You should watch that battle through to the end. Splinterlands does NOT "at a certain point ... just declare a winner".

What DOES happen is that starting with round 20, fatigue sets in. It does 1 point of damage the first round, 2 points of damage the second round, 3 points of damage the third round and so on (all ignoring armor). Needless to say, the match ends pretty quickly after that -- and each player's choice of team is the key to who wins (not random chance). Note, however, that a good portion of the ties in Splinterlands DO come about when a high number of fatigue points are assessed and all the remaining monsters die simultaneously.