I don't understand the dollar auction.

Alice and Bob both have the strategy "bid until you run out of money". Alice has $50, and the auction is currently at $20. If Alice and Bob continue their strategies, Alice has a 50% chance of losing the auction and losing $50, and a 50% chance of winning the auction and losing a smaller amount of money. Her expected loss is more than $25. If she changes her strategy to "never bid", she'll instead lose $20, which clearly isn't as bad. As such, this strategy is unstable, and not a Nash equilibrium.

The only Nash equilibria I can find in it are Alice always bids and Bob never bids, Bob always bids and Alice never bids, and a mixed strategy in which they may-or-may-not bid in any given round.