Converting a multinomial logit model into a binary logit model

Question

As far as I remember, it is possible to convert a multinomial logit model into a binary logit model using restrictions on parameters.

For example, suppose we have three alternatives, say A, B, and C.

Then, the choice probability in a multinomial choice model is $$\Pr[J=A|W]=\frac{1}{1+\exp(\alpha_b+W'\beta_b)+\exp(\alpha_c+W'\beta_c)}, \\[5pt] \Pr[J=B|W]=\frac{\exp(\alpha_b+W'\beta_b)}{1+\exp(\alpha_b+W'\beta_b)+\exp(\alpha_c+W'\beta_c)}, \\[5pt] \Pr[J=C|W]=\frac{\exp(\alpha_c+W'\beta_c)}{1+\exp(\alpha_b+W'\beta_b)+\exp(\alpha_c+W'\beta_c)}.$$

I think, if we impose the restriction, $\beta_b=\beta_c$, and define a new alternative combining alternatives B and C, this multinomial logit model becomes actually a binary logit model.

However, I cannot remember the details (i.e. how to derive the binary logit model).

Thus, is there someone who remember the detail?

Answer 1

Let $$\alpha = -\log(e^{\alpha_b}+e^{\alpha_c})$$ and let $$\beta = -\beta_b = -\beta_c.$$ Then we have $$\exp(\alpha_b + W'\beta_b) + \exp(\alpha_c+W'\beta_c) = (e^{\alpha_b}+e^{\alpha_c})\exp(-W'\beta) = \exp(-(\alpha+W'\beta))$$ giving $$\text{Pr}[J=A\mid W] = \frac{1}{1+\exp(-(\alpha + W'\beta))}.$$ The other outcome is $$\text{Pr}[J\in\{B,C\}\mid W] = \frac{\exp(-(\alpha+W'\beta))}{1+\exp(-(\alpha + W'\beta))}.$$