This appendix provides a formal and complete treatment of the game theoretic model described in Chapter 2. The discussion here is technical and shorn of much of the narrative that is provided in Chapter 2, and hence readers may find it easier to read Chapter 2 closely before engaging with the material that follows below.

Players, strategies, and utility functions

The game is predicated on a policy disagreement between the party leader (L) and the MP, the leader wishing to secure the MP's support for her most preferred policy, and the MP hoping to avoid the electoral consequences of any uncongenial party policy. To structure this policy disagreement, the players have ideal points on the unit interval, such that MP = 0 and L = 1. The leader has the privilege of defining a party policy, x ∈ [0, 1], but can also offer advancement, a ≥ 0, or apply disciplinary sanctions, s ≥ 0, to the MP to induce the MP to support x. In extending advancement or applying sanctions L incurs costs of a2 and s2, respectively. The MP can support or dissent from x, with r ∈ [0, 1] denoting the extent of the MP's dissension. The MP's dissent imposes a direct cost of r2 on the leader and dilutes the policy gains that L obtains by moving x toward her ideal point; i.e., L's utility from party policy is (1 − r)x.

On the MP's side, dissent protects the MP from the electoral impact of x.