A theoretical question has a theoretical answer.
I may have gone a bit overboard, but I thought it was interesting enough to number crunch it.
For an Arena win you get 3 points + the league difference (positive if you beat someone from N higher leagues, negative for N lower leagues), with a minimum of 0 and maximum of 5. So you get 3 Arena points if you beat someone from the same league and more if it was someone from a higher rank (or less if lower).
You start with 1 Arena point in every new league and excess points are lost, so you need at least 9 to rank up. This means that if you always win against opponents who significantly outrank you, you need 2 wins to promote out of all leagues but the second highest (5 points per game), and 3 wins for the last promotion (3 or 4 points per game).
Or, mathematically:
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Required wins
= wins in all lower leagues + wins in second highest league
= 2 * (promotions - 1) + 3
= 2 * promotions + 1
Res Arcana seems to have 4 leagues, so if you start out in Bronze you would need to win at least 7 games. Of course, you won't always play against higher ranked opponents, so a win would probably only give 3 points on average, meaning at least 9 games (3 * promotions).
More realistically, you won't always win, so your expected gain or loss of 3 points is determined by your win rate.
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Expected point gain
= winRate * pointsGained - lossRate * pointsLost
= winRate * 3 - [(1 - winRate) * 3]
= 6 * winRate - 3
This makes sense, since with a win rate of 100% you would always gain 3 points, and with a win rate of 50% you would not make any progress. So, adding it all up:
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Expected games
= (promotions * requiredPoints) / expectedPoints
= (promotions * 9) / (6 * winRate - 3)
For 3 promotions, this would give the following plot:
- y = 27 / (6x - 3) => y = 9 / (2x - 1)
y = number of games, x = win rate - Required games by win rate (4 leagues).png (37.97 KiB) Viewed 546 times
Seems like you've got around 90 games ahead of you. You might want to work on your win rate first.
It will be lower if you play against opponents that are ranked higher than you and you optimise your losses to occur at the start of a new league (since you can't drop below 0), but I think this is a reasonable estimate (unless my math is completely incorrect).