Is there a way to eliminate shilling, spiting, and overbidding in second price auctions? 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.
Overbidding

The second price auction has an equilibrium in which player 1 bids a very large amount and all other players bid zero. None of the losing bidders have a profitable deviation because they would have to pay more than their value to win. People usually ignore this equilibrium because not bidding is weakly dominated. But we should acknowledge that it is indeed an equilibrium because it is relevant for some practical applications. A big challenge, for example, in spectrum auctions is convincing new entrants to show up to bid when everyone expects the big incumbents will win all of the licenses.



One way to eliminate this equilibrium is to make it strictly dominant for players to submit a bid. This can be done, for example, by modifying the auction in the following way:


  1) run a second price auction as normal.
  
  2) with probability $1-\epsilon$, determine the allocation and payments as usual. With probability $\epsilon$, ignore the bid of the highest bidder and give the item to the second highest bidder (who pays the third highest bid). 


Even very small $\epsilon$ will suffice to make is strictly dominant for everyone to bid. Thus, the efficiency loss from this scheme can be kept relatively small.