The purpose of most political elections is to select from the set of candidates a predetermined number $n$ of successful candidates. (Often $n = 1$.) In brief, my question is:

Has it been proved that no voting procedure whatsoever will allow this to be done in a reasonable way?

For comparison, Arrow's theorem states (roughly) that when each voter puts a total order on the set $C$ of candidates, there is no good way to average those orders to produce an overall total order on $C$. This result has of course been enormously influential, spawning many other voting impossibility theorems. However, it does not model the most common real-life situation, where the output we want is a subset of $C$ of cardinality $n$ (the top $n$ candidates, whatever "top" means) rather than a total order on $C$.

Let me emphasize how general my question is. Suppose there are 25 candidates, 3 of whom are to be elected as our representatives. Is there anything whatsoever that voters could be asked to do in the polling booth that could then be processed in some way to select the best 3, in such a way that reasonable conditions hold? By "reasonable conditions" I mean the usual kind that appear in social choice theory, e.g. in Arrow's theorem or the Gibbard- Satterthwaite theorem: non-dictatorship, tactical voting unhelpful, etc.

For instance, perhaps each voter has to mark each candidate out of 10. Or perhaps voters are allowed to make various statements such as "if X is elected but Y is not then Z should be". Or perhaps voters put a partial order on the set of candidates, and whenever they prefer X to Y, they choose a real number specifying how much they prefer X to Y. There are endless possibilities, and my question is whether there's some theorem stating that no matter what voters are required to do, there's no good way to select the top $n$ candidates.

Edit In response to comments, let me emphasize further the generality of this question. Many existing impossibility theorems are of the following form: if each voter provides input of some given type, then there is no good way to produce from it an output of another given type. For instance, in Arrow's theorem, both the input and the output type are "total order on the set of candidates". I'm fixing the output type (cardinality-$n$ subset of the set of candidates, for some fixed $n$) and looking for a theorem stating that no matter what input type is used, there is no good system.

The purpose of most political elections is to select from the set of candidates a predetermined number $n$ of successful candidates. (Often $n = 1$.) In brief, my question is:

Has it been proved that no voting procedure whatsoever will allow this to be done in a reasonable way?

For comparison, Arrow's theorem states (roughly) that when each voter puts a total order on the set $C$ of candidates, there is no good way to average those orders to produce an overall total order on $C$. This result has of course been enormously influential, spawning many other voting impossibility theorems. However, it does not model the most common real-life situation, where the output we want is a subset of $C$ of cardinality $n$ (the top $n$ candidates, whatever "top" means) rather than a total order on $C$.

Let me emphasize how general my question is. Suppose there are 25 candidates, 3 of whom are to be elected as our representatives. Is there anything whatsoever that voters could be asked to do in the polling booth that could then be processed in some way to select the best 3, in such a way that reasonable conditions hold? By "reasonable conditions" I mean the usual kind that appear in social choice theory, e.g. in Arrow's theorem or the Gibbard- Satterthwaite theorem: non-dictatorship, tactical voting unhelpful, etc.

For instance, perhaps each voter has to mark each candidate out of 10. Or perhaps voters are allowed to make various statements such as "if X is elected but Y is not then Z should be". Or perhaps voters put a partial order on the set of candidates, and whenever they prefer X to Y, they choose a real number specifying how much they prefer X to Y. There are endless possibilities, and my question is whether there's some theorem stating that no matter what voters are required to do, there's no good way to select the top $n$ candidates.

Edit In response to comments, let me emphasize further the generality of this question. Many existing impossibility theorems are of the following form: if each voter provides input of some given type, then there is no good way to produce from it an output of another given type. For instance, in Arrow's theorem, both the input and the output type are "total order on the set of candidates". I'm fixing the output type (cardinality-$n$ subset of the set of candidates, for some fixed $n$) and looking for a theorem stating that no matter what input type is used, there is no good system. I don't want to restrict to total orders, or input types that can be derived from total orders. Anything!

The purpose of most political elections is to select from the set of candidates a predetermined number $n$ of successful candidates. (Often $n = 1$.) In brief, my question is:

Has it been proved that no voting procedure whatsoever will allow this to be done in a reasonable way?

For comparison, Arrow's theorem states (roughly) that when each voter puts a total order on the set $C$ of candidates, there is no good way to average those orders to produce an overall total order on $C$. This result has of course been enormously influential, spawning many other voting impossibility theorems. However, it does not model the most common real-life situation, where the output we want is a subset of $C$ of cardinality $n$ (the top $n$ candidates, whatever "top" means) rather than a total order on $C$.

Let me emphasize how general my question is. Suppose there are 25 candidates, 3 of whom are to be elected as our representatives. Is there anything whatsoever that voters could be asked to do in the polling booth that could then be processed in some way to select the best 3, in such a way that reasonable conditions hold? By "reasonable conditions" I mean the usual kind that appear in social choice theory, e.g. in Arrow's theorem or the Gibbard- Satterthwaite theorem: non-dictatorship, tactical voting unhelpful, etc.

For instance, perhaps each voter has to mark each candidate out of 10. Or perhaps voters are allowed to make various statements such as "if X is elected but Y is not then Z should be". Or perhaps voters put a partial order on the set of candidates, and whenever they prefer X to Y, they choose a real number specifying how much they prefer X to Y. There are endless possibilities, and my question is whether there's some theorem stating that no matter what voters are required to do, there's no good way to select the top $n$ candidates.

The purpose of most political elections is to select from the set of candidates a predetermined number $n$ of successful candidates. (Often $n = 1$.) In brief, my question is:

Has it been proved that no voting procedure whatsoever will allow this to be done in a reasonable way?

For comparison, Arrow's theorem states (roughly) that when each voter puts a total order on the set $C$ of candidates, there is no good way to average those orders to produce an overall total order on $C$. This result has of course been enormously influential, spawning many other voting impossibility theorems. However, it does not model the most common real-life situation, where the output we want is a subset of $C$ of cardinality $n$ (the top $n$ candidates, whatever "top" means) rather than a total order on $C$.

Let me emphasize how general my question is. Suppose there are 25 candidates, 3 of whom are to be elected as our representatives. Is there anything whatsoever that voters could be asked to do in the polling booth that could then be processed in some way to select the best 3, in such a way that reasonable conditions hold? By "reasonable conditions" I mean the usual kind that appear in social choice theory, e.g. in Arrow's theorem or the Gibbard- Satterthwaite theorem: non-dictatorship, tactical voting unhelpful, etc.

For instance, perhaps each voter has to mark each candidate out of 10. Or perhaps voters are allowed to make various statements such as "if X is elected but Y is not then Z should be". Or perhaps voters put a partial order on the set of candidates, and whenever they prefer X to Y, they choose a real number specifying how much they prefer X to Y. There are endless possibilities, and my question is whether there's some theorem stating that no matter what voters are required to do, there's no good way to select the top $n$ candidates.

Edit In response to comments, let me emphasize further the generality of this question. Many existing impossibility theorems are of the following form: if each voter provides input of some given type, then there is no good way to produce from it an output of another given type. For instance, in Arrow's theorem, both the input and the output type are "total order on the set of candidates". I'm fixing the output type (cardinality-$n$ subset of the set of candidates, for some fixed $n$) and looking for a theorem stating that no matter what input type is used, there is no good system.