Why are Arryns always so neutral?

Question

The Arryns didn't support anybody in the War of Five Kings, even though Robb Stark, the King in the North was Robin Arryn's (in the books, Robert Arryn) cousin.

After that the only memorable thing they did was helping the Starks retake Winterfell (in the show), but even then it was only because Petyr Baelish is kind of ruling the Vale as a regent.

Whereas all the noble houses are picking a side, the Arryns seem to always be on their own side waiting for everybody else to eventually agree on something or kill each other. Why is that so?

Answer 1

The Arryns have only been neutral during the War of Five Kings

They are not "always so neutral".

• The Arryns were large supporters of Robert Baratheon in during Robert's Rebellion against the Mad King. Ned Stark and Robert Baratheon were wards entrusted to Jon Arryn's care. When King Aerys II called for the heads of Robert and Ned, Jon Arryn was the first to raise the banners in rebellion!

  Instead of granting them fair hearing, King Aerys had them brutally slain, then followed these murders by demanding that Lord Jon Arryn execute his former wards, Robert Baratheon and Eddard Stark. Many now agree that the true start of Robert's Rebellion began with Lord Arryn's refusal and his courageous calling of his banners in the defense of justice.

  - The World of Ice and Fire - The Fall of the Dragons: Robert’s Rebellion

  In more recent years, the importance of the role played by Lord Jon Arryn in Robert's Rebellion cannot be gainsaid. Indeed, it was Lord Jon's refusal to deliver the heads of his wards, Eddard Stark and Robert Baratheon, that began the revolt. Had he done as he was commanded, the Mad King might yet sit the Iron Throne.

  -The World of Ice and Fire - The Vale: House Arryn

• The Arryns supported the Crown during the Blackfyre Rebllions in times earlier than we see in the books/show.

  The Arryns played their part in the wars of the Targaryen kings, and in the Blackfyre rebellions, standing stoutly with the Iron Throne against the Blackfyre Pretenders.

  -The World of Ice and Fire - The Vale: House Arryn

Those keeping the Vale out of the recent wars had ulterior motives

• To make alliances during Robert's Rebellion, Ned Stark was wed to Caitlyn Tully while Lysa Tully was wed to Jon Arryn. This cemented the Baratheon-Stark-Tully-Arryn rebels which were the main force against the Targaryen-Tyrell-Martell loyalists.

• This changed during the War of the Five Kings due to Petyr Baelish manipulating Lysa Arryn (née Tully) into killing her own husband, betraying her sister, and keeping the Vale out of the conflict by design.

• Baelish's master plan is revealed to be "a picture of me [Petyr] on the Iron Throne, and you [Sansa] by my side". This is the only reason the Knights of the Vale rode to side with the Starks in the Battle of the Bastards as Baelish has maneuvered himself to be Lord Regent of House Arryn.

Answer 2

This comes down to who is ruling House Arryn.

Originally the Lord of house Arryn was Jon Arryn, Robert Baratheon's hand.

Jon Arryn mysteriously dies, which is what initially brings Robert to the North, to recruit Ned Stark to be Robert's new hand.

When Jon Arryn mysteriously dies it leaves Lysa left to rule over the Eyrie. Lysa isn't know for her mental stability and becomes even more paranoid when her husband dies; so she locks herself and her child, Robin, in the Eyrie.

When the war of the 5 kings breaks out, Lysa is too fearful and paranoid to get involved, believing she'll be safe if she never leaves the walls of the Eryie, and keeps her army out of the fighting and nearby to protect her.

As the war of the 5 kings comes to a close, Peter Baelish makes his move to gain power by marrying and quickly murdering Lysa, thus leaving weak Robin as Lord of the Eyrie.

Robin is an underdeveloped, winy, child, and has little to no understanding of what's going outside of the Eryie except for what information his mentally ill mother has fed him, or what Peter Baelish told him. This leaves Peter with the power to control which battles the Eryie gets involved in. For the most part Peter sees no gain for him to get involved in any battles until the battle for the North. Which he does in hopes of regaining the trust of Sansa, and thus having a key ally in the North.