## ISIT 2016 Paper

In July, the International Symposium on Information Theory will be held in Barcelona, Spain. One of the papers that will be appearing there is some recent work I have done with Mauro de Freitas, Laurent Clavier, Alban Goupil, Gareth Peters, and Nourddine Azzaoui. We have been considering variations on the additive $\alpha$-stable noise channel $Y = X + N$, where $N$ is an $\alpha$-stable random vector.

These kinds of channels appear in various communication systems including wireless and quite recently molecular. As such, it is interesting to try and compute the capacity of these channels. The special case $\alpha = 2$ with a power constraint has been extensively studied (it is the Gaussian case!), but in general the capacity is not well understood for other values of $\alpha$ with any constraints.

Enter our paper. We considered the case where the noise $N$ is an isotropic $\alpha$-stable random vector. So, the channel is the additive isotropic $\alpha$-stable noise ($AI\alpha SN$) channel. Our results? A quick summary:

1. The optimal input for the $AI\alpha SN$ channel subject to a constraint $\mathbb{E}[|\mathbf{X}|^r] = (\mathbb{E}[|X_1|^r],\mathbb{E}[|X_2|^r])^T \preceq \mathbf{c},~r < \alpha$ exists and is unique.
2. The capacity subject to $\mathbb{E}[|\mathbf{X}|^r] \preceq \mathbf{c},~r < \alpha$, is lower bounded by a function of the form: $\frac{1}{\alpha}\log(1 + Kc_{\min}^{\alpha})$. ($c_{\min}$ is just the minimum of the elements of $\mathbf{c}$ and $K$ is a constant that depends on the noise parameters)

We also had a brief look at the extension to parallel channels, but for that you will need to read the paper.

For the official summary…
Title: Achievable rates for additive isotropic alpha-stable noise channels
Abstract: Impulsive noise arises in many communication systems—ranging from wireless to molecular—and is often modeled via the $\alpha$-stable distribution. In this paper, we investigate properties of the capacity of complex isotropic $\alpha$-stable noise channels, which can arise in the context of wireless cellular communications and are not well understood at present. In particular, we derive a tractable lower bound, as well as prove existence and uniqueness of the optimal input distribution. We then apply our lower bound to study the case of parallel $\alpha$-stable noise channels and derive a bound that provides insight into the effect of the tail index $\alpha$ on the achievable rate.

## Mechanism design for on-demand transport

During 2014 – 2015, one of the key research problems I was working on was how to understand and design on-demand transport mechanisms. On-demand transport is about how an individual or group can get from point A to point B at a time of their choosing. Common examples are taxi services and now new services such as Uber.

Working in a computer science department, primarily with collaborators Michal Jakob and Nir Oren, I was concerned with how passenger journeys are allocated to drivers and how each journey was priced, together called a mechanism, from an algorithmic perspective. I.e., how can these allocations and pricing be done in a computationally efficient way so that passengers get to where they want to go on time, each driver can earn a living, and service providers (e.g., Uber) can make a profit.

The problem of choosing a mechanism is known in economics as the mechanism selection problem, and must account for a range of technical, social and financial issues. For example, can the available computing resources compute the allocation and pricing quickly enough? Or, are passengers or drivers prepared to bid for a journey (a key problem for auction-based approaches)?

We have explored the problem of mechanism selection in on-demand transport by first enumerating the possible mechanisms and evaluating their performance. We observed in this working paper that each approach (e.g., hackney carriages, taxi dispatcher models, and Uber-type approaches) can be differentiated by limitations on communication and financial exchanges.

After enumerating the possibilities, we have begun to explore how the different mechanisms perform in terms of metrics such as the proportion of passengers served, costs of journeys and provider profit. In particular, we have published work in:

(1) Malcolm Egan, Martin Schaefer, Michal Jakob, and Nir Oren, “A double auction mechanism for on-demand transport networks”, in the Proc. PRIMA 2015: Principles and Practice of Multi-Agent Systems,  (2015).

(2) Malcolm Egan, and Michal Jakob, “A profit-aware negotiation mechanism for on-demand transport services”, in the Proc. of the European Conference on Artificial Intelligence (ECAI), (2014)

and now

(3) Malcolm Egan and Michal Jakob, “Market mechanism design for profitable on-demand transport services”, accepted for Transportation Research Part B: Methodological.

A key feature of our article (3) is that we provide and justify an agent-based model for on-demand transport services that captures the preferences of passengers. This means that we do not assume that every passenger will accept whatever they are offered, which is commonly assumed in previous work on on-demand transport mechanisms.

The next step is to continue to study the mechanism selection problem by understanding the requirements and performance of other on-demand transport mechanisms. At the end of the day, we hope that this work will aid new providers and municipalities to decide the kinds of mechanisms they want to support to match the unique economic, social and technical features of their cities. We believe this will provide a means to meet the needs of passengers, drivers, and providers in each city in a sustainable way.

To read more, see the next posts in this series: