3rd place

A single-elimination tournament will determine without doubt who is the best. We assume there is no element of chance involved. It has an advantage over round robin that collusion between the weaker players to fix games cannot prevent the number one player from winning the tournament (Fischer accused the Soviets of this.) Single-elimination will always have one clear winner.

A double-elimination tournament structured with a losers' bracket appears to prevent collusion (I have not proved it), and there will be uniquely one undefeated champion and one runner-up who has lost exactly one game. However, that one loss may not have been to the champion, so there is the confusing possibility that someone who ultimately places worse than second place has actually beaten the runner-up. Can a tournament be structured to avoid this possibility?

Continuing the pattern, a triple elimination tournament will place the top three. The number of games needed becomes very large, while we wish to avoid the same contestants playing against each other more than once.

Is there a better way to design a tournament? We wish to avoid collusion being able to affect the outcome and a clear set of the top few placings at the end.