Competing Mechanisms and Folk Theorems: Two Examples

Abstract

We study competing-mechanism games under exclusive competition: principals first simultaneously post mechanisms, after which 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 complete-information 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-theoremlike 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' 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.

Keeping the Agents in the Dark: Private Disclosures in Competing Mechanisms

“…We now show that the outcome (1)-( 2) for α = 2 3 cannot be supported in any PBE of any game without signals in which principals are constrained to post standard mechanisms, irrespective of the richness of message spaces. That is, private signals are indispensable to support the outcome in ( 1)- (2).…”

“…We now show that the outcome (1)-( 2) for α = 2 3 cannot be supported in any PBE of any game without signals in which principals are constrained to post standard mechanisms, irrespective of the richness of message spaces. That is, private signals are indispensable to support the outcome in ( 1)- (2). To this end, we consider a general competing-mechanism game without signals in which every principal j can only post a standard mechanism φ j : M j → ∆(X j ).…”

“…That is, each agent believes that P2 takes decision x 21 with probability α and decision x 22 with probability 1−α. 2 The result in Claim 1 does not depend on the specific assumptions on the agents' message sets and extends to M i 1 ⊃ Ω i × S i 2 and M i 2 ⊃ Ω i . 3 For any finite set A and each a ∈ A, δ a is the Dirac probability measure over A that assigns probability 1 to a.…”

Quasi Ex-Post Equilibrium in Competing Mechanisms

“…All equilibrium analyses in most applications are consistent with the notion of quasi ex-post equilibrium but their equilibria are special cases. 2 There is multiplicity of continuation equilibria tied to a quasi ex-post equilibrium. For example, consider competition between two principals.…”

“…We show that it does not have to be the case for a robust quasi ex-post equilibrium. It is shown that a robust quasi ex-post equilibrium can be understood in a way that a non-deviating principal punishes a deviating principal with a single EPIC direct mechanism re- 2 In many applications, there are no informational externalities and a non-deviating principal maintains the same dominant-strategy incentive compatible (DIC) direct mechanism on or off the path following a competing principal's deviation. Some of examples are second-price auctions with reserve price (Burdett and Sakovics (1999), McAfee (1993), Peters (1997), Peters and Severinov (1997), Virag (2010)) and fixed-prices (Burdett, Shi, and Wright (2001), Peters (1984Peters ( , 1991).…”