Recall the 2-politician problem: each politician must choose a position (a real number in
the [0,1] interval, let's say), voters are equally distributed across [0,1], each person
votes for the politician closest to her position (dividing the vote
equally if it's a tie), and each politician's payoff is the number of votes she gets
(or here, the length of the subinterval of [0,1]).
(So if the two politicians choose positions $(p_1,p_2)=(0,1)$ or $(.5,.5)$, for example, they split the
votes equally, whereas if they choose positions $(0.1,0.2)$ then the first gets 0.15 of
the votes and the second gets 0.85.)
Now let there be 3 politicians choosing positions $(p_1,p_2,p_3$. Prove that there is no
longer a pure NE. (Hint: do a case
analysis: e.g., why can't a strategy profile of the form $p_1 < p_2 < p_3$ be Nash? why
can't $p_1 = p_2 < p_3$ be Nash?
why can't $p_1=p_2=p_3$ be Nash?)