Essays on robust mechanism design

Chapter 1: Under the average common value function, we select almost uniquely the mechanism that gives the seller the largest portion of the true value in the worst situation among all the direct mechanisms that are feasible, ex-post implementable and individually rational. Chapter 2: Strategy-proof, budget balanced, anonymous, envy-free linear mechanisms assign p identical objects to n agents. The efficiency loss is the largest ratio of surplus loss to efficient surplus, over all profiles of non-negative valuations. The smallest efficiency loss is uniquely achieved by the following simple allocation rule: assigns one object to each of the p−1 agents with the highest valuation, a large probability to the agent with the pth highest valuation, and the remaining probability to the agent with the (p+1)th highest valuation. When “envy freeness” is replaced by the weaker condition “voluntary participation”, the optimal mechanism differs only when p is much less than n. Chapter 3: One group is to be selected among a set of agents. Agents have preferences over the size of the group if they are selected; and preferences over size as well as the “stand-outside” option are single-peaked. We take a mechanism design approach and search for group selection mechanisms that are efficient, strategy-proof and individually rational. Two classes of such mechanisms are presented. The proposing mechanism allows agents to either maintain or shrink the group size following a fixed priority, and is characterized by group strategy-proofness. The voting mechanism enlarges the group size in each voting round, and achieves at least half of the maximum group size compatible with individual rationality.

Optimal static and sequential design : a critical review

The aim of this thesis is to review and augment the theory and methods of optimal experimental design. In Chapter I the scene is set by considering the possible aims of an experimenter prior to an experiment, the statistical methods one might use to achieve those aims and how experimental design might aid this procedure. It is indicated that, given a criterion for design, a priori optimal design will only be possible in certain instances and, otherwise, some form of sequential procedure would seem to be indicated. In Chapter 2 an exact experimental design problem is formulated mathematically and is compared with its continuous analogue. Motivation is provided for the solution of this continuous problem, and the remainder of the chapter concerns this problem. A necessary and sufficient condition for optimality of a design measure is given. Problems which might arise in testing this condition are discussed, in particular with respect to possible non-differentiability of the criterion function at the design being tested. Several examples are given of optimal designs which may be found analytically and which illustrate the points discussed earlier in the chapter. In Chapter 3 numerical methods of solution of the continuous optimal design problem are reviewed. A new algorithm is presented with illustrations of how it should be used in practice. It is shown that, for reasonably large sample size, continuously optimal designs may be approximated to well by an exact design. In situations where this is not satisfactory algorithms for improvement of this design are reviewed. Chapter 4 consists of a discussion of sequentially designed experiments, with regard to both the philosophies underlying, and the application of the methods of, statistical inference. In Chapter 5 we criticise constructively previous suggestions for fully sequential design procedures. Alternative suggestions are made along with conjectures as to how these might improve performance. Chapter 6 presents a simulation study, the aim of which is to investigate the conjectures of Chapter 5. The results of this study provide empirical support for these conjectures. In Chapter 7 examples are analysed. These suggest aids to sequential experimentation by means of reduction of the dimension of the design space and the possibility of experimenting semi-sequentially. Further examples are considered which stress the importance of the use of prior information in situations of this type. Finally we consider the design of experiments when semi-sequential experimentation is mandatory because of the necessity of taking batches of observations at the same time. In Chapter 8 we look at some of the assumptions which have been made and indicate what may go wrong where these assumptions no longer hold.

Conceptualisation, design and development of a primary school based child healthy weight intervention

INTRODUCTION: In common with much of the developed world, Scotland has a severe and well established problem with overweight and obesity in childhood with recent figures demonstrating that 31% of Scottish children aged 2-15 years old were overweight including obese in 2014. This problem is more pronounced in socioeconomically disadvantaged groups and in older children across all economic groups (Scottish Health Survey, 2014). Children who are overweight or obese are at increased risk of a number of adverse health outcomes in the short term and throughout their life course (Lobstein and Jackson-Leach, 2006). The Scottish Government tasked all Scottish Health Boards with developing and delivering child healthy weight interventions to clinically overweight or obese children in an attempt to address this health problem. It is therefore imperative to deliver high quality, affordable, appropriately targeted interventions which can make a sustained impact on children’s lifestyles, setting them up for life as healthy weight adults. This research aimed to inform the design, readiness for application and Health Board suitability of an effective primary school-based curricular child healthy weight intervention. METHODS: the process involved in conceptualising a child healthy weight intervention, developing the intervention, planning for implementation and subsequent evaluation was guided by the PRECEDE-PROCEED Model (Green and Kreuter, 2005) and the Intervention Mapping protocol (Lloyd et al. 2011). RESULTS: The outputs from each stage of the development process were used to formulate a child healthy weight intervention conceptual model then develop plans for delivery and evaluation. DISCUSSION: The Fit for School conceptual model developed through this process has the potential to theoretically modify energy balance related behaviours associated with unhealthy weight gain in childhood. It also has the potential to be delivered at a Health Board scale within current organisational restrictions.