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In a dramatic move he wrote to all party leaders inviting them to talks at noon on Saturday, the day after the results will be known.
None of the parties is putting up enough candidates at the election to win an overall majority so the next administration will inevitably be a minority regime or some sort of cross-party arrangement or coalition.
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The SNP is forecast to be the biggest party, but a renewal of its coalition with Labour is in doubt because Scottish Labour leader Anas Sarwar has made clear his preference that Labour council groups do not enter such formal partnerships.
In his letter, Cllr Day said Edinburgh had “never truly adapted to the reality” of the proportional representation voting system introduced in 2007 and the balance of numbers it produced. “To do so, Edinburgh has to change and has to do politics differently."
He said he remained proud of what the council had achieved over the last five years, including the poverty commission, increased housing provision and improvements to public transport. But he added: “I also feel that this city has been held back by parties, including my own, seeking to use the council as a platform for arguments at other levels of government. If we are all honest with ourselves, we have to admit this has frustrated progress for our city.
"Again, if we are honest, there is more we agree on than we disagree. We all want a city where everyone has opportunity, access to good education, green spaces open to all, services delivered efficiently and effectively, amenities to help those most vulnerable and in need and a city that makes contribution to delivering net zero.
“But a system that seeks to cleave councillors into two groups – those in and those out of administration – promotes division not consensus. It means good ideas get ignored because they come from the wrong quarter. Which is why I would like to explore the possibility of sharing power across parties in the chamber and changing how the council conducts business to foster consensus as the way forward."
He said he hoped the meeting on Saturday could explore whether other parties agreed with him and look at how structures and decision-making processes could be put in place to support a new approach.
He acknowledged his proposal might be “difficult and challenging”. “But I think the people of Edinburgh, as reflected by their votes at the election, want us to work together to create a better Edinburgh for all its citizens.”
Cllr Day told the Evening News he hoped a rainbow coalition would repair what top QC Susanne Tanner had highlighted in her report on the council’s culture as a toxic relationship between some councillors.
He said: “I'm saying to them, rather than deals behind the scenes, let's have a frank open discussion on whether there’s an alternative way to do this. If it could be that idea of a rainbow coalition that brings all parties together to get the best for the city, that's better for all of us.”