A clear and concise title.
A draft costs you 1500 gems. You gain more gems by winning enough matches.
You can keep playing until you’ve either won 5 matches or lost two. To win your gems back (and some packs, but you can’t use packs to draft, so that’s not part of going infinite), you need to win 3, so it would seem that you need a win percentage of 60% to go infinite, but it’s not that simple.
Suppose, you go 5-1 and 1-2. You actually have a better than 60% win percentage, but you only gained 2100 gems, so at this point, you are losing. Clearly, 60% isn’t enough unless you are very consistent in losing your second match after three wins.
Due to the structure of the prices, where the extra gems from the fourth and the fifth wins are very low, you probably need a much higher than 60%.
Here’s a table depicting all the possible outcomes with a win% of 60.
You can see the losses as columns and wins as rows, so whenever you win, move down, and whenever you lose, move to the right. Your expected number of gems won is now the probabilities of each of the end outcomes times the number of gems you’ll get from that particular scenario.
In this case:
(0,1728 * 800) + (0,12825 * 1500) + (0,10368 * 1800) + ((0,07776 + 0,15552) * 2100) = 1022,112 or not nearly enough.
The actual win% you need is roughly 74 (which I found simply by testing with Excel and the table above). That’s is quite high. So high, in fact, that you shouldn’t expect to be able to do it. You can, however, go infinite in constructed, as you can play with gold only, although there’s definite diminishing returns on that.
In that case (74% wins), the table of outcomes looks like this:
This gives you expected gems worth 1507,42, or just enough.
[UPDATE] Fixed a problem in the explanation for how the tables work (I had wins and losses the wrong way around).