabilistic choice model that generalizes the Luce model by incorporating both an item's intrinsic utility and its "salience." The model must: | Flexibly capture substitution patterns that overcome the Independence of Irrelevant Alternatives (IIA) property of the MNL. | Possess a microfoundation rooted in the principle of costly attention. | Furthermore, demonstrate this model's application in an oligopolistic market where firms compete on both price (influencing utility) and advertising (influencing salience), and derive the symmetric equilibrium outcomes for pricing and advertising levels.

Background: The inquiry originates from the limitations of classical discrete choice theory, a cornerstone of microeconomics used to model how individuals choose from a set of alternatives. The workhorse model in this field is the Multinomial Logit (MNL), derived from Luce's choice axiom. While widely used due to its simplicity and tractability, the MNL suffers from a major drawback: the Independence of Irrelevant Alternatives (IIA) property. This property implies that the relative probability of choosing between two options is unaffected by the presence or absence of other alternatives. This leads to empirically unrealistic predictions, famously illustrated by the "red bus, blue bus" paradox. To overcome this, economists have explored more complex models (e.g., nested logit, mixed logit), but these often sacrifice tractability. A more recent approach seeks to build more realistic models from the ground up by incorporating psychological principles, such as costly or limited attention. The idea is that choice is a two-stage process involving not just an evaluation of an item's utility, but also its "salience" or prominence, which determines whether it is even considered in the first place.

Definitions: The choice probability p(x|S) for an item x from a choice set S is composed of two parts: a base probability and a comparative probability transfer. | Base Probability: This component mirrors the Luce model and is defined as the ratio of an item's salience to the total salience in the choice set: m(x) / m(S). | Comparative Probability Transfer: This component adjusts the base probability based on the difference between an item's utility u(x) and the salience-weighted average utility of the set, ū_m(S). This transfer is scaled by the item's own salience: m(x) * [u(x) - ū_m(S)]. | Salience-Weighted Average Utility: ū_m(S) is defined as Σ_{y∈S} u(y)m(y) / Σ_{y∈S} m(y).

Methodology: To derive the microfoundation, formulate an optimization problem for a representative agent who chooses choice probabilities p(x) to maximize their expected utility minus a quadratic cost of attention. The objective function is max [ Σ u(x)p(x) - Σ (p(x)² / (2m(x))) ]. The salience m(x) modulates the cost of attention. | For the oligopoly application, define a firm's profit function: Profit_i = (price_i - marginal_cost) * demand_i - advertising_cost_i. Assume demand is given by the designed choice model, where utility is a function of price (u_i = ũ_i - p_i) and salience m_i is determined by advertising. | To find the symmetric Nash equilibrium, take the first-order conditions of the profit function with respect to both price p_i and salience m_i, and solve the resulting system of equations under the symmetry assumption (p_i = p, m_i = m).

