Rigging an election can be hard. But sometimes it can be easy. If a committee has an agenda of proposals to choose from, where each proposal is compared pairwise using majority rule against another proposal, until a single proposal is victorious, then you can make any arbitrary proposal win. The McKelvey–Schofield chaos theorem tells us that just by manipulating the agenda – adding more proposals and deciding which order to do the pairwise elections in – we can rig the vote. But what exactly does that mean and can it be done in practice?
Hypothetical
Imagine you were the leader of a committee, deciding which budget proposal to use for an upcoming year. Let's say you have an agenda of budget proposals, submitted by members of the committee. A natural way to choose between them is to compare budget proposals in a sequence, head-to-head, choosing the winner using majority rule until there's a single budget proposal left.[1] That's kind of what parliaments do when voting on laws, so it should be fine to use in your committee.
Let's also say that members can submit multiple budget proposals and that you, as the committee leader, can manipulate the agenda, i.e. you submit budget proposals last and select the sequence of head-to-head votes to hold. You can't remove any proposals though, so if someone submits a really good proposal, then it has to be included in at least one of the head-to-head votes.
One might then think that if the committee submits a really good proposal, then that has a high chance of winning. But, if the committee votes in a predictable manner, you will be able to manipulate the agenda such that you can choose any budget as the winner. To understand why, we need to mathematically model how the committee members vote.
Mathematical
Let us assume that all proposals can be defined as a point in n dimensional Euclidean space, En, where n is larger than 1. How a committee votes depends on their utility functions. Let's say each member i in the committee has a utility function Ui:En→R which returns how much that member likes a policy.
The simplest case is in the Euclidean plane (n=2) and when each committee member values proposals proportionally to their Euclidean distance from some ideal proposal. Even this simple case has problems. One can for example encounter Condorcet cycles, where it is unclear which proposal the committee prefer.
Figure 1. An example of a Condorcet cycle. 3 members have ideal proposals (1,2,3) and are considering 3 proposals (A,B,C). Arrows show which would win in a majority vote. (source)
But one might hope that even if Condorcet cycles are possible, they are rare or don't affect votes much. Maybe there's a set of proposals which the committee will clearly support, which we cannot manipulate the agenda to deviate from. Unfortunately Richard McKelvey showed in 1976 that Condorcet cycles are abundant and that it's pretty much always possible to manipulate the agenda.
I won't go through the proof, but one can get an intuition why it would be true.[2] If we consider some specific policy, every committee member will have a circle of proposals they prefer more. Any proposal which is in the intersection of a majority of those circles will beat the original proposal.
Figure 2. Three committee members evaluating a proposal. The white dot is a policy and the lower right box highlights the section of policies which would win over it in a vote. (source)
Every proposal will then have many different proposals it would lose to. By having a carefully chosen series of proposals which win against the previous proposal, we can choose a victorious proposal which is arbitrarily far away from the original proposal.
One might say that real people don't have these kinds of circular utility functions, but Norman Schofield proved similar – but more technical – results in 1978 which apply to a wide range of differentiable utility functions.
Critical
So when can the agenda not be manipulated? Well, if the proposals exist on a one-dimensional political spectrum, then the theorems don't apply.[3] If all utilities have a single peak, then we even have the median voter theorem, which says that the winning proposal will be the proposal preferred by the median voter.
There are also a bunch of technical situations where the theorems don't apply, when there's some kind of equilibrium forcing a single policy to be the winning proposal. But the reason the theorem is named the McKelvey–Schofield chaos theorem is because small changes to utility functions with an equilibrium, makes agenda manipulation work again.
Analytical
So what should the takeaway be? Are these mathematical models realistic? Can these results actually be used to rig an election? Well, it probably wouldn't be as easy in the real world as in the mathematical, as not every committee member's utility function is simple and predictable. People would also probably notice if someone tried to manipulate the agenda. It's more likely that these problems would appear by mistake, than from someone deliberately trying to rig anything.
I think this just highlights how complicated it can be to aggregate a group's preferences. It indicates that Condorcet cycles are common in majority rule, and that one shouldn't be that surprised if they appear. Alternatively, if Condorcet cycles don't appear then people have more complicated behavior or utility functions than in the mathematical models. These ideas may be more applicable if you're trying to aggregate preferences from machines.
