Computations of volumes in five candidates elections

We describe several analytical results obtained in five candidates social choice elections under the assumption of the Impartial Anonymous Culture. These include the Condorcet and Borda paradoxes, as well as the Condorcet efficiency of plurality, negative plurality and Borda voting, including their runoff versions. The computations are done by Normaliz. It finds precise probabilities as volumes of polytopes in dimension 119, using its recent implementation of the Lawrence algorithm

How Likely A Coalition of Voters Can Influence A Large Election?

For centuries, it has been widely believed that the influence of a small coalition of voters is negligible in a large election. Consequently, there is a large body of literature on characterizing the asymptotic likelihood for an election to be influenced, especially by the manipulation of a single voter, establishing an $O(\frac{1}{\sqrt n})$ upper bound and an $\Omega(\frac{1}{n^{67}})$ lower bound for many commonly studied voting rules under the i.i.d.~uniform distribution, known as Impartial Culture (IC) in social choice, where $n$ is the number is voters. In this paper, we extend previous studies in three aspects: (1) we consider a more general and realistic semi-random model, where a distribution adversary chooses a worst-case distribution and then a data adversary modifies up to $\psi$ portion of the data, (2) we consider many coalitional influence problems, including coalitional manipulation, margin of victory, and various vote controls and bribery, and (3) we consider arbitrary and variable coalition size $B$. Our main theorem provides asymptotically tight bounds on the semi-random likelihood of the existence of a size-$B$ coalition that can successfully influence the election under a wide range of voting rules. Applications of the main theorem and its proof techniques resolve long-standing open questions about the likelihood of coalitional manipulability under IC, by showing that the likelihood is $\Theta\left(\min\left\{\frac{B}{\sqrt n}, 1\right\}\right)$ for many commonly studied voting rules. The main technical contribution is a characterization of the semi-random likelihood for a Poisson multinomial variable (PMV) to be unstable, which we believe to be a general and useful technique with independent interest

Collective Decisions on Conditional Topics - An Empirical Study of the Impact of Nonseparable Preferences

Analytical politics investigates collective decision-making in political systems. Such voting behavior in groups takes place in parliaments, committees or the board of local football clubs. It is a frequent object of study for theoretical as well as empirical analysis. Previous contributions have demonstrated well the stabilizing effect of procedural rules, such as agenda-setting or multi-chamber systems, for collective decisions. These rules are applied in many institutions, such as the European Parliament or the German Bundestag. Their main purpose is to ensure reliable policy. Previous work continually used the restrictive assumption of separable preferences. This assumption implies that different aspects of a question do not influence each other. The limited validity of this hypothesis is apparent even in everyday situations. For example, the enjoyment of a delicious meal depends on the combination of food and drink. When choosing between fish and venison for dinner you also have to consider the question of which sort of wine to have with the meal; white with fish, and red with venison. This form of interdependence also occurs in legislation. For example, the savings determined in the Greek budget influence the preferences of the German public for financial assistance to Greece. The assumption of separable preferences is therefore in the critical focus of theoretical research. This literature discusses the impacts of and solutions to nonseparable preferences in detail. The analysis suggests an increased complexity for every decision-making process affected by nonseparable preferences. This complexity leads to difficulties in the operationalization of nonseparable preferences and is one of the reasons that there are too few empirical examinations. In addition, the stabilizing properties of institutional arrangements identified under the assumption of separable preferences are in question. The goal of this study is to close this gap between theory and empiricism. I investigate nonseparable preferences by conducting a laboratory experiment, which allows comprehensive environmental control. This facilitates the operationalization of nonseparable preferences. First, I prove the relevance of nonseparable preferences for analytical research on social interaction. The experiment is therefore completed by empirical case studies. Next, I investigate the effects of nonseparable preferences on collective and individual decision-making in the laboratory. Finally, I assess my contribution with respect to current research in social science and discuss possibilities to more accurately model of human behavior. The dissertation is structured as follows. I start in chapter 1 with the presentation of my research question and design. In chapter 2 the concept of nonseparable preferences is further clarified by means of exemplary case studies. It also discusses the theoretical foundations of nonseparable preferences. My hypotheses are specified along common concepts used in the literature. Based on empirical data the relevance of nonseparable preferences for political science research is demonstrated in chapter 3. Next, chapter 4 presents the design of the laboratory experiment. The effects of nonseparable preferences on collective decision-making are examined in chapter 5. Subsequently, determinants for the motivation function of individuals are scrutinized in chapter 6. In chapter 7 I report the results of the post-experiment survey. All findings are evaluated in chapter 8, where I focus on detailing their usefulness to future research on human behavior. Finally, chapter 9 summarizes the study and lists possibilities to further expand research in this area