The dollar auction is a famous game that shows how seemingly logical choices can end up with very illogical results. Wikipedia says the best strategy is not to play, which is probably true in most instances. But what if all the players are perfect logicians?
The game is played by a group of perfect logicians all of which have, say, exactly $31.41 in their pocket. All the rules are common knowledge, including how much money they all have.
At anytime, any player may bid any amount up to $31.41 as long as it is strictly larger than the current largest bid.
Once the game ends (with either someone bidding \$31.41 or everyone declining to play further), the highest bidder pays their bid and recieves \$1 in return. The second highest bidder pays their bid as well and receives nothing.
Players never do anything to lower their expected winnings, but players would prefer winning over not winning even if the dollar amount is the same (for example, winning with a bid of \$1 is better than not playing at all)
If multiple players bid at the same time, then the bid is given to a random bidder. (Edit: For example, if Alice, Bob and Carol all bid immediately at the start 10, 20, and 30 cents respectively, and Bob is randomly chosen, then the state of the game is Bob has a bid of 20 cents and Alice and Carol have no bid whatsoever)
They are not allowed to collude or even communicate beyond placing their bids.
What happens in the dollar auction described above?