Ranking Sets of Objects: The Complexity of Avoiding Impossibility Results

The problem of lifting a preference order on a set of objects to a preference order on a family of subsets of this set is a fundamental problem with a wide variety of applications in AI. The process is often guided by axioms postulating properties the lifted order should have. Well-known impossibility results by Kannai and Peleg and by Barbera and Pattanaik tell us that some desirable axioms – namely dominance and (strict) independence – are not jointly satisfiable for any linear order on the objects if all non-empty sets of objects are to be ordered. On the other hand, if not all non-empty sets of objects are to be ordered, the axioms are jointly satisfiable for all linear orders on the objects for some families of sets. Such families are very important for applications as they allow for the use of lifted orders, for example, in combinatorial voting. In this paper, we determine the computational complexity of recognizing such families. We show that it is \Pi_2^p-complete to decide for a given family of subsets whether dominance and independence or dominance and strict independence are jointly satisfiable for all linear orders on the objects if the lifted order needs to be total. Furthermore, we show that the problem remains coNP-complete if the lifted order can be incomplete. Additionally, we show that the complexity of these problems can increase exponentially if the family of sets is not given explicitly but via a succinct domain restriction. Finally, we show that it is NP-complete to decide for a family of subsets whether dominance and independence or dominance and strict independence are jointly satisfiable for at least one linear order on the objects.

Evaluating Modeled Intra- to Multidecadal Climate Variability Using Running Mann–Whitney Z Statistics

An analysis method previously used to detect observed intra- to multidecadal (IMD) climate regimes was adapted to compare observed and modeled IMD climate variations. Pending the availability of the more appropriate phase 5 Coupled Model Intercomparison Project (CMIP-5) simulations, the method is demonstrated using CMIP-3 model simulations. Although the CMIP-3 experimental design will almost certainly prevent these model runs from reproducing features of historical IMD climate variability, these simulations allow for the demonstration of the method and illustrate how the models and observations disagree. This method samples a time series’s data rankings over moving time windows, converts those ranking sets to a Mann–Whitney U statistic, and then normalizes the U statistic into a Z statistic. By detecting optimally significant IMD ranking regimes of arbitrary onset and varying duration, this process generates time series of Z values that are an adaptively low-passed and normalized transformation of the original time series. Principal component (PC) analysis of the Z series derived from observed annual temperatures at 92 U.S. grid locations during 1919–2008 shows two dominant modes: a PC1 mode with cool temperatures before the late 1960s and warm temperatures after the mid-1980s, and a PC2 mode indicating a multidecadal temperature cycle over the Southeast. Using a graphic analysis of a Z error metric that compares modeled and observed Z series, the three CMIP-3 model simulations tested here are shown to reproduce the PC1 mode but not the PC2 mode. By providing a way to compare grid-level IMD climate response patterns in observed and modeled data, this method can play a useful diagnostic role in future model development and decadal climate forecasting.

Automated Search for Impossibility Theorems in Social Choice Theory: Ranking Sets of Objects

We present a method for using standard techniques from satisfiability checking to automatically verify and discover theorems in an area of economic theory known as ranking sets of objects. The key question in this area, which has important applications in social choice theory and decision making under uncertainty, is how to extend an agent's preferences over a number of objects to a preference relation over nonempty sets of such objects. Certain combinations of seemingly natural principles for this kind of preference extension can result in logical inconsistencies, which has led to a number of important impossibility theorems. We first prove a general result that shows that for a wide range of such principles, characterised by their syntactic form when expressed in a many-sorted first-order logic, any impossibility exhibited at a fixed (small) domain size will necessarily extend to the general case. We then show how to formulate candidates for impossibility theorems at a fixed domain size in propositional logic, which in turn enables us to automatically search for (general) impossibility theorems using a SAT solver. When applied to a space of 20 principles for preference extension familiar from the literature, this method yields a total of 84 impossibility theorems, including both known and nontrivial new results.