Price of Anarchy when the cost is to society and not to the selfish agents Let's say there is some scarce resource and several agents compete to access this resource. Each agent is able to convert the resources into products that are beneficial to society, but each agent has a different efficiency with which it does so. Therefore, society would like to see the most efficient agent get the resource. However, the agent that pays more gets the resource.

Is it possible to model this scenario in a Price of Anarchy calculation? What would the model look like?

In the introductory examples of Price of Anarchy, I've come across Braess's Paradox and Pigou's example, but these are situations where the cost is incurred by the selfish agents themselves. In my example given above, the cost is incurred by society. That's why I'm not sure how to go about it.
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.