Multiple dependent variables Suppose there are 3 advertisers who want to show an ad on a site. If a user clicks on an ad and makes a purchase the host of the site makes a certain percentage of the purchase. Suppose we have historical data on all 3 ads for a set of individuals, and we have variables: 


*

*conversion(0/1: the user clicked on the ad and made a purchase)    

*gender(male/female)

*the amount of time since the last visit, etc. 


The owner of the site wants to know what ad to show to a particular person with a given set of characteristics(variables in historical data). What would be a good model for this? 
It seems that there is no "single" response variable. I was thinking of building a logistic regression where conversion(0/1) is dependent variable, and type of ad(3 values) is independent variable but I am not sure if this is correct. Do you have any suggestions on what a good modeling paradigm would be?
 A: Logistic regression is a natural choice for the specific situation - it is able to cope with both continuous (time since last visit) and categorical data (gender). Alternatively, you can use some more discrete oriented models such as naive Bayesian classifier or Bayesisan networks. Eventually, some approaches based on local regression (both polynomial/linear and logistic).
As you can see, there is a multitude of applicable approaches and the best can be found by use of testing sets.
Whatever approach is used, in the end you obtain the expected reward of clicking for given inputs and type of ad. Formally, we can write $R(X,A)$ where $A$ is the type of ad and $X$ are all other variables. If your model $R(\cdot,\cdot)$ would be perfect, it would make sense to take
$$
A^{*}(X) = \arg\max_{A}R(X,A)
$$
If you have few or not reliable data, it is worth to experiment randomly and obtain more evidence, i.e. to choose the type with the probability $1/3$ regardless to $X$. This is a well known exploration-exploitation dilemma from the reinforcement learning. There is a flexible tool for that called softmax action selection.
A: 1) logistic model will be biased as % of conversion ( purchase) will be less that .1%. you must sample data in a way that has good percentage of conversion too. 
2) U can't target particular user unless you have user data. ( cookie id etc.).
3) Keeping click/conversion as binary dependent variable will surely help as it might give you CTR/CR if you include all data points. 
