Fractional social choice is a concept in social choice theory that extends traditional voting and decision-making frameworks to incorporate scenarios where preferences can be expressed in a fractional or probabilistic manner. This approach recognizes that individual preferences might not be strictly ranked as in classical voting systems, allowing for a more nuanced representation of societal choices. In conventional social choice mechanisms, individuals provide their preferences in terms of complete rankings or simple majority votes.

Fractional approval voting is a voting method that extends the concept of approval voting, where voters can express approval for multiple candidates. In fractional approval voting, voters can not only approve of a candidate but also indicate varying degrees of approval, effectively allowing voters to allocate fractional values (e.g., from 0 to 1) to each candidate based on their preferences.

Extended sympathy refers to a deeper, more encompassing form of sympathy that goes beyond immediate feelings of pity or compassion. It involves a broader understanding and emotional connection to the experiences, struggles, and pain of others. This type of sympathy often includes: 1. **Empathy**: Understanding and sharing the feelings of another person, putting oneself in their shoes. 2. **Support**: Offering tangible and emotional support, not just in the moment but over time.

An electoral system is a set of rules and processes that govern how votes are cast, counted, and translated into seats in a legislature or the outcome of an election. Electoral systems can significantly influence political processes, party systems, and voter behavior. They determine how representatives are elected, the methods by which votes are aggregated, and how those results translate into political power.

An electoral list is a list of candidates that a political party or coalition presents for an election. It is often used in systems where proportional representation is in place, allowing voters to choose parties rather than individual candidates.

Egalitarian rule refers to a system of governance or societal organization that promotes equality among all individuals. It emphasizes the belief that all people should have equal rights, opportunities, and responsibilities, regardless of their background, status, or any other distinguishing factors. In an egalitarian system: 1. **Political Equality**: Every individual has an equal voice in the political process, such as voting rights and participation in decision-making.

The discursive dilemma refers to a situation in which a group of individuals faces a conflict between collective decision-making and the rational aggregation of individual judgments or beliefs. It often arises in contexts where decisions depend on multiple propositions or issues, leading to potentially inconsistent outcomes if individual judgments are combined in certain ways. The classic example involves a group's decision on whether or not to accept a set of propositions based on the majority opinions of group members.

The term "dictatorship mechanism" can refer to several concepts, typically within political science or game theory, where it suggests a system allowing a single leader or decision-maker to exert control over a group or society. Here are a few interpretations of the term: 1. **Political Dictatorship**: In a political context, a dictatorship mechanism refers to the ways in which a dictator maintains power and controls a state.

Computational social choice is an interdisciplinary field that lies at the intersection of computer science, economics, and political science. It focuses on designing and analyzing algorithms and computational systems for collective decision-making processes, where groups or societies make choices based on the preferences of their individual members. Key aspects of computational social choice include: 1. **Voting Systems**: The study of how different voting procedures can aggregate individual preferences into a collective decision.

Christian List is a prominent philosopher known for his work in areas such as social choice theory, political philosophy, and the philosophy of social science. He has contributed to discussions on various topics including collective decision-making, democracy, and the dynamics of opinion formation. He is known for his examination of how individual preferences can aggregate to form collective decisions and the implications of this process for understanding democratic governance and social cooperation. List has held academic positions at various institutions and has published extensively in both philosophical and interdisciplinary contexts.

Cardinal voting is an electoral system where voters rate each candidate on a scale, rather than simply selecting one candidate or ranking them in order. This allows voters to express their preferences more finely. For example, in a common version of cardinal voting, voters might grade candidates from 0 to 5, where 0 indicates a strong disapproval and 5 indicates strong approval. The overall score for each candidate is calculated by summing the ratings they receive from all voters.

Bayesian regret is a concept used in decision theory and statistics that quantifies the performance of a decision-making strategy in the presence of uncertainty. It measures the difference in expected utility or payoff between the optimal decision (the decision that would yield the highest expected payout if the true state of nature were known) and the decision made by an agent using a specific strategy or approach.

Anonymity in the context of social choice theory refers to a principle that focuses on the treatment of individuals in the decision-making process. Specifically, the anonymity principle states that all individuals should be treated equally and that the preferences of individuals should not be weighted differently based on their identity. In other words, if two individuals swap their preferences, the outcome of the social choice should remain unchanged.

The term "agreeable subset" is not a standard term widely recognized in mathematics or other scientific disciplines. It might refer to a concept in a specific field, study, or context that is not commonly referenced or defined.

Voting theory is a field of study within social choice theory that examines the methods and rules governing voting processes in order to determine how collective decisions are made. It encompasses a range of topics, including the design of voting systems, the analysis of voter preferences, and the aggregation of individual votes into a collective outcome.

The Selberg sieve is a mathematical tool used in number theory, particularly in the field of prime number theory and in the study of additive number theory. It is named after the mathematician A. Selberg, who introduced it as a method for estimating the number of integers that are free of large prime factors or, more generally, to sieve out integers that are not divisible by a specified set of primes.

In the context of sieve theory, the "parity problem" generally refers to questions about the distribution of prime numbers. More specifically, sieve theory involves methods that can help determine how many integers in a given set meet certain criteria, often in relation to being prime or composite. The parity problem in sieve theory can typically involve exploring the even and odd behavior of prime numbers or their residues modulo some integer. One classic observation related to parity and primes is that all prime numbers except for 2 are odd.

The Legendre sieve is a mathematical algorithm used in number theory for finding prime numbers within a certain range. It is based on the idea of sieving out composite numbers from a list of integers by marking the multiples of each prime number. Here's an overview of how the Legendre sieve works: 1. **Initialization**: You start with a range of integers, such as all integers from \( 2 \) to \( n \), where \( n \) is your upper limit.

The Larger Sieve, commonly known in the context of number theory, refers to an advanced mathematical technique used for determining the properties of integers, particularly in relation to prime numbers. It is an extension of the Sieve of Eratosthenes and is particularly useful in analytic number theory and areas dealing with the distribution of prime numbers.