# How to find an optimal strategy in an auction?

I have asked this question in mathematics forum as well but since I have not recieved an appropriate answer yet, I ask it here as well.

Consider an auction of sculptures by four artists: A, B, C and D. The auctioneer introduces each item to be sold in the auction room, and all bidders write their bids on a secretly sealed note that is given to the auctioneer. At the end of each round, the auctioneer announces the highest bidder as the winner, collects payment, and moves on to the next item. There are two different scenarios in this game:

1. If there is a target set of sculptures to be purchased, the auction is won by the first bidder who purchases this set of items, and in each round, the highest bidder pays their own bid. The target set's essential requirement is 3 sculptures by any artist, 2 by another artist, and 1 by another artist. For instance, 3 by artist B, 2 by artist A and 1 by artist D. Or another example, 4 by C, 1 by D and 3 by B (the order is irrelevant).
2. But if there is no target set, the bidding will run until no further items are available to bid on. In this case, in each round, the highest bidder pays the second-highest bid, and the winning bidder is the one who has the largest net worth of sculptures—the sculpture of artist A is worth \$3, artist B is worth \$10, artist C is worth \$21, and artist D is worth \$50.

In both cases, the number of rounds (which is also equal to the number of sculptures) is 200, and the starting budget for every bidder is \\$1000. Also, at the end of each round, the previous round's information is revealed so the bidders can use it for their benefit. The game each bot plays in each round is a one-shot game against the other bots. Each game a bot plays in each round is dependent on the games it has already played. The information includes:

• Details of all bidders, including their names, what they have won, and their remaining budgets.
• A list of the full sculptures order.
• A list of the winners' names of each round till now.
• List of amounts paid for items in the rounds played till now.

I should find an optimal strategy for my bot to beat other bidders (bots). My field is neither mathematics nor economics, and this problem is new to me, so I need help with the theory part based on which I will do the coding implementation. I have searched about the relevant concepts and think that the second case above must be Vickrey auction; however, I am not sure if in the Vickrey auction the remaining budget of each bidder is revealed and if the other information (above bullet points) provided for the bidders in each round, is associated with the Vickrey auction conditions. Also, in the Vickrey auction, it is always optimal for bidders to place a bid that reflects their own valuation of the good. But in this game, the bidders (bots) are provided with more information, so does the truthful bidding idea still apply? Is Nash equilibrium applicable here? Sorry to ask several questions. To encapsulate all, I would appreciate it if someone suggests the correct type of auction in each scenario explained above and helps me with some formulas/tips/references to find the optimal strategy.

Any help is much appreciated.

• A couple details matter for the optimal strategy: How do the bidders value the items being auctioned? Are their values independent or interrelated? How do the other bidders behave? Are they fully rational? Apr 9 at 22:30
• As for reference, Krishna's Auction Theory is classic. See also this related question. Apr 9 at 22:34
• @HerrK. Their values are independent and the answer to the other three questions is that we don't know.
– plpm
Apr 9 at 22:40
• @HerrK. Thank you for the reference suggestion.
– plpm
Apr 10 at 16:42