Derivation of the Logit Choice Probabilities
The following derivation of the logit choice probabilities is based on McFad- den (1974); Train (2009). From (1.1) and (1.2) we get prij = Prob(ik < ij + vij − vik ∀ k ∈ J, k = j). Hence, with (1.3) and (1.4) and since the stochastic utility components are independent, we have prij = +∞∫ −∞ ⎛ ⎝ ∏ k∈J,k =j e−e −(ij+vij−vik)/σ ⎞ ⎠ 1 σ e−ij/σe−e −ij/σ dij. After substituting ˆij = (1/σ)ij and denoting ˆij by z and vij/σ by vˆij for all i ∈ I and j ∈ J for the sake of brevity, prij becomes1 prij = +∞∫ −∞ ⎛ ⎝ ∏ k∈J,k =j e−e −(z+vˆij−vˆik) ⎞ ⎠ e−ze−e−z dz, and thus 1 Note that, for readability reasons, we use two identical expressions, exp and e, for the exponential function. 178 Appendix A. Derivation of the Logit Choice Probabilities prij = +∞∫ −∞ (∏ k∈J e−e −(z+vˆij−vˆik) ) e−z dz = +∞∫ −∞ exp ( −e−z ∑ k∈J e−(vˆij−vˆik) ) e−z dz. Now deﬁne t = e−z. Then prij = 0∫ +∞ − exp ( −t ∑ k∈J e−(vˆij−vˆik) ) dt = +∞∫ 0 exp ( −t ∑ k∈J e−(vˆij−vˆik) ) dt = exp (−t∑k∈J e−(vˆij−vˆik)) −∑k∈J e−(vˆij−vˆik) ∣∣∣∣∣ +∞ 0 = evij/σ∑ k∈J evik/σ as in (1.5).
You are not authenticated to view the full text of this chapter or article.
This site requires a subscription or purchase to access the full text of books or journals.
Do you have any questions? Contact us.Or login to access all content.