The election is taking place under an alternative vote system which allows party members to rank candidates in order of preference.
Mr Kennedy says he expects Labour to honour its pledge but he strenuously denies any suggestion of a pre-election pact between himself and the prime minister to introduce the lesser form of proportional representation - the Alternative Vote system.
His recommendation that Britain switch to an alternative-vote system sank without trace.
The bill paves the way for a referendum on changing the Westminster electoral system from first-past-the-post to the Alternative Vote (AV) system, whereby voters rank candidates in order of preference on the ballot paper.
The home affairs spokesman also rejected the Alternative Vote (AV) system favoured by Mr Blair to replace first-past-the-post.
Constitutional Reform Minister Mark Harper said the coalition wanted to hold the plebiscite on replacing first-past-the-post (FPTP) with the alternative vote (AV) system at an "early opportunity" so voters would know what system they would be using for the 2015 general election.
Mr Cameron, over the shrieks of most of his party, has promised a referendum on introducing an alternative-vote (AV) system allowing voters to rank candidates by preference.
The South East, like the country as a whole, voted overwhelmingly ' No' to changing the voting system from the current first-past-the-post system to the alternative vote.
BBC: Labour's big push in the South East reaps mixed results
Yes, admits a Clegg ally, it now looks disastrous to have sought a referendum on the complicated preferential voting system known as the alternative vote (AV).
MPs are currently debating legislation required to pave the way for a referendum on the way they are elected and whether to change the current first-past-the-post system in favour of an Alternative Vote method.
BBC: Ministers under fire over consultation on AV referendum
The Conservatives agreed to a referendum over whether to replace the first-past-the-post system for electing MPs with the alternative vote (AV) as part of their coalition negotiations.
BBC: Referendum on voting system goes ahead after Lords vote
As well as assembly seats, people have also voted on whether to replace the first-past-the-post system for electing Westminster MPs with the alternative vote (AV).
The same day will see a referendum on changing the electoral system for the House of Commons to the Alternative Vote (AV), in which candidates are ranked in order of preference.
The Conservatives vehemently oppose the alternative vote, arguing that the current first past the post voting system "delivers clear, clean results" even if it does currently seem to work against them.
The bill provides for a referendum on changing the Westminster voting system from first past the post to the alternative vote (AV), making constituencies more equal in population size and cutting the number of MPs from 650 to 600.
Within this mixed system the constituency members should be elected by the alternative vote (AV).
The public will be asked whether they want to replace the existing first-past-the-post system for electing MPs to Westminster with a method known as the alternative vote (AV).
If the coalition works, voters may warm to the idea of power-sharing, and to the proportional voting systems that inevitably give rise to it. (A referendum on the alternative-vote electoral model, which the Lib Dems prefer over Westminster's first-past-the-post system, is part of the deal.) And the Lib Dems got all this despite flopping at the polls.
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