Handful Apps / Articles / A majority can be walked anywhere
McKelvey's chaos theorem says that when no position is stable, a chairman who controls the order of business can walk a majority to almost anywhere, one fair vote at a time. That claim sounds like rhetoric. It is a theorem, and it can be shown with numbers.
Five voters, one vote each, simple majority. Every voter has an ideal position on a two-dimensional map, and prefers whatever is closer to that ideal. No strategy, no logrolling, no bargaining. Each voter simply votes for the nearer of the two proposals in front of them.
The positions, on a map running from −100 to +100 on both axes:
| Voter | Position |
|---|---|
| 1 | (−20, 20) |
| 2 | (−60, −20) |
| 3 | (20, −60) |
| 4 | (60, 40) |
| 5 | (80, 20) |
Today's policy sits at (14, −6), roughly in the middle of the group. The chairman wants to end up at (−24, 92), which is far away in the top left, close to nobody. It is 105 units from where the room starts. No single vote gets there: propose it directly against today's policy and it loses.
It takes three votes. Here they are, with the distance from the destination after each one:
| Vote | Proposal | Carried by | Distance left |
|---|---|---|---|
| Start | (14, −6) | 105 | |
| 1 | (50, −24) | 3 + 4 + 5 | 138 |
| 2 | (−26, −62) | 1 + 2 + 3 | 154 |
| 3 | (−24, 92) | 1 + 4 + 5 | 0 |
Read the last column again. The first vote leaves the room further from the destination than it started. The second leaves it further still, half as far again as at the beginning. Only the third arrives, and it arrives exactly.
Every one of those votes is won honestly. Each proposal beats the one immediately before it, on a straight majority of sincere voters choosing whichever option sits closer to them. Nobody is tricked, nobody votes against their own preference, and no vote is close in a way that depends on turnout or timing. Three different coalitions do the carrying, and no voter is in all three.
The instinct is that a sequence of majority votes should converge on something the majority wants, roughly the middle. That instinct is right only when a stable position exists, one that beats everything else in a head to head vote. Political scientists call that a core.
With these five voters there is no such position. Whatever the room settles on, some majority prefers something else. Once that is true, "the will of the majority" stops naming a single destination, because the majority relation runs in circles: A beats B, B beats C, and C beats A. Preferences are not the problem here. Every voter is perfectly consistent. It is the aggregation that misbehaves.
Richard McKelvey proved in 1976 that this is not a curiosity of odd examples. When no core exists, the sequence of positions reachable through majority votes covers essentially the whole space. Whoever controls the order of business can therefore reach any outcome they like, given enough votes, and every vote along the way is legitimate.
The detour is what makes it work. The second proposal is worse for the eventual winners than the first, but it is better for a different majority, and passing it moves the reference point to somewhere the final proposal can win from. The chairman is not persuading anybody. They are choosing what each vote is against.
Three things, none of which require anyone to behave badly.
The order of business is a power, not a formality. If one person decides which motion is put against which, and in what sequence, they hold something closer to the outcome than any single voter does.
An outcome can be unanimously legitimate and still arbitrary. Auditing the votes will find nothing wrong, because nothing was wrong. Each was carried by a genuine majority.
A stable outcome is a structural fact, not a matter of goodwill. Whether a core exists depends on where people stand and what the voting rule is. Change the rule to a two thirds supermajority, or delete one voter, and the same room can go from walkable to immovable.
The voters here are sincere and myopic. They vote for whichever of the two proposals in front of them is closer, without looking ahead to where the sequence is going. Real committee members sometimes see the trap and vote against their immediate preference to stop it. The setter is assumed to know every position exactly. Votes cost nothing to hold, and patience is unlimited. The policy space has two dimensions, and reachability here is computed over a grid of 101 by 101 positions, so "unreachable" means unreachable at that resolution rather than proven impossible.
None of that rescues the result. Foresight helps the room only if enough members have it, agree about it, and are willing to vote against their own immediate interest to prevent a move whose danger is several steps away.
Every number above came out of Agenda Simulator, which computes these chains rather than illustrating them. The case here is built in: open Learn, choose Load an example, then The 5-voter example. You will get the same three votes and the same coalitions.
Then move a voter and watch the chain change, or switch to a two thirds rule and watch it disappear. The app is free on the Mac App Store, and the user guide explains every reading on the map.