I read a problem in a book describing the 'Winner's curse'. If you were to bid for a 100 USD bank note (auction, 1 opponent), how much would you bid? The special rules are that your first bid is final, all bids have to be made in the same moment and that the bidder who bid the lower amount (lost the auction) has to pay what he bid. The winner doesn't have to pay, but receives the 100 USD plus the bid from the looser. For example, you bid 110 USD for the 100 USD bank note and your opponent bid only 20 USD. Thus he has to pay you what he bid (20 USD). My question: what would an equilibrium strategy look like (if there is one)?
In a case where there are no constraints on how much players can bet, their best should go to infinity. You want to bet more than the second player, and he wants to do the same.
Now let's analyze the situation where there is a constraint on how much each player can bet, and it is public. The player who can bet less should bet 0. At the same time, the other player can bet any amount higher than the limit for the first player. This way, the player with more resources assures his win, while the other participant reduces his losses.