A little operations research into TV programming: simulation and remarks

I wrote a little C++ program implementing the TV scheduling algorithm we devised a few entries ago and used it to do several calculations and get some intuition about how the optimum scheduling looks like.

The figure shows in solid beige an audience curve reaching its maximum around the central slot. The competition has placed its resources (amounting to a total C = 165) according to a sinusoidal function with maxima at slots 3-4 and minima at 8-9 (gray line). We have depicted the resulting OTVS distribution for M = 20, 120 and 220. When our resources are much less than the competition's, the optimum distribution concentrates on the slots of higher profitability from our point or view, i.e. those with higher a_i/c_i. As our resources grow the distribution progressively approaches the limit shape √(a_ic_i). This suggests the following rule of thumb for allocating resources: when you're a weak player, concentrate your resources on the competition's valleys.

What about the competition's optimum behavior, i.e. how to best distribute resources a priori without information on the opponent scheduling? Assuming the solution to OTVS is such that m_i ≥ 0 for all i, we have

S({m_i}) = A − (∑√(a_ic_i))²/(M+C),

where A = ∑a_i. It is then easy to see, using Cauchy's Inequality, that the best policy for the competition is to just follow the audience curve, i.e. set c_i = (C/A)a_i. In this case, the resulting scenario is disappointingly simple: our corresponding counter-scheduling is simply m_i = (M/A)a_i and we obtain our fair share of the audience, S({m_i}) = M/(M+C). It is only when the competition deviates from its natural distribution that we can get more than our money can buy, so to speak.