Order, order!

Louie Leventhall

19 April 2026

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Democracy isn’t perfect. Don’t worry readers, I won’t get too political! All I’m saying is that, mathematically, democracy isn’t perfect.

What I’m referring to is the difficulty in making group decisions: how to convert differing individual desires and wishes into a group preference. You may have experienced this when deciding with friends what movie to watch, or restaurant to go to, or what music piece is the best. Since unanimity, the agreement of everyone, is very rare, voting can be held to gauge the support of different proposals within the group and from there make a single decision.

Since voting is often synonymous with politics, let us imagine a situation where a committee of 10 people are considering three tax proposals. For the sake of simplicity let us call them $A$, $B$ and $C$. Yes, very imaginative. To make a group decision, the simplest, and perhaps most obvious way, would be using majority vote, where the proposal with at least half of the support of the committee is chosen. However, there are three proposals, so a strict binary choice cannot immediately be obtained. What can be done is pairwise majority vote, where each proposal is pitted against each other, thus ensuring a binary choice in each vote.

To illustrate this better we’ll use the standard preference notation where, for the proposals $A$ and $B$, $A$ is preferred more than $B$ is denoted $A \succ B$. For the group preference a subscript $g$ is given. Suppose the preferences of the 10 committee members are as below:

Reading this, three members prefer proposal $A$ over $C$, and proposal $C$ over $B$. Four members prefer proposal $C$ over $B$, and proposal $B$ over $A$. And the final three prefer proposal $B$ over $A$, and proposal $A$ over $C$. From this we can deduce that the group prefers $B$ over $A$ by a vote of 7 to 3, denoted $B \succ_g A$. Success! The committee prefers $B$ to $A$.

But hold on, what about proposal $C$? Looking at our table reveals that the group prefers $C$ over $B$ by a vote of 7 to 3 as well. Uh oh. But are they both still preferred over $A$? Consulting the table again reveals that $A$ is in fact preferred to $C$ by a vote of 6 to 4. So $A$ is preferred to $C$ which is preferred to $B$ which is preferred to $A$ which is $\dots$ ah.

This is called a Condorcet cycle, where no single option can win head-to-head against every other option. To visualise this cyclical nature we thankfully have preference notation: $$\cdots \; A \succ_g C \succ_g B \succ_g A \; \cdots$$

Think of a Condorcet cycle like a game of rock paper scissors

While not a common problem within voting, this intransitivity or inconsistency within group preferences can be exploited. Suppose the chair of the committee is aware of this intransitive relation and supports proposal $C$. Could the chair, by majority vote, somehow guarantee $C$ wins but $B$ and $A$ do not?

Using sequential pairwise majority voting, the chair can pair off alternatives to compete in successive runoffs. Under this method, the chair sets an agenda, an ordered list of alternatives, $A$ then $B$ then $C$. $A$ runs off against $B$ and loses, then $B$ runs off against $C$ and loses leaving $C$ the overall winner. By manipulating the order of alternatives or agenda setting, the chair can influence the final result and is therefore an agenda setter.

Arguably, though, this is not a realistic example of agenda setting.

Firstly, the committee members cannot vote on each of the proposals with respect to their views on other issues. Changing the levels of how much the government takes in taxes, will in the long run affect how much it spends and how much it borrows.

Secondly, the members are unable to express the intensity of their feelings towards a proposal. Consider a voter with the preference $A \succ B \succ C$. On the surface this preference suggests that the member prefers $A$ over $B$ just as much as they prefer $B$ over $C$, which might not be exactly what the member feels.

To address these problems, let us imagine a more general situation where voters can consider more than one political issue and express their exact level of support for a proposal. To address the first point, let us use an $n$-dimensional policy space, which houses all the proposals or policies the voters may wish to consider, each dimension representing a political issue.

Each voter will hold some point within this space to be their ideal policy, the policy they would prefer over all others.

To address the second point, each voter will also hold Euclidean preferences, where a voter’s support of a policy decreases with the distance from an ideal point. So given a choice between two alternatives, they will prefer the alternative closer to their ideal point. What makes this type of preference useful is its simplicity and that being Euclidean, based on distances between policies, preferences can be easily illustrated.

The group preference will be defined by majority vote, where the group prefers one policy over another if a majority prefers one policy over another. Can agenda setting be exerted in this context?

For simplicity, suppose five voters lie in a 2D policy space on the issues of increasing levels of taxation and spending. The status quo point $SQ$ acts as a consensus point that all have initially agreed upon.

Since this is 2D, the voters’ Euclidean preferences can be represented by a circle centred on their ideal points with $SQ$ lying on each curve. Given a choice of $SQ$ or any policy within their circle, a voter prefers the policy within their circle. If a majority of voters prefer a policy, it becomes the new consensus. For instance, any policy in the shaded petal is preferred by a majority coalition over $SQ$. As every policy in a shaded petal can be defeated by a policy in a different petal, the group preference relation is intransitive, just as we saw with the committee example before.

