I have read several articles recently decrying the new 4-3-2-2 draft queue on Magic Online, and calling for a boycott of it. It is written off as a money grab, a cheap tactic, and a strictly worse proposition for all drafters. To begin with, I must say that I agree with the money grab comment to an extent. However, it is misleading to imply that 4-3-2-2 is a worse proposition than 8-4 for all drafters.

For your reference:

• Randy Buehler's introduction
• Tim Aten's article
• Charles Mousseau's first article
• Charles Mousseau's second article

The picture that you get from these articles is pretty clear. The house wins, Wizards makes more profit, and it is no way an improvement to the 8-4 queue for any drafters.

Unfortunately, this picture is not wholly accurate. This is understandable as Tim went into relatively little detail and Charles has based all of his calculations on the presumption that you are trying to beat the game and"go infinite."

Naturally, the 4-3-2-2 drafting queue presents a very bleak picture for those looking to"go infinite." Even if you factor in the value of rares accumulated into the calculations that Charles went through, you still need a pretty amazing record to play ad infinitum in 4-3-2-2. Of course, it isn't particularly likely in an 8-4 queue either.

Calculations and Assumptions:
There is a term in gambling known as expected value. This is the amount that you can statistically expect to gain by making a specific wager. For example, if you bet a dollar that I will flip heads with a fair coin twice in a row, and the bet pays four dollars, your expected value is a dollar. Why? Because there is a 25% chance of me flipping heads two times in a row, and thus you should win four dollars every four times you play (playing four times costs you four dollars). Therefore, by betting a dollar, you expect to recoup your bet exactly. Thus, you have Randy Buehler's statement that every player's expected value is lower in a 4-3-2-2 draft than in an 8-4 draft.

Unfortunately, Randy's statement was not actually true. It was based on the assumption that every player has an equal chance to win every match, and that winning in the second round is no more difficult than winning in the first. If this were the case, then the 4-3-2-2 queue would naturally get no drafters. In fact, even if we preserve the second assumption (and thus, further assume that it is as easy to win in 8-4 as in 4-3-2-2), any player with an expectation of less than 44% to win any given limited match against the average drafter in the queue has a higher expected value in the 4-3-2-2 queue than in the 8-4 queue.

Roughly translated, that means that if you are fifty ratings points lower than the average MODO drafter, 4-3-2-2 gives you a better return than 8-4.

Of course, if this were the case, only pretty darn bad drafters would benefit from the new queue. But now, we can get rid of a couple more assumptions. We'll take on the easy assumption first.

Clearly, for reasons of motivation, intimidation, and other things I will go into below, your average 8-4 drafter should be rated higher (and thus, in theory, harder to beat) than your average 4-3-2-2 drafter. Thus, if you can expect to win 50% of your games in the 4-3-2-2, you can't expect to win the same percentage in 8-4. This drags the comparative expected values even closer together.

First Round Versus Second Round:
Is it as easy to win your second round match as it is to win your first? If people with the following ratings are in a draft....

• 1800
• 1750
• 1700
• 1650
• 1600
• 1550
• 1500

...then the average rating of your potential opponents is 1650. In theory, if you are a 1650 drafter, you are therefore playing against players who have, on average, skills exactly equal to yours - and thus, you would have a 50% chance of advancing past the first round. Of course, the 1800 player has a better than 70% chance of making the second round. Likewise, the other higher-rated drafters have better odds of moving on than their lower-rated counterparts. So in the second round, the average rating of your potential opponents is likely to be larger than 1650, and thus it is less likely you will win in the second round.

Now if you were really determined to make a point, you could go so far as to calculate the expected average rating for a round 2 opponent under the assumption that you made it. I won't do that here. And in actual Magic Online, I can't tell you how much your chances of winning decline in round 2 - I can only tell you that they do.

Our assumptions above told us that if you expect to win 50% of your matches against the current draft table, you can expect to win six packs 25% of the time in 8-4 (50% chance to win round one, 50% chance to win round 2, assume a finals split), giving you an expected return of 1.5 packs per draft, and that you will win two packs 25% of the time, three packs 12.5% of the time, and four packs 12.5% of the time in 4-3-2-2, giving an expected return of 1.375 packs.

But if we factor in the fact that 8-4 queues feature more challenging opposition, the expected value numbers change a little bit. Let's assume that before the new draft queue, you had a 50% chance to win any given match. Now let's assume that 8-4 is tougher to win in than 4-3-2-2, and that you now have a 48% chance to win in 8-4, and a 52% chance to win in 4-3-2-2*. Furthermore, let's assume that our chance to win drops 3% between round 1 and round 2, and again between round 2 and round 3**.

Then we have (0.48 x 0.45 x 6) = 1.296 packs expected in 8-4 and

(.52 x .51 x 2) + (.52 x .49 x .54 x 3) + (.52 x .49 x .46 x 4) = 0.5305 + 0.412776 + .468832 = 1.412108 packs.

Now, naturally these are big assumptions, and the difference may certainly not be that large - but it is obviously quite possible that for someone who was an average online drafter before the change, they actually may have a higher expected value in 4-3-2-2. This depends on the difference in average skill in the two queues and the level of difficulty increase in progressive rounds.

Does this mean things are more fair?

Not really. Wizards still gets more money by having a 4-3-2-2 option than by having only an 8-4 (assuming that the same or a greater number of drafts take place).

Eleven packs/twelve packs per draft is a closed system, so if your expected value is going up when there are fewer packs available, then somebody else's may be going down dramatically. So who are the losers?

