Logit

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The Logit Model is the simplest and most widely used discrete choice method for estimating the probability of a design being chosen over a set of alternative designs. This popularity is most likely due to the model's closed mathematical form and simple evaluation.

The utility U_nj of the decision maker n choosing alternative j can be expressed as:

$\quad U_{nj}=V_{nj}+\varepsilon_{nj}$

where U_nj is the obversed part and ε_nj is the unobserved part in the utility.

The Logit Model states that the probability of a product being chosen instead of alternative options $P j$ is equal to the exponential e raised to the observable portion of the utility of the product $v j$ divided by the sum of the exponential e raised to the observable portion of utility for all of the products in the market, including the product whose probability is being estimated $v k$ .

This is shown below in mathematical form, where the subscripts i and j indicate the alternatives (product design) and k represents all alternatives in the model. According to logit choice probability defined by McFadden, the probability that decison maker n choose alternative i over alternative j can be written as:

$P_{nj} = Prob (V_{ni}+\varepsilon_{ni} > V_{nj}+ \varepsilon_{nj} \forall i \neq j)$

$\quad P_j = \frac{e^{v_j}}{\sum_k e^{v_k}}$

The Logit Model is sometimes summarized with the phrase "us over us plus them," where "us" refers to the design in question and “them” indicates all the other alternative designs in the market.

Assumptions

The Logit Model relies on two assumptions regarding the non-observable factors in the random utility model .

Assumes that the non-observable error effects are independent.
Assumes that each fε follows the extreme value distribution .

$\quad F_{\varepsilon}(\varepsilon)=\exp(-e^{-\varepsilon})$

$\quad f_{\varepsilon}(\varepsilon)=\exp(-\varepsilon) \exp(-e^{-\varepsilon})$

These assumptions are often referred to as "an Independent and identically distributed (IID) Extreme Value distribution of the random unobserved error term."

The downside of the IID assumption is that the model suffers from the independence of irrelevant alternatives (IIA) property.

Example

There are three alternatives, A, B and C, to be chosen by a decision maker without no-choice option. Given the utilities of three alternatives are: v_A=3, v_B=5 and v_C=7. The logit probabillities of three alternatives are:

$P_A = \frac{e^{v_A}}{\sum_{k} e^{v_k}}= \frac{e^3}{e^3+e^5+e^7}=1.59%$