社会决策 Social Choice Theory
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Social Preferences over Two Alternatives
Social aggregation of preferences
Majority voting
Anonymity
Neutrality
Positive Responsiveness
Theorem 13. (May) A social welfare functional F is a majority voting social welfare functional if and only if it is symmetric, neutral and positive responsive.
Arrow's Impossibility Theorem
Social aggregation of preferences - general case
"Pairwise voting"
Independence of Irrelevant Alternatives
May's Conditions Revised
Pareto optimality
Dictatorship
Theorem 14. Arrow's Impossibility Theorem Suppose that the number of alternatives m ≥ 3 and the domain of admissible individual profiles is either A = Rn or A = Pn : Then every social welfare functional that is Paretian and satisfies the pairwise independence axiom is dictatorial.
Restricted Domains
Single Peaked Preferences
Pairwise Majority Voting
Median agent
Theorem 15.
Social Choice Functions
Majority Voting
Plurality Voting Procedures
Simple Plurality: Given a preference pro le p ∈ Pn the simple plurality voting procedure CP selects an alternative that is preferred by most of the voters to all other alternatives,
Plurality with Run-off: CPR is a plurality voting with a run-off between the top two vote-getters; if no alternative receives a majority of votes on the rst ballot.
Plurality elimination procedures: A number of voting procedures have been proposed which involve the sequential elimination of "undesirable" alternatives, until one alternative is able to obtain a majority of first place votes (so called plurality elimination procedures). The most widely used of them is so called Hare's procedure (Hare(1859)), which reduces to the following: Each voter writes down her preference ordering of the n alternatives, and an alternative is declared the winner if a majority of voters rank it first. If no alternative is ranked first by a majority of the voters, the alternative with the smallest number of first place votes is crossed out from all preference orderings, and first place votes are counted again. This is continued until a single winner is selected.
Another elimination procedure was suggested by Coombs (1964). Coombs proposed to eliminate first not the alternatives with the smallest number of the first place votes, but the alternatives with the largest number of last place votes. As in Hare's case, the procedure is to stop when one alternative receives majority support.
Approval Voting Procedures
The approval social choice function requires of each individual that she partition the set X into two subsets, one of which contains the alternatives she approves, while the other of which contains the alternatives she disapproves. An individual casts one vote for each alternative she approves, and no votes for the other alternatives. For each alternative, the number of votes is summed over the set of individuals, and the alternative that receives at least as large a number of votes as any other is the winner.
Strategic Behaviour and Manipulation
Manipulation
Theorem 16. (Gibbard-Satterthwaite) If X contains more than two elements, and C is non-manipulable and yields a single outcome, then C is a dictatorship.
Theorem 17. The majority voting is not manipulable on the domain of single-peaked preferences P≥.
待续，您所看到的只是我的一些思考和感触……开始更新于2004/01/15
参考资料：
Arrow's impossibility theorem
(Redirected from Arrow's paradox)
In voting systems, Arrow's impossibility theorem, or Arrow's paradox demonstrates the impossibility of designing rules for social decision making that obey a number of 'reasonable' criteria.
The theorem is due to the Bank of Sweden Prize ("Nobel prize in Economics") winning economist Kenneth Arrow, who proved it in his PhD thesis and popularized it in his 1951 book Social Choice and Individual Values.
The theorem's content, somewhat simplified, is as follows. A society needs to agree on a preference order among several different options. Each individual in the society has his or her own personal preference order. The problem is to find a general mechanism, called a social choice function, which transforms the set of preference orders, one for each individual, into a global societal preference order. This social choice function should have several desirable properties:
unrestricted domain or universality: the social choice function should create a complete societal preference order from every possible set of individual preference orders. (The vote must have a result.)
non-imposition or citizen sovereignty: every possible societal preference order should be achievable by some set of individual prereference orders. (Every result must be achievable somehow.)
non-dictatorship: the social choice function should not simply follow the preference order of a single individual while ignoring all others.
positive association of social and individual values or monotonicity: if an individual modifies his or her preference order by promoting a certain option, then the societal preference order should change only by (possibly) promoting that same option. (An individual should not be able to hurt a candidate by ranking it higher.)
independence of irrelevant alternatives: if we restrict attention to a subset of options, and apply the social choice function only to those, then the result should be compatible with the outcome for the whole set of options. (Removing some candidates should not have an effect on the relative ranking of the remaining candidates.)
Arrow's theorem says that such a social choice function does not exist if the number of options is at least 3 and the society has at least 2 members.
Another version of Arrow's theorem can be obtained by replacing the monotonicity criterion with that of:
unanimity or Pareto efficiency: if every individual prefers a certain option to another, then so must the resulting societal preference order.
This statement is stronger, because assuming both monotonicity and independence of irrelevant alternatives implies Pareto efficiency.
With a narrower definition of "irrelevant alternatives" which excludes those candidates in the Smith set, some Condorcet methods meet all the criteria.
See also: Gibbard-Satterthwaite theorem, Voting paradox
Arrow's original proof of his impossibility theorem proceeded in two steps: showing the existence of a decisive voter, and then showing that a decisive voter is a dictator. Barbera replaced the decisive voter with the weaker notion of a pivotal voter, thereby shortening the first step, but complicating the second step. I give three brief proofs, all of which turn on replacing the decisive/pivotal voter with an extremely pivotal voter (a voter who by unilaterally changing his vote can move some alternative from the bottom of the social ranking to the top), thereby simplifying both steps in Arrow's proof. My first proof uses almost no notation, while the second uses May's notation and is extremely brief. The third proof is perhaps the most interesting, because along the way to proving the existence of an extremely pivotal voter, it shows that the Arrow axioms guarantee issue neutrality, that is, that every choice must be made by exactly the same process.