# Social Choice Theory

Social choice theory addresses the methods and processes of collective decision making. Applications range from preference aggregation in small committees, voting in democratic elections at the regional and national level to questions of fair resolution of conflicting interests. Important properties of aggregation methods are, for instance, anonymity, unanimity, efficiency, monotonicity and strategy-proofness (incentive compatibility). Our research is devoted to the identification and design of aggregation methods that combine such properties.

Prof. Dr. Clemens Puppe: clemens puppe ∂kit edu

Claudio Kretz: claudio.kretz∂kit.edu

Michael Müller: michael mueller ∂kit edu

# The Liberal Paradox: Rights in Social Choice and Judgment Aggregation

Generally, research on collective decision making is interested in finding aggregation rules which limit the influence of any particular individual or subgroup (cf. anonimity etc.). At the same time, a libertarian would hold that any aspect within what we may consider an individual's private sphere should be decided by herself and herself alone.

As Sen (1970) pointed out, such libertarian demands may be inconsistent with upholding the Pareto principle. To understand his famous liberal paradox, consider a situation in which two individuals A and B have to make a collective choice on the color of their houses (which they can paint either white or yellow). Suppose that both prefer yellow for their own houses but really care more about the color of the other's house color which they would rather have in white.

Denoting social states by (color of A's house, color of B's house), we have

A: (y,w) > (w,w) > (y,y) > (w,y) and B: (w,y) > (w,w) > (y,y) > (y,w).

Given all other aspects are fixed, a libertarian rule should let individuals decide the color of their own houses. In particular, the social ranking would have to respect B's preference for (y,y) over (y,w) and A's preference for (y,w) over (w,w). Thus, socially,

(y,y) > (y,w) > (w,w)

in conflict with the Pareto rule - seeing that both A and B prefer (w,w) to (y,y).

On a related note (cf. Dietrich & List, 2008), granting (expert) rights in judgment aggregation clashes with requiring precedures to respect unanimous consent. For example, in a version of the doctrinal paradox (see below), consider the following sets of individual judgments:

judge 1: {¬A, ¬B, ¬(A Λ B)} judge 2: {A, ¬B, ¬(A Λ B)} judge 3: {¬A, B, ¬(A Λ B)}.

If judge 1 is considered an expert on A, judge 2 an expert on B, - i.e., we grant locally dictatorial decision making power on these issues - we collectively judge A and B to be true. This, however, is inconsistent with unanimous agreement on ¬(A Λ B).

Inspired by the above the literature on rights has dealt with relexations of both the Pareto principle and libertarian claims to avoid the paradox. At the same time, researchers have challenged Sen's consequentialist formulation of rights and proposed a procedural formulation of rights in game forms. While this strand of literature has clarified that there is seldom a problem with the existence of rights per se, Sen's paradox resurfaces as inefficiency of equilibria in games of rights exercise.

Claudio Kretz: claudio.kretz∂kit.edu

# Experimental Voting Theory

We empirically test different voting rules in the allocation of public projects in a laboratory experiment. The mean rule is highly manipulable theoretically and we find indeed that subjects have a strong tendency to play the corresponding Nash equilibrium strategy. In contrast, median-based rules are more difficult to manipulate, and sometimes truth-telling is a weakly dominant strategy. While most subjects play a best response to truth-telling of all other voters, a significant fraction of them does not vote truthfully themselves.

We study the effect on the participation rate of different voting rules in the context of a simple resource allocation problem. While the impact on the social outcome under the mean rule is small for large electorates, it is certain and always greater than zero. By contrast, the impact under the median rule has large variance. We theoretically analyze the individual impact on the social outcome for different preference distributions under the mean and the median rule. For some preference distributions the expected impact on the social outcome is larger under the median rule than under the mean rule.This raises the question whether the differences in the individual impact has implications for the participation rates under these voting rules. In a field experiment, we test whether, and how, voter turnout varies with the voting rule and we elicit beliefs about the impact on the social outcome.

Jana Rollmann: jana.rollmann∂kit.edu

# Strategy-Proofness

Strategy-proofness requires that truth-telling is a (weakly) dominant strategy for all individuals in a collective decision making process. We introduce the concept of robust strategy-proofness on a domain which imposes a stronger requirement for the preferences all other agents have. While in general a strictly stronger requirement, our main result shows that for all tops-only social choice functions strategy-proofness and robust strategy-proofness on minimally rich domains are equivalent.

One important appeal of strategy-proofness is the robustness that it implies. Under a strategy-proof voting rule, every individual has an optimal strategy independently of the behavior of all other voters. In particular, optimal play is robust with respect to the beliefs voters may have about the type and the behavior of the other voters. Following Blin and Satterthwaite (1977), let us call this property "belief-independence". In this paper, we give a number of examples of voting rules that are belief-independent but not strategy-proof. However, we also show that belief-independence implies strategy-proofness under a few natural additional conditions. We also introduce a novel condition of robust strategy-proofness which strengthens strategy-proofness by requiring that no voter can benefit by submitting any unrestricted preference ordering given any unrestricted preference profile for all other voters. Robust strategy-proofness is shown to be a strictly stronger property of a voting rule than the usual notion of strategy-proofness. However, under natural additional conditions the two properties are equivalent.

Michael Müller: michael.mueller∂kit.edu

# Judgement Aggregation

Judgement Aggregation addresses the formation of collective judgements on logically interconnected propositions. The famous doctrinal paradox shows that – just as in voting theory – inconsistencies can easily arise if the aggregation is done by simple issue-wise majority vote:

Prof. Dr. Clemens Puppe: clemens.puppe∂kit.edu

Claudio Kretz: claudio.kretz∂kit.edu

Michael Müller: michael.mueller∂kit.edu

# Single Peaked Preferences

It is proved that, among all restricted preference domains that guarantee consistency (i.e. transitivity) of pairwise majority voting, the single-peaked domain is the only minimally rich and connected domain that contains two completely reversed strict preference orders. It is argued that this result explains the predominant role of single-peakedness as a domain restriction in models of political economy and elsewhere. The main result has a number of corollaries, among them a dual characterization of the single-dipped domain; it also implies that a single-crossing (‘order-restricted’) domain can be minimally rich only if it is a subdomain of a single-peaked domain. The main conclusions are robust as they apply both to strict and weak preference orders.

Prof. Dr. Clemens Puppe: clemens.puppe∂kit.edu

# Condorcet Domains

The Condorcet paradox shows that pairwise majority voting may produce cyclic outcomes on an unrestricted preference domain. For instance consider three voters with their preferences:

Voter 1: a > b > c Voter 2: b > c > a Voter 3: c > a > b

A majority of voters prefers a to b, a majority prefers b to c, and a majority prefers c to a. This leads to the Condorcet cycle

a > b > c > a.

The absence of majority cycles can only be guaranteed under suitable domain restrictions, such that single-peaked preferences, preferences with ‘value-restrictions' or single-crossing preferences – these are so called Condorcet domains. Puppe and Slinko (2015) prove that all (closed) Condorcet domains satisfy a so called intermediateness condition with respect to an appropriate median graph. This property allows us to investigate the Condorcet domains in more detail.

The following figures depict graphical representations of those domains:

# Behavioral Economics

Economic experiments enable the development and testing of new theories that aim at explaining agents’ economic behavior. By using controlled conditions, both in the lab and in the field, deviations from classical assumption in subjects’ behavior can be documented and classified. The importance of such experimental research has grown steadily during the last decades.

Jana Rollmann: jana.rollmann∂kit.edu