Price of Anarchy when the cost is to society and not to the selfish agents
If there is a free market for the product the agent who gets the resource becomes a monopolist. "Produce more efficiently"  is taken to mean "produce every quantity of output for lower cost". That is its usual meaning in economics. Thus if someone different from the most efficient producer by any means gets hold of the resource the best thing that someone could do for himself, is to sell this resource to the most efficient producer, who is going to make more profit than the initial owner could and hence is able to make the initial owner an offer he cannot reject. Hence in the end the product gets produced by the most efficient technology available and the prices for the product are the same in both worlds. The consumers are exactly as well off. There is only a shift of profits between the entrepreneurs.

If the resource is auctioned off to the highest bidder, the most efficient producer will win the auction, as he can outbid all others, because he is going to make the largest profits among all.