We develop a search model of mergers and acquisitions (M&A), intermediated by private equity (PE) funds which may face pressure to sell. The selling pressure leads to the development of a secondary buyout (SBO) market, enabling PE funds to bail each other out. Interestingly, an increase in the number of PE funds can improve each fund’s value, because the enhanced benefits of SBOs can prevail over the reduction in value from narrower buy-sell spreads due to more intense competition. We calibrate the model using data for the US middle market and find that PE funds could lose 64% of their valuation without SBOs. Moreover, the increase in the number of funds from 2000 to 2017 contributes to a 48% increase in fund valuation due to the complementarity among funds. Nevertheless, our model predicts that this mechanism might have peaked in 2021, and more PE funds could decrease their value.

An agent’s preferences depend on an ordered parameter or type. We characterize the set of utility functions with single-crossing differences (SCD) in convex environments. These include preferences over lotteries, both in expected utility and rank-dependent utility frameworks, and preferences over bundles of goods and over consumption streams. Our notion of SCD does not presume an order on the choice space. This unordered SCD is necessary and sufficient for “interval choice” comparative statics. We present applications to cheap talk, observational learning, and collective choice, showing how convex environments arise in these problems and how SCD/interval choice are useful. Methodologically, our main characterization stems from a result on linear aggregations of single-crossing functions.

When does society eventually learn the truth, or take the correct action, via observational learning? In a general model of sequential learning over social networks, we identify a simple condition for learning dubbed excludability . Excludability is a joint property of agents' preferences and their information. We develop two classes of preferences and information that jointly satisfy excludability: (i) for a one‐dimensional state, preferences with single‐crossing differences and a new informational condition, directionally unbounded beliefs; and (ii) for a multi‐dimensional state, intermediate preferences and subexponential location‐shift information. These applications exemplify that with multiple states, “unbounded beliefs” is not only unnecessary for learning, but incompatible with familiar informational structures like normal information. Unbounded beliefs demands that a single agent can identify the correct action. Excludability, on the other hand, only requires that a single agent must be able to displace any wrong action, even if she cannot take the correct action.