subscription math

Ramit Sethi suggests cancelling all your subscriptions and paying for items à la carte to save money. Without considering whether or not this is a good idea in reality, my first thought was, "can we turn this into a cool math problem?".

Using movies as an example, let's assume that you will rent x_i dollars of movies during month i. Before the month, you can choose to either pay x_i (whatever it turns out to be), or pay a c dollar subscription fee to Netflix. Though this is obviously not true in reality, just for simplicity let's assume that whether or not you've subscribed doesn't affect how many movies you will watch that month. So I won't watch more movies just because I subscribed to Netflix.

Now, there is some optimal strategy that, before each month, psychically knows how many movies you're going to watch, and if their total cost is above c, subscribes for the month, otherwise pays à la carte. Of course, you can't use this strategy, because you aren't psychic.

Two questions, nonexistent readers. First, is there any other strategy that we can make any claims about? For instance, is there a strategy that will never pay more than twice what the optimal strategy pays in the long run? I'm pretty certain there is not, since an evil god could make you watch nothing every time you subscribe and $1,000,000 worth of movies every time you don't.

Second, can we add any additional elements to make the problem more interesting, where some strategy is competitive with the optimal one? What if you have to pay a fee every time you switch from subscription to nonsubscription or vice versa? What if the x_i come from a normal distribution?