Hotellings model of candidate question

this is the hint (part of solution)

Question 4. [30 points) Consider a variant of the Hotelling's model of candidate positioning in which each candidate does not care about whether herself wins the election but instead only cares about the position x of the winner. A continuum of citizens is distributed on interval (-1,1] which represent their favorite positions. There are two parties and the only positions they can choose from are -1, -2,0,1,1. Candidate 1's favorite position is - and candidate 2's favorite position is 1. After each candidate chooses a position, each citizens will vote for the candidate closest to her favorite position and the candidate with more than 50% of the votes is the winner. Assume that the distribution of the citizens' favorite positions is symmetric around 0, so that if, for example, one candidate's position is -1 and the other's is 1, or one candidate's position is and the other's is , the candidates tie. (Of course, they also tie when they choose the same position.) The distinctive assumption is that each candidate does not care about winning but instead only cares about the position of the winner. Specifically, if the winner position is , candidate 1's payoff is -1-7-1) and candidate 2's payoff is - 2 - 3 (For any number 2, 1z| denotes the absolute value of z: 1-1 = z if : >0 and | = -z if z 0 and | = -z if z