Game theory games with very counter-intuitive results? I have heard of an interesting game that produces a very counter-intuitive result. It is an auction of a 100 dollar bill, but one in which both the first person in the auction and the second need to pay for the bill (while only the first one gets the bill). What is the most interesting, most often the bill is sold for over 100 USD, sometimes going up to 500 USD. The explanation is that the second-highest bidder doesn't want to lose a lot of money, so they opt to outbid the highest bidder by a bit to minimize their losses.
So my question is, what other game theory games are out there that produce very counter-intuitive results?
 A: These are some standard ones:
Guess 2/3 of the average
Centipede game
Tragedy of the commons
Or pick anything counter-intuitive from this list of games in game theory,
which, perhaps surprisingly, doesn't include the Monty Hall problem and the Sleeping Beauty problem. That would be because these are usually interpreted as a decision-theoretic problems.
One class of interesting games, that definitely go beyond normal intuitions, is that of quantum games. Perhaps somebody knows a few good ones.
A: While it may not qualify as highly counter-intuitive, I always liked the Pirate Game.
A: I did not know Shubik's auction game thanks to @Gugg for putting a reference. It seems pretty interesting though. However, I think some distinctions should be made, counter-intuitive with respect to what? wrto game theoretical (in particular Nash equilibrium) predictions or wrto our predictions.. 
If we consider Nash equilibrium predictions, one can refer to bunch of experiments where the predictions are far from the choices of the people. For example, a recent paper is Arad and Rubinstein's 11-20 money request game. Each of two player chooses an integer amount between 11 and 20. Each player receives his/her amount. Plus a player gets 20 if s/he requests exactly one less than the other. While vast majority of people chose between 17,18 or 19, the Nash equilibrium prediction is far from it. However, I would not say this is against our intuition, besides this can be explained with another phenomenon (level-k reasoning). Basically people do not make very high-level resoning, in this case it is 3. 
You know each solution concept has its assumptions, and they usually produce more than one equilibrium. I think this is one of the toughest challenge of game theory that finding a solution concept which does not produce counter-intuitive results. 

EDIT: 
Okay then that would be a tough question :). I can suggest you one another auction which describes this situation. There are actually 3 types. They are called lowest unique bid, highest unique bid and Penny auctions. Here is the link to wiki. In those type of auctions, auctioneer makes a good profit. First of all, bidders buy some chips, say 1 chip=1 Euro. Using those chips they can place a bid for an expensive phone, but each bid increases the price 1 cent! Eventually the highest bidder wins the object (usually 5€) and winner pays for it. Although it seems the price is very cheap for a phone, auctioneer receives much much more from the other bidders. 
