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Probabilistic Social Choice
Fractional, stochastic, or weighted social choice is a branch of social choice theory in which the collective decision is not a single alternative, but rather a weighted sum of two or more alternatives. For example, if society has to choose between three candidates (A, B, or C), then in standard social choice exactly one of these candidates is chosen. By contrast, in fractional social choice it is possible to choose any linear combination of these, e.g. "2/3 of A and 1/3 of B". A common interpretation of the weighted sum is as a lottery, in which candidate A is chosen with probability 2/3 and candidate B is chosen with probability 1/3. The rule can also be interpreted as a recipe for sharing, for example: * Time-sharing: candidate A is (deterministically) chosen for 2/3 of the time while candidate B is chosen for 1/3 of the time. * Budget-distribution: candidate A receives 2/3 of the budget while candidate B receives 1/3 of the budget. * Fair division with different entitlements ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Social Choice Theory
Social choice theory is a branch of welfare economics that extends the Decision theory, theory of rational choice to collective decision-making. Social choice studies the behavior of different mathematical procedures (social welfare function, social welfare functions) used to combine individual preferences into a coherent whole.Amartya Sen (2008). "Social Choice". ''The New Palgrave Dictionary of Economics'', 2nd EditionAbstract & TOC./ref> It contrasts with political science in that it is a Normative economics, normative field that studies how a society can make good decisions, whereas political science is a Positive economics, descriptive field that observes how societies actually do make decisions. While social choice began as a branch of economics and decision theory, it has since received substantial contributions from mathematics, philosophy, political science, and game theory. Real-world examples of social choice rules include constitution, constitutions and Parliamentary ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Ordering
In probability theory and statistics, a stochastic order quantifies the concept of one random variable being "bigger" than another. These are usually partial orders, so that one random variable A may be neither stochastically greater than, less than, nor equal to another random variable B. Many different orders exist, which have different applications. Usual stochastic order A real random variable A is less than a random variable B in the "usual stochastic order" if :\Pr(A>x) \le \Pr(B>x)\textx \in (-\infty,\infty), where \Pr(\cdot) denotes the probability of an event. This is sometimes denoted A \preceq B or A \le_\mathrm B. If additionally \Pr(A>x) x) for some x, then A is stochastically strictly less than B, sometimes denoted A \prec B. In decision theory, under this circumstance, is said to be first-order stochastically dominant over ''A''. Characterizations The following rules describe situations when one random variable is stochastically less than or equal to another. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zero-sum Game
Zero-sum game is a Mathematical model, mathematical representation in game theory and economic theory of a situation that involves two competition, competing entities, where the result is an advantage for one side and an equivalent loss for the other. In other words, player one's gain is equivalent to player two's loss, with the result that the net improvement in benefit of the game is zero. If the total gains of the participants are added up, and the total losses are subtracted, they will sum to zero. Thus, Fair cake-cutting, cutting a cake, where taking a more significant piece reduces the amount of cake available for others as much as it increases the amount available for that taker, is a zero-sum game if marginal utility, all participants value each unit of cake equally. Other examples of zero-sum games in daily life include games like poker, chess, sport and Contract bridge, bridge where one person gains and another person loses, which results in a zero-net benefit for every ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimax Theorem
In the mathematical area of game theory and of convex optimization, a minimax theorem is a theorem that claims that : \max_ \min_ f(x,y) = \min_ \max_f(x,y) under certain conditions on the sets X and Y and on the function f. It is always true that the left-hand side is at most the right-hand side ( max–min inequality) but equality only holds under certain conditions identified by minimax theorems. The first theorem in this sense is von Neumann's minimax theorem about two-player zero-sum games published in 1928, which is considered the starting point of game theory. Von Neumann is quoted as saying "''As far as I can see, there could be no theory of games ... without that theorem ... I thought there was nothing worth publishing until the Minimax Theorem was proved''". Since then, several generalizations and alternative versions of von Neumann's original theorem have appeared in the literature. Bilinear functions and zero-sum games Von Neumann's original theorem was motivated by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Condorcet Winner
A Condorcet winner (, ) is a candidate who would receive the support of more than half of the electorate in a one-on-one race against any one of their opponents. Voting systems where a majority winner will always win are said to satisfy the Condorcet winner criterion. The Condorcet winner criterion extends the principle of majority rule to elections with multiple candidates. Named after Nicolas de Condorcet, it is also called a majority winner, a majority-preferred candidate, a beats-all winner, or tournament winner (by analogy with round-robin tournaments). A Condorcet winner may not necessarily always exist in a given electorate: it is possible to have a rock, paper, scissors-style cycle, when multiple candidates defeat each other (Rock < Paper < Scissors < Rock). This is called , and is analogous to the counterintuitive [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Expected Value
In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean, mean of the possible values a random variable can take, weighted by the probability of those outcomes. Since it is obtained through arithmetic, the expected value sometimes may not even be included in the sample data set; it is not the value you would expect to get in reality. The expected value of a random variable with a finite number of outcomes is a weighted average of all possible outcomes. In the case of a continuum of possible outcomes, the expectation is defined by Integral, integration. In the axiomatic foundation for probability provided by measure theory, the expectation is given by Lebesgue integration. The expected value of a random variable is often denoted by , , or , with a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximal Lotteries
