# 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?

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Ratio of what: Total output? total welfare? –  FooBar Apr 15 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? –  Shane Apr 15 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. –  Sander Heinsalu Apr 16 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"? –  Alecos Papadopoulos Apr 16 at 3:33

As hinted in a comment, the problem with using the difference instead of the ratio, is that the difference is not free of the units of measurement while the ratio is. And in general we want to not depend on units of measurement (whatever it is that we measure).

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,

$$\text {nPoA} \equiv \frac {\sum_iU_i -\sum_i\log(l_i) }{U_s - \sum_i\log(l_i)}=\frac {\sum_if_i(G)}{f_s(G)}$$

This seems to me as a meaningful concept to reflect the network contribution to utility and how it changes under the two decision-making cases.

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It is difficult to answer your question. The reason is that there is a conceptual problem with the "price of anarchy" literature. The literature uses the sum of utilities, and the ratio of sums of utilities, without reflecting on what it means to sum up utilities, and what it means to divide sums of utilities. In conventional utility theory these are all meaningless operations. Without clearer conceptual foundatinos, it is hard to compare the criterion that the "price of anarchy" literature uses to alternative criteria such as the one that you propose.

Maybe one can interpret the welfare criterion in the "price of anarchy" literature as "relative utilitarian." I would have to do some work, though, to figure out whether there is any meaning to dividing welfare from different outomces when welfare is "relative utilitarian."

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