After the Math Department’s Magic: the Gathering tournament, the five competitors told their colleagues how it turned out. But, mathematicians being what they are, each competitor told only a half-truth: that is, one true and one untrue statement. Here’s what they said:

Allen: Deis came in second. I finished third.
Black: I won the whole thing, and Chang finished second.
Chang: I got third place; poor Black was last.
Deis: I was in second place, and Frayer was fourth.
Frayer: I ended up fourth. Allen was the champion!

What were the true results?