The Duellist's cap is a hat that keeps track of the amount of kills in the Duel Arena of the wearer. The tier 1 cap shows that the player has won 0-9 duels.
The duellist's cap does count your free to play duels, including staked and friendly duels. However, it is important to note that the wildstalker helmet does not track free to play kills.
If a player gets 10 Duel Arena kills, they can turn it into a tier 2 cap by clicking Change-look.