Fractional approval voting
Fractional approval voting is an electoral system using approval ballots (each voter selects one or more candidate alternatives), in which the outcome is "fractional": for each alternative "j" there is a fraction "p" between 0 and 1, such that the sum of "p" is 1. It can be seen as a generalization of approval voting: in the latter, one candidate wins ("p" = 1) and the other candidates lose ("p" = 0). The fractions "p" can be interpreted in various ways, depending on the setting. Examples are: Fractional approval voting is a special case of fractional social  choice in which all voters have "dichotomous preferences". It appears in the literature under many different terms: lottery, sharing, portioning, mixing and distribution. Formal definitions. There is a finite set "C" of "candidates" (also called: "outcomes" or "alternatives"), and a finite set "N" of "n voters" (also called: "agents"). Each voter "i" specifies a subset "A" of "C", which represents the set of candidates that the voter "approves". A "fractional approval voting" rule takes as input the set of sets "A", and returns as output a "mixture" (also called: "distribution" or "lottery") - a vector "p" of real numbers in [0,1 ], one number for each candidate, such that the sum of numbers is 1. It is assumed that each agent "i" gains a utility of 1 from each candidate in his approval set "A", and a utility of 0 from each candidate not in "A". Hence, agent "i" gains from each mixture "p", a utility of formula_1. For example, if the mixture "p" is interpreted as a budget distribution, then the utility of "i" is the total budget allocated to outcomes he likes. Desired properties. Basic properties. Two basic properties of voting rules in general, and fractional-approval-voting rules in  particular, are: Efficiency properties. Pareto-efficiency (PE) means no mixture gives a higher utility to one agent and at least as high utility to all others. Ex-post PE is a weaker property, relevant only for the interpretation of a mixture as a lottery. It means that, after the lottery, no outcome gives a higher utility to one agent and at least as high utility to all others (in other words, it is a mixture over PE outcomes). For example, suppose there are 5 candidates (a,b,c,d,e) and 6 voters with approval sets (ac, ad, ae, bc , bd, be). Selecting any single candidate is PE, so every lottery is ex-post PE. But the lottery selecting c,d,e with probability 1/3 each is not PE, since it gives an expected utility of 1/3 to each voter, while the lottery selecting a,b with probability 1/2 each gives an expected utility of 1/2 to each voter. PE always implies ex-post PE. The opposite is also true in the following cases: Fairness properties. Fairness requirements are captured by variants of the notion of fair share (FS). Individual"-"FS (also called Fair Welfare Share) means that the utility of  each voter "i" is at least 1/"n", that is, at least 1/"n" of the budget is allocated to candidates approved by "i". Individual-Outcome-FS means that the utility of each voter "i" is at least his utility in a lottery that selects a candidate randomly, that is, at least "k"/|"C"|, where "k" is the number of candidates approved by "i". Single-vote-FS (also called faithful) means that, if each voter approves a single candidate, then the fraction assigned to each candidate "j" equals the number of voters who approve "j" divided by "n". Unanimous-FS means that, for  each set "S" of voters with "identical" preferences, the utility of each member in "S" is at least |"S"|/"n." Group-FS (also callde "proportional sharing") means that, for each voter set "S", the total budget allocated to candidates approved by "at least one" member of "S", is at least |"S"|/"n." Average-FS means that, for each voter set "S" with at least one approved candidate in common, the average utility of voters in "S" is at least |"S"|/"n." Core-FS means that, for each voter set "S", there is no other distribution of their |"S"|/"n" budget, which gives all members  of "S" a higher utility. Strategic properties. Several variants of strategyproofness (SP) have been studied for voting rules: A weaker variant of SP is excludable SP. It is relevant in situations where it is possible to exclude voters from using some candidate alternatives. For example, if the candidates are meeting times, then it is possible to exclude voters from participating in the meeting in times which they did not approve. This makes it harder to manipulate, and therefore, the requirement is weaker. Participation properties. Rules should encourage voters to participate in the voting process. Several participation criteria have been studied : A stronger property is required in participatory budgeting settings in which the budget to distribute is donated by the voters themselves: Rules. Utilitarian