$(y_{min}, y_{max}]$ to $( \frac{y_{min} + y_{max}}{2}, y_{max}]$ (the optimal strategy if there are only two contending users).

OSA is an effective contention resolution strategy with the average number of minislots required to resolve contention known to be less than 2.5070 slots, independent of the number of users and channel gain distribution.

Two User Case
-------------

In this section, we will discuss in detail the opportunistic splitting algorithm for the two user case. The example will help us motivate the source coding framework described in the Section \[sec:source\_coding\]. Let $N = 2$ and let $(X_1, X_2)$ correspond to the vector channel gain of the two users in a slot. Define $Y_1 := \min\left(X_1, X_2\right)$ and $Y_2 := \max \left( X_1, X_2 \right)$. Then, $(Y_1, Y_2)$ is the ordered pair of the channel gain values where $0 \leq Y_1 \leq Y_2 \leq 1$.

OSA initializes with $y_{low} = 0, y_{min} = \frac{1}{2}$ and $y_{max} = 1$. In the first minislot, only the user(s) with $\frac{1}{2} < X_i$ transmit a control packet. A success (a single transmission) happens in the first minislot iff ($X_1 \leq \frac{1}{2} < X_2$) or ($X_2 \leq \frac{1}{2} < X_1$), i.e., a success happens iff $Y_1 \leq \frac{1}{2} 