Say you make k products and have a list of potential customers. Assume that you can show each of these potential customers one ad. And that showing them an ad costs nothing. Assume also that you get profits from each of the k products. What is the optimal targeting strategy?

The goal is to maximize profit. If you didn’t know the probability each customer will buy a product if shown an ad or assume it to be the same across products, then the strategy reduces to showing the ad for the most profitable product to everyone.

If we can estimate the probability the customer will buy each product if they are shown an ad for the product well, we can do better. (We assume that customers won’t buy the product if they don’t see the ad for it.)

When k = 1, the optimal strategy is to target everyone. Now let’s solve it for when k = 2. A customer can either buy \(k_0\) or k_1 or buy nothing at all.

If everyone is shown k_0, the profits are p_0*prob_i_0_true where prob_i_0_true gives the true probability of the ith person buying k_0 when shown an ad for it. And if everyone is shown k_1 ad, the profits are p_1*prob_i_1_true. The net utility calc. for each customer = p_1*prob_i_1 – p_0*prob_i_0. (We generally won’t know prob_i but would estimate it from data and hence the subscript true is not there.) You can recover two numbers from this. One is how to target—pick whichever number is bigger for each customer. Another is in what order to target in: sort by absolute value.

**Calculating the Benefits of Targeting**

If you don’t target, the best case scenario is that you earn the bigger of the two numbers: p_1*prob_i_1_true, p_0*prob_i_0_true

If you target well (estimate probabilities well), you will recover something that is as good or better than that.

Since we generally won’t have the true probabilities, the best way to estimate the benefit of targeting is via A/B testing. But if you can’t do that, one estimate =

No Targeting estimate = Larger of p_1*prob_i_1, p_0*prob_i_0

Targeting estimate = Sum over all i

p_1*prob_i_1 if p_1*prob_i_1 > p_0*prob_i_0

p_0*prob_i_0 if p_1*prob_i_1 < p_0*prob_i_0

Similar calculations can be done for deriving estimates of the value of better targeting.