I have a thought auction problem which I would like to know how to properly determine the optimal strategy of players.
Say we have an auction for an item worth W. We have 2 players who alternately declares a bid P higher than the other player's bid. The auction ends when one player decides to pass. Let's say the final two bids are Pi and Pj where Pi > Pj.
Here's the catch, both players will pay his bid to the auctioneer, but only the winner will keep the prize (worth W). In other words, the winner has a net profit of W - Pi, where the loser has -Pj.
Assume that both players want to maximize profit, and would like to suffer the least loss. Both players are greedy. Assume perfect information, both players know the other is greedy.
What would be the outcome of this dilemma?
Extended: what if both players have a maximum loss tolerance of Ti and Tj respectively (and is known to the other player)?
Edit: to provide my thought process.
- Once the auction starts, this clearly has no end. Each player will just keep adding on his bid.
- A player knowing that the auction will only generate a loss, may simply forfeit on first bid, but that only benefits the second player.
- I have a feeling that the first player will never profit from this game.
Extended: what if players may only up his bid by $1 on every turn (cannot place arbitrary bid), and must start from zero?