performance of the contention resolution algorithm as a function of $N$ very well.

Entropy Minimization
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Entropy is a concave function of the distribution. The maximal probability allocation code identifies a local minima in the space of probability distributions. From limited numerical work (not reported in this paper), we conjecture that the maximal probability allocation code is a globally entropy minimizing strategy as well. The following theorem proves the optimality of MPA for the $N = 2$ case.

MPA is a delay minimizing strategy and an entropy minimizing strategy for $N = 2$ case.

Let $(x, 1]$ be the contention range in the first minislot for the delay minimizing strategy. Conditioning on the first minislot, the optimal average delay $D$ can be written as $$D = 2 x (1 - x) + (1 + D) (1 - 2 x (1 - x))$$ The optimal solution of $D$ is $2$ and is obtained at $x = \frac{1}{2}$. We note from Figure \[fig:osa\_vs\_mpa\] that the average delay of MPA is $2$ and hence, MPA is a delay optimal strategy.

Let $\{ p_i \}$ be the discrete distribution that minimizes the entropy of the code for $N = 2$ case. We will define 