  Paradox of choice occurs when permitting new strategies to some players
yields lower payoffs for all players in the new equilibrium via a sequence of
individually rational actions. We consider social network games. In these games
the payoff of each player increases when other players choose the same
strategy. The definition of games on social networks was introduced by K. Apt
and S. Simon. In an article written jointly with E. Markakis, they considered
four types of paradox of choice in such games and gave examples of three of
them. The existence of paradoxical networks of the fourth type was proven only
in a weakened form. The existence of so-called vulnerable networks in the
strong sense remained an open question. In the present paper we solve this open
question by introducing a construction, called a cascade, and use it to provide
uniform examples for all four definitions of paradoxical networks.
