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Kelly Digges played both, but didn't make any comment on how/if it affected gameplay in any meaningful way. He also doesn't list any difference in his What's in a name Multiplayer Variants article. I seem to remember that either Anthony Alongi, or the Ferret commented on the differences between the two, but cannot seem to find their articles.

What effect (if any) does Attack Left vs Attack Right have on MtG multiplayer strategy/gameplay?

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There is a brief note at the end: "We decided that we liked Attack Right slightly better than Attack Left. It was sort of strange, at first, to attack "against" the turn order. Once I got used to it, though, it felt very natural to pass the turn to someone who would then immediately attack me. That meant that I could more or less space out during Matt and Laura's turns." –  Jefromi Jun 16 '12 at 3:33
    
Here's an article about Attack Right‌​, including strategy. Might be helpful as additional reading. –  Alex P Jul 10 '12 at 12:43
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up vote 7 down vote accepted

Advantaging beatdown vs. defense

In "attack left" (unless I'm the last player to go), my turn-2 attack will be against an opponent who has yet to take his second turn; in "attack right" (unless I'm the first player to go), it'll be against the player who has already gotten to take his own turn 2 to fortify his position.

Think of each attacking-player/defending-player pairing as a microcosm of beatdown-vs.-control in a larger game. "Attack left" is a lot like the beatdown deck getting to play first in a duel; "attack right" is a lot like the control player getting to take the first turn. Think about how much that affects the flow of the game in a duel. And, also, note how little it matters when both decks are going for a slow-and-steady strategy, as tends to be common in decks tuned for free-for-all casual multiplayer rather than duels or "attack-one" multiplayer specifically.

Compared to "attack left," "attack right" makes it slightly easier to defend yourself in the early game; that Lingering Souls starts to look a lot better when you realize you'll be able to cast it in time to defend against an aggressive opponent's (non-hasty) two-drops!

The hallmark of the variant is that you're always having to think about both offense and defense, so it would be overly reductive to say "attack right" favors the defender -- everyone is a defender. Rather, it's easier to avoid early damage in "attack right," which offers an enticement to go just a little bit "bigger" without risking an auto-loss to fast aggressive decks.

Sweepers play differently

Here's a niche concern that's big enough to point out specifically.

Let's say you just know one of the other players is going to drop a sweeper on this pass around the board -- maybe because it's obvious (like, say, a Scourglass already in play), maybe just you have strong Magic instincts, maybe you know the game will be over anyway if he or she doesn't. In "attack left," it's desirable to go for an "all-in" attack in the hopes of getting some damage through, since your aggressor's guys will all be dead before they get a chance to attack you anyway. In "attack right," it's much more suicidal, since you're setting yourself up to be wide open when it's time to defend yourself.


Non-strategic gameplay differences

As Kelly says in the linked article,

Instead of ending my turn and waiting all the way around the turn order to be attacked, I passed the turn to someone who immediately attacked me.

That's true, but it doesn't really matter that much on a pure strategic level. Either way, your turn is spent attacking and preparing for your opponent's next assault -- whether that comes before or after a series of actions that mainly don't affect you directly is strategically inconsequential. Rather, the shift between passing the turn and waiting to be attacked vs. passing and immediately being the defending player is psychological rather than strategic.

That said, there's a reason to pick one variant over the other on the this basis, too -- psychological perception still affects gameplay.

In my experience, multiplayer Magic sometimes has problems with board state comprehension, analysis paralysis, and loss of focus. So, when Kelly says,

Once I got used to it, though, it felt very natural to pass the turn to someone who would then immediately attack me. That meant that I could more or less space out during Matt and Laura's turns.

That's non-trivial. It doesn't affect strategy per se, but it does affect gameplay. Making it easier to players to know when they should focus and when they can "space" a bit means less time wasted on "Whose turn is it," "Oh, I had a response for that," and "When did that happen?"

Magic tends towards this cycle:

  1. Decide which resources you're going to commit to attack and defense.
  2. Attack.
  3. Set up for defense (e.g. cast stuff on your second main).
  4. Defend.
  5. Go back to 1.

That's because the main time to make these decisions is at the start of your turn, having just seen your new card. Part of the strategic appeal of MTG is that you need to be adjusting your plans all the time in response to changing circumstances, including on your opponent's turn; but whatever you do then is constrained by the setup decisions you made on your own turn -- which lands and creatures you played, how many dudes you set aside to block, how much untapped mana you have to pay for Instants and abilities, &c.

The attack-defend-wait pattern (in "attack right") fits this decision-making process better. "Attack left" puts more time between making your defensive plans and executing them, which can lead to greater memory and focus issues at the table.

This is all a bit oversimplified, though. The ideal strategic posture is not to "space out" at all, because, even with limited attacks or range of influence, everything that happens on the board is relevant to you. One of the more important "advanced" techniques in "attack-one" multiplayer is using your removal during your aggressor's defense time (thereby forcing him or her to make worse blocks or expend more resources against his/her opponent). Can't really do that if you're only paying attention on your own turn and immediately afterward!

Likewise, if you can avoid cycles of waxing and waning attention, you won't waste everyone's time trying to "catch up" on the board state during your turn.

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