Fractional approval voting

It appears in the literature under many different terms: lottery,[1] sharing,[4] portioning,[3] mixing[5] and distribution.

Ex-post PE is a weaker property, relevant only for the interpretation of a mixture as a lottery.

For example, suppose there are 5 candidates (a,b,c,d,e) and 6 voters with approval sets (ac, ad, ae, bc, bd, be).

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.

Individual-FS[5] (also called Fair Welfare Share[1]) 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[1] 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[3]) 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[5] 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[1]: 2002draft  (also called proportional sharing[4]) 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.

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.

The utilitarian rule has several desirable properties:[1]: Prop.1  it is anonymous, neutral, PE, individual-SP, and preference-monotone.

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.

[1]: Thm.3 in 2002 draft The egalitarian (leximin) rule maximizes the smallest utility, then the next-smallest, etc.

It is anonymous and neutral, and satisfies the following additional properties:[5] For any monotonically increasing function f, one can maximize the sum of f(ui).

It is anonymous, neutral, and satisfies the following additional properties:[1]: Prop.5 A disadvantage of this rule is that it is computationally-hard to find the exact probabilities (see Dictatorship mechanism#Computation).

It is anonymous and neutral, and satisfies the following additional properties: The majoritarian rule[8] aims to concentrate as much power as possible in the hands of few candidates, while still guaranteeing fairness.