If he were elected, there would be a 'tilt' [='shift'] in the political balance of the city council. [=the city council would become more liberal or conservative]
The quarterback's injury could 'tilt' (the outcome of) the game in the other team's favor. [=could make it more likely that the other team will win]
His election 'tilted' the city council to the left/right. [=made the city council more liberal/conservative]
(..) have to do exactly what I say. No, not likely. You're a degenerate creature of darkness. Yeah, yeah, granted but that's it. All I'm asking you to do is just tilt your head a little. You could do that, can't you? Come on. - More, more... - This is stupid. No, no. Come on. Work with on this. (..)2008 Bolt