# Definition of 'Price of anarchy'

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?

• Ratio of what: Total output? total welfare? Apr 15, 2015 at 18:18
• I think the issue is that if you're just using the difference, it won't be normalized. That is, multiply all the payoffs by ten and the difference will increase tenfold whereas the ratio would remain constant. It's hard for me to understand in what scenarios you would benefit from a difference in place of the ratio -- could you elaborate? Apr 15, 2015 at 19:25
• It seems a similar issue arises with ratios, since a strictly increasing transformation of a utility function represents the same preference, but need not keep the ratio constant. For example, add 1 to all payoffs. Money-metric utility may justify a ratio if we want to compare situations with different numbers of agents. With a larger number of agents, the difference would go up as the ratio remains constant. Apr 16, 2015 at 1:06
• Can you please clarify, perhaps writing the functions explicitly, the phrase "I have a number of log based functions in the utilities"? Apr 16, 2015 at 3:33

Although I am not familiar with the literature, I understand that here utility is a fully quantitative, "cardinal" concept. Moreover, there exists the network $G$ whose characteristics generate utility, and then there are the "idiosyncratic" losses, which is the component we want to get out of our sight. Then why not define a $\text {PoA}$ net of idiosyncratic losses, especially since they appear not to depend on the decision making framework, and so they are the same both in the decentralized set up and in the centralized one? I mean,
$$\text {nPoA} \equiv \frac {\sum_iU_i -\sum_i\log(l_i) }{U_s - \sum_i\log(l_i)}=\frac {\sum_if_i(G)}{f_s(G)}$$