Answer: 1. Conceptual Foundation and Motivation for a New Model | Classical discrete choice models, such as the Multinomial Logit (MNL) model derived from Luce's choice axiom, are foundational but suffer from the property of Independence of Irrelevant Alternatives (IIA). IIA implies that the ratio of probabilities of choosing two alternatives is independent of any other alternatives in the choice set. This leads to the "red bus, blue bus" paradox: if a consumer is indifferent between a car and a blue bus (50/50 choice), introducing a nearly identical red bus should intuitively take market share almost exclusively from the blue bus. However, the MNL model predicts the car would retain a 33.3% market share, and each bus would get 33.3%, which is unrealistic. Furthermore, these models predict that a superior product's market share will diminish to zero as an infinite number of inferior products are introduced. | To address these shortcomings, we design the Weighted Linear (WL) model of discrete choice, which is both tractable and more empirically realistic. It characterizes choice probability by two parameters for each alternative x: its intrinsic utility u(x) and its "salience" m(x), which captures prominence or accessibility in a consumer's mind. | 2. The Weighted Linear (WL) Model Formulation | The probability of choosing an alternative x from a choice set S is given by: | p(x|S) = m(x)/m(S) + m(x) * [u(x) - ū_m(S)] | Where: | u(x): A scalar representing the intrinsic utility of alternative x. | m(x): A positive scalar representing the salience of x. Higher m(x) means the alternative is more prominent. | m(S) = Σ_{y∈S} m(y): The total salience of all alternatives in the set. | ū_m(S) = (Σ_{y∈S} u(y)m(y)) / m(S): The salience-weighted average utility in the set S. | This structure elegantly combines two mechanisms: a base probability proportional to salience (a Luce-like term), and a comparative probability transfer reflecting the difference between an item's utility and the weighted-average utility of its competitors, amplified by the item's own salience. | 3. Microfoundation from Costly Attention | The WL model can be derived as the solution to a problem where a representative agent chooses probabilities p(x|S) to maximize expected utility, penalized by the cognitive cost of paying attention. The cost is assumed to be quadratic in probability and inversely related to salience. | Objective Function: max_{p(·|S)} [ Σ_{x∈S} u(x)p(x|S) - Σ_{x∈S} (1 / (2m(x))) * p(x|S)² ] subject to the constraint Σ_{x∈S} p(x|S) = 1. | The Lagrangian is L = Σ u(x)p(x) - Σ (p(x)² / (2m(x))) - λ(Σ p(x) - 1). | The first-order condition with respect to p(x) is: ∂L/∂p(x) = u(x) - p(x)/m(x) - λ = 0. This gives p(x|S) = m(x)[u(x) - λ(S)]. | Summing over all x in S: Σ p(x|S) = 1 = Σ m(x)[u(x) - λ(S)] = Σ m(x)u(x) - λ(S)Σ m(x). | This can be solved for the Lagrange multiplier λ(S): λ(S) = ū_m(S) - 1/m(S). | Substituting λ(S) back into the expression for p(x|S) yields the WL model formula: | p(x|S) = m(x)[u(x) - (ū_m(S) - 1/m(S))] = m(x)/m(S) + m(x)[u(x) - ū_m(S)]. | 4. Resolution of Paradoxes and Model Properties | Overcoming IIA: The cross-price elasticity (change in p(x) due to change in u(y)) is ∂p(x|S)/∂u(y) = -m(x)m(y)/m(S). This depends on the specific properties (m) of alternatives x and y, thus violating IIA. In the "red bus, blue bus" example, the two buses would have similar, high salience, leading to stronger substitution between them than with the less similar car. | Market Share Persistence: Consider a superior product x and a set T_n of n replicas of an inferior product y (with u(x) > u(y)). As n → ∞, the total salience m(S U T_n) also approaches infinity, causing the base probability term to go to zero. However, the weighted average utility ū_m(S U T_n) converges to u(y). Therefore, the limiting market share is: | lim_{n→∞} p(x|S U T_n) = m(x)[u(x) - u(y)] > 0. | The superior product retains a positive market share, determined by its salience and its utility advantage over the sea of inferior competitors. | 5. Application to Oligopolistic Competition | Consider a market with n firms in a symmetric setting where firms have the same exogenous quality ũ, marginal cost k, and advertising cost function C(m_i) = gm_i². Firms compete by setting a price p_i and choosing a salience level m_i. | Utility for consumer: u_i = ũ - p_i. | Firm i's profit: Π_i = (p_i - k) * p(i|N) - gm_i², where N is the set of all n firms. | The demand for firm i is: | p(i|N) = m_i/(Σm_j) + m_i[(ũ-p_i) - (Σm_j(ũ-p_j))/(Σm_j)] | Taking the FOCs of Π_i with respect to p_i and m_i and then imposing symmetry (p_j=p, m_j=m) leads to the following equilibrium results: | Equilibrium Price (Markup): p* - k = 1 / (m*(n-1)) | Equilibrium Salience: m* = (1 / (2g(n-1)²))^(1/3) | This equilibrium has intuitive properties: As the number of firms n increases, the markup p* - k converges to zero. As the cost of advertising g increases, firms choose lower salience levels (m* decreases), and equilibrium markups p* - k increase because firms compete less aggressively on the salience dimension.

Checklist: 1. Correctly identify and explain the limitations of the MNL model, particularly the IIA property and the "red bus, blue bus" paradox.  
2. Accurately formulate the mathematical expression for the Weighted Linear (WL) discrete choice model's probability: p(x|S) = m(x)/m(S) + m(x)[u(x) - ū_m(S)].  
3. Clearly define the distinct roles of the utility parameter u(x) and the salience parameter m(x).  
4. Explain the intuition behind the model's two components: the salience-driven base probability and the utility-comparison-based probability transfer.  
5. Successfully derive the model from its specified microfoundation, i.e., by solving the first-order conditions of the agent's utility-maximization problem with costly attention.  
6. Demonstrate how the model's structure allows for flexible substitution patterns, explaining how it resolves the "red bus, blue bus" paradox.  
7. Explain the mechanism by which the model allows a superior product to retain a non-negligible market share even when a large number of inferior products are introduced.  
8. Correctly set up the profit maximization problem for a firm in the oligopolistic market setting, defining profit as a function of price, salience, and their associated costs.  
9. Derive the symmetric Nash equilibrium strategies for price (p*) and salience (m*).  
10. Analyze the properties of the derived equilibrium, such as how markups (p* - k) and advertising levels behave as the number of competing firms changes.