The intransitivity in voter preferences is something an agenda setter can exploit to their advantage. If they wished to shift the consensus from $SQ$ to $D$, since $D$ lies outside the win set, the union of all shaded petals, the group would reject it. This is denoted with the standard group preference notation, $SQ \succ_g D$. Yet, by proposing alternatives sequentially, the consensus can be gradually shifted from $SQ$ to $D$.

The figures on the right show this shift in action. Suppose the agenda setter proposes policy $A$, which lies in the petal of coalition $\{2,4,5\}$, with the aim of getting the consensus policy closer to $D$. In proposing policy $A$, which lies within the win set, $A$ is preferred by the majority $\{ 2,4,5 \}$ over $SQ$ (step 1). With a new consensus, voter preferences and thus the circle shapes change. In particular, as shown in step 2, it changes in such a way that the agenda setter cannot shift the consensus closer to $D$. However, by proposing a series of varying policies, the agenda setter could engulf the proposal $D$ within the win set. Now suppose the agenda setter proposes $B$, which lies in the petal of coalition $\{1,3,4\}$. With a majority holding $B$ over $A$, the win set changes again as shown in step 3 and the agenda setter can propose policy $C$, which lies within the petal of $\{1,2,3\}$. Once more $C$ holds majority support over $B$ so the win set changes such that it encompasses the policy $D$ as shown in step 4. By proposing more policies from the win set, eventually policy $D$ would be accepted by the group!

Preference notation allows us to write this more nicely:$$\cdots\; D \succ_g C \succ_g B \succ_g A \succ_g SQ \succ_g D \;\cdots$$

Of course, this type of manipulation fails to work if the consensus cannot be moved at all. If a policy is unable to be defeated by another, we call it a core point and the set of all core points, the core.

Consider five voters in a 1D policy space, where $SQ$ lies on voter 3’s ideal point.

Here the group would prefer $SQ$ to $A$ since $SQ$ is supported by the majority coalition $\{1,2,3\}$. In fact no other alternative in this dimension can beat $SQ$. The reason being that every potential majority coalition designed to defeat $SQ$ must contain the support of the median voter, but since $SQ$ lies on the median voter, it could never be defeated! So the median voter’s ideal point forms the core.

Beyond the 1D case though, the core is almost always non-existent. To explain this we must consider how the core can be defined through the convex hull of every majority coalition of voters.

Since every policy outside a winning coalition would be defeated by some point within its convex hull, the common intersection of them must form the core. Yet as the number of dimensions increases, these convex hulls are stretched out across the policy space, making it harder for them to intersect, leaving an empty core.

Under some circumstances, a core can be forced into existence. A Plott configuration is an arrangement of ideal points where pairs of these points are positioned about the median voter’s in order to force a core point. The 1D case is the simplest example of this. To create this configuration in more than one dimension:

• Start by placing $n$ voter ideal points in a straight line.
• If $n$ is odd, the median voter’s ideal point forms the core, if even, add a fictional median voter.

• Split the remaining $n-1$ points into pairs such that the two points lie colinearly on either side of the median voter’s ideal point. The points in the pair need not be equidistant from the median.

• Finally, rotate each pair through the dimensions about the median voter’s ideal point.

• By overlaying every voter’s preference curve, we find that the win set is empty and can confirm the median voter’s ideal point forms a core point.

Note, however, that any slight change in ideal points can cause the configuration, and thus the core, to collapse.

The importance of the core in group decision making was demonstrated by Richard McKelvey who proved that if the core exists, preferences are transitive and if not, they are intransitive. Through this he established his seminal theorem on agenda setting.

The McKelvey chaos theorem states that given a policy space of at least two dimensions and at least three voters, each with Euclidean preferences, then in the absence of the core, there exists an agenda such that one can get from the first item of the agenda to the last in such a way that every item on the agenda will be defeated by the next item by majority vote.

In other words if the core is empty, any policy is reachable! This is what we saw in the 2D example. An agenda setter was able to gradually shift $SQ$ to $D$ since the core was empty.

This is a very powerful result which shows that even a simple binary choice is not beyond manipulation. However, please do not lose your faith in democratic process! This theorem does rely on a few assumptions for it to work:

1. The agenda setter holds complete information on everyone’s preferences.
2. Voters don’t ‘give up’ on assigning preferences and default to being indifferent between policies.
3. Everyone votes in accordance to their true preferences and without collusion.
4. Given the first and final items are fixed on the agenda, any ordering is permissible.
5. No constraint exists on how many items are allowed on the agenda or on how much they may differ from one another.

These conditions are relatively weak and don’t match up exactly with real world decision making.

Yet it provides a useful caution as to the type of manipulation that can occur in group decision making from an agenda setter with a sufficient amount of power. Next time you find yourself in a series of votes, keep an eye on the person leading it. Who knows? Maybe they have an agenda they wish to implement.

Louie Leventhall

Louie is a recent Durham University graduate who did a masters in mathematics and statistics. He currently works as a underwriter. Outside of work he enjoys reading, doing cryptic crosswords and watching baseball.

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