A 12-candidate challenge to a longstanding election theory

A candidate deciding whether to run does not make that choice in isolation. Earlier entrants have already chosen positions, and later entrants may respond. A conjecture associated with economists Martin Osborne and Amoz Kats proposed a strikingly simple outcome for a particular model of this process: the first and last potential candidates enter at the median, while everyone in between stays out.

In a preprint, Dmitry Dagaev presents a mathematical counterexample with 12 potential candidates. The result challenges the conjecture's claim that the predicted outcome is unique. It does not establish that the predicted outcome is impossible, nor does it describe what necessarily happens in a real election.

An election on a line

The model places every voter and candidate on a line running from 0 to 1. Voters are distributed uniformly along it, so equal lengths of the line contain equal shares of voters. Each voter supports a nearest candidate. If several candidates choose the same position, they divide that position's votes equally.

Potential candidates act in a fixed order. Each sees all earlier decisions and gets one chance either to choose a position or stay out. The candidate or candidates with the largest vote share win. They divide a unit payoff equally; an entrant who loses gets a negative payoff, while staying out pays zero. Candidates are therefore motivated by winning office in this model, rather than by a preference for a particular policy.

The relevant solution concept is a subgame-perfect Nash equilibrium. It requires a complete plan for every player, including what they would do after histories that do not occur along the eventual path. At every such point, the remaining plans must form a Nash equilibrium: no player can improve their payoff by changing their own strategy while the others' strategies remain fixed.

That requirement matters because a seemingly attractive entry can invite later responses. A candidate must consider not just the votes available immediately, but also whether subsequent candidates would enter and change the result.

A counterexample built from later responses

Dagaev starts with two positions proposed in an earlier 12-player construction by J.-P. de Vries: the first candidate enters at 1/2, the median, and the second at 1/10. The new paper analyzes the remaining decisions and constructs an equilibrium of that subgame in which the second player receives a payoff of at least 1/10.

One part of the argument considers a third candidate entering at 9/10. The positions 1/10, 1/2 and 9/10 are symmetric around the median. The paper proves that, after those three entries, every subgame-perfect equilibrium of the remaining players gives the candidates at 1/10 and 9/10 the same payoff, of at least 1/10. The proof then considers other possible third-player actions and selects continuation strategies that support a positive payoff for the second player.

The full-game construction uses that positive-payoff subgame to establish an equilibrium outcome different from the conjectured one. Importantly, the paper does not claim that the constructed full-game path necessarily begins with entries at 1/2 and 1/10. If the first player chooses 1/2 in the constructed equilibrium, the second enters; if the first chooses something else, the outcome already differs from the conjectured pattern. The proof leaves the first players' maximizing actions implicit.

Why ties matter

The argument depends on the way candidates can tie for the largest vote share, including at different positions. An entrant may be able to reduce a tied rival's votes without reducing their own, potentially sharing victory with fewer candidates. But that improvement counts only if later players' responses do not undo it.

The paper uses geometric bounds on voter shares and possible entry positions to analyze which earlier candidates remain winners as later candidates act. It also distinguishes properties that hold in every equilibrium of a subgame from cases where it selects an equilibrium with the needed continuation.

The result is a claim about an idealized mathematical game, not a forecast of candidate behavior. The uniform electorate, nearest-candidate voting, fixed entry order and winner-based payoffs are all assumptions. The proof also relies on exact vote-share ties: changing the voter distribution can break them and alter payoffs. The preprint leaves open whether the conjectured median outcome can coexist with the alternative, whether fewer than 12 players suffice, and how the argument extends to more players.