"I have to say the longer and longer I go on, the inconsistency, I face an uphill battle trying to get them more consistent to play their best week in, week out", he said.
The hosts had taken the lead within just six minutes and piled on the misery further throughout an abject first half at the Emirates Stadium for the visitors.
"You set out to block the areas Arsenal like to play in".
"I think that while it was away from home the other night, we studied the Swansea performance where Swansea managed to get a result against them but the Emirates is a different kettle of fish".
In the aftermath of the match, Allardyce lashed out at "unacceptable" performance from his players, calling his side's display 'pathetic'.
"Swansea only had 30% possession but created chances, but I may as well have been talking in a foreign language". "I think that accepting an instruction is part of a player's responsibility at this level of football and I think that we tried to give those instructions with the best information we possibly can with the time available".
"So we'd been basing our plan on that, which is obvious when it's so soon".
"You have to compete and we did not compete in the first half".
Everton did play better in the second half, with Walcott looking unsafe as they tried to peg Arsenal back, but they were fighting a lost cause.
Walcott scored over 100 goals for Arsenal, but is still carries a "what-might-have-been" feeling amongst Arsenal fans.