MY COMMENTARY:
First, we should not call these cards "loyalty cards." Refer to them as frequent shopper cards or continuity of purchase cards, but they have little to do with customer loyalty. The concept of loyalty should be reserved for your family, country, alma mater, etc. In addition, the fact that households average double digit cards suggests little in the way of loyalty.
Imagine your spouse or significant other's surprise at discovering pictures of individuals other than your friends and family in your wallet. It would hardly conjure up an image of loyalty.
Second, if you want to insure continuity of purchase and minimize store or brand switching, I suggest that you concentrate on customer needs that would maintain purchases at a single store. The frozen turkey at Thanksgiving is hardly unique or even desired. For example, what problems might a "loyalty card" ameliorate for a food shoppers? We know that speed of check-out is an important attribute for food shopping. Why not have a special line, like Disney's Fast Pass, in which an individual would be guaranteed a reasonable time and dedicated lane for check-out? Even the airlines, with all of their customer service inadequacies, recognize the value of dedicated frequent flyer telephone numbers and check-in lines. In almost every other business you can make a reservation or appointment for something, e.g., round of golf, dinner, doctor, etc. Why couldn't a frequent shopper be recognized as such and be given something that the shopper really values, namely, time savings?
Third, the real purpose of "frequent shopper" programs is to create a barrier against "brand-switching." In other words, the program should be designed such that a customer has so much invested with the card provider that thoughts of switching are discouraged. This is the acid test. Does your program, whatever you may label it, have such a hold on the customer?