Discrete Choice Model

“If a customer can buy one product from either Tesco, Amazon or Argos, then what is the

probability that they choose Tesco?”

This is easy to generalise to “dependent” products

o Set up a “Discrete Choice” model.o Parameterise model.o Solve all of Tesco’s (stated) problems.

Discrete Choice Model

Market share of Tesco

Market share of Amazon

As a function of price…

Discrete Choice ModelWith N vendors, the market share for vendor i is:

Vectors of parameters

Vector of prices

Alternatively, we can use utility functions based on logistic distributions in a “standard” Discrete Choice Model framework

Discrete Choice Model

Q1. How do we estimate the parameters?

Q2. How do we use parameterised model to maximise profit?

Discrete Choice ModelQ1. How do we estimate the parameters?

Maximum LikelihoodSketch idea:

Discrete Choice ModelQ2. How do we use parameterised model to maximise profit?

Expected profit of vendor i per unit product sold in whole market

Equilibrium: each vendor is self-optimising

Discrete Choice ModelThis system can be solved analytically (or numerically)

Discrete Choice ModelOne quick concrete example to finish:

Three vendors (“red”, “blue” and “green”) all have unit cost £50. Suppose c1 = c2 = c3 = 1/3, but α1 = 1, α2 = 2 and α3 = 3. The prices are initially set to p1 = 100, p2 = 120 and p3 = 150. What happens if all vendors optimise profit?

Discrete Choice ModelEx

pect

ed u

nit p

rofit

(£)

Price (£)

SOLUTION: Prices will converge to the Nash equilibrium defined by p1 = 181.7, p2 = 124.9 and p3 = 108.2.

Discrete Choice ModelOne quick concrete example to finish:Three vendors (“red”, “blue” and “green”) all have unit cost £50. Suppose c1 = c2 = c3 = 1/3, but α1 = 1, α2 = 2 and α3 = 3. The prices are initially set to p1 = 100, p2 = 120 and p3 = 150. What happens if all vendors optimise profit?

The Discrete Choice Model gives rise to a simple method of retrospective evaluation

Forecasting

Test Price Optimisation

Evaluation

Optimisation

Objective Function