We consider an environment where players need to decide whether to buy a certain product (or adopt a technology) or not. The product is either good or bad but its true value is not known to the players. Instead, each player has her own private information on its quality. Each player can observe the previous actions of other players and estimate the quality of the product. A classic result in the literature shows that in similar settings information cascades occur where learning stops for the whole network and players repeat the actions of their predecessors. In contrast to the existing literature on informational cascades, in this work, players get more than one opportunity to act. In each turn, a player is chosen uniformly at random and can decide to buy the product and leave the market or to wait. We provide a characterization of structured perfect Bayesian equilibria (sPBE) with forward-looking strategies through a fixed-point equation of dimensionality that grows only quadratically with the number of players. In particular, a sufficient state for players strategies at each time instance is a pair of two integers, the first corresponding to the estimated quality of the good and the second indicating the number of players that cannot offer additional information about the good to the rest of the players. Based on this characterization we study informational cascades in two regimes. First, we show that for a discount factor strictly smaller than one, informational cascades happen with high probability as the number of players increases. Furthermore, only a small portion of the total information in the system is revealed before a cascade occurs. Secondly, and more surprisingly, we show that for a fixed number of players, as the discount factor approaches one, bad informational cascades are benign when the product is bad, and are completely eliminated when the discount factor equals one.
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