niplav comments on shortplav

[niplav]

Since the Price of Anarchy is the ratio of the best possible strategy to the worst Nash equilibrium $\text{PoA}\left(G\right)=\frac{{\text{max}}_{s\in S}{\sum }_{i\in N}{u}_i\left(s\right)}{{\text{min}}_{s\in \text{NE}\left(G\right)}{\sum }_{i\in N}{u}_i\left(s\right)}$ and the Price of Stability is the ratio of the best possible strategy to the best Nash equilibrium $\text{PoS}\left(G\right)=\frac{{\text{max}}_{s\in S}{\sum }_{i\in N}{u}_i\left(s\right)}{{\text{max}}_{s\in \text{NE}\left(G\right)}{\sum }_{i\in N}{u}_i\left(s\right)}$, one could define the Price of Inadequacy as either the difference between the two $\text{PoI}\left(G\right)=\text{PoA}\left(G\right)-\text{PoS}\left(G\right)$ or as the ratio between the two $\text{PoI}\left(G\right)=\frac{{\text{max}}_{s\in \text{NE}\left(G\right)}{\sum }_{i\in N}{u}_i\left(s\right)}{{\text{min}}_{s\in \text{NE}\left(G\right)}{\sum }_{i\in N}{u}_i\left(s\right)}$.