[EM] Second order proportional representation.

Kristofer Munsterhjelm km_elmet at t-online.de
Wed Sep 14 13:06:50 PDT 2016

On 09/02/2016 07:10 AM, Richard Lung wrote:
> Second order proportional representation.
> First order proportional representation refers to the election of the
> legislature by PR. Second-order PR refers to the composition of the
> executive.
> Transferable voting can prefer candidates from more than one party, to
> decide the democratically prefered majority coalition. This single
> majority government, representing over half the voters in the executive,
> is the rational representation, that is denied by first past the post or
> simple plurality elections.
> But that is still not second-order proportional representation. That
> requires at least a double majority government, which would represent
> two thirds of the voters in the executive. A triple majority government
> would represent three quarters of the voters in the executive. And so
> on. That is second-order proportional representation.

I just read about an interesting idea in this vein by Brams. First you
pick your legislature (using whatever method), then once a majority of
the parties decide to form a government, the parties sequentially claim
ministries according to an order given by a divisor method.

For instance, suppose the following parties are represented in
legislature, with seat counts:

Party A: 55
Party B: 10
Party C: 7
Party D: 1
Party E: 48
Party F: 29
Party G: 10
Party H: 9

Now suppose that, to make things interesting, parties E, F, and H choose
to form government. Furthermore suppose there are 10 ministries, and the
method this legislature uses for the executive ordering is Sainte-Laguë.
Sainte-Laguë would apportion as follows:

First seat: E
Second: F
Third: E
Fourth: F
Fifth: E
Sixth: H
Seventh: E
Eighth: F
Ninth: E
Tenth: E

The share of seats is: 60% E, 30% F, 10% H, compared to the relative
legislature share of 56% E, 34% F, 10% H when only seats belonging to
those three parties are counted.

Anyway, the idea of Brams was to ask the parties (in order) to choose
ministries. So in the assignment above (EFEFEHEFEE), E would first be
asked to pick a ministry/department (and would probably choose the PM).
Then F gets to pick the next, E gets the next after that one, then F, E,
H, and so on.

The point of the procedure is that it would weaken the need to negotiate
for a share of the government. If party X is part of the majority
coalition that decides to form government, how many seats X gets is not
something the coalition can influence. Furthermore, since Sainte-Laguë
works like a proportional ordering, it is fair at every step of the
process (assuming that going first is always beneficial).

Presumably the same idea could be used for more advanced proportional
orderings, but the system would become more opaque.

Unfortunately, Brams then discovered that sometimes it does pay to not
go first, and he suggested that trading might help with it. One
possibility for trading could be that a party, when it's its turn to get
a ministry, proposes to exchange a free ministry seat with one that's
already taken. If the other party (who holds that other ministry seat)
agrees, they switch. Suppose, for instance, that at some point in the
process, E has claimed the ministry of justice, and the ministry of
education is free when F is to choose. F might then propose to E that F
gets the ministry of justice in exchange for E getting the ministry of
education, even though nominally the ministry of justice has already
been claimed.

I suppose Brams' system could be used for supermajority-support
executives, but then the confidence agreement would have to be supported
by a supermajority. Perhaps the system above would make this easier
since less negotiation would be required beforehand, and thus diminish
the chance of a deadlock or, looking at it another way, enable greater
supermajorities to reach a decision about what government to have for
the same risk of deadlock.

> Note that it requires a considerable first-order proportional
> representation to ensure even a single majority government. We saw this
> in Ireland, when the then largest party, Fianna Fail whittled down the
> constituencies to 3 or 4 members, so that they could win a majority of
> seats on only 45% of the votes.

More complex apportionment systems could base the number of executive
seats on support among the voters rather than on legislative support.
However, it may be difficult to use that kind of apportionment, which is
fundamentally party-based, if the legislature is elected using
non-party-list methods like STV.

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