If the playoff system doesn't work, can we come up with a system that will? That brings us back to Arrow's notorious theorem. Arrow made a series of postulates that were so reasonable that every voting system should obey them, and then he proceeded to show that they were incompatible with each other. In other words, there is no voting system that will always satisfy fundamental criteria of fairness.

No. Arrow's theorem only guarantees that you can find a situation in which the election is unfair; it doesn't guarantee bad results in all cases.  Moreover, the theorem is true only when there are three or more candidates. Let's consider the two-candidate case: L and R. Everyone votes for the one who is closest to their preference. In real elections, the positions of the candidates on issues may not be absolutely immutable. Both L and R realize this, so to maximize their chance of winning, they both start shifting toward the middle. They both know that whoever best takes possession of the middle will get the most votes.

By the time the voting takes place, the candidates' positions are almost indistinguishable. Voters complain that they have no real choice. And in some sense this is true. The center has been found-the position where M would have been, had he (or she) run. Both candidates, to get elected, have moved to the position where they best represent the average of all voters. 

Notice how the existence of a primary election can interfere with this center-finding process. A candidate who wins in a primary is often the one who represent the center of his party-a position that tends to be pretty far toward one side or the other of the political spectrum.

If, however, people vote in the primaries for a candidate who "can win"  in the general election rather than one who is closest to their own preference, then the two-party system works well, and results in candidates who are close to the center. If there is sufficient time during the subsequent campaign, the candidates can move even closer to the middle position. Democrats can co-opt Republican issues, and Republicans can co-opt Democratic stands. This leads to a surprising irony. With both candidates moving toward the center, many people ultimately see such little difference that they lose interest in the election. Turnout is low-but for this example, that actually reflects the fact that the election process is working well.