The Black Cats are currently in 19th place in the Premier League table, and Allardyce worries that putting in good performances is all that his squad care about.
Sunderland have improved in recent weeks, but they have taken just four points from their last four matches.
And Allardyce says that the quality of their displays are no longer a concern to him, as he is so desperate to get points on board.
“I don't think players realise just how important it (collecting points) is. If we don't accept the severity of our situation how are we going to cope with it,” Allardyce said.
“Players tend to think that if the performance is good, that's it. It's more than frustrating. We can't keep pussyfooting around. The only way we're going to get out of it is by accepting more responsibility.
“It's a results business... and we have to make sure we get the results we need.”
Sunderland’s displays have improved since the arrival of their five January signings, but they are still in serious trouble.
Allardyce has never been relegated as a manager before, but he faces one of the toughest tests of his career, if he is to keep the Wearsiders in the top tier.
Sunderland are currently one point away from safety, with just 11 matches left in their season.
The Black Cats’ next fixture comes against Crystal Palace later tonight, where a win could take them out of the relegation zone.