I've been playing a lot of Game of Thrones over the last few weeks with my group, and Greyjoy has won the last three games in a row, and done well in all the games, no matter who plays them. After a recent game in which I crushed Lannister and went on to win as Greyjoy, the Lannister player complained that he simply couldn't see any way in which he could have held me off. Upon examining the situation closely, I find myself agreeing with him, but we may be overlooking something.
It looks like Greyjoy can always beat Lannister to Riverrun in the first turn, no matter what, and from that point on, Lannister is in a world of trouble. Now, I know that there are other considerations, such as the potential exposure to Stark aggression such a focused deployment would create, but Stark doesn't always go after Greyjoy, especially if they're not a very aggressive player or if Baratheon is menacing their Eastern fleet.
So is it actually possible for Lannister to take Riverrun if Greyjoy is going for it? Oh, and yes, I'm assuming one is switching the King's Court track as per the FAQ.
The way I see it, Lannister's best bet for getting in there is to March(+1) the Knight and Footman from Lannisport into Riverrun, use the Support(+1) in Stoney Sept, and use the ship to raid Ironman Bay (switching to Defense(+2) if Greyjoy goes for a naval attack instead). The March bonus won't actually help, since Lannister will be forced to move in first, and then defend from Greyjoy (who will be chaining their orders). So they'll take it, and then have a Combat Strength of 5 when they try to defend Riverrun.
But Greyjoy can still win, even if their fleet support is raided (so they might as well attack the Golden Sound). All they have to do is March(-1) their Footman in Greywater Watch to Pyke, forcing Lannister to move first, and then March(+0) the Knight and two Footmen to Riverrun. That gives them a Combat Strength of 4, but they have the Sword, and if both players play their 3, it'll be a tie, which will go to Greyjoy, given his higher position on the Fiefdoms.
I just don't see what Lannister can do to prevent this, and they'll probably lose the Golden Sound too, leaving them in an extremely perilous situation heading into turn two. And God help them if a Muster comes up in turn one (which has happened many times). Can anyone point out a fault in our logic? Otherwise, it feels like the only alternative is to nerf Greyjoy a bit, maybe by removing one of his Footmen.

Kings Landing has a str 5 Neutral Garrison (Not shown for space reasons). You need the plus support in Kingswood or +1 March in Dragonstone to win.

Or

(Muster one ship in port and one ship in Shipbreaker Bay)

Lannister:

(Muster one footman in Lannisport and one ship in the port)

Stark:

(Muster one ship in Shivering Sea and one in Bay of Ice.)

Martell:

(Muster one ship in Sea of Dorne and one Footman in Sunspear)

Greyjoy:

(Attack Golden sound if Lannister did NOT play Defence +2, AND you did not agree to an alliance. Otherwise BURN)

Tyrell:

(Ship has to move to West Summer Sea to insure that it wont be taken by Martell or Grayjoy in round 2.)

Baratheon:

(Muster one ship in Black Water Bay and one Footman in King's Landing.)

or

Kings Landing has a str 5 Neutral Garrison (Not shown for space reasons). You need Support +1 in Blackwater Bay or March +1 in Dragonstone to win.

Lannister:

Muster one Footman in Riverun and one Ship in the golden sound.

Raid order in blackwater to prevent Consolidate power in King's Landing.

Or

(Step 1)

(Step 2)

(Basically, you need Power Tokens to not fall behind and loose all your influence track positions. Moving to Karhold gets you two power tokens a round as long as you keep the footman to place CP order on.)

Muster 1 footman in Winterfell and one ship in Port.

Martell:

Your basically claiming all of Dorne in round 2, which is what you should do. 4 castles, 3 crowns and 4 Supply. Support +1 in Sea of Dorne is very important. If mustering comes, Muster Ships like crazy.

(You have the Valyrian Steel Blade and a nasty deck of House cards. Don't be afraid to use them. Other houses will NOT seek to engage with you. Take Moat Calin in round two with this simple strategy, even if it means to battle with Stark. Use Aeron to see his card, then pay two PT to win, if needs be.)

Tyrell:

(Assuming No muster and no Clash comes up, you're limited to very few choices. Which leads you to have to plan ahead. With Tyrell I find its more important to have Power Tokens(PT) than Castles. Making a Marell Alliance Benefits both, maybe decide on leaving Prince's pass unclaimed by either.)

Baratheon: Keep Throne up until mid game, it makes you not get last on Theifdom and Kings Court Track.

Lannister: Keep Top place on Kings Court.

Stark: Battle for Top on Kings Court.

Martell: Go for Throne and Kings Court

Greyjoy: Keep the top of Theifdom

Tyrell: Go for Theifdom and Kings Court.