rule. The utilitarian rule aims to maximize the sum of utilities, and therefore it distributes the entire budget among the candidates approved by the largest number of voters. In particular, if there is one candidate with the largest number of votes, then this candidate gets 1 (that is, all the budget) and the others get 0, as in single-winner approval voting. If there are some "k" candidates with the same largest number of votes, then the  budget is distributed equally among them, giving 1/"k" to each such candidate and 0 to all others. The utilitarian rule has several desirable properties: it is anonymous, neutral, PE, individual-SP, and preference-monotone. It is also easy to compute. However, it is not fair towards minorities - it violates Individual-FS (as well as all stronger variants of FS). For example, if 51% of the voters approve X and 49% of the voters approve Y, then the utilitarian rule gives all the budget to X and no budget at all to Y, so the 49% who vote for Y get  a utility of 0. In other words, it allows for tyranny of the majority. The utilitarian rule is also not weak-group-SP (and hence not group-SP). For example, suppose there are 3 candidates (a,b,c) and 3 voters, each of them approves a single candidate. If they vote sincerely, then the utilitarian mixture is (1/3,1/3,1/3) so each agent's utility is 1/3. If a "single" voter votes insincerely (say, the first one votes for both a and b), then the mixture is (0,1,0), which is worse for the insincere voter. However, if "two" voters collude and vote  insincerely (say, the first two voters vote for the first two outcomes), then the utilitarian mixture is (1/2, 1/2, 0), which is "better" for both insincere voters. Nash-optimal rule. The Nash-optimal rule maximizes the sum of "logarithms" of utilities. It is anonymous and neutral, and satisfies the following additional properties: The Nash-optimal rule can be computed by solving a convex program. There is another rule, called fair utilitarian, which satisfies similar properties (PE and group-FS) but is easier to compute. Egalitarian rule. The egalitarian (leximin) rule maximizes the smallest utility, then the next-smallest, etc. It  is anonymous and neutral, and satisfies the following additional properties: Other welfarist rules. For any monotonically-increasing function "f", one can maximize the sum of "f"("u"). The utilitarian rule is a special case where f("x")="x", and the Nash rule is a special case where f("x")=log("x"). Every "f"-maximizing rule is PE, and has the following additional properties: Priority rules. A priority rule (also called "serial dictatorhip") is parametrized by a permutation of the voters, representing a priority ordering. It selects an outcome that maximizes the utility of the highest-priority agent; subject to that, maximizes the utility of the second-highest -priorty agent; and so on. Every priority rule is neutral, PE, weak-group-SP, and preference-monotone. However, it is not anonymous and does not satisfy any fairness notion. The random priority rule selects a permutation of the voters uniformly at random, and then implements the priority rule for that permutation. It is anonymous, neutral, and satisfies the following additional properties: A disadvantage of this rule is that it is computationally-hard to find the exact probabilities (see Dictatorship mechanism#Computation). Conditional utilitarian rule. In the conditional utilitarian rule, each agent receives 1/"n" of the total budget. Each agent finds, among  the candidates that he approves, those that are supported by the largest number of other agents, and splits his budget equally among them. It is anonymous and neutral, and satisfies the following additional properties: Majoritarian rule. The majoritarian rule aims to concentrate as much power as possible in the hands of few candidates, while still guaranteeing fairness. It proceeds in rounds. Initially, all candidates and voters are active. In each round, the rule selects an active candidate "c" who is approved by the largest set of active voters, "N". Then, the rule "assigns" these voters "N" to "c", that is , it assumes that voters in "N" voted "only" for "c", and assigns c the fraction "|N"|/n. Then, the candidate "c" and the voters in "N" become inactive, and the rule proceeds to the next round. Note that the conditional-utilitarian rule is similar, except that the voters in "N" do not become inactive. The majoritarian rule is anonymous, neutral, guarantees individual-FS and single-vote-FS. Impossibility results. Some combinations of properties cannot be attained simultaneously. Summary table. In the table below, the number in each cell represents the "strength" of the property: 0 means none (the property is  not satisfied); 1 corresponds to the weak variant of the property; 2 corresponds to a stronger variant; etc.