blue-bus-red-bus

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  • 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: $P_C = \frac{1}{1+exp(-\lambda_1 (C_B - C_C))}$ where $P_B = 1-P_C$; $P_{RB} = \frac{1}{1+\exp(-\lambda_2(C_{BB}-C_{RB})}$; $P_{BB} = 1 - P_{RB}$; $C_B = \frac{-1}{\lambda_2}\log(\exp(-\lambda_2 C_{RB}) + \exp(-\lambda_2 C_{BB}))$; And $\lambda_1$ and $\lambda_2$ 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