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]
(..) I Don't know. He's pretty scared. Maybe we should try taking it forward again. If the trailer tilts any further, We'll lose him, Randolph. It's ok. You're gonna be all right. (..)1993 Free Willy