We study competing-mechanism games under exclusive competition: principals first simultaneously post mechanisms, then agents simultaneously choose to participate and communicate with at most one principal. In this setting, which is common to competing-auction and competitive-search applications, we develop two examples that question the relevance of the folk theorems for competing-mechanism games documented in the literature. The first example shows that there can exist pure-strategy equilibria in which some principal obtains a payoff below her min-max payoff, computed over all principals' decisions. Thus folk-theorem results may have to involve a bound on principals' payoffs that depends on the spaces of messages available to the agents, and not only on the players' available actions. The second example shows that even this nonintrinsic approach is misleading when agents' participation decisions are strategic: there can exist incentive-feasible allocations in which principals obtain payoffs above their min-max payoffs, computed over arbitrary spaces of mechanisms, but which cannot be supported in equilibrium.
 

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