Existence of auctions with both shill-proof and non-shill-proof equilibria

Determine whether there exists a single-item extensive-form auction that admits multiple Perfect Bayesian equilibria such that at least one equilibrium is shill-proof (in the weak sense that shill bidders’ equilibrium strategies induce no shilling and the outcome is invariant to the realization of the set of shill bidders) and at least one equilibrium is not shill-proof; alternatively, prove that such coexistence of shill-proof and non-shill-proof equilibria within the same auction is impossible.

Background

The paper defines weak shill-proofness as an equilibrium property: an auction equilibrium is weakly shill-proof if the equilibrium strategies and outcomes are invariant to which zero-value players (shill bidders) are present. This framing means shill-proofness is not purely a property of the mechanism, but of the mechanism–equilibrium pair.

Immediately after introducing this definition, the authors note a potential phenomenon they have not found an example of: a single auction format could, in principle, admit multiple equilibria where some are shill-proof and others are not. Establishing whether such coexistence can occur (or ruling it out) would clarify the relationship between auction formats and equilibrium selection in the context of shill-proofness.