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