$E(\{ p_i \}) := - \sum_i p_i \log_2(pi)$ as the minimum entropy. Then, conditioning on the first minislot (as before), we have, $$E(\{ p_i \}) = -2 x (1-x) \log(2 x (1-x)) + E(\{ x^2 p_i\}) + E(\{ (1-x)^2 p_i \})$$ where, $$E(\{a p_i\}) = - \sum_i a p_i \log_2(a p_i)$$ Substituting in the above expression and simplifying it, we have, [$$E(\{ p_i \}) = \frac{ - 2 x (1 - x) \log_2(2 x (1 - x)) - x^2 \log(x^2) - (1-x)^2 \log((1-x)^2)}{2 x (1 - x)}$$ ]{} The above expression is minimized at $x = \frac{1}{2}$ and the minimum entropy is $3$. From Figure \[fig:osa\_vs\_mpa\], we note that the entropy of the threshold random variable for MPA is $3$ and hence, MPA is an entropy minimizing strategy as well.

Two Examples {#sec:two_examples}
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In this section, we discuss contention resolution for two different channel scenarios, a constant channel and a correlated channel. We compare the performance of OSA/MPA with the source-coding framework and illustrate the generality of our proposed technique.

Constant Channel
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We consider a downlink wireless channel with $3$ users. We assume that the channel gain is a constant, say $1$ unit, for all users and for 