Fractional approval voting

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In fractional social choice, fractional approval voting refers to a class of electoral systems 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_j between 0 and 1, such that the sum of p_j is 1. It can be seen as a generalization of approval voting: in the latter, one candidate wins (p_j = 1) and the other candidates lose (p_j = 0). The fractions p_j can be interpreted in various ways, depending on the setting. Examples are:

• Time sharing: each alternative j is implemented a fraction p_j of the time (e.g. each candidate j serves in office a fraction p_j of the term).^[1]
• Budget distribution: each alternative j receives a fraction p_j of the total budget.^[2]
• Probabilities: after the fractional results are computed, there is a lottery for selecting a single candidate, where each candidate j is elected with probability p_j.^[1]
• Entitlements: the fractional results are used as entitlements (also called weights) in rules of apportionment,^[3] or in algorithms of fair division with different entitlements.

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,^[1] sharing,^[4] portioning,^[3] mixing^[5] and distribution.^[2]