Britain’s post-election standoff has inched closer to resolution, with the announcement this morning that Prime Minister Gordon Brown will retire from the Labour leadership. But Labour and the Conservatives are still waiting for the Liberal Democrats to decide which way they will jump — or indeed whether they will jump at all.

However, we now know more about just what the Lib Dems have been offered. The Tories have promised a referendum on the adoption of preferential voting (which the British call the alternative vote, or just AV). Labour has offered immediate adoption of preferential voting, and a subsequent referendum at which — apparently, although this hasn’t been officially confirmed — proportional representation would be an option.

British commentary on the issue is often confusing, sometimes implying that preferential voting is a form of proportional representation. It’s not; it’s just a system of single-member districts where voters can rank the candidates instead of (as in first-past-the-post) only voting for one — the familiar system by which we elect our House of Representatives and all our mainland state lower houses.

Proportional representation is quite different: it involves a multi-member system (although there may be single-member districts as well) in which some effort is made to have numbers of seats reflect the proportion of votes won by different parties. The Australian Senate, the Tasmanian House of Assembly and the New Zealand parliament are all examples of ways of doing this.

To see the difference, here is last week’s UK election result with the proportions of vote, the seats actually won, and the seats that would have been won on a pure PR basis (Sainte-Laguë, no threshold):

Party Votes Seats won Seats (PR)
Conservative 36.1% 306 237
Labour 29.0% 258 191
Liberal Democrat 23.0% 57 151
UK Independence Party 3.1% 0 20
British National Party 1.9% 0 13
Scottish National Party 1.7% 6 11
Greens 1.0% 1 6
Sinn Fein 0.6% 5 4
Democratic Unionist Party 0.6% 8 4
Plaid Cymru 0.6% 3 4
Soc. Dem. & Labour Party 0.4% 3 2
Ulster Cons. & Unionists 0.3% 0 2
English Democrats 0.2% 0 1
Alliance 0.1% 1 1
Respect-Unity Coalition 0.1% 0 1
Traditional Unionists 0.1% 0 1
Others 1.3% 1 0

Note that this is a further blow to the theory that first-past-the-post delivers stable majorities but PR doesn’t. Under the actual results, only one plausible combination — Conservatives plus Lib Dems — has a secure majority; under PR there would have been a choice of two, since Labour plus Lib Dems would also have a majority (not unreasonably, since they had 52% of the vote).

Most actual PR systems are not as pure as this — they trade off some of the proportionality in order to provide some local representation and reduce the proliferation of very small parties. But any such system would address two of the big problems with the present system: the gross under-representation of the Lib Dems, and the fact that a party can potentially win a majority with less than 40% of the vote.

Preferential voting doesn’t attempt to reflect the overall proportions of voting strength. It may end up doing so, but it may not; we know from Australian experience that it can produce some highly anomalous results.

Making some assumptions about allocation of preferences, I came up with the following estimate of how last week’s result would have looked under preferential voting:

Conservative 268
Labour 273
Liberal Democrat 77
Scottish National Party 7
Greens 1
Sinn Fein 5
Democratic Unionists 8
Plaid Cymru 4
Ulster Cons. & Unionists 1
Others 2

These numbers are basically educated guesses (the BBC has a different set), and also historical ones based on survey data), but they give the general idea: the Lib Dem total would increase, but still be well short of their proportional entitlement.

Labour, however, would also gain seats and could even overtake the Conservatives, despite winning two million fewer votes.

Preferential voting would also continue the present practice of locking out the far-right parties, UKIP and the BNP. That might seem like an attraction, but it’s unwise to choose a voting system on the basis of whether we like particular parties or not, and giving the far right an additional grievance is probably not the best way to attract their voters back to the mainstream.

So while preferential voting would improve the Lib Dems’ position, any gain for fairness would be slight. Nick Clegg and his party would much prefer to hold out for PR, even in the anemic “AV+” form that was recommended in 1998.

But, of course, Clegg also has to take into account the differing ability of his prospective allies to deliver: the Tories are at least in a position to make good on their promises, whereas an alliance with Labour may well end up producing no reform at all.