Fast and Correct answers will recieve a thumbs up.

1. (20%) Consider two political candidates A and B facing an electorate with ideological positions uniformly distributed between 0 and 1. (To remind you, uniformly distributed means there are equal numbers of voters in the interval between 0.4 and 0.6 as between 0.8 and 1.0 and any interval of the same length.) Candidates want to maximize their vote shares. Each simultaneously and independently of the other announces a position between 0 and 1. A voter chooses to vote for a candidate who is closest to her but will abstain rather than vote for a candidate whose announced position is more than 0.25 distance away. Is there a Nash equilibrium in pure strategies? Explain your answer.