The easiest way to choose a proposal would be to use first-past-the-post, but it would be bad to use in a situation where there can be several very similar voting proposals, because of the spoiler effect.
Rigging an election can be hard. But sometimes it can be easy. If a committee has an agenda of proposals to choose from, where each proposal is compared pairwise using majority rule against another proposal, until a single proposal is victorious, then you can make any arbitrary proposal win. The McKelvey–Schofield chaos theorem tells us that just by manipulating the agenda – adding more proposals and deciding which order to do the pairwise elections in – we can rig the vote. But what exactly does that mean and can it be done in practice?
Hypothetical
Imagine you were the leader of a committee, deciding which budget proposal to use for an upcoming year. Let's say you have an agenda of budget proposals, submitted by members of the committee. A natural way to choose between them is to compare budget proposals in a sequence, head-to-head, choosing the winner using majority rule until there's a single budget proposal left.[1] That's kind of what parliaments do when voting on laws, so it should be fine to use in your committee.
Let's also say that members can submit multiple budget proposals and that you, as the committee leader, can manipulate the agenda, i.e. you submit budget proposals last and select the sequence of head-to-head votes to hold. You can't remove any proposals though, so if someone submits a really good proposal, then it has to be included in at least one of the head-to-head votes.
One might then think that if the committee submits a really good proposal, then that has a high chance of winning. But, if the committee votes in a predictable manner, you will be able to manipulate the agenda such that you can choose any budget as the winner. To understand why, we need to mathematically model how the committee members vote.
Mathematical
Let us assume that all proposals can be defined as a point in n dimensional Euclidean space, En, where n is larger than 1. How a committee votes depends on their utility functions. Let's say each member i in the committee has a utility function Ui:En→R which returns how much that member likes a policy.
The simplest case is in the Euclidean plane (n=2) and when each committee member values proposals proportionally to their Euclidean distance from some ideal proposal. Even this simple case has problems. One can for example encounter Condorcet cycles, where it is unclear which proposal the committee prefer.
But one might hope that even if Condorcet cycles are possible, they are rare or don't affect votes much. Maybe there's a set of proposals which the committee will clearly support, which we cannot manipulate the agenda to deviate from. Unfortunately Richard McKelvey showed in 1976 that Condorcet cycles are abundant and that it's pretty much always possible to manipulate the agenda.
I won't go through the proof, but one can get an intuition why it would be true.[2] If we consider some specific policy, every committee member will have a circle of proposals they prefer more. Any proposal which is in the intersection of a majority of those circles will beat the original proposal.
Every proposal will then have many different proposals it would lose to. By having a carefully chosen series of proposals which win against the previous proposal, we can choose a victorious proposal which is arbitrarily far away from the original proposal.
One might say that real people don't have these kinds of circular utility functions, but Norman Schofield proved similar – but more technical – results in 1978 which apply to a wide range of differentiable utility functions.
Critical
So when can the agenda not be manipulated? Well, if the proposals exist on a one-dimensional political spectrum, then the theorems don't apply.[3] If all utilities have a single peak, then we even have the median voter theorem, which says that the winning proposal will be the proposal preferred by the median voter.
There are also a bunch of technical situations where the theorems don't apply, when there's some kind of equilibrium forcing a single policy to be the winning proposal. But the reason the theorem is named the McKelvey–Schofield chaos theorem is because small changes to utility functions with an equilibrium, makes agenda manipulation work again.
Analytical
So what should the takeaway be? Are these mathematical models realistic? Can these results actually be used to rig an election? Well, it probably wouldn't be as easy in the real world as in the mathematical, as not every committee member's utility function is simple and predictable. People would also probably notice if someone tried to manipulate the agenda. It's more likely that these problems would appear by mistake, than from someone deliberately trying to rig anything.
I think this just highlights how complicated it can be to aggregate a group's preferences. It indicates that Condorcet cycles are common in majority rule, and that one shouldn't be that surprised if they appear. Alternatively, if Condorcet cycles don't appear then people have more complicated behavior or utility functions than in the mathematical models. These ideas may be more applicable if you're trying to aggregate preferences from machines.
The easiest way to choose a proposal would be to use first-past-the-post, but it would be bad to use in a situation where there can be several very similar voting proposals, because of the spoiler effect.
For another visual example, see The Mathematical Danger of Democratic Voting.
Such as the left-right spectrum.