The very best players will in theory prefer to draft 8-4 (and will have a higher expected value there), but since 8-4 will now be tougher to win in, they can expect a slightly lower return. As a trade-off to this, they can expect slightly higher quality competition and thus better practice, so for the most part they probably don't mind the existence of the 4-3-2-2 queue.

The middling-good and mid-1700 players are the ones who really take a beating here. They go from being above-average in the old 8-4 queue to being quite average in the new one, and meanwhile still get lower expected value from the 4-3-2-2 queue than they did in the old (and possibly the new) 8-4 queue. No matter which path they choose, it is a worse choice than used to be available to them. Of course, the 1800+ room could help remedy the situation for them, and we are still only discussing this under the assumption than expected value is the only consideration.

What's More Important Than Money? Sustainability And Odds Vs. Stakes
Charles Mousseau based most of his calculations on the goal of"going infinite" or beating the game, a feat that most players will not achieve in either queue, but which is always harder in 4-3-2-2 (or even, as Charles explains, in 5-3-2-2), because it presupposes that you are one of the best players at the table. While beating the game is a legitimate consideration in things like blackjack, it really should not be in Magic Online - except, perhaps, as a matter of pride or a very low budget. Why? Because the stakes are set, and they are low.

Reading Charles' comparisons to blackjack, do you ever wonder how dealers, pit bosses, and the mysterious men behind the cameras spot a card counter? Obviously there are tell-tale signs. While card counting is not nearly as complicated as the layman tends to think it is, it is still a remarkable feat of memory and concentration. And to take advantage of counting, a counter's bets and other decisions will fluctuate. But how can they tell? With 500 people in a casino at blackjack tables, how do they single out the pro?

Part of the answer is the stakes. Even a quite good card counter can only push his odds of winning slightly above 50%. Because of this, playing"for a living" at blackjack really only makes sense if the stakes are high enough. Thus, someone who is serious about making money counting cards is unlikely to be found at the 5-15 dollar tables, because while they would expect to win in the long run, the winnings would not be worth the investment of time involved.

The stakes are set in Magic Online. If you're fast and lucky in an 8-4 queue, and don't split, you can win at most eight packs for an investment of three packs, two tickets, and approximately three hours. That comes out to roughly 5-6 tickets (dollars) per hour. And that assumes you are always fast and always win.

In short, going infinite isn't exactly a profitable endeavor. So while it's nice and great to brag about, it isn't exactly necessary unless you are addicted to playing and cannot afford to spend any money to continue.

But what if beating the game isn't your goal? What if you are a realist, and you just want to play the game on a certain budget, either because:

• You enjoy the game and just want to compete
• You want some quality practice

Or, more likely:

• All of the above

In other words, let's say you want to play in a PTQ in 3 weeks (or practice up for your first Pro Tour in a few weeks). You have X dollars to spend (or X product available), and can get in three drafts a week. So your goal is to make your money/product/winnings stretch enough to get in those nine drafts of practice. If X is high enough, then your best choice is your highest expected value game - which may well be the 8-4 queue. But with a limited supply of packs, that may not be the case.

Let's look at another example from Charles' article: Short-pay blackjack. Why would a short-pay blackjack table be packed to the rafters while a full-pay one sat empty? Most gamblers are recreational, and many of them are on relatively tight budgets. In addition, the short-pay tables also tend to be the lower-stakes tables. So even though a regular pay $100 table is a better theoretical bet than a short-pay $10 table, if you have only $200 to lose, chances are quite reasonable that you would lose your entire bankroll in a matter of minutes (or even seconds!) at a $100 table. Meanwhile, at the $10 table, you could play for hours. So if you are there for entertainment, being able to get in more hands is a bigger value to you than having a higher expected value on each hand.

Likewise, if you have only nine packs and six tickets to your name, and can't invest any more for a while, then a run of misfortune drafting online can leave you busted after only nine drafts. Now, this can happen in 4-3-2-2 as well (and it could be argued that the practice/entertainment has less value against lesser competition), but it is much easier for a middling-to-good player to get going and get that set number of drafts in using 4-3-2-2, even if they could in theory make more in 8-4"if they get going."

Conclusion
It's a shame that the new drafting queue gives out one pack less prize than the traditional queue. It should be possible to make drafts more accessible and less costly for newer players without cutting into the prize payout (with separate rooms and flatter payouts still totaling twelve packs) - although since new accounts start at 1600, and players can have multiple accounts, this would have to be done carefully. And at the end of the day, it is true that Wizards are setting themselves up for a higher profit margin by reducing the prize level. But in our haste to condemn the new queue, it is important to note that it has its uses - and that contrary to what some would have you believe, sometimes it is the right choice for you.

If people want to demand better, I won't argue. If they want to petition for 5-3-2-2 or even 4-3-2-2-1 (with the one being randomly assigned to a round one loser), I'll sign it. But in absence of such an improvement, the new queue should not be treated as the end of the world, nor should it be presented as always being a rip-off compared to 8-4. Someone's gonna lose that pack, but that should not always matter to you.

Until next time (whenever that is),

Mark Acheson
Nevyn on various posting forums (Nevyn_ on MODO)

* - This would make the average drafter in an 8-4 queue roughly 28 rating points higher than the average drafter in a 4-3-2-2 queue, assuming my half-assed calculations are correct.

** - This would mean that the average drafter in the second round is roughly 20 rating points higher than one from the first round, again assuming my half-assed calculations are correct.