  In many resource allocation problems, a centralized controller needs to award
some resource to a user selected from a collection of distributed users with
the goal of maximizing the utility the user would receive from the resource.
This can be modeled as the controller computing an extremum of the distributed
users' utilities. The overhead rate necessary to enable the controller to
reproduce the users' local state can be prohibitively high. An approach to
reduce this overhead is interactive communication wherein rate savings are
achieved by tolerating an increase in delay. In this paper, we consider the
design of a simple achievable scheme based on successive refinements of scalar
quantization at each user. The optimal quantization policy is computed via a
dynamic program and we demonstrate that tolerating a small increase in delay
can yield significant rate savings. We then consider two simpler quantization
policies to investigate the scaling properties of the rate-delay trade-offs.
Using a combination of these simpler policies, the performance of the optimal
policy can be closely approximated with lower computational costs.
