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The Royals have won eight in a row and achieved full-on meme status with a fifteen of twenty stretch to start August, carried by their various Rally Mantises. It’s starting to feel like the playoffs might be a real possibility. Most projection sites still have the odds as long, and you can get more info about that from Shaun here.
Dreaming about Royals playoff chances right now involves passing multiple teams whether they aim for the wild card or the division. The Royals won't play all of those teams. Even when the Royals are still scheduled to play a team ahead of them there are not always enough remaining games for the Royals to do everything themselves. So the Royals will have to count on some other teams to do some of the work for them. Which teams are those?
National League
This one is pretty obvious. Royals fans can generally root for whoever they want as most games featuring an NL team will have no impact on them. If an NL team plays a team ahead of the Royals, root for the NL team because an NL team win does not otherwise affect the Royals playoff chances. The only NL team the Royals face for the rest of the season is Miami/Florida starting tonight; it would be nice if they'd go ahead and roll over and give the Royals a sweep.
American League
Teams behind the Royals
There are a bunch of teams who have won fewer games than the Royals. Much like the NL you can root for whomever you want as long as they’re not playing the Royals or a team ahead of the Royals. Teams in this category include the Twins, White Sox, Angels, Athletics, and Rays. If you want a more detailed look at these, MrAndersonmm has an excellent FanPost on the topic.
Teams ahead of/tied with/near to Royals
Texas, Seattle, Houston, Toronto, Boston, Baltimore, New York, Cleveland, and Detroit are all ahead of, tied with, or very near to the Royals. This is the group where things get a bit more complicated.
If any of these teams are playing any of the teams behind the Royals, root against these teams. But what about if these teams are playing each other? The quick answer is to root for lots of extra innings and blown leads. The longer answer requires things be broken down a bit. Keep in mind that all of this assumes the Royals will also take care of their own business, winning most or all of their remaining games.
Texas
This one is actually pretty easy: Root for Texas against any of the others. Texas has a pretty strong 7 game lead in the West and the best record in the American League. Therefore, it benefits the Royals far more for the Rangers to hang on to that division and punish the other wild card hopefuls. If Texas lets Seattle or Houston pass them, then both teams would likely end up with 90+ wins, bad news for the Royals. The Rangers have series remaining against Cleveland, Seattle, and Houston during which they can really help out the Royals by helping themselves.
Seattle
Alternately, Royals and Royals fans should be rooting against the Mariners. The Mariners aren’t close enough to winning the division to remove them from the wild card race that way. They’re also far enough ahead of the Royals, with no remaining head-to-head games between them, that the Royals will need someone to take the Mariners out for them.
Houston
The Astros are currently a half game ahead of the Royals. It's still probably safe enough to root for them against the other teams in competition - so long as they lose to the other teams and the Royals win, they won't be an issue.
Toronto, Baltimore, Boston
These three teams get listed together because they’re all bunched together at the top of the AL East and the AL Wild Card. It seems very likely that one of them will win the division and at least one will get a wild card spot. There are two methods that seem entirely reasonable for choosing the teams to root for/against out of this trio.
Because the Royals still have three games remaining against Boston and none remaining against the other two, you might choose to root against them. If the Royals sweep the Red Sox in their series - an unlikely result, but then this is all an exercise in the unlikely, anyway - they would cut the distance between them and the Red Sox in half all by themselves. A little help from other teams and the Royals could knock Boston out, entirely.
If you don't think the Royals can beat the Red Sox - or you are reading this a week after it was posted and you already know the Royals didn't beat the Red Sox - then it makea more sense to root against Baltimore. Baltimore is already behind Boston and they have the toughest remaining schedule of the three by winning percentage.
New York
The Yankees are currently tied with the Royals in the wild card standings, but the Royals control their own destiny in this matchup because the two teams still go up against each other three more times. It may go against every fiber of your being, but if you want the Royals to make the playoffs their best chance lies in the Yankees beating all the other teams in the running as well as getting swept in KC.
Detroit
The Tigers are currently ahead of the Royals in both the wild card and the division by two games. So they might seem like an obvious choice to root against when they play other contenders. However, the Royals have six more games against them so the boys in blue don’t need anyone’s help to pass them. That means we are going to spend a lot of time rooting for the Tigers, unfortunately.
Cleveland
This initially seemed like the hardest choice to make. The Cleveland baseball team owns the second best record in the AL, and a commanding eight game lead over the Royals in the division. Rooting for Cleveland would mean giving up any hope for the safest playoff path, single game randomness being what it is. But rooting against Cleveland when they play other contenders would mean sacrificing hope for a wild card spot in order to make a run at the increasingly difficult to achieve division title.
After examining the Indians schedule, however, there is some good news and some bad news. The good news is that the Indians don’t play any of the teams on the "Root Against" list for the reason of the season, so we can just go ahead and keep rooting against them without fear. The bad news is that not playing against any of the teams on that list means their competition is not as stiff as it could be, so the road to the Royals winning the division requires even more luck than it might have otherwise.
Cleveland will play four against the Astros, four against the Rangers, and have multiple series remaining against both the Royals and Tigers; it is not impossible for the Royals to catch them if Cleveland loses most of those games. Unfortunately, Cleveland hasn’t lost more than three in a row at any point this season. They also continue to exhibit the same luck the Royals had last year with late inning heroics and amazing come-back wins.
Even if the Royals can't pull it off, it sure is a lot more fun and a lot harder to complain about draft positioning and lack of deadline deals when the team wins enough to keep things interesting. Scoreboard watching will keep us all entertained during the interminable delays while the Royals or their opponents employ shifts or deploy their relievers.