We decided to have Random Battles be Random. This increases the diversity of experience as you will encounter a wide variety of play and skill variance.
Also, MMR systems tend to move a player to a point where they have a 50/50 win rate even when pushing at their absolute utmost. When not, the result is a string of losses until the MMR has rebalanced to that different norm. There are positives and negatives to any system, but the constant uphill battle can and does lead to burnout as feelings of progress stall out.
For players that crave an MMR system, Clan Battles has a point-based ranking structure which comes close to the concept. Your team will escalate to the point where it finds itself steady-stating amongst its peers. Clan Battles is a constant high-effort game-mode, so burnout isn't uncommon over the duration of a season.
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Ranked Battles use the Random Battles Matchmaker, but with a restricted player pool. Players in a league can only be matched against other players in that same league. Qualifying for higher leagues makes for a system where the player pool reduces to only those who have qualified to the league being played. There is a significant reduction in player pool size with each move up a league.