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Suppose two players play the following game:
\begin{array}{cc} & L & R \\ U & 1,1 & 0,0 \\ D & 0,0 & 4,4 \end{array}
Is there any way to compare the top-left Nash equilibrium with the bottom-right one? Is there any way to distinguish between two equilibria and to define which one is "better"?
Yes there absolutely is. It is generally agreed upon that utility is (at least) ordinal. That means I can compare utility levels for a single person (not necessarily across people) and the numbers have meaning in this sense. So for a single person:
$U(down, right) = 4 > 1 = U(Up, Left)$.
This means (Down,Right) is strictly better for each agent than (Up, Left). You will notice this satisfies the definition of a pareto improvement. So (Down,Right) pareto dominates (Up, Left).
It is also generally agreed that a good way to compare outcomes in economics is pareto efficiency (no person can be made better off without someone being made worse off).
Hence we can safely say that the equilibrium with (Down,Right) each is comparable and better than (Up, Left). This is generally accepted, not controversial and doesn't require any assumptions we dislike.
Note that it is not necessary to assume here that utility is cardinal. That would mean we can compare utility levels across agents. E.g. for the statement: if U(agent 1)=4, U(agent 2)= 1, then U(agent 1)>(agent 2). Another example: saying the sum of utilities for everyone is larger under policy A than under policy B and hence A is better than B requires cardinal utility. This is a wrong assumption, but sometimes used (for example in public and welfare economics) to make comparisons. It is however controversial to use, since it is actually wrong. Public economists see it as a necessary evil. What everyone accepts is that utility is ordinal and that is all we need to compare the two equilibria.
The answer is most certainly yes. You could compare NE on any criterion that you find relevant. A few examples: You could say that $u_1(U,L) < u_1(D,R)$, or $u_2(U,L) < u_2(D,R)$. You could define $V = \alpha u_1 + (1-\alpha) u_2$ ($\alpha \in [0,1]$) and call it a weighted utilitarian social welfare function. Then obviously $V(U,L)<V(D,R)$ for any $\alpha \in [0,1]$. Obviously in other games, one player might prefer one NE and another might prefer another, but not so here.
If your question is whether there is a standard way of comparing two NE, I would say that it really all depends on the context.
