Fractional social choice

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Fractional, stochastic, or weighted social choice is a branch of social choice theory in which the collective decision is not a single alternative, but rather a weighted sum of two or more alternatives.^[1] For example, if society has to choose between three candidates (A, B, or C), then in standard social choice exactly one of these candidates is chosen. By contrast, in fractional social choice it is possible to choose any linear combination of these, e.g. "2/3 of A and 1/3 of B".^[2]

A common interpretation of the weighted sum is as a lottery, in which candidate A is chosen with probability 2/3 and candidate B is chosen with probability 1/3. The rule can also be interpreted as a recipe for sharing, for example:

• Time-sharing: candidate A is (deterministically) chosen for 2/3 of the time while candidate B is chosen for 1/3 of the time.
• Budget-distribution: candidate A receives 2/3 of the budget while candidate B receives 1/3 of the budget.
• Fair division with different entitlements can also be used to divide a heterogeneous resource between candidates A and B, with their entitlements being 2/3 and 1/3.