One of the classes that I most enjoy teaching is my “Party & Parliament” class. The class introduces students to game theory, but it does so in the context of parliamentary politics. Every year for a final assignment, I find an ongoing political event and ask the students to develop a model that helps to explain or illuminate the event in question. This year I asked students to consider the abortive intra-party rebellion against Premier Selinger in Manitoba (which I wrote about briefly yesterday). The rebellion has been little short of a débâcle: 5 ministers resigned from Selinger’s cabinet, the premier nonetheless refused to resign, and the party appears hopelessly adrift. I asked my students to come up with a model to explain why parties attack and try to displace their own leaders when doing so looks to be costly and risky?

My students came up with some great answers. Some wrote about the ingrained risk aversion of backbenchers, others wrote down signalling games. I always try to write down an answer too, at the very least to help me gauge how the assignment is. My answer centred on the mixed motives of party leaders and challengers: while they want their party to win elections (and changing the leader may improve the party’s odds of winning), they also value the leadership in and of itself. Consequently, a challenger may find it worthwhile to rebel against a sitting leader, and a sitting leader may find it worthwhile to fight to keep the leadership, even if the infighting damages the party’s electoral prospects.

I modelled the situation as a two-player game between an incumbent (**I**) and a challenger (**C**). Both players covet the party leadership, *L*, which they value at 1. Both players also want to improve their party’s electoral prospects. The party wins the next election with probability Pr(Win|**I**) if the incumbent retains the leadership and probability Pr(Win|**C**) if the challenger takes over the party leadership. Whilst many revolts are sparked by the party’s unpopularity under the current leader combined with a concomitant belief that changing the leader improve the party’s popularity, I make no assumption as to whether Pr(Win|**C**) > Pr(Win|**I**) or the converse. Observe, however, that the law of iterated expectations implies that the unconditional probability of the party winning the next election, Pr(W) , is a weighted average of Pr(Win|**I**) and Pr(Win|**C**), i.e., Pr(W)=*q*Pr(Win|**I**)+(1-*q*)Pr(Win|**C**), where *q* is the probability that **I** remains the leader.

The players have two strategies available to them: they can either 1) *confront* or 2)* defer* to their opponent. I assume that if one player confronts while the other defers, the player who confronts obtains the leadership. If both players defer, the incumbent leader remains in place. If both players confront, then they each obtain the leadership with a probability of ½. Confrontation is costly to all involved because of the wider damage infighting does to the party’s reputation. If one player confronts, the players pay a cost, *k*>0; if both confront, they pay 2*k*. Given these assumptions, the players’ utility functions can be defined as:

(1) Pr(*j*)*L*+ Pr(Win|*l*=*j*)-*k*∑*c*(*j*)

where Pr(*j*) is the probability that player *j* obtains the leadership; *L*=1 is the value of the leadership; Pr(Win|*l*=*j*) is the probability that the party wins the election given that *j* is the leader; *k* is the cost of confrontation; and *c*(*j*) is an indicator variable that takes on a value of 1 if player *j* confronts. A normal form representation of the game is shown below.

C |
|||

Confront |
Defer |
||

I | Confront | ½+Pr(Win)-2k, ½+Pr(Win)-2k |
1+Pr(Win|I)-k, Pr(Win|I)-k |

Defer | Pr(Win|C)-k, 1+Pr(Win|C)-k |
1+Pr(Win|I), Pr(Win|I) |

Are there conditions under which both players confront each other in equilibrium? This seems closest to the Manitoba situation, with Selinger fighting to keep the leadership despite an open revolt by five of his ministers. It is also the most puzzling outcome as it seems to impose the heaviest costs on both players. Confrontation is a strictly dominant strategy for the challenger whenever,

(2) ½–*k*>[Pr(Win|**I**)-Pr(Win|**C**)]/2

If the inequality above holds, the incumbent responds with confrontation provided that:

(3) ½–*k*>[Pr(Win|**C**)-Pr(Win|**I**)]/2

A confront-confront equilibrium hinges on satisfying the Equations 2 and 3 simultaneously. This is possible provided that *k* is not too large. To see this, assume that the party’s electoral prospects improve under the incumbent’s (continued) leadership (i.e., Pr(Win|**I**)-Pr(Win|**C**)>0). If so, then [Pr(Win|**I**)-Pr(Win|**C**)]/2 > [Pr(Win|**C**)-Pr(Win|**I**)]/2 and any *k* that satisfies Equation 2 also satisfies Equation 3. Conversely, if the party’s electoral prospects improve under the challenger’s leadership, [Pr(Win|**I**)-Pr(Win|**C**)]/2 < [Pr(Win|**C**)-Pr(Win|**I**)]/2 and any *k* that satisfies Equation 3 also satisfies Equation 2.

This result suggests that leadership rebellions are not driven solely by the prospect of electoral defeat under the current leadership. Indeed, Equations 2 and 3 can be satisfied even if the party’s electoral prospects are somewhat worse under the challenger than under the incumbent. The reason that fights over the party leadership can break out even when a change in the leadership has little effect on the party’s electoral prospects is that the party leadership is such a valuable prize. The more valuable rivals consider the party leadership to be, the more they are willing to mount challenges, even if this diminishes the party’s electoral prospects. The only brake on this behaviour is the wider damage that the infighting does to the party’s brand.*

© C.Kam 2015

*I suspect that confrontation would be less likely if I added a neutral backbencher to the model who cares only about the party’s electoral prospects. If the backbencher always threw their support behind the leadership candidate who most improved only party’s overall electoral prospects, then mutual confrontation would be less likely. But there could be uncertainty as to whether the party’s electoral prospects were better under the incumbent or challenger, or the challenger might promise a side-payment to the backbencher, e.g., a cabinet position.

There’s a variety of work on the dynamics of party leadership. Patrick Weller had a paper a few years back that argued that the manner of the prime minister’s exit depended on the rules the party used to select leaders. Other papers examine the relationship between party leader’s survival in office and their electoral success. My colleague, Ben Nyblade, is also doing work on party leadership.