In the show, Tyrion devises a deception to figure out who Cersei's spy is. It turned out that Pycelle was the spy, and Tyrion ended up marrying Myrcella to Dorne. But what if it turned out that Varys or Littlefinger was the spy? Was Tyrion's marriage plan for Myrcella dependent on who the rat was or was Myrcella going to be wed to Dorne no matter the circumstances? In other words, was there ever a possibility that Myrcella was going to be wed to either Theon Greyjoy or Robin Arryn?
I dont think Tyrion made the marriage plans known to any other guy except Pycell. Yes, not even the prospective groom's houses. Simple reason being, if the Princess was promised to each of the 3 houses, that promised would have to be kept. So 2 of the 3 houses would turn against Lannister, which were as yet sort of neutral.
Tyrion's choice of the 3 prospective houses were perfect. Dorne, Greyjoys & Arryns. Dorne was resentful of the Lannister after the King's Landing sacking & killing of Elia. So this betrothal would bring them back into the fold. A move of reconciliation. Greyjoys were up in rebellion & declared themselves Kings. They didnt support anyone but themselves but this proposal could calm them & bring them into the fold. Arryns were wary of the Lannisters after Roberts & Jon Arryns death & it would be good to get their support through a betrothal.
These three houses were chosen because, Robb Stark & Stannis were coming to Kings Landing & these 3 houses were as yet neutral to both sides. So in order to tilt one of the houses in his favour, Tyrion made those choices