Hy y2 y3 ye yf y3555

withp12 = 3lnH(yuy2,0)/<5 In yx, ^ = 3 In H^, y2, y3)/d In yt, and so on. The parameters of the model are jl5 y2, y3, and $, with the normalization j3 = 1.

To form an MNP model with a similarity structure mimicing figure 5.3, we assume the multivariate normal random utility vector (w,, u2, u3) has u3 independent of u1 and u2. Imposing the trinary condition (5.53) implies a common variance for u1 and u2. Then this model also has three independent parameters.

Table 5.1 compares the multinomial probabilities from these three models for a selection of values of the binary probabilities. For the MNP model both the exact probabilities and approximate values obtained by the Clark method are given. Appendix 5.21 gives computational formulas.

The most striking feature of table 5.1 is the closeness of the multinomial probabilities predicted by HEB A, TEV, and MNP. The absolute deviation of HEBA and TEV for these cases is at most 0.0074, and the maximum relative deviation is 6 percent. The absolute deviation of TEV and MNP is at most 0.016. The relative deviation of TEV and MNP can rise to 22 percent for small probabilities but for probabilities over 0.1 is under 6 percent. We conclude that at least for simple preference trees such as figure 5.3, these models are for all practical purposes indistinguishable. Cases 4,5, and 6 parallel the red bus/blue bus example, with case 4 corresponding to high similarity of the bus alternatives and case 6 to low similarity. All three models generate the intuitively plausible multinomial probabilities for these cases.

The Clark approximation to the MNP probabilities is quite inaccurate in a few cases, with absolute deviations as high as 0.1 and relative deviations

Table 5.1

A comparison of HEBA, TEV, and MNP choice probabilities for a simple preference tree

Case

Nonsimilarity

Pi 2

Pi 3

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