Note that this assumes that candidates care about nothing else but their own election. When there is an odd number of candidates in one dimension, there is no equilibrium. Therefore there never is an equilibrium. Page 41 Any point outside the core of the game could be beaten by at least one alternative. The instability can come from the efforts to introduce this alternative.
Unbeatable options creates stability since there are no incentives to compete.
If any price above marginal cost was proposed, we could always find at least one firm willing to undercut that price, therefore leaving zero sales and thus zero profits to those committed to the higher price. This person might sell something and then buy it back at a higher price. You are better trading with him than having him in charge of your assets. The President can set the agenda through indirectly trough party affiliation and the media. The House Speaker and Committee chairs directly set agendas.