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A social choice function, sometimes called a voting system in the context of politics, is a rule that takes an individual's complete and transitive preferences over a set of outcomes and returns a single chosen outcome (or a set of tied outcomes). We can think of this subset as the winners of an election, and compare different social choice ...
A VCG mechanism implements a utilitarian social-choice function - a function that maximizes a weighted sum of values (also called an affine maximizer). Roberts' theorem proves that, if: The agents' valuation functions are unrestricted (each agent can have as value function any function from to ), and -
The upper-left space depicts the type space and the upper-right space X the space of outcomes. The social choice function () maps a type profile to an outcome. In games of mechanism design, agents send messages in a game environment .
When agents have general preferences represented by cardinal utility functions, the utilitarian social-choice function (selecting the outcome that maximizes the sum of the agents' valuations) is not strongly-monotonic but it is weakly monotonic. Indeed, it can be implemented by the VCG mechanism, which is a truthful mechanism with money.
Arrow's theorem assumes as background that any non-degenerate social choice rule will satisfy: [15]. Unrestricted domain – the social choice function is a total function over the domain of all possible orderings of outcomes, not just a partial function.
A random social choice function (RSCF) takes as input the set of voters' preference relations. It returns as output a "mixture" - a vector p of real numbers in [0,1], one number for each candidate, such that the sum of numbers is 1.
A social-choice-function is a function that maps a set of individual preferences to an optimal social outcome. An example function is the utilitarian rule, which says "give the item to a person that values it the most". We denote a social choice function by Soc and its recommended outcome given a set of preferences by Soc(Prefs). A mechanism is ...
Less informally, the social choice function is the function mapping each environment S of available social states (at least two) for any given set of orderings (and corresponding social ordering R) to the social choice set, the set of social states each element of which is top-ranked (by R) for that environment and that set of orderings.