Can the price of anarchy be infinite? It seems that the price of anarchy for a normal-form game can be infinite, or undefined, when the social welfare of the "worst" equilibrium is 0. For example, consider this game: \begin{matrix} & a & b & c \\\ A& 10,10 & 0,0 & 0,0 \\\ B& 0,0 & 1,1 & 0,0 \\\ C& 0,0 & 0,0 & 0,0 \end{matrix} It seems that the worst equilibrium occurs for $s_e = (C,c)$, and the best Pareto-optimal strategy is at $s_p = (A,a)$. Then PoA using the utilitarian welfare function would be $W(s_p)/W(s_e) = 20/0 = \infty$. I know there is something wrong with my understanding of this (and probably my example too) - please point it out.

As far as I know, there is nothing wrong with your understanding or example. It's correct that the price of anarchy in the game is $20/0 = \infty$.

But usually we're not interested in studying such games, so this issue usually doesn't arise. A different way in which we can get an unbounded price of anarchy that does come up often is to have a sequence of games of increasing size (say increasing number of players), and for each one, the POA is larger than the last. Then we say that such a "game" (really a family of games) has an unbounded POA.