Hi,
Depending on your objective:
If you mean winning the projects - there is no definite answer:
If you mean quality - MBB is better unless you don;t work with ex-mbb partners in Delloite
If you mean competitive for resume - MBB is definitely better
If you mean internal competition within the company among employees - There is no competition in any of these companies unless you are on the pre-partner level
I will try to go straight to the point, Deloitte as the other firms which are mainly focus on audit and advisory (EY, KPMG, PWC), sometimes and in some offices have excellent consulting divisions, however on average MBB have to be better otherwise they would not justify on the market the huge gap of their fees.
In my experience, the big 4 tend to offer consulting services on a more operational, less strategic level. Usually they combine their expertise in other areas (IT, finance, transfer pricing, etc.) with consulting services in order to be better positioned for offerings.
They operate on a lower price level than MBB and the strategy consulting capabilities are very dependant on the country, office or team.