Necessary and sufficient conditions for pairwise majority decisions on path-connected domains

In this paper, we consider choice functions that are unanimous, anonymous, symmetric, and group strategy-proof and consider domains that are single-peaked on some tree. We prove the following three results in this setting. First, there exists a unanimous, anonymous, symmetric, and group strategy-proof choice function on a path-connected domain if and only if the domain is single-peaked on a tree and the number of agents is odd. Second, a choice function is unanimous, anonymous, symmetric, and group strategy-proof on a single-peaked domain on a tree if and only if it is the pairwise majority rule (also known as the tree-median rule) and the number of agents is odd. Third, there exists a unanimous, anonymous, symmetric, and strategy-proof choice function on a strongly path-connected domain if and only if the domain is single-peaked on a tree and the number of agents is odd. As a corollary of these results, we obtain that there exists no unanimous, anonymous, symmetric, and group strategy-proof choice function on a path-connected domain if the number of agents is even.

The median rule in judgement aggregation

A judgement aggregation rule takes the views of a collection of voters over a set of interconnected issues and yields a logically consistent collective view. The median rule is a judgement aggregation rule that selects the logically consistent view which minimizes the average distance to the views of the voters (where the “distance” between two views is the number of issues on which they disagree). In the special case of preference aggregation, this is called the Kemeny rule. We show that, under appropriate regularity conditions, the median rule is the unique judgement aggregation rule which satisfies three axioms: Ensemble Supermajority Efficiency, Reinforcement, and Continuity. Our analysis covers aggregation problems in which the consistency restrictions on input and output judgements may differ. We also allow for issues to be weighted, and provide numerous examples in which issue weights arise naturally.

ADAPTIVE SELECTION OF NEURAL NETWORKS FOR A COMMITTEE DECISION

To improve recognition results, decisions of multiple neural networks can be aggregated into a committee decision. In contrast to the ordinary approach of utilising all neural networks available to make a committee decision, we propose creating adaptive committees, which are specific for each input data point. A prediction network is used to identify classification neural networks to be fused for making a committee decision about a given input data point. The jth output value of the prediction network expresses the expectation level that the jth classification neural network will make a correct decision about the class label of a given input data point. The proposed technique is tested in three aggregation schemes, namely majority vote, averaging, and aggregation by the median rule and compared with the ordinary neural networks fusion approach. The effectiveness of the approach is demonstrated on three well known real data sets and also applied to fault identification of the actuator valve at one sugar factory within the DAMADICS RTN.

Constructing Legal Rules on Appellate Courts

Appellate courts make policy, not only by hearing cases themselves, but by establishing legal rules for the disposition of future cases. The problem is that such courts are generally multimember, or collegial, courts. If different judges prefer different rules, can a collegial court establish meaningful legal rules? Can preferences that take the form of legal rules be aggregated? I use a “case-space” model to show that there will exist a collegial rule that captures majoritarian preferences, and to show that there will exist a median rule even if there is no single median judge. I show how collegial rules can differ from the rules of individual judges and how judicial institutions (such as appellate review and the power to write separate opinions) affect the stability and enforceability of legal rules. These results are discussed in light of fundamental debates between legal and political perspectives on judicial behavior.

SELECTING NEURAL NETWORKS FOR A COMMITTEE DECISION

To improve recognition results, decisions of multiple neural networks can be aggregated into a committee decision. In contrast to the ordinary approach of utilizing all neural networks available to make a committee decision, we propose creating adaptive committees, which are specific for each input data point. A prediction network is used to identify classification neural networks to be fused for making a committee decision about a given input data point. The jth output value of the prediction network expresses the expectation level that the jth classification neural network will make a correct decision about the class label of a given input data point. The proposed technique is tested in three aggregation schemes, namely majority vote, averaging, and aggregation by the median rule and compared with the ordinary neural networks fusion approach. The effectiveness of the approach is demonstrated on two artificial and three real data sets.