We conduct a laboratory experiment to compare the fairness and intensity of round-robin tournaments with three symmetric players, a single prize, and two alternative match formats. Matches are either organized as lottery contests or all-pay auctions. Contrary to theory, we find no significant differences in intensity and fairness between the two match formats. The main reason is the behavior of the player acting in the final two matches in APA-tournaments. Rather than being severely discouraged when having to compete first against the loser of the first match, subjects try to exploit a perceived negative psychological momentum in such situations. This leads them into a dissipation trap: an effort-intense, final-like last match which significantly reduces their payoffs, but also levels the playing field for the other two players and thus enhances the fairness of APA-tournaments.