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The thesis comprises of two essays. Although the two essays deal with somewhat different situations and use different approaches, both of them essentially examine the problem of making decisions that affect some group of individuals. The first essay is on moral hazard and looks at the principal's problem in a principal-agent(s) free-rider problem in which, unlike most existing work, the principal is not precluded from participating in the production process. Furthermore, there are no uncertainties, but moral hazard is caused by joint production which renders the action of each individual in the production process unobservable. A multi-stage extensive game in which only the principal can propose the output sharing rule determines both the set of individuals who actually participate in the joint production process and the output sharing rule. The main conclusion we draw in the first essay is that, when designing the optimal output sharing rule, the principal need not look for any output sharing rule more sophisticated than the linear or piecewise linear rules we frequently observe. We also characterize the condition under which the principal chooses to take part in production, and conclude that the issue of mitigation of moral hazard and sustainability of efficiency crucially hinges on whether the principal actually participates in production or not. More concretely, we show that moral hazard dissipates completely whenever the principal does not participate in production, however, even then she does not achieve as much welfare as in the First Best situation if her best option in the First Best situation is to take part in production. The second essay is in stochastic social choice theory. In a paper published in 1986 in Econometrica, Pattanaik and Peleg formulated stochastic analogues for each of Arrow's axioms and concluded that the stochastic social choice functions that satisfy their axioms are essentially randon dictatorships when individuals have strict preferences. More precisely, there is a unique weight associated with each individual such that the vector of these individual weights has the properties of a probability distribution over the set of individuals, and, given any preference profile and any feasible set, the probability that a feasible alternative is chosen is equal to the sum of the weights of those individuals who have this alternative as their best feasible alternative. We extend the analysis of Pattanaik and Peleg by allowing individuals to have weak preferences. As in their paper, it turns out that the probabilistic versions of Arrow's condition simply that there are individual weights. However, now, given a preference profile and a feasible set, we partition the society so that any two individuals from different elements of the partition have no common best feasible alternatives, but the set of best feasible alternatives of each individual in an element of the partition overlaps with that of some other individual in the same element. Using this partition, it is shown that the only restriction on the stochastic social choice function is that the sum of the weights of all individuals belonging to the same element in the partition is equal to the probability that some alternative which is best in the feasible set for one of these individuals is chosen. When everyone has unique best feasible alternatives, the rules characterized here reduce to those of Pattanaik and Peleg.

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Recent research has suggested disrupted interactions between brain regions may contribute to some of the symptoms of motor disorders such as Parkinson’s Disease (PD). It is therefore important to develop models for inferring brain functional connectivity from data obtained through non-invasive imaging technologies, such as functional magnetic resonance imaging (fMRI). The complexity of brain activities as well as the dynamic nature of motor disorders require such models to be able to perform complex, large-scale, and dynamic system computation. Traditional models proposed in the literature such as structural equation modeling (SEM), multivariate autoregressive models (MAR), dynamic causal modeling (DCM), and dynamic Bayesian networks (DBNs) have all been suggested as suitable for fMRI data analysis. However, they suffer from their own disadvantages such as high computational cost (e.g. DBNs), inability to deal with non-linear case (e.g. MAR), large sample size requirement (e.g. SEM), et., al. In this research, we propose applying Probabilistic Boolean Network (PBN) for modeling brain connectivity due to its solid stochastic properties, computational simplicity, robustness to uncertainty, and capability to deal with small-size data, typical for fIVIRI data sets. Applying the proposed PBN framework to real fMRI data recorded from PD subjects enables us to identify statistically significant abnormality in PD connectivity by comparing it with normal subjects. The PBN results also suggest a mechanism of evaluating the effectiveness of L-dopa, the principal treatment for PD. In addition to PBNs’ promising application in inferring brain connectivity, PBN modeling for brain ROTs also enables researchers to study dynamic activities of the system under stochastic conditions, gaining essential information regarding asymptotic behaviors of ROTs for potential therapeutic intervention in PD. The results indicate significant difference in feature states between PD patients and normal subjects. Hypothesizing the observed feature states for normal subject as the desired functional states, we further explore possible methods to manipulate the dynamic network behavior of PD patients in the favor of the desired states from the view of random perturbation as well as intervention. Results identified a target ROT with the best intervention performance, and that ROl is a potential candidate for therapeutic exercise.

In this work we study the space of group homomorphisms Hom(Γ,G) from a geometric and simplicial point of view. The case in which the source group is a free abelian group of rank n is studied in more detail since this space can be identified with the space of commuting n-tuples of elements from G. This latter case is of particular interest when the target is a Lie group. The simplicial approach allows us to to construct a family of spaces that filters the classifying space of a group by filtering group theoretical information of the given group. Namely, we use the lower central series of free groups to construct a family of simplicial subspaces of the bar construction of the classifying space of a group. The first layer of this filtration is studied in more detail for transitively commutative (TC) groups.

Past research has demonstrated that individual and group judgments are subject to systematic biases. Although much effort has been devoted to the debiasing of individual judgments, no corresponding work to date has been found on the debiasing of group judgments. Complicating this research gap is the fact that group and team work is gaining increasing importance in organizational settings. The current study examines the usefulness of group support systems (GSS) in addressing two important judgment biases, namely, representativeness bias and availability bias. Representativeness bias refers to the bias incurred in posterior-probability estimation by not properly utilizing information sources such as base rate. Availability bias occurs when events of higher availability to the memory are correspondingly judged as occurring more frequently. The formation of a judgment is seen from the perspective of an information integration process. Two orthogonal dimensions of information integration -- interpersonal and intrapersonal -- are involved in group judgments. Interpersonal information integration concerns the aspect of information sharing among group members, and can be supported with the electronic communication channel of GSS. Intrapersonal information integration deals with the information processing capacities and capabilities of individuals, and is supportable using cognitive-support tools of GSS. A laboratory experiment with a 2x2 factorial design was conducted. One hundred and twenty subjects took part in the experiment. They were randomly allocated to 40 groups. Two experimental tasks, designed to examine the two judgment biases of interest, were solved by each group. Data pertaining to both processes and outcomes were collected and analyzed. Representativeness bias was reduced by the use of cognitive support, in the form of a problem representation tool. Groups with the problem representation tool made fewer references to diagnostic information versus base rate, leading to the use of more correct strategies which combined these two information sources. The use of the problem representation tool was found to be responsible for causing this chain of events. On the other hand, electronic communication did not lead to a similar change in the pattern of group processes, and, correspondingly, did not reduce the representativeness bias. Although electronic communication is capable of improving the interpersonal aspect of information integration, the representativeness bias is primarily a result of cognitive limitations, and benefits little from improved communication among group members. Availability bias was reduced by both cognitive support and communication support. Cognitive support, in the form of electronic brainstorming, increased the information search scope of issues, especially those issues of relatively low availability to the memory. Electronic communication allows parallel input and has a lower social presence than verbal communication. These features helped to reduce the extent of groupthink and widened the range of alternative solutions proposed. Some interaction effects were observed on group members’ perceptions of the group process. For example, communication medium had an effect on group members’ satisfaction in groups without cognitive support, but not those with cognitive support. Correspondingly, cognitive support affected some perceptual variables in verbally-communicating groups, but not electronically-communicating groups. Examples of such effects include an increase in perceived socio-emotional behavior and perceived informal leadership.

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