The classical definition of Price of anarchy is the ratio of decentralised optimal utility to the centralised optimal utility. This concept is used to quantify how inefficient is it when agents behave selfishly, in comparison with a situation where a central planner takes optimal actions.
My question: why cannot Price of anarchy be defined as the difference between the two solutions? In my setting, I have a number of log based functions in the utilities. For example, in a network setting defined on a graph $G$, the central planner's utility function is $U_s=f_s(G)+\sum_i \log(l_i)$ and the individual firm's utility is $U_i=f_i(G)+\log(l_i)$, where $l_i$ are loss terms for node $i$. Having a difference as PoA helps me eliminate idiosyncratic losses and I am left with $f_s(G)-\sum_i f_i(G)$, and I have a means to compare dependence of PoA on network structure.
I am greatly benefited by having a difference in place of ratio. Is it fine to define that way? Is there some other term for the difference?