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]
(..) He will do whatever he must to win. (URGING HORSES) Now, when you cut first, they'll tend to put you up on one wheel. So, you have to learn how to tilt. It's all about balance, Judah. It's a dangerous maneuver, of course, but with courage and practice, it can be mastered. This is the spina. (..)2016 Ben-Hur