Maximal lotteries are a probabilistic voting rule that use ranked ballots and returns a lottery over candidates that a majority of voters will prefer, on average, to any other.P. C. Fishburn. ''Probabilistic social choice based on simple voting comparisons''. Review of Economic Studies, 51(4):683–692, 1984. In other words, in a series of repeated head-to-head matchups, voters will (on average) prefer the results of a maximal lottery to the results produced by any other voting rule. Maximal lotteries satisfy a wide range of desirable properties: they elect the Condorcet winner with probability 1 if it exists and never elect candidates outside the Smith set. Moreover, they satisfy reinforcement,F. Brandl, F. Brandt, and H. G. SeedigConsistent probabilistic social choice Econometrica. 84(5), pages 1839-1880, 2016. participation,F. Brandl, F. Brandt, and J. HofbauerWelfare Maximization Entices Participation Games and Economic Behavior. 14, pages 308-314, 2019. and independenc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Score Voting
Score voting, sometimes called range voting, is an electoral system for single-seat elections. Voters give each candidate a numerical score, and the candidate with the highest average score is elected. Score voting includes the well-known approval voting (used to calculate approval ratings), but also lets voters give partial (in-between) approval ratings to candidates. Usage Political use Historical A crude form of score voting was used in some elections in ancient Sparta, by measuring how loudly the crowd shouted for different candidates. This has a modern-day analog of using clapometers in some television shows and the judging processes of some athletic competitions. Beginning in the 13th century, the Republic of Venice elected the Doge of Venice using a multi-stage process with multiple rounds of score voting. This may have contributed to the Republic's longevity, being partly responsible for its status as the longest-lived democracy in world history. Score voting w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Borda Count
The Borda method or order of merit is a positional voting rule that gives each candidate a number of points equal to the number of candidates ranked below them: the lowest-ranked candidate gets 0 points, the second-lowest gets 1 point, and so on. The candidate with the most points wins. The Borda count has been independently reinvented several times, with the first recorded proposal in 1435 being by Nicholas of Cusa (see History below), but is named after the 18th-century French mathematician and naval engineer Jean-Charles de Borda, who re-devised the system in 1770. The Borda count is well-known in social choice theory both for its pleasant theoretical properties and its ease of manipulation. In the absence of strategic voting and strategic nomination, the Borda count tends to elect broadly-acceptable options or candidates (rather than consistently following the preferences of a majority); when both voting and nomination patterns are completely random, the Borda count gener ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Dictatorship
A random ballot or random dictatorship is a randomized electoral system where the election is decided on the basis of a single randomly-selected ballot. A closely-related variant is called random serial (or sequential) dictatorship, which repeats the procedure and draws another ballot if multiple candidates are tied on the first ballot. Random dictatorship was first described in 1977 by Allan Gibbard, who showed it to be the unique social choice rule that treats all voters equally while still being strategyproof in all situations. Its application to elections was first described in 1984 by Akhil Reed Amar. The rule is rarely, if ever, proposed as a genuine electoral system, as such a method (in Gibbard's words) "leaves too much to chance". However, the rule is often used as a tiebreaker to encourage voters to cast honest ballots, and is sometimes discussed as a thought experiment. Random dictatorship and random serial dictatorship The dictatorship rule is obviously unfair, b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Participation Criterion
The participation criterion is a voting system criterion that says candidates should never lose an election as a result of receiving too many votes in support. More formally, it says that adding more voters who prefer ''Alice'' to ''Bob'' should not cause ''Alice'' to lose the election to ''Bob''. Voting systems that fail the participation criterion exhibit the no-show paradox, where a voter is effectively disenfranchised by the electoral system because turning out to vote could make the result worse for them; such voters are sometimes referred to as having negative vote weights, particularly in the context of German constitutional law, where courts have ruled such a possibility violates the principle of one man, one vote. Positional methods and score voting satisfy the participation criterion. All deterministic voting rules that satisfy pairwise majority-rule can fail in situations involving four-way cyclic ties, though such scenarios are empirically rare, and the random ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strategyproofness
In mechanism design, a strategyproof (SP) mechanism is a game form in which each player has a weakly- dominant strategy, so that no player can gain by "spying" over the other players to know what they are going to play. When the players have private information (e.g. their type or their value to some item), and the strategy space of each player consists of the possible information values (e.g. possible types or values), a truthful mechanism is a game in which revealing the true information is a weakly-dominant strategy for each player. An SP mechanism is also called dominant-strategy-incentive-compatible (DSIC), to distinguish it from other kinds of incentive compatibility. A SP mechanism is immune to manipulations by individual players (but not by coalitions). In contrast, in a group strategyproof mechanism, no group of people can collude to misreport their preferences in a way that makes every member better off. In a strong group strategyproof mechanism, no group of people can c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
