In a 2nd price option, if all the player choice their bids independently, the optimal strategy is to bid their valuation. However, in some situations, their decisions can effect each other.
For example, an over bidder could announce that they will bid \$1000000 for a painting worth \$1000. This would cause all the bidders to not bid (since now there is no point), and the over bidder pays $0 since there were no other bids.
A solution to the over bidder is a spiter, who predicts the bid of and underbids the highest bidder by 1 cent, forcing them to pay their bid. However, the spiter presents a new problem. If the players know their is a spiter, they will bid like they would in a first price auction, since they know they will end up paying whatever they bid. Even worse is a shill, someone who colludes with auctioneer to underbid the highest bidder by a cent. If they accidentally win (if they overestimated the highest bid), the auctioneer can secretly refund the purchase. If they succeed, however, they can share the extra revenue with the auctioneer, meaning that they have an incentive to succeed.
My question is, are there any modifications you can make to the auction to solve this? Of course, you could just say "no over bidders, spiters, or shills allowed", but I would prefer a mathematical solution.