We study the price of stability in undirected network design games with fair cost sharing. Our work provides multiple new pieces of evidence that the true price of stability, at least for special subclasses of games, may be a constant. We make progress on this long-outstanding problem, giving a bound of O(log log log n) on the price of stability of undirected broadcast games (where n is the number of players). This is the first progress on the upper bound for this problem since the O(log log n) bound of [Fiat et al. 2006] (despite much attention, the known lower bound remains at 1.818, from [Bilò et al. 2010]). Our proofs introduce several new techniques that may be useful in future work. We provide further support for the conjectured constant price of stability in the form of a comprehensive analysis of an alternative solution concept that forces deviating players to bear the entire costs of building alternative paths. This solution concept includes all Nash equilibria and can be viewed as a relaxation thereof, but we show that it preserves many properties of Nash equilibria. We prove that the price of stability in multicast games for this relaxed solution concept is Θ(1), which may suggest that similar results should hold for Nash equilibria. This result also demonstrates that the existing techniques for lower bounds on the Nash price of stability in undirected network design games cannot be extended to be super-constant, as our relaxation concept encompasses all equilibria constructed in them.