Among the electoral system types to be considered by Canada’s upcoming reform-proposal process is the mixed-member proportional system (MMP). What might we expect Canada’s effective number of seat-winning parties (N_{s}) to be under MMP?

As noted in the previous post on comparing the Alternative Vote to FPTP, Canada has had almost exactly the N_{s}, on average, that we should expect it to have, given its assembly size (around 2.6). Thus I will start from the premise here that Canada would continue to have about what we expect if it had MMP staring in 2019 (or whenever). That is, recent elections the country are neither surprisingly under- or over-fragmented, so there is no reason to think the country would over-shoot the expectation under a new system. (It might be more realistic that it would under-shoot, but that depends on how much we believe there is pent-up demand for new parties or for growth of existing smaller ones. I will leave that aside here.)

The answer to this “what if” depends on the precise MMP model. What Li and Shugart (2016) have shown is that a minor addition to the Seat Product Model (of Taagepera, 2007) captures two-tier compensatory systems well. MMP is a type of two-tier compensatory system, so let’s apply that model to a hypothetical Canadian MMP. The formula is:

N_{s}=2.5^{t}(MS_{B})^{1/6}.

In this formula, MS_{B} is the “basic-tier seat product”, defined as the mean magnitude of the basic tier, times the total number of seats in that tier. For a typical MMP system, we retain M=1 in the basic tier, so the basic-tier seat product is simply the number of seats elected in that tier. The *t* in the formula stands for “tier ratio”, which is the share of the total assembly that is elected in the compensatory tier. This is an exponent on a constant term that is empirically determined to be 2.5. (Determined from the broader set of cases on which this is tested.*)

If a country’s electoral system is “simple”, meaning there is just a single tier, as with FPTP, then the above formula reduces to Taagepera’s (2007) seat product model:

N_{s}=(MS)^{1/6}.

(In a system with no upper tier, *t*=0, and MS_{B}=MS.)

For purposes of estimating, I am going to assume the assembly size will remain the same, currently S=338. I will further assume that it would be politically difficult to reduce the number of districts (ridings) in the basic tier of an MMP system too much below the current number (which is, of course, 338). I will adopt two thirds of the current number as my estimate of “not too much”. In such a scenario, we get a basic tier of 225 seats, giving us *t*=.33:

N_{s}=2.5^{.33}(225)^{1/6}=1.35(2.47)=3.33.

I did not plan my scenario to get to three-and-a-third, but it has a nice ring to it. And seems pretty reasonable. For comparison purposes, this is not much different form what Canada had in 2006 (3.22) or Germany, an actual MMP system, had in 1998 (3.31).

Based on another formula in Taagepera (2007), which is empirically very accurate, we can also derive an expectation for the seat share of the largest party (s_{1}):

s_{1} = N_{s}^{-.75}.

For our hypothetical MMP system in Canada, this implies the largest party with just over 40% of the seats.

We can tinker with the scenarios. For instance, suppose the assembly size were increased to 400, with half the seats in the basic tier. Then we get:

N_{s}=2.5^{.5}(200)^{1/6}=1.58(2.42)=3.82.

As would be expected intuitively, the fragmentation of the House goes up due to the larger compensation tier, and in spite of the basic-tier seat product being correspondingly lower. This scenario would have an expected s_{1}=.366.

Note that I have ignored thresholds here. My ongoing research with Taagepera suggests that thresholds matter, but unless the threshold is very high (more than 5%) or the seat product, MS, is extremely high, the value of the threshold has much less impact than the parameters discussed here.

In conclusion, under common Mixed-Member Proportional (MMP) designs, Canada could expect its House to have an effective number of parties ranging from 3.33 to 3.8, compared to a recent average of around 2.6 under FPTP. Its largest party could be expected to have around 36% to 40% of the seats, on average, compared to majorities or nearly so in recent elections.

* Note that this parameter’s empirical derivation raises the risk that it could be an artifact of the particular sample we have, and thus not reliable for determining an expectation value, as I am doing here. Fortunately, unless it is off by a wide amount, it does not make a large difference. For instance, 2^{.33} =1.26, whereas 3^{.33} =1.44. This variance in our prediction is much less than “normal” fluctuation from election to election in many countries.