On the Gaussian Many-to-One X Channel

In this paper, the Gaussian many-to-one X channel, which is a special case of general multiuser X channel, is studied. In the Gaussian many-to-one X channel, communication links exist between all transmitters and one of the receivers, along with a communication link between each transmitter and its corresponding receiver. As per the X channel assumption, transmission of messages is allowed on all the links of the channel. This communication model is different from the corresponding many-to-one interference channel (IC). Transmission strategies which involve using Gaussian codebooks and treating interference from a subset of transmitters as noise are formulated for the above channel. Sum-rate is used as the criterion of optimality for evaluating the strategies. Initially, a $3 \times 3$ many-to-one X channel is considered and three transmission strategies are analyzed. The first two strategies are shown to achieve sum-rate capacity under certain channel conditions. For the third strategy, a sum-rate outer bound is derived and the gap between the outer bound and the achieved rate is characterized. These results are later extended to the $K \times K$ case. Next, a region in which the many-to-one X channel can be operated as a many-to-one IC without loss of sum-rate is identified. Further, in the above region, it is shown that using Gaussian codebooks and treating interference as noise achieves a rate point that is within $K/2 -1$ bits from the sum-rate capacity. Subsequently, some implications of the above results to the Gaussian many-to-one IC are discussed. Transmission strategies for the many-to-one IC are formulated and channel conditions under which the strategies achieve sum-rate capacity are obtained. A region where the sum-rate capacity can be characterized to within $K/2-1$ bits is also identified.Comment: Submitted to IEEE Transactions on Information Theory; Revised and updated version of the original draf

Almost Budget Balanced Mechanisms with Scalar Bids For Allocation of a Divisible Good

This paper is about allocation of an infinitely divisible good to several rational and strategic agents. The allocation is done by a social planner who has limited information because the agents' valuation functions are taken to be private information known only to the respective agents. We allow only a scalar signal, called a bid, from each agent to the social planner. Yang and Hajek [Jour. on Selected Areas in Comm., 2007] as well as Johari and Tsitsiklis [Jour. of Oper. Res., 2009] proposed a scalar strategy Vickrey-Clarke-Groves (SSVCG) mechanism with efficient Nash equilibria. We consider a setting where the social planner desires minimal budget surplus. Example situations include fair sharing of Internet resources and auctioning of certain public goods where revenue maximization is not a consideration. Under the SSVCG framework, we propose a mechanism that is efficient and comes close to budget balance by returning much of the payments back to the agents in the form of rebates. We identify a design criterion for {\em almost budget balance}, impose feasibility and voluntary participation constraints, simplify the constraints, and arrive at a convex optimization problem to identify the parameters of the rebate functions. The convex optimization problem has a linear objective function and a continuum of linear constraints. We propose a solution method that involves a finite number of constraints, and identify the number of samples sufficient for a good approximation.Comment: Accepted for publication in the European Journal of Operational Research (EJOR

Learning to detect an oddball target with observations from an exponential family

The problem of detecting an odd arm from a set of K arms of a multi-armed bandit, with fixed confidence, is studied in a sequential decision-making scenario. Each arm's signal follows a distribution from a vector exponential family. All arms have the same parameters except the odd arm. The actual parameters of the odd and non-odd arms are unknown to the decision maker. Further, the decision maker incurs a cost for switching from one arm to another. This is a sequential decision making problem where the decision maker gets only a limited view of the true state of nature at each stage, but can control his view by choosing the arm to observe at each stage. Of interest are policies that satisfy a given constraint on the probability of false detection. An information-theoretic lower bound on the total cost (expected time for a reliable decision plus total switching cost) is first identified, and a variation on a sequential policy based on the generalised likelihood ratio statistic is then studied. Thanks to the vector exponential family assumption, the signal processing in this policy at each stage turns out to be very simple, in that the associated conjugate prior enables easy updates of the posterior distribution of the model parameters. The policy, with a suitable threshold, is shown to satisfy the given constraint on the probability of false detection. Further, the proposed policy is asymptotically optimal in terms of the total cost among all policies that satisfy the constraint on the probability of false detection