of possible 'states of the world' or 'alternatives'. Society wishes to choose a single state from . For example, in a single-winner election, may represent the set of candidates; in a resource allocation setting, may represent all possible allocations. Let be a finite set, representing a collection of individuals. For each , let be a utility function, describing the amount of happiness an individual i derives from each possible state. A social choice rule is a mechanism which uses the data to select some element(s) from which are 'best' for society. The question of what 'best' means is the basic question of social choice theory. The following rules are most common: The utilitarian rule - also called the max-sum rule - aims to maximize the sum of utilities, thus maximizing the efficiency. The egalitarian rule - also called the max-min rule - aims to maximize the smallest utility, thus maximizing the fairness. The proportional-fair rule - sometimes called the max-product rule - aims to balance between the previous two rules, attaining a balance between efficiency and fairness. Functions A social choice function or a voting rule takes an individuals' complete and transitive preferences over a set of candidates (also called alternatives), and returns some subset of (possible singular) the candidates. We can think of this subset as the winners of an election. This is different from social welfare function, which returns a linear order of the set of alternatives as opposed to simply selecting some subset. We can compare different social choice functions based on which axioms or mathematical properties they fulfill. For example, Instant-runoff voting satisfies the Independence of clones criterion, whereas the Borda count does not; conversely, Borda Count satisfies the Monotonicity criterion whereas IRV does not. Theorems Arrow's impossibility theorem is what often comes to mind when one thinks about impossibility theorems in voting. However, Arrow was concerned with social welfare functions, not social choice functions. There are several famous theorems concerning social choice functions. The Gibbard–Satterthwaite theorem states that all non-dictatorial voting rules that is resolute (it always returns a single winner no matter what the ballots are) and non-imposed (every alternative could be chosen) with more than three alternatives (candidates) is manipulable. That is, a voter can cast a ballot that misrepresents their preferences to obtain a result that is more favorable to them under their sincere preferences. The Campbell-Kelley theorem states that, if there