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I am estimating sales with data on customer and store locations and attributes using a Huff-style model, where sales decay with drive time and increase with attractiveness of the store.

One hypothetical instance of the problem is illustrated below. The 3 pushpins represent stores and the white flag represents the site where the customers live.** The size of the pins represents the attractiveness of the store (like floor space). The problem I am having is that I would expect red store's sales at the site to be lower than the the green store's even though they are just as far away and have identical attractiveness, because the purple store is somewhat in between the the red store and the site. I would like to translate this intuition into rigorous (but tractable) math so that I add it into my statistical model. I am also having trouble figuring out what this is called in the literature (other than the n-body gravity problem in physics).

Example

** For the daltonic folks, the red store is in the upper left corner. The green store is in the bottom right. Purple store is to the left of the site.

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  • $\begingroup$ If red and green have a wider variety of items than purple (rather than just more inventory of the same items) wouldn't this be as expected? $\endgroup$ – Michelle Mar 10 '12 at 2:37
  • $\begingroup$ I am not sure how that explains why sales at the red store are less than green. Can you elaborate? $\endgroup$ – Dimitriy V. Masterov Mar 10 '12 at 4:37
  • $\begingroup$ I'm suggesting the opposite, that the sales could be the same (or close enough) at red and green. Green should also lose sales to purple, assuming it is distance-to-store that is the deciding factor. $\endgroup$ – Michelle Mar 10 '12 at 5:29
  • $\begingroup$ If I plot the actual data, the sales do seem to behave the way I am describing. My predictions for the red store tend to be too high, so think this is a real concern. I completely agree with you on the effect of the purple store. $\endgroup$ – Dimitriy V. Masterov Mar 10 '12 at 17:46
  • $\begingroup$ Consumer psychology can play an important part in this type of analysis. Back to spatial factors, is there anything about the roading (e.g. traffic patterns, congestion, roundabouts vs. uncontrolled intersections vs. traffic lights) that could help explain? $\endgroup$ – Michelle Mar 10 '12 at 18:30
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One of the assumptions of the Huff model (which we call multinomial logit in economics) is Independence of Irrelevant Alternatives. IIA says that the ratio of red store to green store sales is independent of the existence and characteristics of all other alternatives --- it only depends on red and green store characteristics. Your intuition is that this assumption should be violated in this application.

What you want is one of the alternatives to multinomial logit which relaxes the IIA assumption. There are a number of these, including multinomial probit, nested logit, and logit models using the generalized extreme value distribution, sometimes called generalized logit. There are large literatures in this in both Industrial Organization and in Marketing.

Though these models are all defined at the individual level, they can be estimated with data at the market level (like total sales at each store in your example). There is a nice free online book by Kenneth Train. Actually, there are two.

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You are right, the original model seems to calculate individual attractiveness and probabilities but does not take into account the interactions. Let's say you calculate P(i,j) Using the definition of huff-model from - http://www.directionsmag.com/articles/retail-trade-area-analysis-using-the-huff-model/123411 then to translate your intuition - P'(i,j) = P(i,j) * [k / [sum(A(i,j')]] where k is some constant. sum(A(i,j')) is the attractiveness (numerator of earlier formula) of all stores except this one. This will introduce dampening and penalize a store for being close to other desirable stores.

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  • $\begingroup$ This is interesting. Can I ask where k comes from? Also, unless I am missing something, the position of the purple store has no effect on the value of $P'_{i,red}$. That seems counter-intuitive. $\endgroup$ – Dimitriy V. Masterov Mar 12 '12 at 15:36
  • $\begingroup$ A(i,purple) will be high, so P'(i,red) will be lower than if purple was further away. $\endgroup$ – Saurabh Mar 12 '12 at 15:44

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