If Mr Livingstone stops his posturing, he could prove to be an effective mayor for London

Although it came unusually close to doing so before the last general election, The Independent does not normally presume to advise its readers to vote for a particular party, and it is not going to do so on the day of the local elections. Nevertheless, they constitute the most important test of public opinion since May 1997, including a brave experiment to give London a mayor directly elected by its five million voters. It is an innovation we support because it will rekindle interest and accountability in local government. The result will matter far beyond the capital, and not only because the winner will be the first of several big-city mayors.

The undisputed front runner is Ken Livingstone, the candidate who has best understood - and is by temperament best suited to - the new personality politics that a mayoral election requires. Mr Livingstone has seen some quite serious reverses in his campaign over the past few weeks, not least the revelation that he failed to disclose large earnings in the Commons register of interests and some rather ill-judged remarks comparing the World Trade Organisation to Adolf Hitler. But he appears to have survived almost intact as the main repository, across an extraordinarily wide spectrum of opinion, of a mid-term protest vote against the Blair government.

There may be good reasons to vote for Ken Livingstone but, oddly, that is not one of them. This is not a by-election at which a protest vote can be made; whoever wins will be an important figure, and any competent politician - which Mr Livingstone certainly is - will be able to use his exceptional mandate to carry influence beyond the scope of his formal powers. How he represents London will, further, have an effect on how this international city is seen globally, from tourists to inward investors.

The policies matter. In Mr Livingstone's case, is he really going to fight every inch of the way through the courts to reverse the Government's policy of a public-private partnership for the Underground? And will it necessarily help London if he does? How will the as-yet-unspecified congestion charges work? And what if he fails to secure additional government funding for public transport? It is hard to believe that Gordon Brown feels well disposed towards a man who - in the pages of this newspaper - has called for his removal.

Nevertheless, Mr Livingstone has one real advantage over Frank Dobson, the Labour candidate, which is that he so obviously wants the job. Mr Dobson is a decent man, and he might make a much better mayor than he has been a candidate. But until Downing Street and others prevailed upon him to "save Labour in London", Mr Dobson had made it clear that he did not want the mayoralty. Mr Dobson was an innovative leader of Camden council and a much better Secretary of State for Health than his critics expected. We also believe that Mr Dobson would put London's interests first. But he has failed to convince the electorate of that; and he has not persuaded voters that his heart is in it.

Susan Kramer, the Liberal Democrat, combines to a rare degree a developed liberal conscience with real business and financial experience. There would be a great deal to be said for a woman mayor. But it is hard to escape the conclusion that it was a big error on the part of Simon Hughes, a much more credible potential candidate, not to run. Given the abysmal disarray into which both the main parties have plunged, he might well have won.

It would be entertaining, to say the least, if Steve Norris became, as he will if he wins, the most powerful Tory in Britain. While William Hague continues to mouth off in the political saloon bar, Mr Norris has provided an almost statesmanlike contrast. Unlike his party leader, he has opposed Section 28 and has an inclusive attitude to the concerns of the gay community. He has some of the panache needed for the job. He has convincingly balanced conventional Tory emphasis on law and order with a real sensitivity to the deep anatagonisms toward the police triggered by the Stephen Lawrence case. He was a perfectly competent London minister. But his strong - and arguably rather opportunistic - opposition to the euro is a serious drawback.

In the wider local elections outside London, however, we have a far greater disquiet about the Conservatives. It does not implicate Mr Norris, because he has been commendably reluctant to follow the irresponsible example on race and asylum set by Mr Hague. The danger is that likely Tory gains will be attributed within the party to the race card that Mr Hague has played so vehemently. There is a strong case for denying the right wing of the Tory party the benefit of that argument and for reminding it of the success of the comparatively liberal Mr Norris, if we are to prevent the party from drifting towards a "Little England" constituency.

As for the Greens, we do not feel London is ready for Green governance but, for the sake of more pluralist politics and London's chronically clogged environment, it would be good to see one or two Greens in the Greater London Assembly.

The likely winner tomorrow, however, will almost certainly be Mr Livingstone. There are two roles that he can play upon taking office as London mayor. The first is that of unofficial leader of the opposition to Tony Blair, full of futile posturing against New Labour, capitalism and globalisation. This would be true to his last period running London. Certainly, his attacks on the Chancellor Gordon Brown reinforce the fear that he will use the mayoralty to refight old Labour wars. Such a stance would, however, damage not the Government but the interests of London.

But there is another possible course for a Livingstone mayorship: one where he deploys his considerable gifts for the practical benefit of Londoners. In an era of competing international cities, Mr Livingstone's extraordinary capacity for public relations could be used to boost London on the world stage. If this was combined with his commendable social liberalism, and if he could forsake the interests of the unions for those of the consumers when it comes to public transport, he could yet prove his detractors wrong. The opportunity is there; now Mr Livingstone must live up to the expectations of his electorate.