Multi-issue voting is a setting in which several issues have to be decided by voting. Multi-issue voting raises several considerations, that are not relevant in single-issue voting.

The first consideration is attaining fairness both for the majority and for minorities. To illustrate, consider a group of friends who decide each evening whether to go to a movie or a restaurant. Suppose that 60% of the friends prefer movies and 40% prefer restaurants. In a one-time vote, the group will probably accept the majority preference and go to a movie. However, making the same decision again and again each day is unfair, since it satisfies 60% of the friends 100% of the time, while the other 40% are never satisfied. Considering this problem as multi-issue voting allows attain a fair sequence of decisions by going 60% of the evenings to a movie and 40% of the evenings to a restaurant. The study of fair multi-issue voting mechanisms is sometimes called fair public decision making. The special case in which the different issues are decisions in different time-periods, and the number of time-periods is not known in advance, is called perpetual voting.

The second consideration is the potential dependence between the different issues. For example, suppose the issues are two suggestions for funding public projects. A voter may support funding each project on its own, but object to funding both projects simultaneously, due to its negative influence on the city budget. If there are only few issues, it is possible to ask each voter to rank all possible combinations of candidates. However, the number of combinations increases exponentially in the number of issues, so it is not practical when there are many issues. The study of this setting is sometimes called combinatorial voting.

There are several issues to be decided on. For each issue t, there is a set C_t of candidates or alternatives to choose from. For each issue t, a single candidate from C_t should be elected. Voters may have different preferences regarding the candidates. The preferences can be numeric (cardinal ballots) or ranked (ordinal ballots) or binary (approval ballots). In combinatorial settings, voters may have preferences over combinations of candidates.

A multi-issue voting rule is a rule that takes the voters' preferences as an input, and returns the elected candidate for each issue. Multi-issue voting can take place offline or online:

With cardinal ballots, each voter assigns a numeric utility to each alternative in each round. The total utility of a voter is the sum of utilities he assigns to the elected candidates in each round.

Conitzer, Freeman and Shah studied multi-issue voting with offline cardinal ballots (they introduced the term public decision making). They focus on fairness towards individual agents. A natural fairness requirement in this setting is proportional division, by which each agent should receive at least 1/n of their maximum utility. Since proportionality might not be attainable, they suggest three relaxations:

These relaxations make sense when the number of voters is small and the number of issues is large, so a difference of one issue is small w.r.t. 1/n. They show that the Maximum Nash Welfare solution (maximizing the product of all agents' utilities) satisfies or approximates all three relaxations. They also provide polynomial time algorithms and hardness results for finding allocations satisfying these axioms, with or without Pareto efficiency.