Optimal Protocols for 2-Party Contention Resolution
Dingyu Wang
TL;DR
The paper addresses the symmetry-breaking problem of contention resolution on a shared channel under acknowledgement-based feedback, focusing on the simplest nontrivial case of two devices. It formalizes a rigorous model and proves the existence of optimal policies for reasonable cost metrics, then derives exact optimal protocols for the avg, min, and max latency objectives. The key contributions are explicit probability sequences and closed-form performance costs: $\mathbb{E}[X_1]=\sqrt{3/2}+3/2\approx2.72474$ for avg, $\mathbb{E}[\min(X_1,X_2)]=2$, and $\mathbb{E}[\max(X_1,X_2)]=1/\gamma\approx3.33641$ with $\gamma\approx0.299723$. These results provide the first precise optimal strategies for two-party contention under ack-based feedback and establish a foundation for exploring larger $n$ and more complex metrics in future work.
Abstract
\emph{Contention Resolution} is a fundamental symmetry-breaking problem in which $n$ devices must acquire temporary and exclusive access to some \emph{shared resource}, without the assistance of a mediating authority. For example, the $n$ devices may be sensors that each need to transmit a single packet of data over a broadcast channel. In each time step, devices can (probabilistically) choose to acquire the resource or remain idle; if exactly one device attempts to acquire it, it succeeds, and if two or more devices make an attempt, none succeeds. The complexity of the problem depends heavily on what types of \emph{collision detection} are available. In this paper we consider \emph{acknowledgement-based protocols}, in which devices \underline{only} learn whether their own attempt succeeded or failed; they receive no other feedback from the environment whatsoever, i.e., whether other devices attempted to acquire the resource, succeeded, or failed. Nearly all work on the Contention Resolution problem evaluated the performance of algorithms \emph{asymptotically}, as $n\rightarrow \infty$. In this work we focus on the simplest case of $n=2$ devices, but look for \underline{\emph{precisely}} optimal algorithms. We design provably optimal algorithms under three natural cost metrics: minimizing the expected average of the waiting times ({\sc avg}), the expected waiting time until the first device acquires the resource ({\sc min}), and the expected time until the last device acquires the resource ({\sc max}).