How much would you bid for a dollar? There's a catch - you have to pay your highest bid even if you don't win the dollar.

Generally it goes like this. You bid one cent, because you stand to gain 99c which is clear profit. Unfortunately, someone else bids 2c against you. So you go to 3c. Eventually you get to 99c and the other fellow bids 100. What do you do? Well, if you bid 101c you lose 1c even if you win, but if you don't bid you lose 99c regardless. So you mitigate you loses by bidding the 101c. The other fellow reasons similarly. Experimental subjects have bid as much as 3 or 4 dollars before sanity is restored.

Advertising is similar to a dollar auction in that you have to pay for the adverts even if they don't bring in new business. Also with ResearchAndDevelopment, you have to pay your boffins even if they don't invent a new product. However, the winner is not necessarily the person who bids, as it were, the highest on advertising or spends the most on R&D. In these activities, the product or service being advertised, market being targeted, the thing being developed, the occurrence of research breakthroughs, all these other external factors come into play.

Consider also the cold war arms race.

''My recollection is that in the experiments mentioned above it was only the second-highest bidder who had to pay even though he didn't get the dollar.''

The solution is for the first bidder to bid $0.99. No one else will then bid. If the first bidder bids less than $0.99, it is still suboptimal for anyone else to bid (as the bid race will begin), so in theory, the first bidder should bid $0.01 and then no one else should bid. However, since that leaves room that someone might bid, $0.99 is still the optimal first bid.
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Even the $0.99 bid suffers from the same game theory assumption that the $0.01 approach assumes - game theory assumes your opponent is intelligent and looks out for their own best interests.  If you remove that assumption, then you have the problem that other players may (a) be malicious (b) stupid, or otherwise have goals that do not focus on their maximization of their own profit.

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The interesting part of this is ''why even enter?''

 A STRANGE GAME.
 THE ONLY WINNING MOVE IS
 NOT TO PLAY.
-- WOPR (WarGames)''

''Because your psychology professor wants you to. Because you're dim enough to think the other people involved might let the bidding stop before it gets to $1.00. Any other suggestions?''

Another suggesstion - (Psychology class requirement (proceeds to charity of course)): If all must bid and pay what they bid even if they lose) The likely winner is the auction(charity), If the bidding goes to minimize individual losses, the first bidder bids $.01, the second $.02, and the nth $ n cents. Then no one else bids, the auction gets n+(n_1)+(n-2)...+$01, and gives the dollar to the winner(loser - if the class has more than 100)or to the winner(winner - if the class has less than 100) (If the class has less than 14 then the auctioneer(charity) and all but the 13th player lose). If it is the psychology professor's dollar, he wouldn't initiate the game for a class less than 14. (This would be a good fund raising gimmick for a class larger than 13.) 
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And what if it's just one round of sealed bids?

''The optimal bet would be $0.99 then because it would be worthless to bet >= $1.00, so no one would bother. Thus, the best bet you can make without losing is $0.99. However, since everyone is going to bid that much, why even bother?''

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The DollarAuction can be converted into an iterated PrisonersDilemma-type of problem by having multiple auctions with the same participants. Each auction has only one round of sealed bids, with results announced before the next auction.

Suppose there are two bidders participating in 10 consecutive auctions ($10.00). If the minimum winning bid is $0.01, then the bidders (collectively) can walk away with up to $9.90. They will only get that amount if they agree to split the auctions.

Suppose the bidders were Alice and Bob. If one person (like Bob) is ''very'' trusting, they could agree to split the winnings after the experiment is over. (Bob would not bid every round, letting Alice always get $0.99 each round. After the experiment is over, they could split the cash.)

In cases of less trust, Alice and Bob could agree to exchange bids for 1 penny. Alice would first get $0.99, then Bob would get $0.99, etc. Note that it is very much in Alice's favor in this case. Suppose Bob is trusting on the first round, and Alice wins $0.99. Then Alice shows her evil side and bids $0.02 on the next round. Even if Bob always bids $0.99 afterwards, Alice got away with $1.97. (Bob could defect on the first round, but he could only get $0.98 before Alice realizes Bob isn't nice.)

Also, even if Alice is trustworthy for the first 9 rounds, there is no (rational) reason not to defect on the last round (Bob's turn to win). An interesting experiment would be to iterate for a random number of rounds, so that the bidders don't know which is the last round.

There is an excellent description and discussion of this in the book MoralCalculations by LaszloMero, ISBN:0387984194. The book is quite good.
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The DollarAuction might make a good charity event after all. Instead of putting a dollar up for auction, put some trinket worth about a dollar. The top two bidders must pay, but only the top bidder gets the trinket. Would this work the same? Not 100% sure, but I think it would.