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One of the most prominent characteristics of multilateral organizations is that they do not "sprint forth full blown"; they grow. Although this is well known, relatively few attempts have been made to explain it at a general level or to explore its implications. In this paper we show why states that desire to create a multilateral organization or agreement might be attracted to a strategy that involves admitting potential members sequentially based on their preferences. Such a "sequential construction" strategy can generate an unusual kind of structure-induced equilibrium that dramatically mitigates the breadth-depth trade-off and increases the level of cooperation a multilateral is able to attain. We evaluate these claims with data drawn from the history of the European Union and twenty environmental multilaterals.