all time slots. The objective of the contention resolution algorithm is to identify a user from the set of $3$ users (akin to a distributed medium access problem). Suppose that the users pick a uniform random variable, $X_i$, in $[0,1]$ independent of the other users. Then, OSA can be used to resolve contention among the $3$ users by identifying the user with the largest value of $X_i$; this is a popular strategy to apply OSA for discrete channel distributions. The average number of slots required to resolve contention using OSA is then $2.12$ slots (obtained from simulations).

The OSA, in every slot, attempts to identify a $y$ such that the probability of a unique user in the interval $(y,1]$ is maximized. Here, in this example, we note that it is more appropriate to identify a $y$ that maximizes the probability of success either in $(y,1]$ or in $[0,y)$. The following algorithm is a contention resolution strategy optimized for this problem.

Initialize: $y_{low} = 0, y_{min} = 1 - \frac{1}{3}, y_{max} = 1$ Initialize: $f = 0, k = 1$ $f = (0,1,e)$ feedback for interval $(y_{min},y_{max}]$ $k = k + 1$ $f = (0,1,e)$ feedback from interval $[y_{low}, y_{min})$ $y_{low} 