blue-bus-red-bus

Metadata
- Author: Mayberry 1973
- Tags: #statistics
A famous example depicting the shortcomings of N-way structure for alternatives that are correlated
Consider a city where 50% of travelers choose car (C) and 50% choose bus (B)
- Assume: $C_C = C_B$
- In a stroke of genius, the manager paint half the buses red (RB) and half the buses blue (BB), while maintaining the same level of performance: $C_{RB} = C_{BB} = C_C$
- Thus, the probability of the car using logit formulation is now: $P_C = \frac{\exp(-\beta C_C)}{\exp(-\beta C_C) + \exp(-\beta C_{RB}) + \exp(-\beta C_{BB})}$
- Which given the above assumptions, reduces the probability of a user choosing a car to decrease of 50% to 33%
Applying the nested structure to this problem gives a more reasonable result: $P_C = \frac{1}{1+exp(-\lambda_1 (C_B - C_C))}$ $P_{C} = \frac{1}{1 + e x p ( - λ _{1} ( C _{B} - C _{C} ))}$ where $P_B = 1-P_C$ $P_{B} = 1 - P_{C}$ ; $P_{RB} = \frac{1}{1+\exp(-\lambda_2(C_{BB}-C_{RB})}$ $P_{R B} = \frac{1}{1 + e x p ( - λ _{2} ( C _{B B} - C _{R B} )}$ ; $P_{BB} = 1 - P_{RB}$ $P_{B B} = 1 - P_{R B}$ ; $C_B = \frac{-1}{\lambda_2}\log(\exp(-\lambda_2 C_{RB}) + \exp(-\lambda_2 C_{BB}))$ $C_{B} = \frac{- 1}{λ _{2}} lo g (exp (- λ_{2} C_{R B}) + exp (- λ_{2} C_{B B}))$ ; And $\lambda_1$ $λ_{1}$ and $\lambda_2$ $λ_{2}$ are the primary and secondary split parameters
- If $C_B = C_C$ , this model will correctly assign 50% to the car and 25% to each bus options