# Is there a name for the ratio between the optimal social-welfare equilibrium and the worst social-welfare equilibrium of a strategic game?

Suppose you have a $n$ players strategic game, and assume that the "social-welfare"(SW) of the game is defined as the sum of payoffs to the players.

Two well known measures about the "efficiency" of the game are:

• Price of Stability (PoS) - What is the ratio between the optimal SW (for any combination of player strategies) and the SW of the best Nash equilibrium. (or, in other words, how much would society pay for the fact that players aim at maximizing their own profit, Given we can select which equilibrium will be played).

• Price of Anarchy (PoA) - What What is the ratio between the optimal SW (for any combination of player strategies) and the SW of the worst Nash equilibrium. (or, in other words, how much would society pay for the fact that players aim at maximizing their own profit, assuming we might end up in the worst equilibrium possible (SW-wise)).

I'm interested in a bit different measure $M$, which is the ratio between the SW of the best equilibrium and the SW of the worst equilibrium.

In other words, $M=\frac{PoS}{PoA}$.

Is there a known name for the measure $M$?

The measure aims to capture the potential of forcing the players into playing a specific equilibrium in a repeated game, that is, assume you could mediate between the players and compensate the players that lose money if this equilibrium is played, is it worth doing? How much can we gain?

• I do not know a name for that measure. And a search of web did not yield much. I am to curious too if someone has already examined this measure and has given a name to it. – Jimmy R. Nov 23 '14 at 20:41