The Two Strongest of Twenty

Twenty players have distinct, fixed strengths (a stronger player always beats a weaker one, and the order is transitive). You do not know the strengths; the only operation is to pick two players and play one match, which reveals who is stronger. You may choose matches adaptively. You must determine who is the strongest and who is the second strongest. What is the smallest number of matches that is guaranteed to suffice, whatever the hidden strengths are?

• combinations
• combinatorial games
• permutations
• pigeonhole

Source: Kislitsyn (1964) and Schreier on selecting the two largest elements; Knuth, TAOCP vol. 3. Statement written for AxiomIQ.