Yesterday’s article was a card-specific look at the London Mulligan in Modern. However, we are able to utilize multivariate probability to gauge the impact it would have on a sixty card level. We are able to formulate the recipe for a passable hand and then calculate the frequency with which it occurs under each ruleset. Then we can compare those results to see how much it actually changes things. Today we are looking at Tron as the community is already calling for the ban hammer.

**How it Impacts Sample Sizes**

The main thing that I think most of us are missing when we theory-craft, is how often and in what ways this rules change would come up. Obviously a change in mulligan rules has no effect on your seven card hands; apart from shrinking the power level gap between mulligans. But have we actually considered the impact on six card hands? It is surprisingly small when you think about it. Under the Vancouver mulligan rule you draw six cards and then you if you keep you scry the seventh card. Under the London mulligan rule you draw seven cards and then you must place one on the bottom. Once mulligans are complete, the number of cards in the sample was seven either way. The difference is that under Vancouver rules you have one option to bottom, or top, and under London rules you have seven options to bottom with no option to top. So a London mulligan is generally a little more powerful in this case.

The difference becomes significant when we begin to look at five and four card hands. In these situations a London mulligan has a larger sample of cards, always seven, in addition to greater card selection when compared to a Vancouver mulligan. A London five is still generally worse than a Vancouver six but it is much better than a Vancouver five. This became evident as we calculated the multivariate probabilities for today’s topics. The sample size for the first mulligan under either rule is still seven but a London mulligan will always be seven while a Vancouver mulligan sample shrinks by one for each mulligan after the first. The one difference that sampling cannot account for is the difference between opening hand and the top card of the library. If one of the components in a passable hand must be played on the first turn, there is a chance that it is the scry portion of a Vancouver mulligan sample and will not be playable. This issue does not occur with a London mulligan so this is an advantage that should be considered when viewing the results of our calculations.

**Tron Decks
**The deck that is generating the most fear in the wake of this ruleset is Tron. The deck is known for aggressively mulling to a hand that can assemble the three Urza lands and cast a massive early threat. So being able to pair down to those exact four cards with your choice of seven should be easy right? For our calculation, the acceptable outcome is the assembly of Tron Land A + Tron Land B + Tron Land C or a Tutor + Threat within nine cards; an opening seven with two draws signifying Turn 3. The number of these in the deck are four, four, twelve, and fourteen respectively.

To find the difference between the Vancouver and London rules we need to calculate the odds of not assembling this in an opening hand, the first mulligan, and the second mulligan. By multiplying the failure odds for each, we can find the chance of total failure. The sample size for both the opening hand and first mulligan will be nine; as under either ruleset there are seven card looked at before play begins and two draws after. The true difference comes in at the second mulligan when the Vancouver sample size drops to eight while the London holds fast at nine.

**Results:**

Under the Vancouver rules, if you are willing to mull non-suitable hands up to two times you will be able to assemble Tron with a threat 85.8% of the time.

Under the London rules, if you are willing to mull non-suitable hands up to two times you will be able to assemble Tron with a threat 87.8% of the time.

So when employing this mulliganstrategy with a Tron deck, the difference between the old and proposed rules is only 2%; equivalent to one out of fifty games. So it does matter but not nearly as much as would be expected. This is because as far as the calculation goes, the opening hand and first mulligan have identical probabilities across each ruleset and do not diverge until the second mulligan. But again, if one of the components in a passable hand must be played on the first turn, there is a chance that it is the scry portion of a Vancouver mulligan sample and will not be playable. What if the Tron player is willing to take an additional mulligan down to four cards though?

Under the Vancouver rules, if you are willing to mull non-suitable hands up to three times you will be able to assemble Tron with a threat 90.6% of the time.

Under the London rules, if you are willing to mull non-suitable hands up to three times you will be able to assemble Tron with a threat 93.9% of the time.

If you employ this strategy, the difference is 3.3% so it is a bit more relevant than before. However, it is still quite small and I cannot honestly recommend mulling any acceptable five card hand in hopes of pulling this powerful four card hand together. Card quantity is still relevant and one of the tutor options, Sylvan Scrying, requires an additional card in the recipe to actually function.

**Wrap-Up
**Consistent with our conclusions from yesterday’s article, this ruleset does not seem to have the massive impact that you would expect. It clearly improves Tron decks, but it improves mulligans for all decks. It will not benefit all decks equally though. Our current perception is that it more strongly benefits deck that value card quality over quantity. We will put this to the test in the coming days but we would like your feedback. Please join our discussion group and let us know what deck you would like us to run the numbers on. All we need is the archetype and the recipe for a passable hand as exemplified by Tron today. Tomorrow we will be presenting predictions for the new Modern product being announced on Thursday. Until then my friends.